Contractive and Nonexpansive Mappings

#### **Chapter 2**

## Coupled Fixed Points for ð Þ *φ*, *ψ* -Contractive Mappings in Partially Ordered Modular Spaces

*Tayebe Lal Shateri*

#### **Abstract**

The Banach contraction principle is the most famous fixed point theorem. Many authors presented some new results for contractions in partially ordered metric spaces. Fixed point theorems in modular spaces, generalizing the classical Banach fixed point theorem in metric spaces, have been studied extensively by many mathematicians. The aim of this paper is to determine some coupled fixed point theorems for nonlinear contractive mappings in the framework of a modular space endowed with a partial order. Our results are generalizations of the fixed point theorems due to M. Mursaleen, S.A. Mohiuddine and R.P. Agarwal.

**Keywords:** coupled fixed point, contraction, modular space, partially ordered modular space

#### **1. Introduction**

In 1922, Banach established the most famous fundamental fixed point theorem, so-called the Banach contraction principle [1], which has played an important role in various fields of applied mathematical analysis. Fixed point theory is one of the most important theory in mathematics. The Banach contraction mapping principle has many applications to very different type of problems arise in different branches. Many authors have obtained many interesting extensions and generalizations (cf. [2–8]).

The more generalization was given by Nakano [9] in 1950 based on replacing the particular integral form of the functional by an abstract one. This functional was called modular. In 1959, this idea, which was the basis of the theory of modular spaces and initiated by Nakano, was refined and generalized by Musielak and Orlicz [10]. Modular spaces have been studied for almost 40 years and there is a large set of known applications of them in various parts of analysis. For more details about modular spaces, we refer the reader to [11, 12].

Fixed point theorems in modular spaces, generalizing the classical Banach fixed point theorem in metric spaces, have been studied extensively by many mathematicians, see [13–18].

The author [19] has investigated some coupled coincidence and coupled common fixed point theorems for mixed g-monotone nonlinear contractive mappings in partially ordered modular spaces.

The aim of this paper is to determine some coupled fixed point theorems for ð Þ *φ*, *ψ* - contractive mappings in the framework of partially ordered complete modular spaces. Our results are generalizations of the fixed point theorems due to M. Mursaleen, S.A. Mohiuddine and R.P. Agarwal [20]. First, we recall some basic definitions and notations about modular spaces from [11].

**Definition 1.1.** Let X be a vector space over ð Þ ¼ or ℂ . A functional *ρ* : X ! ½ � 0, ∞ is said to be modular if for all *x*,*y* ∈ X, (i) *ρ*ð Þ¼ *x* 0 if and only if *x* ¼ 0, (ii) *ρ α*ð Þ¼ *x ρ*ð Þ *x* for every *α*∈ such that ∣*α*∣ ¼ 1, (iii) *ρ α*ð Þ *x* þ *βy* ≤*ρ*ð Þþ *x ρ*ð Þ*y* if *α*,*β* ≥0 and *α* þ *β* ¼ 1. **Definition 1.2.** If in Definition 1.1, ð Þ *iii* is replaced by

$$
\rho(a\mathbf{x} + \beta \mathbf{y}) \le \alpha^\* \rho(\mathbf{x}) + \beta^\* \rho(\mathbf{y}), \tag{1}
$$

for *α*, *β* ≥0, *α* þ *β* ¼ 1 with an *s*∈ð � 0, 1 , then we say that *ρ* is an *s*-convex modular, and if *s* ¼ 1, *ρ* is said to be a convex modular.

Let *ρ* be a modular, we define the corresponding modular space, i.e. the vector space X*<sup>ρ</sup>* given by

$$\mathcal{X}\_{\rho} = \{ \mathfrak{x} \in \mathcal{X} : \quad \rho(\lambda \mathfrak{x}) \to \mathbf{0} \quad \text{as} \ \lambda \to \mathbf{0} \}. \tag{2}$$

The modular space X*<sup>ρ</sup>* is a normed space with the Luxemburg norm, defined by

$$\|\mathbf{x}\|\_{\rho} = \inf \left\{ \lambda > \mathbf{0}; \quad \rho\left(\frac{\mathbf{x}}{\lambda}\right) \le \mathbf{1} \right\}.\tag{3}$$

**Definition 1.3.** We say a function modular *ρ* satisfies the Δ2–condition if there exists *κ* >0 such that for any *x*∈ X*ρ*, we have *ρ*ð Þ 2*x* ≤*κρ*ð Þ *x* .

**Definition 1.4.** Let X*<sup>ρ</sup>* be a modular space and suppose f g *xn* and *x* are in X*ρ*. Then.


*Remark* 1.5. ð*ii*) A *ρ*-convergent sequence is *ρ*-cauchy if and only if *ρ* satisfies the Δ2–condition. ð Þ *ii ρ*ð Þ *:x* is an non-decreasing function, for any *x*∈ X. Fro this, let 0<*a*<*b*, putting *y* ¼ 0 in ð Þ *iii* of Definition 1.1 implies that

$$
\rho(a\mathbf{x}) = \rho\left(\frac{a}{b}b\mathbf{x}\right) \le \rho(b\mathbf{x}),
$$

for all *x*∈ X. Also, if *ρ* is a convex modular on X and ∣*α*∣ ≤1, then *ρ α*ð Þ *x* ≤*αρ*ð Þ *x* and *<sup>ρ</sup>*ð Þ *<sup>x</sup>* <sup>≤</sup> <sup>1</sup> <sup>2</sup> *ρ*ð Þ 2*x* for all *x*∈ X.

*Coupled Fixed Points for (*φ*,* ψ*)-Contractive Mappings in Partially Ordered Modular Spaces DOI: http://dx.doi.org/10.5772/intechopen.108695*

We end this section with a notion of a coupled fixed point introduced by Bhaskar and Lakshmikantham [5].

**Definition 1.6.** An element ð Þ *x*, *y* ∈ X � X is called a coupled fixed point of the mapping *F* : X � X ! X if

$$F(\mathbf{x}, \boldsymbol{\chi}) = \mathbf{x}, \quad F(\boldsymbol{\chi}, \boldsymbol{\chi}) = \boldsymbol{\chi}.$$

#### **2. Coupled fixed point theorems for nonlinear** ð Þ *φ***,** *ψ* **-contractive type mappings**

In this section, we establish some coupled fixed point results by considering ð Þ *φ*, *ψ* contractive mappings on modular spaces endowed with a partial order. We assume that *ρ* satisfies the Δ2-condition with *κ* <1.

Let Ψ<sup>0</sup> be the family of non-decreasing functions *ψ* : ½ Þ! 0, þ∞ ½ Þ 0, þ∞ such that

$$\sum\_{n=1}^{\infty} \psi^n(t) < \infty, \ \psi^{-1}(\{0\}) = \{0\}, \ \psi(t) < t \quad \text{and} \ \lim\_{r \to t^+} \psi(r) < t, \ \text{for all} t > 0. \tag{4}$$

The following results are generalizations of the fixed point theorems due to M. Mursaleen, S.A. Mohiuddine and R.P. Agarwal [20] in partially ordered modular spaces.

**Definition 2.1.** Let <sup>X</sup>*ρ*, <sup>≤</sup> � � be a partially ordered modular space and *<sup>F</sup>* : <sup>X</sup> � <sup>X</sup> ! X be a mapping. Then a map *F* is said to be ð Þ *φ*, *ψ* -contractive if there exist two functions *<sup>φ</sup>* : <sup>X</sup><sup>2</sup> � <sup>X</sup><sup>2</sup> ! ½ Þ 0, <sup>∞</sup> and *<sup>ψ</sup>* <sup>∈</sup> <sup>Ψ</sup> and there exist *<sup>α</sup>*, *<sup>β</sup>* <sup>&</sup>gt;0 with *<sup>α</sup>* <sup>&</sup>gt;*<sup>β</sup>* such that

$$\rho((\mathbf{x},\ \mathbf{y}),(\mathbf{z},\ \mathbf{w})) \rho(a(F(\mathbf{x},\ \ \mathbf{y}) - F(\mathbf{z},\ \ \mathbf{w}))) \le \nu \left( \frac{\rho(\beta(\mathbf{x} - \mathbf{z})) + \rho(\beta(\mathbf{y} - \mathbf{w}))}{2} \right) \tag{5}$$

for all *x*, *y*, *z*, *w* ∈ X with *x*≥*z* and *y*≤ *w*.

**Definition 2.2.** Let *<sup>F</sup>* : <sup>X</sup> � <sup>X</sup> ! <sup>X</sup> and *<sup>φ</sup>* : <sup>X</sup><sup>2</sup> � <sup>X</sup><sup>2</sup> ! ½ Þ 0, <sup>∞</sup> be two mappings. Then *F* is called *φ*-admissible if

$$\varphi((\mathbf{x},\ \mathbf{y}),(\mathbf{z},\ \mathbf{w})) \ge \mathbf{1} \Rightarrow \varphi((F(\mathbf{x},\ \ \mathbf{y}),\ F(\mathbf{y},\ \ \mathbf{x})),(F(\mathbf{z},\ \ \mathbf{w}),\ F(\mathbf{w},\ \ \mathbf{z}))) \ge \mathbf{1} \tag{6}$$

for all *x*, *y*, *z*, *w* ∈ X.

In the following theorem, we give some requirements that a *φ*-admissible mapping has a coupled fixed point.

**Theorem 2.3.** *Let* ð Þ X, ≤, *ρ be a complete ordered modular function space. Let F* : X � X ! X *be a* ð Þ *φ*, *ψ -contractive mapping having the mixed monotone property of* X*. Suppose that.*

i. *F is φ-admissible,*

ii. *there exist x*0,*y*<sup>0</sup> ∈ X *such that x*<sup>0</sup> ≤*F x*0, *y*<sup>0</sup> � � *and y*<sup>0</sup> <sup>≥</sup> *F y*0, *<sup>x</sup>*<sup>0</sup> � �*, also*

$$
\varphi\left(\begin{pmatrix}\mathbf{x}\_{0},\ \boldsymbol{y}\_{0}\end{pmatrix},\begin{pmatrix}F\begin{pmatrix}\mathbf{x}\_{0},\ \boldsymbol{y}\_{0}\end{pmatrix},\ F\begin{pmatrix}\boldsymbol{y}\_{0},\ \boldsymbol{x}\_{0}\end{pmatrix}\right)\right)\geq\mathbf{1} \quad\text{and}\quad\varphi\left(\begin{pmatrix}\boldsymbol{y}\_{0},\ \boldsymbol{x}\_{0}\end{pmatrix},\begin{pmatrix}F\begin{pmatrix}\boldsymbol{y}\_{0},\ \ \boldsymbol{x}\_{0}\end{pmatrix},\ F\begin{pmatrix}\boldsymbol{x}\_{0},\ \ \boldsymbol{y}\_{0}\end{pmatrix}\right)\right)\geq\mathbf{1},\tag{7}
$$

iii. *if x*f g*<sup>n</sup> and yn* � � *are sequences in* X *such that*

$$\rho\left(\left(\mathbf{x}\_{n},\ y\_{n}\right),\left(\mathbf{x}\_{n+1},\ y\_{n+1}\right)\right) \ge \mathbf{1} \quad and \quad \rho\left(\left(y\_{n},\ \mathbf{x}\_{n}\right),\left(y\_{n+1},\ \mathbf{x}\_{n+1}\right)\right) \ge \mathbf{1} \tag{8}$$

for all *n* and lim *<sup>n</sup>*!<sup>∞</sup>*xn* ¼ *x* and lim *<sup>n</sup>*!<sup>∞</sup>*yn* ¼ *y*, then

$$\rho\left(\left(\mathbf{x}\_{n},\ \boldsymbol{y}\_{n}\right),\left(\mathbf{x},\ \boldsymbol{y}\right)\right) \geq \mathbf{1} \quad \text{and} \quad \rho\left(\left(\boldsymbol{y}\_{n},\ \boldsymbol{x}\_{n}\right),\left(\boldsymbol{y},\ \boldsymbol{x}\right)\right) \geq \mathbf{1}.\tag{9}$$

*Then F has a coupled fixed point. Proof.* Let *x*0,*y*<sup>0</sup> ∈ X be such that

$$\varphi\left(\begin{pmatrix}\mathbf{x}\_{0},\ \mathbf{y}\_{0}\end{pmatrix},\begin{pmatrix}F(\mathbf{x}\_{0},\ \ \mathbf{y}\_{0}),\ F(\mathbf{y}\_{0},\ \ \mathbf{x}\_{0})\end{pmatrix}\right) \geq \mathbf{1} \quad \text{and} \quad \varphi\left(\begin{pmatrix}\mathbf{y}\_{0},\ \ \mathbf{x}\_{0}\end{pmatrix},\begin{pmatrix}F(\mathbf{y}\_{0},\ \ \mathbf{x}\_{0}),\ F(\mathbf{x}\_{0},\ \ \mathbf{y}\_{0})\end{pmatrix}\right) \geq \mathbf{1} \quad \text{(10)}$$

and *x*<sup>0</sup> ≤*F x*0, *y*<sup>0</sup> and *<sup>y</sup>*<sup>0</sup> <sup>≥</sup>*F y*0, *<sup>x</sup>*<sup>0</sup> . Put *<sup>x</sup>*<sup>1</sup> <sup>¼</sup> *F x*0, *<sup>y</sup>*<sup>0</sup> and *<sup>y</sup>*<sup>1</sup> <sup>¼</sup> *F y*0, *<sup>x</sup>*<sup>0</sup> . Let *x*2,*y*<sup>2</sup> ∈ X be such that *x*<sup>2</sup> ¼ *F x*1, *y*<sup>1</sup> and *<sup>y</sup>*<sup>2</sup> <sup>¼</sup> *F y*1, *<sup>x</sup>*<sup>1</sup> . Continuing this process, we can construct two sequences f g *xn* and *yn* in X such that

$$\mathbf{x}\_{n+1} = F(\mathbf{x}\_n, \boldsymbol{y}\_n) \quad \text{and} \quad \boldsymbol{y}\_{n+1} = F(\boldsymbol{y}\_n, \boldsymbol{x}\_n) \quad (n \ge 0). \tag{11}$$

Using the mathematical induction, we will show that

$$
\infty\_n \le \varkappa\_{n+1} \quad \text{and} \quad \varkappa\_n \ge \varkappa\_{n+1} \quad (n \ge 0). \tag{12}
$$

By assumption, (12) hold for *n* ¼ 0. Now suppose that (12) hold for some fixed *n*≥ 0. Then by the mixed monotone property of *F*, we have

$$\mathbf{x}\_{n+2} = F(\mathbf{x}\_{n+1}, \mathbf{y}\_{n+1}) \ge F(\mathbf{x}\_n, \mathbf{y}\_{n+1}) \ge F(\mathbf{x}\_n, \mathbf{y}\_n) = \mathbf{x}\_{n+1} \tag{13}$$

and

$$\mathbb{E}\left(\mathbf{y}\_{n+1} = F(\mathbf{y}\_{n+1}, \boldsymbol{\kappa}\_{n+1}) \le F(\mathbf{y}\_n, \boldsymbol{\kappa}\_{n+1}) \le F(\mathbf{y}\_n, \boldsymbol{\kappa}\_n) = \mathbf{y}\_{n+1}.\tag{14}$$

Hence (12) hold for *<sup>n</sup>*≥0. If for some *<sup>n</sup>*, *xn*þ1, *yn*þ<sup>1</sup> <sup>¼</sup> *xn*, *yn* , then *F xn*, *yn* <sup>¼</sup> *xn* and *F yn*, *xn* <sup>¼</sup> *yn*, and so *<sup>F</sup>* has a coupled fixed point. Thus we assumed that *xn*þ1, *yn*þ<sup>1</sup> 6¼ *xn*, *yn* for all *n* ≥0. Since *F* is *φ*-admissible, we have

$$\rho\left(\begin{pmatrix}\mathbf{x}\_{0},\ \mathbf{y}\_{0}\end{pmatrix},\begin{pmatrix}\mathbf{x}\_{1},\ \mathbf{y}\_{1}\end{pmatrix}\right) = \rho(\begin{pmatrix}\begin{pmatrix}\mathbf{x}\_{0},\ \mathbf{y}\_{0}\end{pmatrix},\begin{pmatrix}F(\mathbf{x}\_{0},\ \mathbf{y}\_{0}),\ F(\mathbf{y}\_{0},\ \mathbf{x}\_{0})\end{pmatrix}) \geq \mathbf{1} \tag{15}$$

hence

$$\rho\left(\left(F\left(\mathbf{x}\_{0},\ \boldsymbol{y}\_{0}\right),\ F\left(\mathbf{y}\_{0},\ \boldsymbol{\varkappa}\_{0}\right)\right),\left(F\left(\mathbf{x}\_{1},\ \boldsymbol{y}\_{1}\right),\ F\left(\mathbf{y}\_{1},\ \boldsymbol{\varkappa}\_{1}\right)\right)\right) = \rho\left(\left(\mathbf{x}\_{1},\ \boldsymbol{y}\_{1}\right),\ \left(\mathbf{x}\_{2},\ \boldsymbol{y}\_{2}\right)\right) \geq \mathbf{1}.\tag{16}$$

Therefore by induction we get

$$\rho\left(\left(\mathbf{x}\_{n},\ \boldsymbol{y}\_{n}\right),\left(\mathbf{x}\_{n+1},\ \boldsymbol{y}\_{n+1}\right)\right) \geq \mathbf{1} \quad \text{and} \quad \rho\left(\left(\boldsymbol{y}\_{n},\ \boldsymbol{x}\_{n}\right),\left(\boldsymbol{y}\_{n+1},\ \boldsymbol{x}\_{n+1}\right)\right) \geq \mathbf{1} \tag{17}$$

for all *n* ∈ . Since *F* is ð Þ *φ*, *ψ* -contractive, using (35) and (17), we obtain

$$\begin{split} \rho(a(\mathbf{x}\_{n} - \mathbf{x}\_{n+1})) &= \rho\left(a(F(\mathbf{x}\_{n-1}, \ y\_{n-1}), F(\mathbf{x}\_{n}, \ y\_{n}))\right) \\ &\leq \rho\left(\left(\left(\mathbf{x}\_{n-1}, \ y\_{n-1}\right), \ \left(\mathbf{x}\_{n}, \ y\_{n}\right)\right)\right) \rho\left(\alpha(F(\mathbf{x}\_{n-1}, \ y\_{n-1}), F(\mathbf{x}\_{n}, \ y\_{n}))\right) \\ &\leq \rho\left(\frac{\rho\left(\beta(\mathbf{x}\_{n-1} - \mathbf{x}\_{n})\right) + \rho\left(\beta\left(y\_{n-1} - y\_{n}\right)\right)}{2}\right), \end{split} \tag{18}$$

$$\begin{split} \rho\left(a(\boldsymbol{y}\_{n}-\boldsymbol{y}\_{n+1})\right) &= \rho\left(a(\boldsymbol{F}(\boldsymbol{y}\_{n-1},\ \boldsymbol{x}\_{n-1}),\ \boldsymbol{F}(\boldsymbol{y}\_{n},\ \boldsymbol{x}\_{n}))\right) \\ &\leq \rho\left(\left(\left(\boldsymbol{y}\_{n-1},\ \boldsymbol{x}\_{n-1}\right),\ \left(\boldsymbol{y}\_{n},\ \ \boldsymbol{x}\_{n}\right)\right)\right) \rho\left(a(\boldsymbol{F}(\boldsymbol{y}\_{n-1},\ \ \boldsymbol{x}\_{n-1}),\ \ \boldsymbol{F}(\boldsymbol{y}\_{n},\ \ \boldsymbol{x}\_{n}))\right) \\ &\leq \rho\left(\frac{\rho\left(\rho\left(\boldsymbol{y}\_{n-1}-\boldsymbol{y}\_{n}\right)\right)+\rho\left(\rho\left(\boldsymbol{x}\_{n-1}-\boldsymbol{x}\_{n}\right)\right)}{2}\right). \end{split} \tag{19}$$

$$\frac{\rho(a(\mathbf{x}\_n - \mathbf{x}\_{n+1})) + \rho\left(a(\mathbf{y}\_n - \mathbf{y}\_{n+1})\right)}{2} \le \varphi\left(\frac{\rho(\beta(\mathbf{x}\_{n-1} - \mathbf{x}\_n)) + \rho\left(\beta(\mathbf{y}\_{n-1} - \mathbf{y}\_n)\right)}{2}\right) \tag{20}$$

$$\frac{\rho\left(a(\mathbf{x}\_n - \mathbf{x}\_{n+1})\right) + \rho\left(a(\mathbf{y}\_n - \mathbf{y}\_{n+1})\right)}{2} \le \nu^n \left(\frac{\rho\left(\beta(\mathbf{x}\_0 - \mathbf{x}\_1)\right) + \rho\left(\beta\left(\mathbf{y}\_0 - \mathbf{y}\_1\right)\right)}{2}\right) \tag{21}$$

$$\sum\_{n\geq N} \mu^n \left( \frac{\rho(\beta(\mathbf{x}\_0 - \mathbf{x}\_1)) + \rho\left(\beta\left(\mathbf{y}\_0 - \mathbf{y}\_1\right)\right)}{2} \right) < \frac{\varepsilon}{2}.\tag{22}$$

$$\begin{split} \rho(\beta(\mathbf{x}\_{n} - \mathbf{x}\_{m})) &\leq \rho(a(\mathbf{x}\_{n} - \mathbf{x}\_{n+1})) + \rho(a\_{0}\beta(\mathbf{x}\_{n+1} - \mathbf{x}\_{m})) \\ &\leq \rho(a(\mathbf{x}\_{n} - \mathbf{x}\_{n+1})) + \kappa \rho(\beta(\mathbf{x}\_{n+1} - \mathbf{x}\_{m})) \\ &\leq \rho(a(\mathbf{x}\_{n} - \mathbf{x}\_{n+1})) + \rho(a(\mathbf{x}\_{n+1} - \mathbf{x}\_{n+2})) + (a\_{0}\beta(\mathbf{x}\_{n+2} - \mathbf{x}\_{m})) \\ &\leq \rho(a(\mathbf{x}\_{n} - \mathbf{x}\_{n+1})) + \rho(a(\mathbf{x}\_{n+1} - \mathbf{x}\_{n+2})) + (\beta(\mathbf{x}\_{n+2} - \mathbf{x}\_{m})) \\ &\leq \cdots \end{split} \tag{23}$$

$$\leq \sum\_{i=n}^{m-1} \rho(a(\mathbf{x}\_{i} - \mathbf{x}\_{i+1})), \tag{24}$$

$$\rho\left(\boldsymbol{\beta}\left(\boldsymbol{y}\_{n}-\boldsymbol{y}\_{m}\right)\right)\leq\sum\_{i=n}^{m-1}\rho\left(a\left(\boldsymbol{y}\_{i}-\boldsymbol{y}\_{i+1}\right)\right).\tag{24}$$

$$\begin{split} \frac{\rho(\boldsymbol{\beta}(\mathbf{x}\_{n} - \boldsymbol{x}\_{m})) + \rho(\boldsymbol{\beta}(\mathbf{y}\_{n} - \boldsymbol{y}\_{m}))}{2} &\leq \sum\_{i=n}^{m-1} \frac{\rho(\boldsymbol{a}(\mathbf{x}\_{i} - \boldsymbol{x}\_{i+1})) + \rho(\boldsymbol{a}(\boldsymbol{y}\_{i} - \boldsymbol{y}\_{i+1}))}{2} \\ &\leq \sum\_{i=n}^{m-1} \boldsymbol{y}^{n} \left( \frac{\rho(\boldsymbol{\beta}(\mathbf{x}\_{0} - \boldsymbol{x}\_{1})) + \rho(\boldsymbol{\beta}(\mathbf{y}\_{0} - \boldsymbol{y}\_{1}))}{2} \right) \\ &< \frac{\varepsilon}{2} . \end{split} \tag{25}$$

Consequently

$$
\rho\left(\beta(\mathbf{x}\_n - \mathbf{x}\_m)\right) \le \rho\left(\beta(\mathbf{x}\_n - \mathbf{x}\_m)\right) + \rho\left(\beta\left(\mathbf{y}\_n - \mathbf{y}\_m\right)\right) < \varepsilon \tag{26}
$$

and

$$
\rho\left(\beta(\mathbf{y}\_n - \mathbf{y}\_m)\right) \le \rho\left(\beta(\mathbf{x}\_n - \mathbf{x}\_m)\right) + \rho\left(\beta(\mathbf{y}\_n - \mathbf{y}\_m)\right) < \varepsilon,\tag{27}
$$

therefore f g *xn* and *yn* are cauchy sequences in complete modular space ð Þ <sup>X</sup>, *<sup>ρ</sup>* , and so f g *xn* and *yn* are convergent in ð Þ <sup>X</sup>, *<sup>ρ</sup>* . Thus there exist *<sup>x</sup>*, *<sup>y</sup>*<sup>∈</sup> <sup>X</sup> such that

$$\lim\_{n \to \infty} \mathfrak{x}\_n = \mathfrak{x} \quad \text{and} \quad \lim\_{n \to \infty} \mathfrak{y}\_n = \mathfrak{y}.$$

Now from (17) and hypothesis ð Þ *iii* , we get

$$\rho\left(\left(\mathbf{x}\_{n},\ \boldsymbol{y}\_{n}\right),\left(\mathbf{x}\_{n+1},\ \boldsymbol{y}\_{n+1}\right)\right) \geq \mathbf{1} \quad \text{and} \quad \rho\left(\left(\boldsymbol{y}\_{n},\ \boldsymbol{x}\_{n}\right),\left(\boldsymbol{y}\_{n+1},\ \boldsymbol{x}\_{n+1}\right)\right) \geq \mathbf{1} \tag{28}$$

for all *n* ∈ . From (28) and the condition ð Þ *iii* of the modular *ρ* we obtain

$$\begin{split} \rho\left(\rho(\mathsf{F}(\mathsf{x},\ \mathsf{y})-\mathsf{x})\right) &\leq \rho\left(a\left(\mathsf{F}(\mathsf{x},\ \mathsf{y})-\mathsf{F}(\mathsf{x}\_{n},\ \mathsf{y}\_{n})\right)\right) + \rho\left(a\_{0}\rho(\mathsf{x}\_{n+1}-\mathsf{x})\right) \\ &\leq \rho\left(\left(\mathsf{x}\_{n},\ \mathsf{y}\_{n}\right),\left(\mathsf{x},\ \mathsf{y}\right)\right)\rho\left(a\left(\mathsf{F}(\mathsf{x},\ \mathsf{y})-\mathsf{F}(\mathsf{x}\_{n},\ \mathsf{y}\_{n})\right)\right) + \rho\left(a\_{0}\rho(\mathsf{x}\_{n+1}-\mathsf{x})\right) \\ &\leq \rho\left(\frac{\rho\left(\rho(\mathsf{x}\_{n}-\mathsf{x})\right)+\rho\left(\rho\left(\mathsf{y}\_{n}-\mathsf{y}\right)\right)}{2}\right) + \rho\left(a\_{0}\rho(\mathsf{x}\_{n+1}-\mathsf{x})\right) \\ &< \frac{\rho\left(\rho(\mathsf{x}\_{n}-\mathsf{x})\right)+\rho\left(\rho\left(\mathsf{y}\_{n}-\mathsf{y}\right)\right)}{2} + \rho\left(a\_{0}\rho(\mathsf{x}\_{n+1}-\mathsf{x})\right) \end{split} \tag{29}$$

similarly, we get

$$\begin{split} \rho\left(\rho(\mathcal{F}(\textbf{y},\ \textbf{x})-\textbf{y})\right) &\leq \rho\left(a(\mathcal{F}(\textbf{y},\ \textbf{x})-\textbf{F}(\textbf{y}\_{n},\ \textbf{x}\_{n}))\right) + \rho\left(a\_{0}\rho(\textbf{y}\_{n+1}-\textbf{y})\right) \\ &\leq \rho\left(\left(\textbf{y}\_{n},\ \textbf{x}\_{n}\right),\left(\textbf{y},\ \textbf{x}\right)\right)\rho\left(a(\mathcal{F}(\textbf{y},\ \textbf{x})-\textbf{F}(\textbf{y}\_{n},\ \textbf{x}\_{n}))\right) + \rho\left(a\_{0}\rho\left(\textbf{y}\_{n+1}-\textbf{y}\right)\right) \\ &\leq \rho\left(\frac{\rho\left(\rho\left(\textbf{y}\_{n}-\textbf{y}\right)\right)+\rho\left(\beta\left(\textbf{x}\_{n}-\textbf{x}\right)\right)}{2}\right) + \rho\left(a\_{0}\beta\left(\textbf{y}\_{n+1}-\textbf{y}\right)\right) \\ &< \frac{\rho\left(\beta\left(\textbf{y}\_{n}-\textbf{y}\right)\right)+\rho\left(\beta\left(\textbf{x}\_{n}-\textbf{x}\right)\right)}{2} + \rho\left(a\_{0}\beta\left(\textbf{y}\_{n+1}-\textbf{y}\right)\right). \end{split} \tag{30}$$

Taking the limit as *n* ! ∞, we obtain

$$
\rho(\beta(F(\mathbf{x},\ \ y) - \mathbf{x})) = \mathbf{0} \quad \text{and} \quad \rho(\beta(F(\mathbf{y},\ \ \mathbf{x}) - \mathbf{y})) = \mathbf{0}.\tag{31}
$$

Therefore *F x*ð Þ¼ , *<sup>y</sup> <sup>x</sup>* and *F y*ð Þ¼ , *<sup>x</sup> <sup>y</sup>*, that is *<sup>F</sup>* has a coupled fixed point. □

*Remark* 2.4. If in Theorem 2.3, we replace the property ð Þ *iii* with the continuity of *F*, then the result holds that is *F* has a coupled fixed point. In fact, since *F* is continuous and *xn*þ<sup>1</sup> ¼ *F xn*, *yn* and *yn*þ<sup>1</sup> <sup>¼</sup> *F yn*, *xn* , we get

$$\mathfrak{x} = \lim\_{n \to \infty} \mathfrak{x}\_n = \lim\_{n \to \infty} F(\mathfrak{x}\_{n-1}, \mathfrak{y}\_{n-1}) = F(\mathfrak{x}, \mathfrak{y}) \tag{32}$$

*Coupled Fixed Points for (*φ*,* ψ*)-Contractive Mappings in Partially Ordered Modular Spaces DOI: http://dx.doi.org/10.5772/intechopen.108695*

and

$$\mathcal{Y} = \lim\_{n \to \infty} \mathcal{Y}\_n = \lim\_{n \to \infty} F(\mathcal{Y}\_{n-1}, \mathcal{X}\_{n-1}) = F(\mathcal{Y}, \mathcal{X}).\tag{33}$$

Hence *F* has a coupled fixed point.

As the proof of Theorem 2.3, one can prove the following theorem. **Theorem 2.5.** *In addition to the hypothesis of Theorem 2.3, suppose that for every* ð Þ *x*, *y* ,ð Þ *z*, *w in* X � X*, there exists s*ð Þ , *t in* X � X *such that*

$$
\rho((\mathfrak{x},\ \mathfrak{y}),(\mathfrak{s},\ t)) \ge \mathbf{1} \quad \text{and} \quad \rho((\mathfrak{z},\ \mathfrak{w}),(\mathfrak{s},\ t)) \ge \mathbf{1},\tag{34}
$$

and assume that ð Þ *s*, *t* is comparable to ð Þ *x*, *y* and ð Þ *z*, *w* . Then *F* has a unique coupled fixed point.

If we put *ψ*ðÞ¼ *t mt* for *m* ∈½ Þ 0, 1 in Theorem 2.3, we obtain the following corollary.

**Corollary 2.6.** *Let* ð Þ X, ≤, *ρ be a complete ordered modular function space. Let F* : X � X ! X *be a* ð Þ *φ*, *ψ -contractive mapping having the mixed monotone property of* X*. Suppose that there exist α*,*β* >0 *with α*>*β such that*

$$\rho((\mathbf{x},\ \mathbf{y}),(\mathbf{z},\ \mathbf{w})) \rho(a(F(\mathbf{x},\ \ \mathbf{y}) - F(\mathbf{z},\ \ \mathbf{w}))) \le \frac{m}{2} (\rho(\beta(\mathbf{x}-\mathbf{z})) + \rho(\beta(\mathbf{y}-\mathbf{w}))) \tag{35}$$

for all *x*, *y*, *z*, *w* ∈ X with *x*≥*z* and *y*≤ *w*. Also if.

ð Þ *ii there exist x*0, *y*<sup>0</sup> ∈ X *such that x*<sup>0</sup> ≤*F x*0, *y*<sup>0</sup> *and y*<sup>0</sup> <sup>≥</sup> *F y*0, *<sup>x</sup>*<sup>0</sup> *, also*

$$
\rho\left(\left(\mathbf{x}\_{0},\ \boldsymbol{y}\_{0}\right),\left(\mathcal{F}\left(\mathbf{x}\_{0},\ \boldsymbol{y}\_{0}\right),\ \mathcal{F}\left(\boldsymbol{y}\_{0},\ \boldsymbol{x}\_{0}\right)\right)\right) \geq \mathbf{1} \quad \text{and} \quad \rho\left(\left(\boldsymbol{y}\_{0},\ \boldsymbol{x}\_{0}\right),\ \left(\mathcal{F}\left(\boldsymbol{y}\_{0},\ \boldsymbol{x}\_{0}\right),\ \mathcal{F}\left(\boldsymbol{x}\_{0},\ \boldsymbol{y}\_{0}\right)\right)\right) \geq \mathbf{1},\tag{36}
$$

ð Þ *ii if x*f g*<sup>n</sup> and yn are sequences in* X *such that*

$$\rho\left(\left(\mathbf{x}\_{n},\ y\_{n}\right),\left(\mathbf{x}\_{n+1},\ y\_{n+1}\right)\right) \ge \mathbf{1} \quad \text{and} \quad \rho\left(\left(y\_{n},\ \mathbf{x}\_{n}\right),\left(y\_{n+1},\ \mathbf{x}\_{n+1}\right)\right) \ge \mathbf{1} \tag{37}$$

for all *n* and lim *<sup>n</sup>*!<sup>∞</sup>*xn* ¼ *x* and lim *<sup>n</sup>*!<sup>∞</sup>*yn* ¼ *y*, then

$$
\rho\left(\left(\mathbf{x}\_n,\ \boldsymbol{y}\_n\right),\left(\mathbf{x},\ \mathbf{y}\right)\right) \ge \mathbf{1} \quad \text{and} \quad \rho\left(\left(\boldsymbol{y}\_n,\ \boldsymbol{x}\_n\right),\left(\boldsymbol{y},\ \boldsymbol{x}\right)\right) \ge \mathbf{1} \tag{38}
$$

*Then F has a coupled fixed point.*

#### **3. Conclusion**

In the present paper, nonlinear contractive mappings in the framework of a modular space endowed with a partial order have been given, then some well-known coupled fixed point theorems in ordered metric spaces are extended to these mappings in modular spaces endowed with a partial order.

#### **2010 Mathematics Subject Classification:**

Primary 47H10, 54H25; Secondary 46A80

*Fixed Point Theory and Chaos*

### **Author details**

Tayebe Lal Shateri Department of Mathematics and Computer Sciences, Hakim Sabzevari University, Sabzevar, Iran

\*Address all correspondence to: t.shateri@hsu.ac.ir; shateri@ualberta.ca

© 2022 The Author(s). Licensee IntechOpen. This chapter is distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

*Coupled Fixed Points for (*φ*,* ψ*)-Contractive Mappings in Partially Ordered Modular Spaces DOI: http://dx.doi.org/10.5772/intechopen.108695*

#### **References**

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[2] Abbas M, Nazir T, Radenović S. Fixed points of four maps in partially ordered metric spaces. Applied Mathematics Letters. 2011;**24**:1520-1526

[3] Agarwal RP, El-Gebeily MA, ÓRegan D. Generalized contractions in partially ordered metric spaces. Applicable Analysis. 2008;**87**:109-116

[4] Ahmad J, Arshad M, Vetro P. Coupled coincidence point results for ϕψ-contractive mappings in partially ordered metric spaces. Georgian Mathematical Journal. 2014;**21** (2):1-13

[5] Bhaskar TG, Lakshmikantham V. Fixed point theorems in partially ordered metric spaces and applications. Nonlinear Analysis. 2006;**65**:1379-1393

[6] Choudhury BS, Kundu A. A coupled coincidence point result in partially ordered metric spaces for compatible mappings. Nonlinear Analysis. 2010;**73**: 2524-2531

[7] Ćirić LB, Cakić N, Rajović M, Ume JS. Monotone generalized nonlinear contractions in partially ordered metric spaces. Fixed Point Theory and Applications. 2008;**2008**:131294

[8] Luong NV, Thuan NX. Coupled fixed points in partially ordered metric spaces and application. Nonlinear Analysis. 2011;**74**:983-992

[9] Nakano H. Modulared semi-ordered linear spaces. In: Tokyo Math. Book Ser. Vol. 1. Tokyo: Maruzen Co.; 1950

[10] Musielak J, Orlicz W. On Modular Spaces. Studia Mathematica. 1959;**18**:49- 56

[11] Koslowski WM. Modular function spaces. New York, Basel: Dekker; 1988

[12] Musielak J. Orlicz Spaces and Modular Spaces, Lecture Notes in Math. Berlin: Springer; 1983. p. 1034

[13] Arandelović ID. On a fixed point theorem of Kirk. Journal of Mathematical Analysis and Applications. 2005;**301**(2):384-385

[14] Ćirić LB. A generalization of Banach's contraction principle. Proceedings of the American Mathematical Society. 1974;**45**(2):267- 273

[15] Edelstein M. On fixed and periodic points under contractive mappings. Journal of the London Mathematical Society. 1962;**37**(1):74-79

[16] Kuaket K, Kumam P. Fixed point of asymptotic pointwise contractions in modular spaces. Applied Mathematics Letters. 2011;**24**:1795-1798

[17] Reich S. Fixed points of contractive functions. Bollettino dell'Unione Mathematica Italiana. 1972;**4**(5):26-42

[18] Wang X, Chen Y. Fixed points of asymptotic pointwise nonexpansive mappings in modular spaces. Applications of Mathematics. 2012; **2012**). Article ID 319394:6

[19] Shateri TL. Coupled fixed points theorems for non-linear contractions in partially ordered modular spaces. International Journal of Nonlinear

Analysis and Applications. 2020;**11**(2): 133-147

[20] Mursaleen M, Mohiuddine SA, Agarwal RP. Coupled fixed point theorems for αψ-contractive type mappings in partially ordered metric spaces. Fixed Point Theory and Applications. 2012;**2012**. DOI: 10.1186/ 1687-1812-2012-228

#### **Chapter 3**

## Common Fixed Points of Asymptotically Quasi-Nonexpansive Mappings in Cat(0) Spaces

*Jamnian Nantadilok and Buraskorn Nuntadilok*

#### **Abstract**

In this manuscript, we investigate and approximate common fixed points of two asymptotically quasi-nonexpansive mappings in CAT(0) spaces. Suppose is a CAT (0) space and *C* is a nonempty closed convex subset of . Let *T*1,*T*<sup>2</sup> : *C* ! *C* be two asymptotically quasi-nonexpansive mappings, and ¼ *F T*ð Þ<sup>1</sup> ∩ *F T*ð Þ<sup>2</sup> ≔ f g *x*∈*C* : *T*1*x* ¼ *T*2*x* ¼ *x* 6¼ ∅. Let f g *α<sup>n</sup>* ,f g *β<sup>n</sup>* be sequences in [0,1]. If the sequence {*xn*} is generated iteratively by *xn*þ<sup>1</sup> <sup>¼</sup> ð Þ <sup>1</sup> � *<sup>α</sup><sup>n</sup> xn* <sup>⊕</sup> *<sup>α</sup>nT<sup>n</sup>* <sup>1</sup> *yn*,*yn* ¼ 1 � *β<sup>n</sup>* ð Þ*xn* ⊕ *βnT<sup>n</sup>* <sup>2</sup>*xn*,*n* ≥1 and *x*<sup>1</sup> ∈*C* is the initial element of the sequence (A). We prove that {*xn*} converges strongly to a common fixed point of *T*<sup>1</sup> and *T*<sup>2</sup> if and only if lim*<sup>n</sup>*!∞*d x*ð Þ¼ *<sup>n</sup>*, <sup>0</sup>*:* (B). Suppose f g *<sup>α</sup><sup>n</sup>* and f g *<sup>β</sup><sup>n</sup>* are sequences in ½ � *<sup>ε</sup>*, 1 � *<sup>ε</sup>* forsome *ε*∈ð Þ 0, 1 . If X is uniformly convex and if either *T*<sup>2</sup> or *T*<sup>1</sup> is compact, then {*xn*} converges strongly to some common fixed point of *T*<sup>1</sup> and *T*2. Our results extend and improve the related results in the literature. We also give an example in support of our main results.

**Keywords:** asymptotically quasi-nonexpansive mappings, uniformly *L*-Lipschitzian mappings, fixed points, banach spaces, CAT(0) spaces

#### **1. Introduction**

Let *C* be a nonempty subset of a real normed linear space X. Let *T* : *C* ! *C* be a self-mapping of *C*. Then *T* is said to be.


d. symptotically quasi-nonexpansive with sequence f g *kn* ⊂ ½ Þ 0, ∞ if *F T*ð Þ 6¼ ∅, lim*n*!<sup>∞</sup>*kn* <sup>¼</sup> 1 and <sup>∥</sup>*Tnx* � *<sup>p</sup>*∥ ≤ *kn*∥*<sup>x</sup>* � *<sup>p</sup>*<sup>∥</sup> for all *<sup>x</sup>*∈*C*,*p*∈*F T*ð Þ and *<sup>n</sup>* <sup>≥</sup>1.

It is clear that a nonexpansive mapping with *F T*ð Þ 6¼ ∅ is quasi-nonexpansive and an asymptotically nonexpansive mapping with *F T*ð Þ 6¼ ∅ is asymptotically quasinonexpansive. The converses are not true in general. The mapping *T* is said to be uniformly ð Þ *L*, *γ* -Lipschitzian if there exists a constant *L* >0 and *γ* >0 such that <sup>∥</sup>*Tnx* � *<sup>T</sup>ny*∥ ≤*L*∥*<sup>x</sup>* � *<sup>y</sup>*∥*<sup>γ</sup>* for all *<sup>x</sup>*,*y*∈*<sup>C</sup>* and *<sup>n</sup>*≥1.

The following example shows that there is a quasi-nonexpansive mapping which is not a nonexpansive mapping.

**Example 1.1**. (see [1]) Let *<sup>C</sup>* <sup>¼</sup> <sup>1</sup> and define a mapping *<sup>T</sup>* : *<sup>C</sup>* ! *<sup>C</sup>* by

$$T\mathbf{x} = \begin{cases} \frac{x}{2} & \text{, if } x \neq \mathbf{0} \\ \mathbf{0} & \text{, if } x = \mathbf{0} \end{cases}$$

Then *T* is quasi-nonexpansive but not nonexpansive.

It is easy to see that a nonexpansive mapping is an asymptotically nonexpansive mapping with the sequence f g¼ *kn* f g1 *:*

It is easy to see that a quasi-nonexpansive mapping is an asymptotically quasinonexpansive mapping with the sequence f g *kn* ¼ f g1 *:*

In 1972, Goebel and Kirk [2] introduced the class of asymptotically nonexpansive maps as a significant generalization of the class of nonexpansive maps. They proved that if the map *T* : *C* ! *C* is asymptotically nonexpansive and *C* is a nonempty closed convex bounded subset of a uniformly convex Banach space *X*, then *T* has a fixed point. In [3], Goebel and Kirk extended this result to the broader class of uniformly ð Þ *L*, 1 -Lipschitzian mappings with *L*<*λ* and, where *λ* is sufficiently near 1 (but greater than 1).

Iterative approximation of fixed points of nonexpansive mappings and their generalizations (asymptotically nonexpansive mappings, etc.) have been investigated by a number of authors (see, [4–21] for examples) via the Mann iterates or the Ishikawatype iteration.

Later, in 2001 Khan and Takahashi [22] studied the problem of approximating common fixed points of two asymptotically nonexpansive mappings. In 2002, Qihou [23] also established a strong convergence theorem for the Ishikawa-type iterative sequences with errors for a uniformly ð Þ *L*, *γ* -Lipschitzian asymptotically nonexpansive self-mapping of a nonempty compact convex subset of a uniformly convex Banach space.

Recently, in 2005 Shahzad and Udomene [24] investigated the approximation of common fixed points of two asymptotically quasi-nonexpansive mappings in Banach spaces. More precisely, they obtained the following results.

**Theorem 1.2.** [24] *Let C be a nonempty closed convex subset of a real Banach space X. Let T*1,*T*<sup>2</sup> : *C* ! *C be two asymptotically quasi-nonexpansive mappings with sequences* f g *un* ,f g *vn* <sup>⊂</sup> ½ Þ 0, <sup>∞</sup> *such that* <sup>P</sup><sup>∞</sup> *<sup>n</sup>*¼<sup>1</sup>*un* <sup>&</sup>lt; <sup>∞</sup> *and* <sup>P</sup><sup>∞</sup> *<sup>n</sup>*¼<sup>1</sup>*vn* <sup>&</sup>lt; <sup>∞</sup>*, and*

 ¼ *F T*ð Þ<sup>1</sup> ∩ *F T*ð Þ<sup>2</sup> ≔ f g *x*∈*C* : *T*1*x* ¼ *T*2*x* ¼ *x* 6¼ ∅*. Let x*<sup>1</sup> ∈*C be arbitrary, define the sequence x*f g*<sup>n</sup> iteratively by the iteration*

$$\begin{aligned} \boldsymbol{\omega}\_{n+1} &= (\mathbf{1} - \boldsymbol{\alpha}\_{n}) \boldsymbol{\omega}\_{n} + \boldsymbol{\alpha}\_{n} T\_{1}^{\boldsymbol{n}} \boldsymbol{\uprho}\_{n} \\ \boldsymbol{\uprho}\_{n} &= (\mathbf{1} - \boldsymbol{\beta}\_{n}) \boldsymbol{\uprho}\_{n} + \boldsymbol{\beta}\_{n} T\_{2}^{\boldsymbol{n}} \boldsymbol{\uprho}\_{n}, \end{aligned} \tag{1}$$

*Common Fixed Points of Asymptotically Quasi-Nonexpansive Mappings in Cat(0) Spaces DOI: http://dx.doi.org/10.5772/intechopen.107186*

for all *n* ≥1, where f g *α<sup>n</sup>* and f g *β<sup>n</sup>* are sequences in 0, 1 ½ �. Then.


**Theorem 1.3.** [24] *Let C be a nonempty closed convex subset of a real Banach space X. Let T*1,*T*<sup>2</sup> : *C* ! *C be two asymptotically quasi-nonexpansive mappings with sequences* f g *un* ,f g *vn* <sup>⊂</sup> ½ Þ 0, <sup>∞</sup> *such that* <sup>P</sup><sup>∞</sup> *<sup>n</sup>*¼<sup>1</sup>*un* <sup>&</sup>lt; <sup>∞</sup> *and* <sup>P</sup><sup>∞</sup> *<sup>n</sup>*¼<sup>1</sup>*vn* <sup>&</sup>lt; <sup>∞</sup>*, and* <sup>¼</sup> *F T*ð Þ<sup>1</sup> <sup>∩</sup> *F T*ð Þ<sup>2</sup> 6¼ <sup>∅</sup>*. Let* f g *α<sup>n</sup>* ,f g *β<sup>n</sup>* ⊂½ � 0, 1 *. Define the sequence x*f g*<sup>n</sup> as in (1) and x*<sup>1</sup> ∈*C is the initial element of the sequence. Then x*f g*<sup>n</sup> converges strongly to a common fixed point of T*<sup>1</sup> *and T*2⇔ lim inf *<sup>n</sup>*!∞*d x*ð Þ¼ *<sup>n</sup>*, 0*:*.

**Theorem 1.4.** [24] *Let X be a real uniformly convex Banach space and C a nonempty closed convex subset of X. Let T*1,*T*<sup>2</sup> : *C* ! *C be two uniformly continuous asymptotically quasi-nonexpansive mappings with sequences u*f g*<sup>n</sup>* ,f g *vn* <sup>⊂</sup>½ Þ 0, <sup>∞</sup> *such that* <sup>P</sup><sup>∞</sup> *n*¼1 P *un* <sup>&</sup>lt; <sup>∞</sup>*,* <sup>∞</sup> *<sup>n</sup>*¼<sup>1</sup>*vn* <sup>&</sup>lt; <sup>∞</sup>*, and* <sup>¼</sup> *F T*ð Þ<sup>1</sup> <sup>∩</sup> *F T*ð Þ<sup>2</sup> 6¼ <sup>∅</sup>*. Let* f g *<sup>α</sup><sup>n</sup> and* f g *<sup>β</sup><sup>n</sup> be sequences in* ½ � *<sup>ε</sup>*, 1 � *<sup>ε</sup> forsome ε*∈ð Þ 0, 1 *. Define the sequence x*f g*<sup>n</sup> as in* (1) *and x*<sup>1</sup> ∈*C is the initial element of the sequence. Assume, in addition, that either T*<sup>2</sup> *or T*<sup>1</sup> *is compact. Then x*f g*<sup>n</sup> converges strongly to a common fixed point of T*<sup>1</sup> *and T*2*.*

#### **2. Preliminaries**

In this section, we present some basic facts about the CAT(0) spaces and hyperbolic spaces with some useful results which are required in the sequel. The connection between CAT(0) spaces and hyperbolic spaces presented here would help, at least for beginners, to appreciate the main results presented in this manuscript.

#### **2.1 CAT(0) spaces**

Let ð Þ *X*, *d* be a metric space. A geodesic path joining *x*∈*X* to *y* ∈*X* (or, more briefly, a geodesic from *x* to *y*) is a map *ω* : ½ �! 0, *a X*, 0, ½ � *a* ⊂ *R* such that *ω*ð Þ¼ 0 *x*,*ω*ð Þ¼ *a y*, and *d*ð Þ¼ *ω*ð Þ *m* , *ω*ð Þ *n* ∣*m* � *n*∣ for all *m*,*n*∈½ � 0, *a* . In particular, *ω* is an isometry and *d x*ð Þ¼ , *y a*. The image *α* of *ω* is called a geodesic (or metric) segment joining *x* and *y*. A unique geodesic segment from *x* to *y* is denoted by ½ � *x*, *y* . The space ð Þ *X*, *d* is called to be a geodesic space if every two points of *X* are joined by a geodesic, and *X* is said to be uniquely geodesic if there is exactly one geodesic joining *x* and *y* for each *x*,*y* ∈*X*. If *Y* ⊆*X* then *Y* is said to be convex if *Y* includes every geodesic segment joining any two of its points. If ð Þ *X*, *d* is a geodesic metric space, a *geodesic triangle* Δð Þ *a*1, *a*2, *a*<sup>3</sup> consists of three points *a*1,*a*2,*a*<sup>3</sup> in X (the vertices of Δ) and a geodesic segment between each pair of vertices (the edges of Δ). A comparison triangle for geodesic triangle Δð Þ *a*1, *a*2, *a*<sup>3</sup> in ð Þ *X*, *d* is a triangle Δð Þ *a*1, *a*2, *a*<sup>3</sup> ≔ Δð Þ *a*1, *a*2, *a*<sup>3</sup> in the Euclidean plane <sup>2</sup> satisfying *d*<sup>2</sup> *ai*, *aj* � � <sup>¼</sup> *d ai*, *aj* � � for *i*,*j*∈1,2,3. Such a triangle always exists (See [25]).

**Definition 2.1.** A geodesic space ð Þ *X*, *d* is said to be a CAT(0) space if for any geodesic triangle <sup>Δ</sup> <sup>⊂</sup> *<sup>X</sup>* and *<sup>a</sup>*,*b*<sup>∈</sup> <sup>Δ</sup> we have *d a*ð Þ , *<sup>b</sup>* <sup>≤</sup>*<sup>d</sup> <sup>a</sup>*, *<sup>b</sup>* � � where *<sup>a</sup>*,*b*<sup>∈</sup> <sup>Δ</sup>.

**Remark 2.2.** Any complete, simply connected Riemannian manifold having nonpositive sectional curvature is a CAT(0) space. Other examples of CAT(0) spaces include pre-Hilbert spaces, R-trees, Euclidean buildings, and the complex Hilbert ball with a hyperbolic metric, (see [25–27] for example).

**Definition 2.3.** A geodesic triangle Δð Þ *p*, *q*, *r* in ð Þ *X*, *d* is said to satisfy the *CAT*ð Þ 0 *inequality* if for any *u*,*v*∈ Δð Þ *p*, *q*, *r* and for their comparison points *u*, *v*∈ Δð Þ *p*, *q*, *r* , one has

$$d(u,v) \le d\_{\mathbb{R}^2}(\overline{u}, \overline{v})\,.$$

For other equivalent definitions and basic properties of *CAT*ð Þ 0 spaces, we refer the readers to standard texts, such as ref. [25].

Note that if *x*,*a*1,*a*<sup>2</sup> are points of *CAT*ð Þ 0 space and if *a*<sup>0</sup> is the midpoint of the segment ½ � *<sup>a</sup>*1, *<sup>a</sup>*<sup>2</sup> (we write *<sup>a</sup>*<sup>0</sup> <sup>¼</sup> <sup>1</sup> <sup>2</sup> *a*<sup>1</sup> ⊕ <sup>1</sup> <sup>2</sup> *a*2), then the *CAT*ð Þ 0 inequality implies

$$d(\mathbf{x}, a\_0)^2 = d\left(\mathbf{x}, \frac{1}{2}a\_1 \oplus \frac{1}{2}a\_2\right) \le \frac{1}{2}d(\mathbf{x}, a\_1)^2 + \frac{1}{2}d(\mathbf{x}, a\_2)^2 - \frac{1}{4}d(a\_1, a\_2)^2 \tag{2}$$

The inequality (2) is called the **CN inequality of Bruhat and Tits** [28]. We refer readers to some brilliant known CAT(0) space results in [29–33] and references therein.

We now collect some useful facts about CAT(0) spaces, which will be used frequently in the proof of our main results.

**Lemma 2.4.** *(See* [31]*) Let X*ð Þ , *d be a CAT(0) space.*

i. *For x*1,*x*<sup>2</sup> ∈*X and α*∈½ � 0, 1 *, there exists a unique point y*∈½ � *x*1, *x*<sup>2</sup> *such that*

$$d(\mathbf{x}\_1, \mathbf{y}) = ad(\mathbf{x}\_1, \mathbf{x}\_2) \, and \, d(\mathbf{x}\_2, \mathbf{y}) = (1 - a)d(\mathbf{x}\_1, \mathbf{x}\_2). \tag{3}$$

*We write y* ¼ ð Þ 1 � *α x*<sup>1</sup> ⊕ *αx*<sup>2</sup> *for the unique point y satisfying* (3)*.*

ii. *For x*,*y*,*z*∈*X and α* ∈½ � 0, 1 , *we have*

$$d((1-a)\mathfrak{x}\oplus a\mathfrak{y},z)\leq (1-a)d(\mathfrak{x},z)+ad(\mathfrak{y},z).$$

iii. *For x*,*y*,*z*∈*X and α* ∈½ � 0, 1 *we have*

$$d((
\mathbf{1} - a)\mathbf{x} \oplus a\mathbf{y}, \mathbf{z})^2 \le (\mathbf{1} - a)d(\mathbf{x}, \mathbf{z})^2 + ad(\mathbf{y}, \mathbf{z})^2 - a(\mathbf{1} - a)d(\mathbf{x}, \mathbf{y})^2.$$

**Lemma 2.5.** *(See* [34]*) Let* f g *α<sup>n</sup>* ,f g *β<sup>n</sup> be two sequences such that.*

$$\text{i. } 0 \le a\_n \beta\_n < 1,$$

$$\text{ii. } \beta\_n \to 0 \text{ and } \sum a\_n \beta\_n = \infty.$$

*Let <sup>γ</sup>*f g*<sup>n</sup> be a nonnegative real sequence such that* <sup>P</sup>*αnβ<sup>n</sup>* <sup>1</sup> � *<sup>β</sup><sup>n</sup>* ð Þ*γ<sup>n</sup> is bounded. Then γ*f g*<sup>n</sup> has a subsequence that converges to zero.*

**Lemma 2.6.** *(see,* [17]*). Let* f g *λ<sup>n</sup> and* f g *σ<sup>n</sup> be sequences of nonnegative real numbers such that <sup>λ</sup><sup>n</sup>*þ<sup>1</sup> <sup>≤</sup>*λ<sup>n</sup>* <sup>þ</sup> *<sup>σ</sup>n,* <sup>∀</sup> *<sup>n</sup>*≥<sup>1</sup> *and* <sup>P</sup><sup>∞</sup> *<sup>n</sup>*¼<sup>1</sup>*σ<sup>n</sup>* <sup>&</sup>lt; <sup>∞</sup>*: Then* lim*<sup>n</sup>*!<sup>∞</sup>*λ<sup>n</sup> exists. Moreover, if there exists a subsequence λnj* n o *of* f g *<sup>λ</sup><sup>n</sup> such that <sup>λ</sup>nj* ! <sup>0</sup> *as j* ! <sup>∞</sup>, *then <sup>λ</sup><sup>n</sup>* ! <sup>0</sup> *as n* ! <sup>∞</sup>*:.*

*Common Fixed Points of Asymptotically Quasi-Nonexpansive Mappings in Cat(0) Spaces DOI: http://dx.doi.org/10.5772/intechopen.107186*

#### **2.2 Hyperbolic spaces**

In this section, we recall some notions of hyperbolic spaces. This class of spaces contains the class of CAT(0) spaces (See [35, 36]).

**Definition 2.7.** (See [36]) Let ð Þ *X*, *d* be a metric space and W : *X* � *X* � ½ �! 0, 1 *X* be a mapping satisfying:-.

W1. *d z*ð Þ , Wð Þ *x*, *y*, *α* ≤ ð Þ 1 � *α d z*ð Þþ , *x αd z*ð Þ , *y* ,. W2. *d*ðWð Þ *x*, *y*, *α* , Wð Þ *x*, *y*, *β* Þ ¼ ∣*α* � *β*∣*d x*ð Þ , *y* , W3. Wð Þ¼ *x*, *y*, *α* Wð Þ *y*, *x*, 1ð Þ � *α* ,. W4. *d*ð Þ Wð Þ *x*, *z*, *α* , Wð Þ *y*, *w*, *α* ≤ ð Þ 1 � *α d x*ð Þþ , *y αd z*ð Þ , *w* . for all *x*,*y*,*z*,*w* ∈*X*,*α*,*β* ∈½ � 0, 1 *:* We call the triple ð Þ *X*, *d*, W a **hyperbolic space**. It follows from (W1.) that, for each *x*,*y*∈*X* and *α*∈ ½ � 0, 1 ,

$$d(\mathbf{x}, \mathcal{W}(\mathbf{x}, \ y, \ a)) \le ad(\mathbf{x}, \mathcal{y}),\\d(\mathbf{y}, \mathcal{W}(\mathbf{x}, \ y, \ a)) \le (1 - a)d(\mathbf{x}, \mathcal{y})\tag{4}$$

In fact, we can get that (see [33]),

$$d(\mathbf{x}, \mathcal{W}(\mathbf{x}, \ y, \ a)) = ad(\mathbf{x}, \mathcal{y}), \\ d(\mathbf{y}, \mathcal{W}(\mathbf{x}, \ y, \ a)) = (1 - a)d(\mathbf{x}, \mathcal{y}).\tag{5}$$

Similar to (3), we can also use the notation 1ð Þ � *α x* ⊕ *αy* for such a point Wð Þ *x*, *y*, *α* in **hyperbolic space**.

A mapping *η* : ð Þ� 0, ∞ ð �! 0, 2 ð � 0, 1 providing such a *δ* ≔ *η*ð Þ *r*, *ε* forgiven *r*>0 and *ε*∈ð � 0, 2 is called a **modulus of uniform convexity**.

**Definition 2.8.** (See [37, 38]) Let ð Þ *X*, *d* be a hyperbolic metric space. X is said to be uniformly convex whenever *δ*ð Þ *r*, *ε* >0, for any *r*>0 and *ε*>0, where

$$d(r,e) = \inf\left\{1 - \frac{1}{r}d\left(\frac{1}{2}\chi \oplus \frac{1}{2}\mathcal{y},\ a\right) : d(\infty,a) \le r, \, d(\mathcal{y},a) \le r, \, d(\infty,\mathcal{y}) \ge r\varepsilon\right\}$$

for any *a*∈*X*.

Note that if *X* is a uniformly convex hyperbolic space, then for every *s*≥0 and *ε*>0, there exists *η*ð Þ *s*, *ε* > 0 such that *δ*ð Þ *r*, *ε* >*η*ð Þ *s*, *ε* > 0 for any *r*>*s*. One can see that *δ*ð Þ¼ *r*, 0 0. Moreover *δ*ð Þ *r*, *ε* is an increasing function of *ε*.

The following result is very useful which is an analog of Shu ([15], Lemma 1.3). It can be applied to a CAT(0) space as well.

**Lemma 2.9.** *(See* [33, 39]*) Let X*ð Þ , *d be a uniformly convex hyperbolic space. Let* f g *xn* , *yn be sequences in X and c*<sup>∈</sup> ½ Þ 0, <sup>þ</sup><sup>∞</sup> *be such that* limsup*<sup>n</sup>*!<sup>∞</sup>*d x*ð Þ *<sup>n</sup>*, *<sup>a</sup>* <sup>≤</sup> *<sup>c</sup>*,limsup*<sup>n</sup>*!<sup>∞</sup>*d yn*, *<sup>a</sup>* <sup>≤</sup> *<sup>c</sup>*, *and* lim*<sup>n</sup>*!<sup>∞</sup>*<sup>d</sup>* ð Þ <sup>1</sup> � *<sup>α</sup><sup>n</sup> xn* <sup>⊕</sup> *<sup>α</sup>nyn*, *<sup>a</sup>* <sup>¼</sup> *<sup>c</sup>*, *where <sup>α</sup><sup>n</sup>* <sup>∈</sup>½ � *<sup>a</sup>*, *<sup>b</sup>* , *with* <sup>0</sup><*a*≤*b*<1*. Then* lim*<sup>n</sup>*!<sup>∞</sup>*d xn*, *yn* <sup>¼</sup> <sup>0</sup>*:*.

Inspired and motivated by Shahzad and Udomene [24], the purpose of this paper is to establish common fixed point theorems for two asymptotically quasi-nonexpansive mappings in the setting of CAT(0) spaces. Our results significantly extend and improve the results obtained by Shahzad and Udomene in ref. [24], as well as the related results in the existing literature.

#### **3. Main results**

In this section, we let X denote a CAT(0) space and *C* be a nonempty closed convex subset of a CAT(0) space X. Let *T*1,*T*<sup>2</sup> : *C* ! *C* be two asymptotically quasinonexpansive mappings with sequences *k*ð Þ*<sup>i</sup> n* n o <sup>⊂</sup>½ Þ 1, <sup>∞</sup> satisfying P<sup>∞</sup> *<sup>n</sup>*¼<sup>1</sup> *<sup>k</sup>*ð Þ*<sup>i</sup> <sup>n</sup>* � 1 � �<sup>&</sup>lt; <sup>∞</sup>,ð Þ *<sup>i</sup>* <sup>¼</sup> 1, 2 , respectively. Put *kn* <sup>¼</sup> max *<sup>k</sup>*ð Þ<sup>1</sup> *<sup>n</sup>* , *<sup>k</sup>*ð Þ<sup>2</sup> *n* n o, then obviously P<sup>∞</sup> *<sup>n</sup>*¼1ð Þ *kn* � <sup>1</sup> <sup>&</sup>lt; <sup>∞</sup>. From now on we will take this sequence f g *kn* for both *<sup>T</sup>*<sup>1</sup> and *<sup>T</sup>*2. Recall that *F T*ð Þ¼ f g *x* : *Tx* ¼ *x* and ≔ *F T*ð Þ<sup>1</sup> ∩ *F T*ð Þ¼ <sup>2</sup> f g *x*∈*C* : *T*1*x* ¼ *T*2*x* ¼ *x* . Following ref. [24], we introduce the following iterative scheme in the setting of CAT (0) space. Starting from arbitrary *x*<sup>1</sup> ∈*C*,

$$\begin{aligned} \boldsymbol{\chi}\_{n+1} &= (\mathbf{1} - \boldsymbol{\alpha}\_{n}) \boldsymbol{\chi}\_{n} \oplus \boldsymbol{\alpha}\_{n} T\_{1}^{\boldsymbol{n}} \boldsymbol{\chi}\_{n} \\ \boldsymbol{\chi}\_{n} &= (\mathbf{1} - \boldsymbol{\beta}\_{n}) \boldsymbol{\chi}\_{n} \oplus \boldsymbol{\beta}\_{n} T\_{2}^{\boldsymbol{n}} \boldsymbol{\chi}\_{n}, \end{aligned} \tag{6}$$

for all *n* ≥1, where f g *α<sup>n</sup>* and f g *β<sup>n</sup>* are sequences in 0, 1 ½ �.

**Lemma 3.1.** *Let* ð Þ , *d be a CAT(0) space and C a nonempty closed convex subset of* X*. Let T*1,*T*<sup>2</sup> : *C* ! *C be two asymptotically quasi-nonexpansive mappings and* ¼ *F T*ð Þ<sup>1</sup> ∩ *F T*ð Þ<sup>2</sup> 6¼ ∅*. Let* f g *α<sup>n</sup> and* f g *β<sup>n</sup> be sequences in* [0,1]. *Define the sequence x*f g*<sup>n</sup> by iteration* (6)*. Then.*


**Proof:** (i). Taking *<sup>p</sup>* <sup>∈</sup>. Let *yn* <sup>¼</sup> <sup>1</sup> � *<sup>β</sup><sup>n</sup>* ð Þ*xn* <sup>⊕</sup> *<sup>β</sup>nT<sup>n</sup>* <sup>2</sup>*xn*. From (6) and by using Lemma 2.4(ii) we get

$$\begin{split} d(x\_{n+1}, p) &= d\left( (1 - a\_n) \mathbf{x}\_n \oplus a\_n T\_1^n \mathbf{y}\_n, p \right) \\ &\le (1 - a\_n) d(\mathbf{x}\_n, p) + a\_n d\left( T\_1^n \mathbf{y}\_n, p \right) \\ &\le (1 - a\_n) d(\mathbf{x}\_n, p) + a\_n k\_n d\left( \mathbf{y}\_n, p \right) \\ &= (1 - a\_n) d(\mathbf{x}\_n, p) + a\_n k\_n d\left( (1 - \beta\_n) \mathbf{x}\_n \oplus \beta\_n T\_2^n \mathbf{x}\_n, p \right) \\ &\le (1 - a\_n) d(\mathbf{x}\_n, p) + a\_n k\_n \left[ (1 - \beta\_n) d(\mathbf{x}\_n, p) + \beta\_n d\left( T\_2^n \mathbf{x}\_n, p \right) \right] \\ &\le (1 - a\_n) d(\mathbf{x}\_n, p) + a\_n k\_n \left[ (1 - \beta\_n) d(\mathbf{x}\_n, p) + \beta\_n k\_n d(\mathbf{x}\_n, p) \right] \\ &= \left( \left( 1 - a\_n + a\_n k\_n - a\_n k\_n \beta\_n + a\_n \beta\_n k\_n^2 \right) d(\mathbf{x}\_n, p) \right) \\ &\le \left( \left( 1 + a\_n k\_n + a\_n \beta\_n k\_n^2 \right) d(\mathbf{x}\_n, p) \right) \\ &= (1 + \gamma\_n) d(\mathbf{x}\_n, p) \end{split} \tag{7}$$

where *<sup>γ</sup><sup>n</sup>* <sup>¼</sup> *<sup>α</sup>nkn* <sup>þ</sup> *<sup>α</sup>nβnk*<sup>2</sup> *<sup>n</sup>* with <sup>P</sup><sup>∞</sup> *<sup>n</sup>*¼<sup>1</sup>*bn* <sup>&</sup>lt; <sup>∞</sup>*:*

i. We know that 1 þ *x*≤ exp ð Þ *x* , for all *x*≥0. Notice that for any *n*,*m* ≥1,

$$\begin{split} d(\boldsymbol{x}\_{n+m},\boldsymbol{p}) &\quad \leq (1+b\_{n+m-1})d(\boldsymbol{x}\_{n+m-1},\boldsymbol{p}) \\ &\quad \leq \exp\left(b\_{n+m-1}\right)d\left(\boldsymbol{x}\_{n+m-1,\boldsymbol{p}}\right) \\ &\quad \leq \exp\left(b\_{n+m-1}+b\_{n+m-2}\right)d\left(\boldsymbol{x}\_{n+m-2,\boldsymbol{p}}\right) \\ &\vdots \\ &\leq \exp\left(\sum\_{k=n}^{n+m-1}b\_{k}\right)d(\boldsymbol{x}\_{n},\boldsymbol{p}). \end{split} \tag{8}$$

*Common Fixed Points of Asymptotically Quasi-Nonexpansive Mappings in Cat(0) Spaces DOI: http://dx.doi.org/10.5772/intechopen.107186*

Taking *<sup>K</sup>* <sup>¼</sup> exp <sup>P</sup><sup>∞</sup> *<sup>k</sup>*¼*<sup>n</sup>bk* � �*:* Then 0 <*K* < ∞, we obtain

$$d(\mathfrak{x}\_{n+m}, p) \le \mathcal{K}d(\mathfrak{x}\_n, p) \tag{9}$$

where *p* ∈*:* This completes our proof.

**Theorem 3.2.** *Let* ð Þ , *d be a complete CAT(0) space and C a nonempty closed convex subset of* X*. Let T*1,*T*<sup>2</sup> : *C* ! *C be two asymptotically quasi-nonexpansive mappings (T*<sup>1</sup> *and T*<sup>2</sup> *need not be continuous), and* ¼ *F T*ð Þ<sup>1</sup> ∩ *F T*ð Þ<sup>2</sup> ¼6 ∅*. Let* f g *α<sup>n</sup>* ,f g *β<sup>n</sup> be sequences in* [0,1]*. From arbitrary x*<sup>1</sup> ∈*C, define the sequence x*f g*<sup>n</sup> by iteration* (6)*. Then* f g *xn converges strongly to a common fixed point of T*<sup>1</sup> *and T*<sup>2</sup> *if and only if* lim*n*!<sup>∞</sup>*d x*ð Þ¼ *<sup>n</sup>*, <sup>0</sup>*:*

**Proof:** The necessary conditions are obvious. We shall only prove the sufficient condition. By Lemma 3.1, we have *d x*ð Þ *<sup>n</sup>*þ1, *p* ≤ 1 þ *γ<sup>n</sup>* ð Þ*d x*ð Þ *<sup>n</sup>*, *p* for all *n* ≥1 and *p*∈*:* Therefore,

$$d(\mathfrak{x}\_{n+1}, \mathbb{F}) \le (1 + \chi\_n) d(\mathfrak{x}\_n, \mathbb{F})\.$$

Since P<sup>∞</sup> *<sup>n</sup>*¼<sup>1</sup>*γ<sup>n</sup>* <sup>&</sup>lt; <sup>∞</sup> and lim inf *<sup>n</sup>*!∞*d x*ð Þ¼ *<sup>n</sup>*, 0, from Lemma 2.6 we deduce that lim*<sup>n</sup>*!∞*d x*ð Þ¼ *<sup>n</sup>*, 0. Next, we show that the sequence f g *xn* is Cauchy. Since lim*<sup>n</sup>*!∞*d x*ð Þ¼ *<sup>n</sup>*, 0, given any *<sup>ε</sup>*<sup>&</sup>gt; 0, there exists a positive number *<sup>N</sup>*<sup>0</sup> such that *d x*ð Þ *<sup>n</sup>*, <sup>&</sup>lt; *<sup>ε</sup>* <sup>4</sup>*<sup>K</sup>* for all *n*≥ *N*0, where *K* > 0 is the constant in Lemma 3.1(2). So we can find *<sup>q</sup>*∈ such that *d xN*<sup>0</sup> ð Þ , *<sup>q</sup>* <sup>≤</sup> *<sup>ε</sup>* <sup>3</sup>*<sup>K</sup>*. Again by Lemma 3.1(2), we have that

$$\begin{split}d(\boldsymbol{x}\_{n+m},\boldsymbol{x}\_{n}) &\quad \leq d(\boldsymbol{x}\_{n+m},\boldsymbol{q}) + d(\boldsymbol{x}\_{n},\boldsymbol{q}) \\ &\leq \mathcal{K}d(\boldsymbol{x}\_{N\_{0}},\boldsymbol{q}) + \mathcal{K}d(\boldsymbol{x}\_{N\_{0}},\boldsymbol{q}) \\ &= 2\mathcal{K}d(\boldsymbol{x}\_{N\_{0}},\boldsymbol{q}) < \varepsilon.\end{split} \tag{10}$$

for all *n* ≥ *N*<sup>0</sup> and *m* ≥ 1. This implies that f g *xn* is Cauchy and so is convergent since is complete. Hence, f g *xn* is a Cauchy sequence in a closed convex subset C of a CAT(0) space , therefore, it must converge to a point in *<sup>C</sup>*. Let lim*<sup>n</sup>*!<sup>∞</sup>*xn* <sup>¼</sup> *<sup>q</sup>*<sup>0</sup> *:*

Now, lim*<sup>n</sup>*!<sup>∞</sup>*d x*ð Þ¼ *<sup>n</sup>*, 0 yields that *d q*<sup>0</sup> ð Þ¼ , 0. Since the set of fixed points of

asymptotically nonexpansive mappings is closed, we have *q*<sup>0</sup> ∈. This completes our proof. **Lemma 3.3.** *Let* ð Þ , *d be a CAT(0) space and C a nonempty closed convex subset of* X*. Let T*1,*T*<sup>2</sup> : *C* ! *C be two uniformly continuous asymptotically quasi-*

*nonexpansive mappings, and* ¼ *F T*ð Þ<sup>1</sup> ∩ *F T*ð Þ<sup>2</sup> 6¼ ∅*. Let* f g *α<sup>n</sup> and* f g *β<sup>n</sup> be sequences in* ½ � *ε*, 1 � *ε forsome ε*∈ð Þ 0, 1 *. From arbitrary x*<sup>1</sup> ∈*C, define the sequence x*f g*<sup>n</sup> by iteration* (6)*. Then*

$$\lim\_{n \to \infty} d\left(\mathbf{x}\_n, T\_2^n \mathbf{x}\_n\right) = \lim\_{n \to \infty} d\left(\mathbf{x}\_n, T\_1^n \mathbf{x}\_n\right) = \lim\_{n \to \infty} d\left(\mathbf{x}\_n, T\_1^n y\_n\right) = \mathbf{0}.\tag{11}$$

*Proof:* Let *<sup>p</sup>*<sup>∈</sup> . Then, by Lemma 3.1(1) and Lemma 2.6 lim*<sup>n</sup>*!<sup>∞</sup>*d x*ð Þ *<sup>n</sup>*, *<sup>p</sup>* exists. Suppose lim*<sup>n</sup>*!<sup>∞</sup>*d x*ð Þ¼ *<sup>n</sup>*, *<sup>p</sup> <sup>r</sup>*. If *<sup>r</sup>* <sup>¼</sup> 0, then by the continuity of *<sup>T</sup>*<sup>1</sup> and *<sup>T</sup>*<sup>2</sup> the conclusion follows. Now suppose *r*>0. We claim

$$\lim\_{n \to \infty} d\left(\mathbf{x}\_n, T^n\_1 \mathbf{y}\_n\right) = \lim\_{n \to \infty} d\left(\mathbf{x}\_n, T^n\_1 \mathbf{x}\_n\right) = \lim\_{n \to \infty} d\left(\mathbf{x}\_n, T^n\_2 \mathbf{x}\_n\right) = \mathbf{0}.\tag{12}$$

From *yn* <sup>¼</sup> <sup>1</sup> � *<sup>β</sup><sup>n</sup>* ð Þ*xn* <sup>⊕</sup> *<sup>β</sup>nT<sup>n</sup>* <sup>2</sup>*xn*. Since f g *xn* is bounded, there exists *R*> 0 such that *xn* � *p*,*yn* � *p*∈*BR*ð Þ 0 for all *n*≥ 1. Using Lemma 2.4(iii), we have that

$$\begin{split} d\left(y\_{n},p\right)^{2} &= d\left(\left(1-\beta\_{n}\right)\mathbf{x}\_{n}\oplus\beta\_{n}T\_{2}^{n}\mathbf{x}\_{n},p\right)^{2} \\ &\leq \left(1-\beta\_{n}\right)d\left(\mathbf{x}\_{n},p\right)^{2}+\beta\_{n}d\left(T\_{2}^{n}\mathbf{x}\_{n},p\right)^{2}-\beta\_{n}\left(\mathbf{1}-\beta\_{n}\right)d\left(\mathbf{x}\_{n},T\_{2}^{n}\mathbf{x}\_{n}\right)^{2} \\ &\leq \left(1-\beta\_{n}\right)d\left(\mathbf{x}\_{n},p\right)^{2}+\beta\_{n}k\_{n}^{2}d\left(\mathbf{x}\_{n},p\right)^{2}-\beta\_{n}\left(1-\beta\_{n}\right)d\left(\mathbf{x}\_{n},T\_{2}^{n}\mathbf{x}\_{n}\right)^{2} \\ &\leq \left(1+\beta\_{n}\left(k^{2}-1\right)\right)d\left(\mathbf{x}\_{n},p\right)^{2}\leq d\left(\mathbf{x}\_{n},p\right)^{2}. \end{split} \tag{13}$$

Again by Lemma 2.4(iii), it follows that

$$\begin{split} d(\mathbf{x}\_{n+1},\boldsymbol{p})^2 &= d\left( (\mathbf{1}-a\_n)\mathbf{x}\_n \oplus a\_n T\_1^n \boldsymbol{y}\_n, \boldsymbol{p} \right)^2 \\ &\leq (\mathbf{1}-a\_n)d(\mathbf{x}\_n,\boldsymbol{p})^2 + a\_n d\left( T\_1^n \mathbf{x}\_n,\boldsymbol{p} \right)^2 - a\_n (\mathbf{1}-a\_n) d\left( \mathbf{x}\_n, T\_1^n \boldsymbol{y}\_n \right)^2 \\ &\leq (\mathbf{1}-a\_n)d(\mathbf{x}\_n,\boldsymbol{p})^2 + a\_n k\_n^2 d(\mathbf{x}\_n,\boldsymbol{p})^2 - a\_n (\mathbf{1}-a\_n) d\left( \mathbf{x}\_n, T\_1^n \boldsymbol{y}\_n \right)^2. \end{split} \tag{14}$$

Equivalently

$$\begin{split} \left(a\_{n}(\mathbf{1}-a\_{n})d\left(\mathbf{x}\_{n},T^{n}\_{\mathbf{1}}\mathbf{y}\_{n}\right)^{2} \quad \leq & \left(\mathbf{1}+a\_{n}\left(\mathbf{k}^{2}\_{n}-\mathbf{1}\right)\right)d\left(\mathbf{x}\_{n},p\right)^{2}-d\left(\mathbf{x}\_{n+1},p\right)^{2} \\ \leq & \left(\mathbf{1}+\left(\mathbf{k}^{2}\_{n}-\mathbf{1}\right)\right)d\left(\mathbf{x}\_{n},p\right)^{2}-d\left(\mathbf{x}\_{n+1},p\right)^{2} \\ = & d\left(\mathbf{x}\_{n},p\right)^{2}-d\left(\mathbf{x}\_{n+1},p\right)^{2}. \end{split} \tag{15}$$

Summing up the first m term of the above inequality, we get

$$\sum\_{n=1}^{m} a\_n (1 - a\_n) d\left(\mathbf{x}\_n, \, T\_1^n \boldsymbol{\uprho}\_n\right)^2 \le d(\mathbf{x}\_1, \, p)^2 - d(\mathbf{x}\_{m+1}, \, p)^2 < \infty \tag{16}$$

for all *m* ≥ 1*:* Now (16) implies that

$$\sum\_{n=1}^{\infty} a\_n (1 - a\_n) d\left(\mathfrak{x}\_n, \, T\_1^n \mathfrak{y}\_n\right)^2 < \infty. \tag{17}$$

Since 0 <sup>≤</sup>*αn*ð Þ <sup>1</sup> � *<sup>α</sup><sup>n</sup>* <sup>&</sup>lt; 1, *d xn*, *<sup>T</sup><sup>n</sup>* <sup>1</sup>*xn* � �<sup>2</sup> ! 0 as *<sup>n</sup>* ! <sup>∞</sup>. Therefore, we obtain

$$\lim\_{n \to \infty} d\left(\kappa\_n, T^n\_{\mathbf{1}} \mathcal{Y}\_n\right) = \mathbf{0}. \quad (\* \,) \tag{18}$$

Since *T*<sup>1</sup> is asymptotically quasi-nonexpansive, we can get that *d T<sup>n</sup>* <sup>1</sup> *yn*, *<sup>p</sup>* � �Þ≤*knd yn*, *<sup>p</sup>* � � for all *<sup>n</sup>*∈ℕ. From (13), we have that

$$\limsup\_{n \to \infty} d\left(T\_1^n y\_n, p\right) \le r. \tag{19}$$

Similarly, we get

$$\limsup\_{n \to \infty} d\left(T\_2^n \kappa\_n, p\right) \le r. \tag{20}$$

One can see that

*Common Fixed Points of Asymptotically Quasi-Nonexpansive Mappings in Cat(0) Spaces DOI: http://dx.doi.org/10.5772/intechopen.107186*

$$\lim\_{n \to \infty} \sup d(\mathfrak{x}\_n, p) \le \lim\_{n \to \infty} d(\mathfrak{x}\_n, p) = r. \tag{21}$$

Since *T*<sup>1</sup> is asymptotically quasi-nonexpansive, we get

$$\begin{array}{ll}d(\boldsymbol{x}\_{n},\boldsymbol{p}) & \leq d\left(\boldsymbol{x}\_{n},\,T^{\boldsymbol{u}}\_{1}\mathcal{Y}\_{n}\right) + d\left(T^{\boldsymbol{u}}\_{1}\mathcal{Y}\_{n},\,\boldsymbol{p}\right) \\ & \leq d\left(\boldsymbol{x}\_{n},\,T^{\boldsymbol{u}}\_{1}\mathcal{Y}\_{n}\right) + k\_{n}d\left(\boldsymbol{y}\_{n},\,\boldsymbol{p}\right). \end{array} \tag{22}$$

Taking the limit inferior to above inequality and from (18), we obtain

$$r \le \liminf\_{n \to \infty} d(y\_n, p). \tag{23}$$

On the other hand, by Lemma 2.4(ii) we have

$$\begin{split} d(\boldsymbol{y}\_n, \boldsymbol{p}) &= d\left( (\mathbb{1} - \boldsymbol{\beta}\_n) \boldsymbol{\kappa}\_n \oplus \boldsymbol{\beta}\_n T\_2^n \boldsymbol{\kappa}\_n, \boldsymbol{p} \right) \\ &\leq (\mathbb{1} - \boldsymbol{\beta}\_n) d(\boldsymbol{\kappa}\_n, \boldsymbol{p}) + \boldsymbol{\beta}\_n d\left( T\_2^n \boldsymbol{\kappa}\_n, \boldsymbol{p} \right) \\ &= [(\mathbb{1} + \boldsymbol{\beta}\_n (\boldsymbol{k}\_n - \mathbf{1})] d(\boldsymbol{\kappa}\_n, \boldsymbol{p}) \end{split} \tag{24}$$

which implies

$$\limsup\_{n \to \infty} d(y\_n, p) \le r. \tag{25}$$

This gives

$$\lim\_{n \to \infty} d\left( (\mathbf{1} - a\_n) \boldsymbol{\kappa}\_n \oplus a\_n T\_2^n \boldsymbol{\kappa}\_n, p \right) = r. \tag{26}$$

Using (20), (21), (26), and Lemma 2.9, we obtain

$$\lim\_{n \to \infty} d\left(\mathfrak{x}\_n, \, T\_2^n \mathfrak{x}\_n\right) = \mathbf{0}. \,\, (\* \,) \tag{27}$$

From (23) and (25), we obtain

$$\lim\_{n \to \infty} d(y\_n, p) = r.\tag{28}$$

On the other hand, consider

$$\begin{split}d(\mathfrak{x}\_{n+1},p) &= d\left((\mathfrak{1}-a\_{n})\mathfrak{x}\_{n}\oplus a\_{n}T\_{1}^{\mathfrak{u}}\mathfrak{y}\_{n},p\right) \\ &\leq (\mathfrak{1}-a\_{n})d(\mathfrak{x}\_{n},p) + a\_{n}k\_{n}d(\mathfrak{y}\_{n},p). \end{split} \tag{29}$$

This implies

$$\lim\_{n \to \infty} d\left( (\mathbf{1} - a\_n) \mathbf{x}\_n \oplus a\_n T\_\mathbf{1}^n \mathbf{y}\_n, p \right) = r. \tag{30}$$

From (19), (21), (30), and by Lemma 2.9, we also obtain

$$\lim\_{n \to \infty} d\left(\mathfrak{x}\_n, \, T\_{\mathbf{1}}^n \mathfrak{y}\_n\right) = \mathbf{0}.\tag{31}$$

Next, we show lim*<sup>n</sup>*!<sup>∞</sup> *d xn*, *<sup>T</sup><sup>n</sup>* <sup>1</sup>*xn* <sup>¼</sup> <sup>0</sup>*:*. Consider

$$\begin{aligned} d\left(\mathfrak{x}\_{n},\boldsymbol{y}\_{n}\right) &\quad \leq d\left(\mathfrak{x}\_{n},(\mathbbm{1}-\beta\_{n})\mathfrak{x}\_{n}\oplus\beta\_{n}T\_{2}^{\mathfrak{u}}\mathfrak{x}\_{n}\right) \\ &\leq (\mathbbm{1}-\beta\_{n})d(\mathfrak{x}\_{n},\boldsymbol{x}\_{n}) + \beta\_{n}d\left(\mathfrak{x}\_{n},T\_{2}^{\mathfrak{u}}\mathfrak{x}\_{n}\right) \to 0 \text{ as } n \to \infty \end{aligned} \tag{32}$$

and

$$d\left(T\_1^n \mathbf{x}\_n, \mathbf{x}\_n\right) \quad \le d\left(T\_1^n \mathbf{x}\_n, T\_1^n \mathbf{y}\_n\right) + d\left(T\_1^n \mathbf{y}\_n, \mathbf{x}\_n\right). \tag{33}$$

Since *T*<sup>1</sup> is uniformly continuous and *d xn*, *yn* ! 0 as *<sup>n</sup>* ! <sup>∞</sup>, it follows from (32) and (33) that

$$\lim\_{n \to \infty} d\left(T\_1^n \mathfrak{x}\_n, \mathfrak{x}\_n\right) = \mathbf{0}. \ (\* \ )$$

Our proof is finished.

**Theorem 3.4.** *Let* ð Þ , *d be a CAT(0) space and C a nonempty closed convex subset of . Let T*1,*T*<sup>2</sup> : *C* ! *C be two uniformly continuous asymptotically quasi-nonexpansive mappings, and* ¼ *F T*ð Þ<sup>1</sup> ∩ *F T*ð Þ<sup>2</sup> 6¼ ∅*. Let* f g *α<sup>n</sup> and* f g *β<sup>n</sup> be sequences in* ½ � *ε*, 1 � *ε forsome ε*∈ ð Þ 0, 1 *. From arbitrary x*<sup>1</sup> ∈*C, define the sequence x*f g*<sup>n</sup> by the recursion* (6)*. Assume, in addition, that either T*<sup>2</sup> *or T*<sup>1</sup> *is compact. Then x*f g*<sup>n</sup> converges strongly to some common fixed point of T*<sup>1</sup> *and T*2*.*

**Proof:** By Lemma 3.3, we have

$$\lim\_{n \to \infty} d\left(\mathbf{x}\_n, T\_1^n \mathbf{x}\_n\right) = \mathbf{0} = \lim\_{n \to \infty} d\left(\mathbf{x}\_n, T\_2^n \mathbf{x}\_n\right) \tag{34}$$

and also

$$\lim\_{n \to \infty} d\left(\mathfrak{x}\_n, \, T\_1^n \mathfrak{y}\_n\right) = \mathbf{0}.\tag{35}$$

If *T*<sup>2</sup> is compact, then there exists a subsequence *Tnk* <sup>2</sup> *xnk* of *T<sup>n</sup>* <sup>2</sup>*xn* such that *Tnk* <sup>2</sup> *xnk* ! *<sup>p</sup>* as *<sup>k</sup>* ! <sup>∞</sup> for some *<sup>p</sup>*∈*<sup>C</sup>* and so *<sup>T</sup>nk*þ<sup>1</sup> <sup>2</sup> *xnk* ! *T*2*p* as *k* ! ∞. From (34), we have *xnk* ! *<sup>p</sup>* as *<sup>k</sup>* ! <sup>∞</sup>. Also, by (35) we get that *<sup>T</sup>nk* <sup>1</sup> *ynk* ! *p* as *k* ! ∞. Consider

$$\begin{split}d\left(\mathfrak{x}\_{n\_k+1},\mathfrak{x}\_{n\_k}\right) &= d\left((\mathbbm{1}-\mathfrak{a}\_{n\_k})\mathfrak{x}\_{n\_k}\oplus\mathfrak{a}\_{n\_k}T\_1^{n\_k}\mathfrak{y}\_{n\_k},\mathfrak{x}\_{n\_k}\right) \\ &\leq d\left(\mathfrak{x}\_{\mathfrak{x}\_k},T\_1^{n\_k}\mathfrak{y}\_{n\_k}\right). \end{split} \tag{36}$$

From (35) and (36), it follows that *xnk*þ<sup>1</sup> ! *p* as *k* ! ∞. Again, from (35), we have *Tnk*þ<sup>1</sup> <sup>1</sup> *ynk* ! *T*1*p*.

Next, we show that *p*∈. Notice that

$$\begin{split} d(p, T\_2 p) \le d\left(p, \mathbf{x}\_{n\_k+1}\right) &+ d\left(\mathbf{x}\_{n\_k+1}, T\_2^{n\_k+1} \mathbf{x}\_{n\_k+1}\right) \\ &+ d\left(T\_2^{n\_k+1} \mathbf{x}\_{n\_k+1}, T\_2^{n\_k+1} \mathbf{x}\_{n\_k}\right) + d\left(T\_2^{n\_k+1} \mathbf{x}\_{n\_k}, T\_2 p\right) \end{split} \tag{37}$$

Since *T*<sup>2</sup> is uniformly continuous, taking the limit as *k* ! ∞, and using (34) we obtain that *p* ¼ *T*2*p* and so *p*∈ *F T*ð Þ<sup>2</sup> *:* Notice that

*Common Fixed Points of Asymptotically Quasi-Nonexpansive Mappings in Cat(0) Spaces DOI: http://dx.doi.org/10.5772/intechopen.107186*

$$\begin{split} d(p, T\_1 p) \le d\left(p, \mathbf{x}\_{n\_k+1}\right) &+ d\left(\mathbf{x}\_{n\_k+1}, T\_1^{n\_k+1} \mathbf{x}\_{n\_k+1}\right) \\ &+ d\left(T\_1^{n\_k+1} \mathbf{x}\_{n\_k+1}, T\_1^{n\_k+1} \mathbf{x}\_{n\_k}\right) + d\left(T\_1^{n\_k+1} \mathbf{x}\_{n\_k}, T\_1 p\right). \end{split} \tag{38}$$

Letting *k* ! ∞, we also obtain that *p* ¼ *T*1*p* and hence *p*∈*F T*ð Þ<sup>1</sup> . Therefore *p* ∈. Hence, by Lemma 2.6, *xn* ! *<sup>p</sup>*∈, since lim*n*!<sup>∞</sup>*d x*ð Þ *<sup>n</sup>*, *<sup>p</sup>* exists. If *<sup>T</sup>*<sup>1</sup> is compact, then

essentially the same arguments as above our result follow. This completes the proof. We give the following example in support of our main results.

**Example 3.5.** Let <sup>¼</sup> <sup>1</sup> *:* and *C* ¼ ½ � 0, 1 , a closed convex subset of and define *T*1,*T*<sup>2</sup> : *C* ! *C* by

$$T\_1 x = \begin{cases} \frac{x}{2} & \text{, if } x \in \left[0, \ \frac{1}{2}\right] \\\\ 0 & \text{, if } x \in \left(\frac{1}{2}, \ 1\right] \end{cases}$$

and

$$T\_2 \mathfrak{x} = \begin{cases} \mathfrak{x} & \text{, if } \mathfrak{x} \in \left[ \mathbf{0}, \ \frac{1}{2} \right] \\ \mathbf{1} & \text{, if } \mathfrak{x} \in \left( \frac{1}{2}, \ \mathbf{1} \right] . \end{cases}$$

Then, *T*1,*T*<sup>2</sup> are asymptotically quasi-nonexpansive but not nonexpansive with ¼ f g0 6¼ ∅. For

$$d\left(T\_1^{\mathfrak{n}}\mathfrak{x},\ T\_1^{\mathfrak{n}}\mathfrak{y}\right) \le \frac{1}{2^{\mathfrak{n}}}d(\mathfrak{x},\mathcal{y}) \le d(\mathfrak{x},\mathcal{y}), \forall \mathfrak{x}, \mathcal{y} \in \left[0,\ \frac{1}{2}\right].$$

And

$$d\left(T\_1^\eta \mathfrak{x}, T\_1^\eta \mathfrak{y}\right) = 0 \le d(\mathfrak{x}, \mathfrak{y}), \forall \mathfrak{x}, \mathfrak{y} \in \left(\frac{1}{2}, 1\right].$$

Hence, *T*<sup>1</sup> is asymptotically quasi-nonexpansive. Similarly, we can show that *T*<sup>2</sup> is asymptotically quasi-nonexpansive.

Define a sequence f g *xn* as in (6) by starting from arbitrary *x*<sup>1</sup> ∈*C*,

$$\begin{aligned} \boldsymbol{\chi}\_{n+1} &= (\mathbf{1} - \boldsymbol{\alpha}\_{n}) \boldsymbol{\chi}\_{n} + \boldsymbol{\alpha}\_{n} T\_{1}^{\sf n} \boldsymbol{\chi}\_{n} \\ \boldsymbol{\mathcal{Y}}\_{n} &= (\mathbf{1} - \boldsymbol{\beta}\_{n}) \boldsymbol{\chi}\_{n} + \boldsymbol{\beta}\_{n} T\_{2}^{\sf n} \boldsymbol{\chi}\_{n}, \end{aligned} \tag{39}$$

for all *<sup>n</sup>* <sup>≥</sup>1, where f g *<sup>α</sup><sup>n</sup>* and f g *<sup>β</sup><sup>n</sup>* are sequences in 0, 1 ½ �. Taking *<sup>α</sup><sup>n</sup>* <sup>¼</sup> <sup>1</sup> <sup>2</sup> ¼ *βn:* Next, we construct a sequence f g *xn* . Starting from *x*<sup>1</sup> ¼ 1, we get

$$y\_1 = \frac{1}{2}\mathbf{x}\_1 + \frac{1}{2}T\_2\mathbf{x}\_1 = \frac{1}{2}(\mathbf{1}) + \frac{1}{2}T\_2(\mathbf{1}) = \frac{1}{2}(\mathbf{1}) + \frac{1}{2}\left(\frac{\mathbf{1}}{2}\right) = \frac{3}{4},$$

we get

$$\infty\_2 = \frac{1}{2}\mathbb{x}\_1 + \frac{1}{2}T\_1\mathbb{y}\_1 = \frac{1}{2}(1) + \frac{1}{2}T\_1\left(\frac{3}{4}\right) = \frac{1}{2} + \frac{1}{2}(0) = \frac{1}{2} = 0.5.$$

$$\mathbb{y}\_2 = \frac{1}{2}\mathbb{x}\_2 + \frac{1}{2}T\_2^2\mathbb{x}\_2 = \frac{1}{2}\left(\frac{1}{2}\right) + \frac{1}{2}T\_2^2\left(\frac{1}{2}\right) = \frac{1}{4} + \frac{1}{2}\left(\frac{1}{2}\right) = \frac{1}{2},$$

we get

$$\mathbf{x}\_{3} = \frac{1}{2}\mathbf{x}\_{2} + \frac{1}{2}T\_{2}^{2}\mathbf{y}\_{2} = \frac{1}{2}\left(\frac{1}{2}\right) + \frac{1}{2}T\_{1}^{2}\left(\frac{1}{2}\right) = \frac{1}{4} + \frac{1}{2}\left(\frac{1}{8}\right) = \frac{5}{16} = 0.3125.$$

$$\mathbf{y}\_{3} = \frac{1}{2}\mathbf{x}\_{3} + \frac{1}{2}T\_{2}^{3}\mathbf{x}\_{3} = \frac{1}{2}\left(\frac{5}{16}\right) + \frac{1}{2}T\_{2}^{3}\left(\frac{5}{16}\right) = \frac{5}{32} + \frac{1}{2}\left(\frac{5}{16}\right) = \frac{5}{16},$$

we get

$$\mathbf{x}\_{4} = \frac{1}{2}\mathbf{x}\_{3} + \frac{1}{2}T\_{1}^{3}\mathbf{y}\_{3} = \frac{1}{2}\left(\frac{5}{16}\right) + \frac{1}{2}T\_{1}^{3}\left(\frac{5}{16}\right) = \frac{5}{32} + \frac{1}{2}\left(\frac{5}{128}\right) = \frac{45}{256} = 0.1757.$$

$$\mathbf{y}\_{4} = \frac{1}{2}\mathbf{x}\_{4} + \frac{1}{2}T\_{2}^{4}\mathbf{x}\_{4} = \frac{1}{2}\left(\frac{45}{256}\right) + \frac{1}{2}T\_{2}^{4}\left(\frac{45}{256}\right) = \frac{45}{512} + \frac{1}{2}\left(\frac{45}{256}\right) = \frac{45}{256},$$

we get

$$\mathbf{x}\_5 = \frac{1}{2}\mathbf{x}\_4 + \frac{1}{2}T\_1^4\\\mathbf{y}\_4 = \frac{1}{2}\left(\frac{45}{256}\right) + \frac{1}{2}T\_1^4\left(\frac{45}{256}\right) = \frac{45}{512} + \frac{1}{2}\left(\frac{45}{4096}\right) = \frac{765}{8192} = 0.09333.$$

Proceeding in a similar method, we will get a sequence f g *xn* that converges to 0, the common fixed point of *T*<sup>1</sup> and *T*2, that is, we obtain the sequence

$$1, \frac{1}{2}, \frac{5}{16}, \frac{45}{256}, \frac{765}{8192}, \dots \text{.} \varkappa\_n \to 0.$$

**Corollary 3.6.** *Let be a CAT(0) space and C a nonempty* **compact** *convex subset of* X*. Let T*1,*T*<sup>2</sup> : *C* ! *C be two uniformly continuous asymptotically quasi-nonexpansive mappings, and* ¼ *F T*ð Þ<sup>1</sup> ∩ *F T*ð Þ<sup>2</sup> 6¼ ∅*. Let* f g *α<sup>n</sup> and* f g *β<sup>n</sup> be sequences in* ½ � *ε*, 1 � *ε forsome ε*∈ ð Þ 0, 1 *. From arbitrary x*<sup>1</sup> ∈*C, define the sequence x*f g*<sup>n</sup> by iteration* (6)*. Assume, in addition, that either T*<sup>2</sup> *or T*<sup>1</sup> *is compact. Then, x*f g*<sup>n</sup> converges strongly to some common fixed point of T*<sup>1</sup> *and T*2*.*

**Corollary 3.7.** *Let be a CAT(0) space and C a nonempty* **compact** *convex subset of . Let T* : *C* ! *C be two uniformly continuous asymptotically quasi-nonexpansive mappings with sequences k*f g*<sup>n</sup>* <sup>⊂</sup>½ Þ 0, <sup>∞</sup> *such that* <sup>P</sup><sup>∞</sup> *<sup>n</sup>*¼<sup>1</sup>*kn* <sup>&</sup>lt; <sup>∞</sup>*. Let* f g *<sup>α</sup><sup>n</sup> and* f g *<sup>β</sup><sup>n</sup> be sequences in* ½ � *ε*, 1 � *ε forsome ε*∈ð Þ 0, 1 *. From arbitrary x*<sup>1</sup> ∈*K, define the sequence x*f g*<sup>n</sup> by the iteration*

$$\begin{aligned} \boldsymbol{\varkappa}\_{n+1} &= (\mathbf{1} - \boldsymbol{a}\_n) \boldsymbol{\varkappa}\_n \oplus \boldsymbol{a}\_n T^\mathfrak{n} \boldsymbol{\jmath}\_n \\ \boldsymbol{\varkappa}\_n &= (\mathbf{1} - \boldsymbol{\beta}\_n) \boldsymbol{\varkappa}\_n \oplus \boldsymbol{\beta}\_n T^\mathfrak{n} \boldsymbol{\varkappa}\_n, \end{aligned} \tag{40}$$

with *n*≥1*:* Then, f g *xn* converges strongly to some fixed point of T.

*Common Fixed Points of Asymptotically Quasi-Nonexpansive Mappings in Cat(0) Spaces DOI: http://dx.doi.org/10.5772/intechopen.107186*

**Corollary 3.8.** *Let* X *be a* **Hibert** *space and C a nonempty closed convex subset of* X*. Let T*1,*T*<sup>2</sup> : *C* ! *C be two uniformly continuous asymptotically quasi-nonexpansive mappings, and* ¼ *F T*ð Þ<sup>1</sup> ∩ *F T*ð Þ<sup>2</sup> ≔ f g *x*∈*K* : *T*1*x* ¼ *T*2*x* ¼ *x* 6¼ ∅*. Let* f g *α<sup>n</sup> and* f g *β<sup>n</sup> be sequences in* ½ � *ε*, 1 � *ε forsome ε*∈ ð Þ 0, 1 *. From arbitrary x*<sup>1</sup> ∈*C, define the sequence x*f g*<sup>n</sup> by the iterative scheme* (6)*. Assume, in addition, that either T*<sup>2</sup> *or T*<sup>1</sup> *is compact. Then, x*f g*<sup>n</sup> converges strongly to some common fixed point of T*<sup>1</sup> *and T*2*.*

#### **4. Conclusions**

In this chapter, we establish strong convergence results for two asymptotically quasi-nonexpansive mappings *T*1,*T*<sup>2</sup> in the setting of CAT(0) spaces via the sequence f g *xn* generated iteratively by arbitrary *<sup>x</sup>*<sup>1</sup> <sup>∈</sup>*C*, *xn*þ<sup>1</sup> <sup>¼</sup> ð Þ <sup>1</sup> � *<sup>α</sup><sup>n</sup> xn* <sup>⊕</sup> *<sup>α</sup>nT<sup>n</sup>* <sup>1</sup> *yn*, *yn* ¼ <sup>1</sup> � *<sup>β</sup><sup>n</sup>* ð Þ*xn* <sup>⊕</sup> *<sup>β</sup>nT<sup>n</sup>* <sup>2</sup>*xn*,*n* ≥1*:* We obtained the following results:-.


$$\lim\_{n \to \infty} d\left(\mathfrak{x}\_n, T^n\_2\mathfrak{x}\_n\right) = \lim\_{n \to \infty} d\left(\mathfrak{x}\_n, T^n\_1\mathfrak{x}\_n\right) = \lim\_{n \to \infty} d\left(\mathfrak{x}\_n, T^n\_1\mathfrak{y}\_n\right) = \mathbf{0}.$$

This lemma extends and improves Theorem 3.3 in [24].

d. Theorem 3.4, it is proved that If *T*1,*T*<sup>2</sup> : *C* ! *C* be two uniformly continuous asymptotically quasi-nonexpansive mappings. Suppose, in addition, that either *T*<sup>2</sup> or *T*<sup>1</sup> is compact. Then, f g *xn* converges strongly to some common fixed point of *T*<sup>1</sup> and *T*2. This theorem significantly extends and improves Theorem 1.3 (See [24], Theorem 3.4).

As consequence, we obtain Corollaries 3.6, 3.7, and 3.8. All of our results remain true for the subclass of asymptotically nonexpansive mappings.

#### **Acknowledgements**

The authors would like to thank the referee for his/her useful comments on the improvement of this chapter. The second author wishes to thank the faculty of Science of Maejo University for moral support in the writing of this manuscript.

#### **Competing interests**

The authors declare that they have no competing interests.

*Fixed Point Theory and Chaos*

#### **Author details**

Jamnian Nantadilok<sup>1</sup> \*† and Buraskorn Nuntadilok2†

1 Faculty of Science, Department of Mathematics, Lampang Rajabhat University, Lampang, Thailand

2 Faculty of Science, Department of Mathematics, Maejo University, Chiang Mai, Thailand

\*Address all correspondence to: jamnian2010@gmail.com

† These authors contributed equally.

© 2022 The Author(s). Licensee IntechOpen. This chapter is distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

*Common Fixed Points of Asymptotically Quasi-Nonexpansive Mappings in Cat(0) Spaces DOI: http://dx.doi.org/10.5772/intechopen.107186*

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#### **Chapter 4**

## Iterative Algorithms for Common Solutions of Nonlinear Problems in Banach Spaces

*Getahun Bekele Wega*

#### **Abstract**

The purpose of this manuscript is to construct an iterative algorithm for approximating a common solution of variational inequality problem and *g*-fixed point problem of pseudomonotone and Bregman relatively *g*-nonexpansive mappings, respectively, and prove strong convergence of a sequence generated by the proposed method to a common solution of the problems in real reflexive Banach spaces. The assumption that the mapping is Lipschitz monotone mapping is dispensed with. In addition, we give an application of our main result to find a minimum point of a convex function in real reflexive Banach spaces. Finally, we provide a numerical example to validate our result. Our results extend and generalize many results in the literature.

**Keywords:** common solution, Bregman relatively *g*-nonexpansive, *g*-fixed point, monotone mapping, pseudomonotone mapping, variational inequality

#### **1. Introduction**

Let *E* be a real Banach space with its dual space *E*<sup>∗</sup> . Let *C* be a nonempty, closed, and convex subset of *<sup>E</sup>*. A mapping *<sup>G</sup>* : *<sup>C</sup>* ! *<sup>E</sup>*<sup>∗</sup> is said to be monotone provided that for all points *p* and *q* in *C*,

$$
\langle \mathbf{G}p - \mathbf{G}q, p - q \rangle \ge \mathbf{0}.\tag{1}
$$

It is called *α*-strongly monotone if there exists a positive real number *α* such that for all points *p* and *q* in *C*,

$$\langle \mathbf{G}p - \mathbf{G}q, p - z \rangle \ge \left|| \mathbf{G}p - \mathbf{G}q \right||^2. \tag{2}$$

We remark that *<sup>α</sup>*-strongly monotone is *<sup>α</sup>*�<sup>1</sup>�Lipschitz monotone mapping. A mapping *<sup>G</sup>* : *<sup>C</sup>* ! *<sup>E</sup>*<sup>∗</sup> is called pseudomonotone mapping provided that for all points *<sup>p</sup>* and *q* in *C*,

$$
\langle Gp, p - q \rangle \ge 0 \quad \text{implies} \quad \langle Gq, p - q \rangle \ge 0. \tag{3}
$$

From inequalities (1)-(3) above, we can observe that the class of pseudomonotone mappings contains the classes of monotone and *α*-strongly monotone mappings. Let *<sup>G</sup>* : *<sup>C</sup>* ! *<sup>E</sup>*<sup>∗</sup> be a mapping. The variational inequality problem (VIP) introduced by Hartman and Stampacchia [1] in 1966 is mathematically formulated as the problem of finding a point *z* in *C* such that for all points *p* in *C*,

$$
\langle \mathrm{Gz}, p - z \rangle \ge 0. \tag{4}
$$

We denote the solution set of problem (4) by *VIP C*ð Þ , *V* . This problem contains, as special cases, many problems in the fields of applied mathematics, such as mechanics, physics, engineering, the theory of convex programming, and the theory of control. Consequently, considerable research efforts have been devoted to methods of finding approximate solutions of variational inequality problems in several directions for different classes of mappings (see, e.g., [2–10]).

Several authors have also studied, different iterative algorithms for approximating a common solution of VIP and fixed point problem of Lipschitz monotone and nonexpansive mappings, respectively (see, e.g., [4, 7, 11–15]).

In 2003, Takahashi and Tododa [13] introduced an iterative algorithm for finding a common solution for VIP and fixed point problem of *α*-strongly monotone and nonexpansive mappings, respectively, in Hilbert spaces setting. Under certain conditions, they proved that the sequence generated by their proposed method converges weakly to a common solution.

In 2005, Iiduka and Takashi [3] studied an iterative scheme for finding a common solution of VIP and fixed point problem of *α*-strongly monotone and nonexpansive mappings, respectively, in Hilbert spaces setting. They proved that the sequence generated by their proposed scheme converges strongly to a common solution provided that the control sequences satisfy appropriate conditions.

In 2016, Zhang and Yuan [16] established an algorithm for approximating a common solution of VIP and fixed point problem for a finite family of *α*-inverse strongly monotone and nonexpansive mappings, respectively, in the Hilbert spaces setting. They proved strong convergence of the sequence proposed by their method.

In space, more general than Hilbert spaces, Tufa and Zegeye [17] introduced an iterative algorithm for approximating a common solution of VIP and fixed point problem of Lipschitz monotone and relatively nonexpansive mappings, respectively in real 2-uniformly convex and uniformly smooth Banach spaces. They proved that the sequence generated by their algorithm converges strongly to a common solution of the problems. A mapping *<sup>T</sup>* : *<sup>C</sup>* ! *<sup>E</sup>*<sup>∗</sup> is said to be relatively nonexpansive if *F T*ð Þ 6¼ <sup>∅</sup>, *<sup>ϕ</sup>*ð Þ *<sup>z</sup>*, *Tu* <sup>≤</sup>*ϕ*ð Þ *<sup>z</sup>*, *<sup>u</sup>* <sup>∀</sup> *<sup>u</sup>* <sup>∈</sup>*C*, *<sup>z</sup>*<sup>∈</sup> *F T*ð Þ and *F T* ^ð Þ¼ *F T*ð Þ, where *F T*ð Þ, is the set of fixed points of *<sup>T</sup>* and *F T* ^ð Þ is the set of asymptotical fixed point of *<sup>T</sup>*.

Recently, Wega and Zegeye [18] introduced an iterative scheme for approximating a common solution of VIP and *g*-fixed point problem (GFP) of Lipschitz monotone and Bregman relatively *g*-nonexpansive mappings, respectively in real reflexive Banach spaces and obtained strong convergence results. A mapping *<sup>T</sup>* : *<sup>C</sup>* ! *<sup>E</sup>*<sup>∗</sup> is said to be Bregman relatively *g*-nonexpansive (BRGN) if *F T*ð Þ 6¼ ∅, *Dg*ð Þ *<sup>z</sup>*, *Tu* <sup>≤</sup> *Dg*ð Þ *<sup>z</sup>*, *<sup>u</sup>* <sup>∀</sup> *<sup>u</sup>*∈*C*, *<sup>z</sup>*∈*F T*ð Þ and *<sup>F</sup>*^*<sup>g</sup>* ð Þ¼ *<sup>T</sup> Fg* ð Þ *<sup>T</sup>* , where *Fg*ð Þ *<sup>T</sup>* , is the set of *<sup>g</sup>*-fixed points of *<sup>T</sup>* and *<sup>F</sup>*^*<sup>g</sup>* ð Þ *<sup>T</sup>* is the set of asymptotical *<sup>g</sup>*-fixed point of *<sup>T</sup>*, where *<sup>g</sup>* is a convex function of *E* satisfies certain conditions. A point *z* in *C* is said to be *g*-fixed point of *T* provided that *Tz* ¼ ∇ *gz*.

*Iterative Algorithms for Common Solutions of Nonlinear Problems in Banach Spaces DOI: http://dx.doi.org/10.5772/intechopen.106547*

Motivated and inspired by the above results, it is our purpose in this book chapter to construct an iterative algorithm, which converge strongly to a common element of the set of VIP solutions of continuous pseudomonotone and the set GFPP of BRGN mappings in real reflexive Banach spaces. In addition, we give an application of our main result to find a minimum point of a convex function and provide a numerical example to validate our main result. Our results extend and generalize many results in the literature.

Now, we recall some definitions that we will need in the sequel.

Hereafter in this paper let *E* be a real reflexive Banach space with its dual space *E*<sup>∗</sup> , *C* be a nonempty, convex and closed subset of *E* and let G be a family of proper, lower semi-continuous and convex functions on *E*.

Let *g* be an element of G. The domain of *g*, *dom g*, is given by *dom g* ¼ f g *<sup>p</sup>*<sup>∈</sup> *<sup>E</sup>* : *g p*ð Þ<sup>&</sup>lt; <sup>∞</sup> , the Fenchel conjugate of *<sup>g</sup>* at *<sup>p</sup>*<sup>∗</sup> , *<sup>g</sup>* <sup>∗</sup> *<sup>p</sup>*<sup>∗</sup> ð Þ, is given by *<sup>g</sup>* <sup>∗</sup> *<sup>p</sup>*<sup>∗</sup> ð Þ¼ *sup p*<sup>∗</sup> h i , *<sup>p</sup>* � *g p*ð Þ : *<sup>p</sup>*<sup>∈</sup> *E and p*<sup>∗</sup> <sup>∈</sup>*E*<sup>∗</sup> f g, the subdifferential of *<sup>g</sup>* at *<sup>p</sup>*, *<sup>∂</sup>g p*ð Þ, is given by *<sup>∂</sup>g p*ð Þ¼ *<sup>p</sup>*<sup>∗</sup> <sup>∈</sup>*E*<sup>∗</sup> : *g q*ð Þ≥*g p*ð Þþ *<sup>p</sup>*<sup>∗</sup> f g h i , *<sup>q</sup>* � *<sup>p</sup>* , <sup>∀</sup>*p*<sup>∈</sup> *<sup>E</sup>* , the right-hand derivative of *g* at *u* in the direction of *q*, *g*<sup>0</sup> ð Þ *p*, *q* , is given by:

$$\lg'(p,q) = \lim\_{s \to 0\_+} \frac{\mathbf{g}(p+sq) - \mathbf{g}(p)}{s},\tag{5}$$

and the gradient of *g*, at *p* is a linear function, ∇ *g*, is given by h i ∇*g p*ð Þ, *q* ¼ *g*<sup>0</sup> ð Þ *p*, *q* . The function *g* is called:


G*a*^teaux differentiable function *g* is called Legendre if *g* <sup>∗</sup> is G*a*^teaux differentiable, both *int dom g* and *int dom g* <sup>∗</sup> are nonempty, *dom* <sup>∇</sup> *<sup>g</sup>* <sup>¼</sup> *int dom g* and *dom* <sup>∇</sup> *<sup>g</sup>* <sup>∗</sup> <sup>¼</sup> *int dom g* <sup>∗</sup> .

**Remark 1.1** <sup>∇</sup> *<sup>g</sup>* <sup>∗</sup> <sup>¼</sup> ð Þ <sup>∇</sup> *<sup>g</sup>* �<sup>1</sup> (see, [19]) provided that *<sup>g</sup>* is Legendre function and the gradient of Legendre function *<sup>g</sup>* defined by *g u*ð Þ¼ k k*<sup>u</sup> <sup>p</sup> <sup>p</sup>* is coinciding with the generalized duality map, that is, ∇*g* ¼ *Jp*, where 1ð Þ <*p*, *q* < ∞ and *q* is a conjugate of *p* (see, e.g., [20]).

The Bregman distance with respect to *g* (see, e.g., [21]) is a function *Dg* : *dom g* � *int dom g* ! ½ Þ 0, ∞ defined by:

$$D\_{\mathbb{g}}(q, p) = \mathbf{g}(q) - \mathbf{g}(p) - \langle \nabla \mathbf{g}(p), q - p \rangle,\tag{6}$$

where *g* is G*a*^teaux differentiable. The Bregman projection with respect to *g* at *p* in *int dom g* onto *C* is denoted by *P<sup>g</sup> <sup>C</sup><sup>p</sup>* defined by *Dg <sup>P</sup><sup>g</sup> <sup>C</sup>p*, *<sup>p</sup>* <sup>¼</sup> inf *Dg* ð Þ *<sup>q</sup>*, *<sup>p</sup>* : <sup>∀</sup>*<sup>q</sup>* <sup>∈</sup>*<sup>C</sup>* .

**Remark 1.2** We note that the Bregiman distance is not distance in the usual sense. However, it has the following properties (see, e.g., [22–24]):

i. The three point identity:

$$D\_{\mathfrak{g}}(p,q) + D\_{\mathfrak{g}}(q,w) - D\_{\mathfrak{g}}(p,q) = \langle \nabla \ \mathfrak{g}(w) - \nabla \ \mathfrak{g}(q), p - q \rangle \tag{7}$$

for all *q*∈*dom g* and *p*, *w* ∈*int dom g*.

ii. The four point identity:

$$D\_{\mathcal{S}}(q, p) + D\_{\mathcal{S}}(q, z) - D\_{\mathcal{S}}(w, p) + D\_{\mathcal{S}}(w, z) - \langle \nabla \ \mathbf{g}(z) - \nabla \ \mathbf{g}(p), q - w \rangle,\tag{8}$$

for all *q*, *w* ∈ *dom g* and *p*, *z*∈*int dom g*.

**Lemma 1.3** Let *g* be a totally convex and Gáteaux differentiable on *int domg*. Let *p*∈ *int domg*. Then, the *P<sup>g</sup> <sup>c</sup>* from *E* onto *C* is a unique point with the following properties [25]:

$$\begin{aligned} \text{i. } \langle \nabla \ g(p) - \nabla \ g(z), q - z \le 0 \text{ if and only if } z = P\_C^\sharp p, \forall q \in \mathcal{C}. \\\\ \text{ii. } D\_\mathfrak{g}(p, q) \ge D\_\mathfrak{g}\left(q, P\_C^\sharp p\right) + D\_\mathfrak{g}\left(P\_C^\sharp p, p\right), \forall q \in \mathcal{C}. \end{aligned}$$

Let *<sup>g</sup>* be a Legendre and *Vg* : *<sup>E</sup>* � *<sup>E</sup>*<sup>∗</sup> ! ½ Þ 0, <sup>∞</sup> be a function defined by:

$$W\_{\mathfrak{g}}(p, p^\*) = \mathfrak{g}(p) - \langle p^\*, p \rangle + \nabla \ \mathfrak{g}^\*(q^\*), \forall p \in E, p^\* \in E^\*. \tag{9}$$

Then, *Vg* is nonnegative which satisfies (see, e.g., [26])

$$V\_{\mathfrak{F}}(p, p^\*) = D\_{\mathfrak{F}}(p, \nabla \ g^\*(p^\*)) \tag{10}$$

and

$$V\_{\mathcal{S}}(p, p^\*) \le V\_{\mathcal{S}}(p, p^\* + q^\*) - \langle q^\*, \nabla \ g^\*(p^\*) - p \rangle,\tag{11}$$

for all *p* ∈*E* and *p*<sup>∗</sup> ∈*E*<sup>∗</sup> .

**Lemma 1.4** If *g* is lower, convex, semi-convex proper function, then *g* <sup>∗</sup> is a weak<sup>∗</sup> lower semi-convex and proper function and hence, we have

$$D\_{\mathcal{S}}\left(w, \nabla \ g^\*\left(\sum\_{i=1}^N s\_i \nabla \mathcal{g}\left(p\_i\right)\right)\right) \le \sum\_{i=1}^N s\_i D\_{\mathcal{S}}\left(w, p\_i\right), \tag{12}$$

for all *w* in *E*, where *pi* � �<sup>⊆</sup> *<sup>E</sup>* and f g*si* <sup>⊆</sup>ð Þ 0, 1 with <sup>P</sup>*<sup>N</sup> <sup>i</sup>*¼<sup>1</sup>*si* <sup>¼</sup> 1 [27]. A G*a*^teaux differentiable function *g* is called.

i. Uniformly convex function (see, [28]), provided that for all *p* and *q dom g s*∈½ � 0, 1 , we have

$$\lg(sp + (\mathbf{1} - s)q) \le \lg(p) + (\mathbf{1} - s)\lg(p) - (\mathbf{1} - s)s\phi(\|p - q\|),\tag{13}$$

where *ϕ* is a function that is increasing and vanishes only at zero.

i. Strongly convex with constant *α* >0 for all *u* and *q* elements of *domg* (see, [29])

$$\alpha \langle \nabla \ g(p) - \nabla \ g(p), p - q \rangle \ge \alpha ||p - q||^2. \tag{14}$$

ii. Totaly convex if *ν<sup>g</sup>* ð Þ¼ *p*, *s* inf f g *<sup>p</sup>*∈*E*:k*p*�*q*k¼*<sup>s</sup> Dg* ð Þ *q*, *p* > 0, for all *p*∈ *E* and *s*> 0.

We note that *g* is uniformly convex if and only if *g* is totally convex on bounded subsets of *E* (see, [25], Theorem 2.10 p. 9). Moreover, the class of uniformly convex function functions contains the class of strongly convex functions.

**Lemma 1.5** Let *E* be a Banach space and *r*>0 be a constant. Let *g* : *E* ! be a continuous convex function that is uniformly convex on bounded subsets of *E*. Then,

$$\log\left(\sum\_{k=0}^{n}\beta\_{k}u\_{k}\right)\leq\sum\_{k=0}^{n}\beta\_{k}\mathbf{g}\left(u\_{k}\right)-\beta\_{i}\beta\_{j}\rho\_{r}\left(||u\_{i}-u\_{j}||\right),\tag{15}$$

<sup>∀</sup>0≤*i*,*j*<sup>≤</sup> *<sup>n</sup>*, *uk* <sup>∈</sup>*Br*, *<sup>β</sup><sup>k</sup>* <sup>∈</sup>ð Þ 0, 1 with <sup>P</sup>*<sup>n</sup> <sup>k</sup>*¼<sup>0</sup>*β<sup>k</sup>* <sup>¼</sup> 1, where *<sup>ρ</sup><sup>r</sup>* is the gauge of uniform convexity of *g* [30].

**Lemma 1.6** Let *g* be a total convex G*a*^ teaux differentiable such that *dom g* ¼ *E*. Then, for each *<sup>x</sup>*<sup>∗</sup> <sup>∈</sup>*E*<sup>∗</sup> f g<sup>0</sup> ,~*y*<sup>∈</sup> *<sup>E</sup>*,*x*<sup>∈</sup> *<sup>H</sup>*<sup>þ</sup> and *<sup>x</sup>*<sup>~</sup> <sup>∈</sup> *<sup>H</sup>*�, it holds that

$$D\_{\mathcal{g}}(\tilde{\mathbf{x}}, \boldsymbol{\kappa}) \ge D\_{\mathcal{g}}(\tilde{\mathbf{x}}, \boldsymbol{z}) + D\_{\mathcal{g}}(\boldsymbol{z}, \boldsymbol{\kappa}), \tag{16}$$

where *<sup>z</sup>* <sup>¼</sup> *argminy*<sup>∈</sup> *<sup>H</sup>Dg*ð Þ *<sup>y</sup>*, *<sup>x</sup>* and *<sup>H</sup>*� <sup>¼</sup> *<sup>y</sup>*<sup>∈</sup> *<sup>E</sup>* : *<sup>x</sup>*<sup>∗</sup> f g h i , *<sup>y</sup>* � <sup>~</sup>*<sup>y</sup>* <sup>≤</sup><sup>0</sup> , *<sup>H</sup>* <sup>¼</sup> *<sup>y</sup>*∈*<sup>E</sup>* : *<sup>x</sup>*<sup>∗</sup> f g h i , *<sup>y</sup>* � <sup>~</sup>*<sup>y</sup>* <sup>¼</sup> <sup>0</sup> and *<sup>H</sup>*<sup>þ</sup> <sup>¼</sup> *<sup>y</sup>*∈*<sup>E</sup>* : *<sup>x</sup>*<sup>∗</sup> f g h i , *<sup>y</sup>* � <sup>~</sup>*<sup>y</sup>* <sup>≥</sup> <sup>0</sup> .

### **2. An iterative algorithm for a common solution of variational inequality and** *g*�**fixed problems**

In this section, let *E* be a real reflexive Banach space with its dual space *E*<sup>∗</sup> . Let *C* be a nonempty, closed, and convex subset of *E*. Let *g* : *E* ! �ð � ∞, þ∞ ∈G be a uniformly Fr^*e*chet differentiable Legendre which is bounded, uniformly convex, and strongly coercive on bounded subsets of *E*. We denote the family of such functions by Gð Þ *E* .

In the sequel, we shall make use of the following assumptions.

#### **Assumption:**

\*\*A1) Let  $l \in (0, 1)$ ,  $\mu > 0$  and  $\left[\beta \in \underline{\beta}, \overline{\beta}\right] \subset \left(0, \frac{1}{\mu}\right)$ .

\*\*A2) Let  $\{a\_{\pi}\} \subset (0, c)$  with the properties  $\lim\_{n \to \infty} a\_n = 0$ . and  $\sum\_{n=1}^{\infty} a\_n = \infty$ , where  $c > 0$ .

**Algorithm 1:** For any *x*0,*v*∈*C*, define an algorithm by.

**Step 1.** Compute

$$\mathbf{y}\_n = \nabla \ g^\* \left[ \nabla \ \mathbf{g} \mathbf{x}\_n - \beta \mathbf{G} \mathbf{x}\_n \right] \text{ and } \ d(\mathbf{y}\_n) = \mathbf{x}\_n - P\_C^\mathbf{g} \mathbf{y}\_n. \tag{17}$$

If *d yn* � � <sup>¼</sup> 0 and <sup>∇</sup> *gxn* � *Txn* <sup>¼</sup> 0, then stop and *xn* <sup>∈</sup> <sup>Ω</sup>. Otherwise, **Step 2.** Compute *pn* ¼ *xn* � *τnd yn* � �,

where *τ<sup>n</sup>* ¼ *l j <sup>n</sup>* and *j <sup>n</sup>* is the smallest nonnegative integer *j* satisfying

$$\left< \left( \mathbf{G} \mathbf{x}\_n - \mathbf{G} p\_n, \ \ d \left( \mathbf{y}\_n \right) \right) \right> \le \mu D\_\mathbf{g} \left( P\_\mathbf{C}^\mathbf{g} \mathbf{y}\_n, \ \mathbf{x}\_n \right). \tag{18}$$

**Step 3.** Compute

$$\begin{cases} \boldsymbol{a}\_{n} = \boldsymbol{P}\_{p\_{n}}^{\boldsymbol{f}} \nabla \ \text{g}^{\*} \left( \nabla \ \text{g} \boldsymbol{\chi}\_{n} - \beta \text{G} \boldsymbol{p}\_{n} \right), \\ \boldsymbol{r}\_{n} = \nabla \ \text{g}^{\*} \left( \boldsymbol{\eta}\_{n,1} \nabla \ \text{g} \boldsymbol{\chi}\_{n} + \boldsymbol{\eta}\_{n,2} T \boldsymbol{\chi}\_{n} + \boldsymbol{\eta}\_{n,3} \nabla \ \text{g} \boldsymbol{u}\_{n} \right), \\ \boldsymbol{\chi}\_{n+1} = \boldsymbol{P}\_{C}^{\boldsymbol{\xi}} \ \text{g}^{\*} \left( \boldsymbol{\alpha}\_{n} \nabla \ \text{g} \boldsymbol{v} + (\mathbf{1} - \boldsymbol{a}\_{n}) \nabla \ \text{g} \boldsymbol{r}\_{n} \right), \end{cases} \tag{19}$$

where *g* ∈ Gð Þ *E* , *Pn* ¼ *p* ∈*C* : *Gpn*, *p* � *pn* � � <sup>¼</sup> <sup>0</sup> � �, *un* <sup>¼</sup> *<sup>P</sup><sup>g</sup> <sup>C</sup>an* and *η<sup>n</sup>*,*<sup>i</sup>* � �⊂½ Þ *<sup>ε</sup>*, 1 <sup>⊂</sup>ð Þ 0, 1 , for *<sup>i</sup>* <sup>¼</sup> 1,2,3 such that <sup>P</sup><sup>3</sup> *<sup>i</sup>*¼<sup>1</sup>*η<sup>n</sup>*,*<sup>i</sup>* <sup>¼</sup> 1, <sup>∀</sup>*<sup>n</sup>* <sup>≥</sup>0. **Step 4.** Set *n* ≔ *n* þ 1 and go to **Step 1**.

We shall need the following Lemmas in the sequel.

**Lemma 1.7** Assume that f g *xn* and *yn* � � are sequences generated by Algorithm 1. Then, the search rule in Step 2 is well defined.

**Proof:** Since *l* ∈ð Þ 0, 1 and *G* is continuous on *C*, we have

$$
\langle \left( G\boldsymbol{x}\_{n} - \boldsymbol{G}\boldsymbol{p}\_{n}, \boldsymbol{d}\left( \boldsymbol{y}\_{n} \right) \right) \rangle \to \mathbf{0} \tag{20}
$$

as *<sup>j</sup>* ! <sup>∞</sup>. On the other hand, the fact that *Dg <sup>P</sup><sup>g</sup> <sup>C</sup>yn*, *xn* � �> 0, there exists a nonnegative integer *j <sup>n</sup>* satisfying the inequality in Step 2, and the claim holds.

**Lemma 1.8** Assume that f g *xn* and *yn* � � are sequences generated by Algorithm 1. Then, we have:

$$\left< \mathbf{G} \mathbf{x}\_n, d\left(\mathbf{y}\_n\right) \right> \geq \frac{1}{\beta} D\_{\xi} \left( P\_{\mathcal{O}}^{\mathcal{S}} \mathbf{y}\_n, \mathbf{x}\_n \right) \tag{21}$$

**Proof:** From (17), we have:

$$
\nabla \text{ g} \mathbf{y}\_n = \nabla \text{ g} \mathbf{x}\_n - \beta \mathbf{G} \mathbf{x}\_n,\tag{22}
$$

which implies:

$$
\nabla \text{ gx}\_n - \nabla \text{ gy}\_n = \beta \text{Gx}\_n. \tag{23}
$$

Thus, from (23), (17), and (7), we get:

$$
\langle \left( \mathbf{G} \mathbf{x}\_n, d(\mathbf{y}\_n) \right) \rangle = \frac{1}{\beta} \langle \nabla \ \mathbf{g} \mathbf{x}\_n - \nabla \ \mathbf{g} \mathbf{y}\_n, \mathbf{x}\_n - P\_C^\mathbf{g} \mathbf{y}\_n \rangle \tag{24}
$$

$$\mathcal{I} = \frac{1}{\beta} \left[ D\_{\mathcal{S}} \left( P\_{\mathcal{C}}^{\mathcal{S}} y\_n, \ \mathbf{x}\_n \right) + D\_{\mathcal{S}} \left( \mathbf{x}\_n, \ \mathcal{Y}\_n \right) - D\_{\mathcal{S}} \left( P\_{\mathcal{C}}^{\mathcal{S}} y\_n, \ \mathcal{Y}\_n \right) \right] \tag{25}$$

$$\geq \frac{1}{\beta} D\_{\mathbb{g}} \left( P\_{\mathcal{O}}^{\mathbb{g}} \mathcal{y}\_n, \varkappa\_n \right), \tag{26}$$

and hence the assertion hold.

*Iterative Algorithms for Common Solutions of Nonlinear Problems in Banach Spaces DOI: http://dx.doi.org/10.5772/intechopen.106547*

**Lemma 1.9** Suppose the assumption (A1) holds. Let *<sup>G</sup>* : *<sup>C</sup>* ! *<sup>E</sup>*<sup>∗</sup> be a continuous pseudomonotone mapping. Then, *Gpn*, *xn* � *pn* ≥*τ<sup>n</sup>* <sup>1</sup> *<sup>β</sup>* � *μ Dg Pg <sup>C</sup>yn*, *xn* . In particular, if *d yn* 6¼ 0, then *Gpn*, *xn* � *pn* >0.

**Proof:** Using Step 2 of the algorithm we know that

$$
\langle \langle \mathcal{G}p\_n, \boldsymbol{\varkappa}\_n - \boldsymbol{p}\_n \rangle = \langle \mathcal{G}p\_n, \boldsymbol{\varkappa}\_n - \left(\boldsymbol{\varkappa}\_n - \boldsymbol{\pi}\_n d\left(\boldsymbol{\jmath}\_n\right)\right) \rangle \tag{27}
$$

$$=\mathfrak{r}\_n \langle \mathsf{G}p\_n, d(\mathsf{y}\_n) \rangle. \tag{28}$$

On the other hand, from (18), we have:

$$\left< \left( G\boldsymbol{\kappa}\_{n} - G\boldsymbol{p}\_{n}, d\left( \boldsymbol{\mathcal{y}}\_{n} \right) \right) \leq \mu \boldsymbol{D}\_{\mathcal{g}}\left( \boldsymbol{P}\_{\mathcal{C}}^{\mathcal{G}} \boldsymbol{\mathcal{y}}\_{n}, \boldsymbol{\kappa}\_{n} \right) \tag{29}$$

which implies that

$$<\langle \mathcal{G}p\_n, d(\boldsymbol{y}\_n) \rangle \ge \langle \mathcal{G}\boldsymbol{\kappa}\_n, d(\boldsymbol{y}\_n) \rangle - \mu \mathcal{D}\_{\mathfrak{E}}(\mathcal{P}\_{\mathcal{G}}^{\mathfrak{E}}\boldsymbol{y}\_n, \boldsymbol{\kappa}\_n). \tag{30}$$

From (30) and Lemma 8, we get:

$$
\langle \left( G p\_n, d(y\_n) \right) \rangle \ge \left( \frac{1}{\beta} - \mu \right) D\_\mathcal{S} \left( P\_\mathcal{G}^\mathcal{Y} y\_n, \varkappa\_n \right). \tag{31}
$$

Combining (28) and (31), we obtain:

$$\langle \!\langle G p\_n, \varkappa\_n - p\_n \rangle \ge \tau\_n \left( \frac{1}{\beta} - \mu \right) D\_{\lg} \left( P\_C^{\lg} \wp\_n, \varkappa\_n \right), \tag{32}$$

and the proof is complete.

**Theorem 1.10** Suppose the Assumptions (A1) and (A2) hold. Let *<sup>G</sup>* : *<sup>C</sup>* ! *<sup>E</sup>*<sup>∗</sup> and *<sup>T</sup>* : *<sup>C</sup>* ! *<sup>E</sup>*<sup>∗</sup> be continuous pseudomonotone and BRGN mappings, respectively, with <sup>Ω</sup> <sup>¼</sup> *VI C*ð Þ , *G* ∩ *Fg*ð Þ *T* ¼6 ∅. Then, the sequens f g *xn* generated by Algorithm 1 is bounded.

**Proof:** Let *<sup>x</sup>*<sup>∗</sup> <sup>¼</sup> *<sup>P</sup><sup>g</sup>* <sup>Ω</sup>ð Þ*<sup>v</sup>* and *wn* <sup>¼</sup> <sup>∇</sup> *<sup>g</sup>* <sup>∗</sup> *<sup>α</sup>n*<sup>∇</sup> *gv* <sup>þ</sup> ð Þ <sup>1</sup> � *<sup>α</sup><sup>n</sup>* <sup>∇</sup> *grn* . We note that from Lemma 1.3 (i), we obtain

$$
\langle \mu - \varkappa^\*, \nabla \text{ g} \nu - \nabla \text{ g} \varkappa^\* \rangle \le 0, \forall \mu \in \Omega. \tag{33}
$$

Now, for each *n*≥ 0, define the sets: *P*� *<sup>n</sup>* ¼ f g *p*∈*C* : h i *Gxn*, *p* � *xn* ≤0 , *Pn* ¼ f g *p*∈*C* : h i *Gxn*, *p* � *xn* ¼ 0 , and *P*<sup>þ</sup> *<sup>n</sup>* <sup>¼</sup> f g *<sup>p</sup>*∈*<sup>C</sup>* : h i *Gxn*, *<sup>p</sup>* � *xn* <sup>≥</sup> <sup>0</sup> . Let *<sup>x</sup>*<sup>∗</sup> <sup>∈</sup> <sup>Ω</sup>, from definition of *<sup>G</sup>*, we have *Gx*<sup>∗</sup> , *<sup>y</sup>* � *<sup>x</sup>*<sup>∗</sup> h i<sup>≥</sup> 0, which implies that *Gy*, *<sup>y</sup>* � *<sup>x</sup>*<sup>∗</sup> h i<sup>≥</sup> 0 for all *y*∈*C*, and hence, *x*<sup>∗</sup> ∈*P*� *<sup>n</sup>* for all *n* ≥0. Moreover, from Lemma 9, we have *Gpn*, *xn* � *pn* >0, which implies that *xn* ∈*P*<sup>þ</sup> *<sup>n</sup>* and *xn* ∉ *P*� *<sup>n</sup>* for all *n*≥0. Now, from Lemma 1.6, we get:

$$D\_{\mathcal{g}}(\boldsymbol{\kappa}^\*, \ \boldsymbol{a}\_n) + D\_{\mathcal{g}}(\boldsymbol{a}\_n, \ \boldsymbol{\kappa}\_n) \le D\_{\mathcal{g}}(\boldsymbol{\kappa}^\*, \ \boldsymbol{\kappa}\_n). \tag{34}$$

Since *un* <sup>¼</sup> *<sup>P</sup><sup>g</sup> <sup>C</sup>an*, from Lemma 1.3, we get:

$$D\_{\mathcal{g}}(\boldsymbol{\mathfrak{x}}^{\*}, \ u\_{n}) + D\_{\mathcal{g}}(u\_{n}, \ a\_{n}) \leq D\_{\mathcal{g}}(\boldsymbol{\mathfrak{x}}^{\*}, \ a\_{n}).\tag{35}$$

Substituting (35) into (34), we obtain:

$$D\_{\mathcal{g}}(\boldsymbol{\varkappa}^\*, \ \boldsymbol{u}\_n) + D\_{\mathcal{g}}(\boldsymbol{u}\_n, \ \ \boldsymbol{a}\_n) + D\_{\mathcal{g}}(\boldsymbol{a}\_n, \ \ \boldsymbol{\varkappa}\_n) \le D\_{\mathcal{g}}(\boldsymbol{\varkappa}^\*, \ \ \ \ \boldsymbol{\varkappa}\_n), \tag{36}$$

which implies that

$$D\_{\mathcal{g}}(\boldsymbol{\mathfrak{x}}^{\*}, \ \boldsymbol{u}\_{n}) \leq D\_{\mathcal{g}}(\boldsymbol{\mathfrak{x}}^{\*}, \ \boldsymbol{\mathfrak{x}}\_{n}) - D\_{\mathcal{g}}(\boldsymbol{u}\_{n}, \ \boldsymbol{a}\_{n}) - D\_{\mathcal{g}}(\boldsymbol{a}\_{n}, \ \boldsymbol{\mathfrak{x}}\_{n}).\tag{37}$$

Using the same techniques of proof of Theorem 3.2 pp. 64 of [18] from (19), (9), (10), and Lemma 1.5, we get:

$$D\_{\mathbf{g}}(\mathbf{x}^\*, \ \cdot \ \mathbf{r}\_n) \le \eta\_{n,1} D\_{\mathbf{g}}(\mathbf{x}^\*, \ \cdot \ \mathbf{x}\_n) + \eta\_{n,2} D\_{\mathbf{g}}(\mathbf{x}^\*, \ \cdot \ \nabla \ \mathbf{g}^\* T \mathbf{x}\_n) + \eta\_{n,3} D\_{\mathbf{g}}(\mathbf{x}^\*, \ \cdot \ \mathbf{u}\_n) \tag{38}$$

$$-\eta\_{n,1}\eta\_{n,2}\rho\_r^\*\left(||\nabla\ \text{gx}\_n - T\text{x}\_n||\right).\tag{39}$$

From inequalities (37) and (39) above and assumption on *T*, we get:

$$D\_{\mathbf{g}}(\mathbf{x}^\*, \ \cdot \ \mathbf{z}\_n) \le \eta\_{n,1} D\_{\mathbf{g}}(\mathbf{x}^\*, \ \cdot \ \mathbf{x}\_n) + \eta\_{n,2} D\_{\mathbf{g}}(\mathbf{x}^\*, \ \cdot \ \mathbf{x}\_n) + \eta\_{n,3} D\_{\mathbf{g}}(\mathbf{x}^\*, \ \cdot \ \mathbf{z}\_n) \tag{40}$$

$$\leq D\_{\mathcal{g}}(\boldsymbol{\infty}^\*, \ \boldsymbol{\infty}\_n) - \eta\_{n,3} \left[ D\_{\mathcal{g}}(\boldsymbol{u}\_n, \ \boldsymbol{a}\_n) + D\_{\mathcal{g}}(\boldsymbol{a}\_n, \ \boldsymbol{\infty}\_n) \right] \tag{41}$$

$$-\eta\_{n,1}\eta\_{n,2}\rho\_r^\* \left( ||\nabla \text{ gx}\_n - T\mathbf{x}\_n|| \right) \tag{42}$$

$$\leq D\_{\mathfrak{g}}(\mathfrak{x}^\*, \ , \ \varkappa\_n). \tag{43}$$

Now, from (19), Lemma 1.3 (ii), Lemma 1.4, and (44), we obtain:

*Dg <sup>x</sup>*<sup>∗</sup> ð Þ , *xn*þ<sup>1</sup> <sup>≤</sup> *Dg <sup>x</sup>*<sup>∗</sup> , <sup>∇</sup> *<sup>g</sup>* <sup>∗</sup> *<sup>α</sup>n*<sup>∇</sup> *gv* <sup>þ</sup> ð Þ <sup>1</sup> � *<sup>α</sup><sup>n</sup>* <sup>∇</sup> *grn* (44)

$$\leq a\_n D\_\mathcal{g}(\boldsymbol{\varkappa}^\*, \ \boldsymbol{\nu}) + (\mathbf{1} - a\_n) D\_\mathcal{g}(\boldsymbol{\varkappa}^\*, \ \boldsymbol{\varkappa}\_n) \tag{45}$$

$$\leq \max\left\{ D\_{\mathcal{g}}(\boldsymbol{\pi}^\*, \quad \boldsymbol{v}), D\_{\mathcal{g}}(\boldsymbol{\pi}^\*, \quad \boldsymbol{\pi}\_n) \right\},\tag{46}$$

and by induction, we get:

$$D\_{\xi}(\mathbf{x}^\*, \ \varkappa\_n) \le \max \left\{ D\_{\xi}(\mathbf{x}^\*, \ \ v), D\_{\xi}(\mathbf{x}^\*, \ \varkappa\_0) \right\}. \tag{47}$$

Hence, the sequence *Dg <sup>x</sup>*<sup>∗</sup> ð Þ , *xn* is bounded. Thus, by Lemma 7 in ref. [31], the sequence f g *xn* is bounded and so are f g *an* , f g *un* , f g *rn* , *Gpn* , and f g *Txn* .

**Theorem 1.11** Suppose the Assumptions (A1) and (A2) hold. Let *<sup>G</sup>* : *<sup>C</sup>* ! *<sup>E</sup>*<sup>∗</sup> and *<sup>T</sup>* : *<sup>C</sup>* ! *<sup>E</sup>*<sup>∗</sup> be continuous pseudomonotone and BRGN mappings, respectively with Ω ¼ *VI C*ð Þ , *G* ∩ *Fg* ð Þ *T* 6¼ ∅. Then, the sequens f g *xn* generated by Algorithm 1 converge strongly to an element *<sup>x</sup>*<sup>∗</sup> <sup>¼</sup> *<sup>P</sup><sup>g</sup>* <sup>Ω</sup>ð Þ*v* .

**Proof:** From Theorem 1.10 above, we know that the sequence f g *xn* is bounded. Let *<sup>x</sup>*<sup>∗</sup> <sup>¼</sup> *<sup>P</sup><sup>g</sup>* <sup>Ω</sup>ð Þ*v* . Now, using the same techniques of proof of Theorem 2 of ref. [32], we get:

$$D\_{\mathcal{g}}(\mathbf{x}^\*, \ \mathbf{x}\_{n+1}) \le (1 - a\_n) D\_{\mathcal{g}}(\mathbf{x}^\*, \ \mathbf{x}\_n) + a\_n \|\nabla \ \mathbf{g}\nu - \nabla \ \mathbf{g}\mathbf{x}^\*\| \|\mathbf{x}\_n - \mathbf{w}\_n\|\tag{48}$$

$$+a\_n \langle \nabla \text{ g}\boldsymbol{\nu} - \nabla \text{ g}\boldsymbol{\kappa}^\*, \boldsymbol{\kappa}\_n - \boldsymbol{\kappa}^\* \rangle. \tag{49}$$

Furthermore, from (19), Lemma 1.3 (ii), and Lemma 1.4, we have:

*Dg <sup>x</sup>*<sup>∗</sup> ð Þ , *xn*þ<sup>1</sup> <sup>≤</sup> *Dg <sup>x</sup>*<sup>∗</sup> , <sup>∇</sup> *<sup>g</sup>* <sup>∗</sup> *<sup>α</sup>n*<sup>∇</sup> *gv* <sup>þ</sup> ð Þ <sup>1</sup> � *<sup>α</sup><sup>n</sup>* <sup>∇</sup> *grn* (50)

$$\leq a\_n \ D\_\mathbf{g}(\boldsymbol{\pi}^\*, \ \boldsymbol{v}) + (\mathbf{1} - a\_n) \ \boldsymbol{D}\_\mathbf{g}(\boldsymbol{\pi}^\*, \ \boldsymbol{r}\_n). \tag{51}$$

*Iterative Algorithms for Common Solutions of Nonlinear Problems in Banach Spaces DOI: http://dx.doi.org/10.5772/intechopen.106547*

Thus, from (51) and (43), we get:

$$D\_{\mathcal{g}}(\mathbf{x}^\*, \ \mathbf{x}\_{n+1}) \le a\_n \ D\_{\mathcal{g}}(\mathbf{x}^\*, \ \ \nu) + (\mathbf{1} - a\_n) \ D\_{\mathcal{g}}(\mathbf{x}^\*, \ \ \mathbf{x}\_n) \tag{52}$$

$$- \left( \mathbf{1} - a\_n \right) \ \eta\_{n,3} \left[ D\_\mathbf{g} \left( u\_n, \quad a\_n \right) + D\_\mathbf{g} \left( a\_n, \quad \varkappa\_n \right) \right] \tag{53}$$

$$-(\mathbf{1} - a\_n)\eta\_{n,1}\eta\_{n,2}\rho\_r^\*\left(||\nabla\ \mathbf{g}\mathbf{x}\_n - T\mathbf{x}\_n||\right),\tag{54}$$

Now, to complete the proof we use the following two cases:

**Case 1.** Assume that there exists *<sup>n</sup>*<sup>0</sup> <sup>∈</sup><sup>ℕ</sup> such that the sequence *Dg <sup>x</sup>*<sup>∗</sup> ð Þ , *xn* is decreasing for all *<sup>n</sup>*<sup>≥</sup> *<sup>n</sup>*0. It then follows that the sequence *Dg <sup>x</sup>*<sup>∗</sup> ð Þ , *xn* converges, and hence, *Dg <sup>x</sup>*<sup>∗</sup> ð Þ� , *xn Dg <sup>x</sup>*<sup>∗</sup> ð Þ! , *xn*þ<sup>1</sup> 0 as *<sup>n</sup>* ! <sup>∞</sup>. Thus, from (53), we obtain:

$$\lim\_{n \to \infty} \left[ D\_{\xi}(u\_n, \quad a\_n) + D\_{\xi}(a\_n, \quad x\_n) \right] = 0,\tag{55}$$

and

$$\lim\_{n \to \infty} \rho\_r^\* \left( \left\| T\mathbf{x}\_n - \nabla \text{ gx}\_n \right\| = \mathbf{0}.\tag{56}$$

Hence, from (55) and Lemma 2.4 of [33] p. 15, we get:

$$\lim\_{n \to \infty} ||u\_n - a\_n|| = \lim\_{n \to \infty} ||x\_n - a\_n|| = \mathbf{0}.\tag{57}$$

From (56) and property of *ρ* <sup>∗</sup> *<sup>r</sup>* , we obtain:

$$\lim\_{n \to \infty} \|T\mathbf{x}\_n - \nabla \text{ gx}\_n\| = \mathbf{0}.\tag{58}$$

From (58) and the fact that ∇ *g* <sup>∗</sup> is uniformly continuous on bounded subsets of *E*<sup>∗</sup> , we obtain:

$$\lim\_{n \to \infty} \|\nabla \text{ g}^\* T \mathfrak{x}\_n - \mathfrak{x}\_n\| = \mathbf{0}.\tag{59}$$

Moreover, from (19) and Lemma 1.4, we get:

$$D\_{\mathbf{g}}(\mathbf{x}\_{n},\ \left.w\_{n}\right) = D\_{\mathbf{g}}\left(\mathbf{x}\_{n},\ \ \nabla\ \mathbf{g}^{\*}\left[a\_{n}\ \ \nabla\ \ g\nu + (\mathbf{1} - a\_{n})\ \ \nabla\ \ g\mathbf{r}\_{n}\right]\right) \tag{60}$$

$$\leq a\_n \ D\_\mathbf{g}(\mathbf{x}\_n, \ \upsilon) + (\mathbf{1} - a\_n) \ D\_\mathbf{g}(\mathbf{x}\_n, \ \upsilon\_n) \tag{61}$$

$$\mathbf{x} = a\_n \ D\_\mathbf{g}(\mathbf{x}\_n, \ \nu) + (\mathbf{1} - a\_n) \ \begin{bmatrix} \eta\_{n,1} D\_\mathbf{g}(\mathbf{x}\_n, \ \mathbf{x}\_n) + \eta\_{n,2} D\_\mathbf{g}(\mathbf{x}\_n, \ \nabla \ \mathbf{g}^\* \ \mathbf{T} \mathbf{x}\_n) \end{bmatrix} \tag{62}$$

þð Þ 1 � *α<sup>n</sup> η<sup>n</sup>*,3*Dg*ð Þþ *xn*, *un η<sup>n</sup>*,4*Dg* ð Þ *xn*, *vn* (63)

Thus, from Lemma 2.4 of [33] p. 15, (57), (59), and (61), we get:

$$\lim\_{n \to \infty} D\_G(\mathfrak{x}\_n, \ \left. w\_n \right\vert = \mathbf{0}, \tag{64}$$

which implies that

$$\lim\_{n \to \infty} ||x\_n - w\_n|| = 0.\tag{65}$$

Now, since f g *xn* is bounded in *C* there exists *u* ∈*C* and a subsequence *xnk* � � such that *xnk* � � converges weakly to *u* and

$$\limsup\_{n \to \infty} \langle \mathfrak{x}\_n - \mathfrak{x}^\*, \nabla \text{ g}v - \nabla \text{ g}\mathfrak{x}^\* \rangle = \lim\_{k \to \infty} \langle \mathfrak{x}\_{\mathfrak{n}\_k} - \mathfrak{x}^\*, \nabla \text{ g}v - \nabla \text{ g}\mathfrak{x}^\* \rangle. \tag{66}$$

From (58) and definition of *T*, we have *u*∈ *Fg*ð Þ *T* . Next, we prove that *u*∈*VI C*ð Þ , *G* . Since *an* ∈*Pn*then we can get:

$$\mathbf{0} = \left\langle \mathbf{G}p\_{n\_k}, a\_{n\_k} - p\_{n\_k} \right\rangle \tag{67}$$

$$= \left\langle Gp\_{n\_k}, a\_{n\_k} - \boldsymbol{\varkappa}\_{n\_k} \right\rangle + \left\langle Gp\_{n\_k}, \boldsymbol{\varkappa}\_{n\_k} - p\_{n\_k} \right\rangle \tag{68}$$

which implies that

$$\left\langle \left\langle Gp\_{n\_k}, \mathbf{x}\_{n\_k} - p\_{n\_k} \right\rangle \right\rangle = \left\langle Gp\_{n\_k}, \mathbf{x}\_{n\_k} - a\_{n\_k} \right\rangle \le ||Gp\_{n\_k}|| ||\mathbf{x}\_{n\_k} - a\_{n\_k}||.\tag{69}$$

From (57), (69) and the fact that the sequence *Gpn* � � is bounded, we get:

$$\lim\_{k \to \infty} \left\langle G p\_{n\_k}, \varkappa\_{n\_k} - p\_{n\_k} \right\rangle = \mathbf{0}.\tag{70}$$

Now, we prove

$$\lim\_{k \to \infty} \left|| P\_C^{\mathcal{E}} y\_{n\_k} - \varkappa\_{n\_k} \right|| \quad = \mathbf{0}.\tag{71}$$

From (70), Lemma 1.9 and Lemma 2.4 of [33] p. 15, we get:

$$\lim\_{k \to \infty} \pi\_{n\_k} \| P\_C^{\mathbf{g}} \boldsymbol{\upchi}\_{n\_k} - \boldsymbol{\upchi}\_{n\_k} \| \ = \mathbf{0}.\tag{72}$$

First, consider the case when liminf *<sup>k</sup>*!<sup>∞</sup>*τnk* >0. In this case, there is a constant *τ* > 0 such that *τnk* ≥*τ* >0 for all *k*∈ℕ. Thus, we have:

$$||P\_{\mathcal{O}}^{\mathcal{g}}y\_{n\_k} - \boldsymbol{\chi}\_{n\_k}|| \ = \frac{1}{\tau\_{n\_k}}\boldsymbol{\tau}\_{n\_k}||P\_{\mathcal{O}}^{\mathcal{g}}y\_{n\_k} - \boldsymbol{\chi}\_{n\_k}|| \ \leq \frac{1}{\tau}\boldsymbol{\tau}\_{n\_k}||P\_{\mathcal{O}}^{\mathcal{g}}y\_{n\_k} - \boldsymbol{\chi}\_{n\_k}||.\tag{73}$$

Thus, from (72) and (73), we obtain:

$$\lim\_{k \to \infty} \left\| P\_C^{\mathcal{E}} \mathcal{y}\_{n\_k} - \mathfrak{x}\_{n\_k} \right\| \, \, = \mathbf{0}.\tag{74}$$

Second, we consider the case when liminf *<sup>k</sup>*!<sup>∞</sup>*τnk* ¼ 0. In this case, we take a subsequence *nkj* n o of f g *nk* , if necessary, we assume without loss of generality that

$$\lim\_{k \to \infty} \tau\_{n\_k} = 0 \quad \text{and} \quad \lim\_{k \to \infty} ||\varkappa\_{n\_k} - P\_C^\mathbf{g} \boldsymbol{\jmath}\_{n\_k}|| = \boldsymbol{a} > \mathbf{0}. \tag{75}$$

Consider *p*<sup>0</sup> *nk* <sup>¼</sup> <sup>1</sup> *<sup>l</sup> <sup>τ</sup>nkPg <sup>C</sup>ynk* <sup>þ</sup> <sup>1</sup> � <sup>1</sup> *<sup>l</sup> τnk* � �*xnk* . Then, from (75), we have: *Iterative Algorithms for Common Solutions of Nonlinear Problems in Banach Spaces DOI: http://dx.doi.org/10.5772/intechopen.106547*

$$\lim\_{k \to \infty} ||\boldsymbol{x}\_{n\_k} - \boldsymbol{p}'\_{n\_k}|| \quad = \lim\_{k \to \infty} \frac{1}{l} \boldsymbol{\tau}\_{n\_k} ||\boldsymbol{x}\_{n\_k} - \boldsymbol{P}^{\boldsymbol{g}}\_{\mathcal{O}} \boldsymbol{y}\_{n\_k}|| \quad = \mathbf{0}.\tag{76}$$

From the search rule in Step 2 and the definition of *p*<sup>0</sup> *nk* , we get:

$$
\left\langle \left( \mathbf{G} \mathbf{x}\_{n\_k} - \mathbf{G} \mathbf{p}'\_{n\_k}, \mathbf{x}\_{n\_k} - \mathbf{P}\_{\mathbf{C}}^{\mathbf{g}} \mathbf{y}\_{n\_k} \right) > \mu \mathbf{D}\_{\mathbf{g}} \left( \mathbf{P}\_{\mathbf{C}}^{\mathbf{g}} \mathbf{y}\_{n\_k}, \mathbf{x}\_{n\_k} \right) . \right\rangle . \tag{77}$$

Using (76), (77), and Lemma 2.4 of [33] p. 15, and the fact that *G* is uniformly continuous on bounded subsets of *C*, we obtain:

$$\lim\_{k \to \infty} \|P\_C^{\mathcal{E}} \mathcal{Y}\_{n\_k} - \mathcal{x}\_{n\_k} \|\:\! = \mathbf{0},$$

which is a contradiction to (75). Therefore, the equality in (71) holds. Combining Lemma 1.3 and 17, we get:

$$\left\langle \left( \mathbf{G} \mathbf{x}\_{n\_k}, \mathbf{z} - P\_{\mathbf{C}}^{\mathbf{f}} \mathbf{z}\_{n\_k} \right) \right\rangle \geq \left\langle \nabla \ \mathbf{g} \mathbf{x}\_{n\_k} - \nabla \ \mathbf{g} P\_{\mathbf{C}}^{\mathbf{f}} \mathbf{y}\_{n\_k}, \mathbf{z} - P\_{\mathbf{C}}^{\mathbf{f}} \mathbf{y}\_{n\_k} \right\rangle, \forall \mathbf{z} \in \mathbf{C},\tag{78}$$

which implies that

$$\left\langle \left\langle \mathbf{G} \boldsymbol{\pi}\_{n\_k}, \boldsymbol{y} - \boldsymbol{\pi}\_{n\_k} \right\rangle \geq \left\langle \mathbf{G} \boldsymbol{\pi}\_{n\_k}, P^{\mathbf{g}}\_{\complement\mathbf{G}} \boldsymbol{y}\_{n\_k} - \boldsymbol{\pi}\_{n\_k} \right\rangle \right. \tag{79}$$

$$+\left\langle \nabla \text{ g\r}\_{n\_k} - \nabla \text{ gP}\_{\mathcal{O}}^{\text{g}} \text{y}\_{n\_k}, \mathbf{z} - P\_{\mathcal{O}}^{\text{g}} \text{y}\_{n\_k} \right\rangle, \forall \mathbf{z} \in \mathbf{C}.\tag{80}$$

Thus, from (80), (78) and the fact that ∇ *g* is uniformly continuous, we obtain:

$$\liminf\_{k \to \infty} \left< \mathcal{G} \boldsymbol{\omega}\_{n\_k}, \boldsymbol{z} - \boldsymbol{\omega}\_{n\_k} \right> \geq \mathbf{0}, \forall \boldsymbol{z} \in \mathbf{C}. \tag{81}$$

Moreover, let f g *ξ<sup>k</sup>* be a sequence of decreasing numbers such that f g *ξ<sup>k</sup>* ! 0 as *k* ! ∞ and *w* be an arbitrary element of *C*. Using inequality (81), we can find a large enough *Nk* such that

$$
\left\langle G\boldsymbol{\mathfrak{x}}\_{n\_k}, \boldsymbol{w} - \boldsymbol{\mathfrak{x}}\_{n\_k} \right\rangle + \xi\_k \ge \mathbf{0}, \forall k \ge \mathcal{N}\_k. \tag{82}
$$

From (82) and the fact that *Gxnk* 6¼ 0, we get:

$$
\left< G\boldsymbol{x}\_{n\_k}, \xi\_k d\_k + w - \boldsymbol{x}\_{n\_k} \right> \ge \mathbf{0}, \forall k \ge N\_k,\tag{83}
$$

for some *dk* ∈*C* satisfying *Gxnk* , *dk* � � <sup>¼</sup> 1. In addition, from the definition of *<sup>G</sup>* and inequality (83), we have:

$$<\langle G(w + \xi\_n d\_k w), w + \xi\_k d\_k w - \varkappa\_{n\_k} \rangle \ge 0, \forall k \ge N\_k,\tag{84}$$

which implies that

$$\left\langle Gw, w - \mathfrak{x}\_{n\_k} \right\rangle \ge \left\langle Gw - G(w + \xi\_k d\_k w), w + \xi\_k d\_k w - \mathfrak{x}\_{n\_k} \right\rangle \tag{85}$$

�*ξk*h i *Gw*, *dk* ,∀*k*≥ *Nk*, (86)

Since *ξ<sup>k</sup>* ! 0 as *k* ! ∞ and *G* is continuous, then from inequality (86), we obtain:

$$
\langle \mathcal{G}w, w - u \rangle = \liminf\_{k \to \infty} \left\langle \mathcal{G}w, w - \mathfrak{x}\_{n\_k} \right\rangle \ge \mathbf{0}, \forall w \in \mathbf{C}. \tag{87}
$$

Thus, *u*∈*VI C*ð Þ , *G* , and hence, *u* ∈ Ω. It follows Lemma 1.3 (i), that

$$\limsup\_{n \to \infty} \langle \mathfrak{x}\_n - \mathfrak{x}^\*, \nabla \text{ g}v - \nabla \text{ g}\mathfrak{x}^\* \rangle = \lim\_{k \to \infty} \langle \mathfrak{x}\_{\mathfrak{n}} - \mathfrak{x}^\*, \nabla \text{ g}v - \nabla \text{ g}\mathfrak{x}^\* \rangle \tag{88}$$

$$= \langle \mu - \mathbf{x}^\*, \nabla \text{ g}v - \nabla \text{ g}\mathbf{x}^\* \rangle \le \mathbf{0}.\tag{89}$$

Therefore, from (49), (65), (89), and Lemma 2.5 of [34] p. 243, we conclude that *Dg <sup>x</sup>*<sup>∗</sup> ð Þ! , *xn* 0 as *<sup>n</sup>* ! <sup>∞</sup>. Hence, by Lemma 2.4 of [33] p. 15, *xn* ! *<sup>x</sup>*<sup>∗</sup> as *<sup>n</sup>* ! <sup>∞</sup>. **Case 2.** Suppose that there exists a subsequence f g *ni* of f g*n* such that

$$D\_{\mathfrak{g}}(\boldsymbol{\pi}^\*, \boldsymbol{\pi}\_{n\_i}) < D\_{\mathfrak{g}}(\boldsymbol{\pi}^\*, \boldsymbol{\pi}\_{n\_i+1}), \forall i \in \mathbb{N}.\tag{90}$$

Then, by Lemma 3.1 of [35] p. 904, there exists a nondecreasing sequence f g *mk* in the set of natural numbers such that *mk* ! <sup>∞</sup> as *<sup>k</sup>* ! <sup>∞</sup>, *Dg <sup>x</sup>*<sup>∗</sup> , *xmk* <sup>≤</sup> *Dg <sup>x</sup>*<sup>∗</sup> , *xmk*þ<sup>1</sup> and *Dg <sup>x</sup>*<sup>∗</sup> ð Þ , *xk* <sup>≤</sup> *Dg <sup>x</sup>*<sup>∗</sup> , *xmk*þ<sup>1</sup> for all *k* elements of the set of natural numbers. Thus, from (53), we obtain:

$$\lim\_{k \to \infty} ||u\_{m\_k} - a\_{m\_k}|| = \lim\_{k \to \infty} ||T\mathbf{x}\_{m\_k} - \nabla \text{ gx}\_{m\_k}|| = \mathbf{0}.\tag{91}$$

Moreover, following the methods in Case 1 above, we get:

$$\lim\_{k \to \infty} ||\omega\_{m\_k} - \omega\_{m\_k}||,\tag{92}$$

and

$$\limsup\_{k \to \infty} \langle \langle \mathfrak{x}\_{m\_k} - \mathfrak{x}^\*, \nabla \text{ g}\nu - \nabla \text{g}\mathfrak{x}^\* \rangle \le 0. \tag{93}$$

In addition, from (49) and inequality (90) above, we obtain:

$$D\_{\mathcal{g}}\left(\mathbf{x}^\*, \mathbf{x}\_{m\_k}\right) \le \left||\mathbf{x}\_{m\_k} - r\_{m\_k}||\nabla\_{\mathbf{g}}\mathbf{y} - \nabla\_{\mathbf{g}}\mathbf{g}\mathbf{x}^\*||\tag{94}$$

$$+\langle \mathfrak{x}\_{m\_k} - \mathfrak{x}^\*, \nabla \text{ g}\boldsymbol{\nu} - \nabla \mathfrak{g} \boldsymbol{\mathfrak{x}}^\* \rangle. \tag{95}$$

Therefore, from (92), (93), and (95), we obtain lim *<sup>k</sup>*!<sup>∞</sup>*Dg <sup>x</sup>*<sup>∗</sup> , *xmk* <sup>¼</sup> 0. But from inequality (53), we obtain that lim *<sup>k</sup>*!<sup>∞</sup>*Dg <sup>x</sup>*<sup>∗</sup> , *xmk*þ<sup>1</sup> <sup>¼</sup> 0, which implies that lim *<sup>k</sup>*!<sup>∞</sup>*Dg <sup>x</sup>*<sup>∗</sup> ð Þ¼ , *xk* 0. Thus, by Lemma 2.4 of [33] p. 15 *xk* ! *<sup>x</sup>*<sup>∗</sup> as *<sup>k</sup>* ! <sup>∞</sup>.

We remark that the proof of Theorem 11 provides the following result for a common point in the solution set of VIP and the set of *g*�fixed point of continuous monotone and BRGN, mappings, respectively.

**Theorem 1.12** Suppose the Assumptions (A1) and (A2) hold. Let *<sup>G</sup>* : *<sup>C</sup>* ! *<sup>E</sup>*<sup>∗</sup> and *<sup>T</sup>* : *<sup>C</sup>* ! *<sup>E</sup>*<sup>∗</sup> be continuous monotone and BRGN mappings, respectively with <sup>Ω</sup> <sup>¼</sup> *VI C*ð Þ , *G* ∩ *Fg* ð Þ *T* 6¼ ∅. Then, the sequens f g *xn* generated by Algorithm 1 converge strongly to an element *<sup>x</sup>*<sup>∗</sup> <sup>¼</sup> *<sup>P</sup><sup>g</sup>* <sup>Ω</sup>ð Þ*v* .

*Iterative Algorithms for Common Solutions of Nonlinear Problems in Banach Spaces DOI: http://dx.doi.org/10.5772/intechopen.106547*

If in Algorithm 1, we put *<sup>C</sup>* <sup>¼</sup> *<sup>E</sup>*, then *Pg <sup>C</sup>* is reduced to the identity mapping in *E* and *VI C*ð Þ¼ , *<sup>G</sup> <sup>G</sup>*�<sup>1</sup> ð Þ 0 . Thus, we get the following Algorithm 2 for a common point in the set of zeros and the set of *g*�fixed point of continuous pseudomonotone and BRGN mappings, respectively.

**Algorithm 2:** For any *x*0,*v*∈ *E*, define an algorithm by.

**Step 1.** Compute

$$\mathbf{y}\_n = \nabla \text{ g}^\* \left[ \nabla \text{ g} \mathbf{x}\_n - \beta \mathbf{G} \mathbf{x}\_n \right] \text{ and } \; d(\mathbf{y}\_n) = \mathbf{x}\_n - \mathbf{y}\_n. \tag{96}$$

If *d yn* � � <sup>¼</sup> 0 and <sup>∇</sup> *gxn* � *Txn* <sup>¼</sup> 0, then stop and *xn* <sup>∈</sup> <sup>Ω</sup>. Otherwise, **Step 2.** Compute *pn* ¼ *xn* � *τnd yn* � �,

where *τ<sup>n</sup>* ¼ *l j <sup>n</sup>* and *j <sup>n</sup>* is the smallest nonnegative integer *j* satisfying

$$d\left<\mathbf{G}\mathbf{x}\_n - \mathbf{G}p\_n, \ \left.d\left(\mathbf{y}\_n\right)\right> \le \mu \mathbf{D}\_\mathbf{g}\left(\mathbf{y}\_n, \ \mathbf{x}\_n\right). \tag{97}$$

**Step 3.** Compute

$$\begin{cases} \mathfrak{u}\_{n} = \operatorname{P}\_{\operatorname{P}\_{n}}^{\operatorname{f}} \nabla \text{ g}^{\*} \left( \nabla \ \operatorname{g} \boldsymbol{\chi}\_{n} - \beta \operatorname{G} \boldsymbol{p}\_{n} \right), \\ r\_{n} = \nabla \operatorname{g}^{\*} \left( \eta\_{n,1} \nabla \ \operatorname{g} \boldsymbol{\chi}\_{n} + \eta\_{n,2} \operatorname{T} \boldsymbol{\chi}\_{n} + \eta\_{n,3} \nabla \ \operatorname{g} \boldsymbol{u}\_{n} \right), \\ \boldsymbol{\chi}\_{n+1} = \nabla \ \operatorname{g}^{\*} \left( a\_{n} \nabla \ \operatorname{g} \boldsymbol{v} + (\mathbf{1} - a\_{n}) \nabla \ \operatorname{g} \boldsymbol{r}\_{n} \right), \end{cases} \tag{98}$$

where *g* ∈ Gð Þ *E* , *Pn* ¼ *p* ∈*C* : *Gpn*, *p* � *pn* � � <sup>¼</sup> <sup>0</sup> � �, and *<sup>η</sup><sup>n</sup>*,*<sup>i</sup>* � �<sup>⊂</sup> ½ Þ <sup>ϵ</sup>, 1 <sup>⊂</sup>ð Þ 0, 1 , for *<sup>i</sup>* <sup>¼</sup> 1,2,3 such that <sup>P</sup><sup>3</sup> *<sup>i</sup>*¼<sup>1</sup>*η<sup>n</sup>*,*<sup>i</sup>* <sup>¼</sup> 1, <sup>∀</sup>*<sup>n</sup>* <sup>≥</sup>0. **Step 4.** Set *n* ≔ *n* þ 1 and go to **Step 1**.

**Corollary 1.13** Suppose the Assumptions (A1) and (A2) hold. Let *<sup>G</sup>* : *<sup>E</sup>* ! *<sup>E</sup>*<sup>∗</sup> and *<sup>T</sup>* : *<sup>E</sup>* ! *<sup>E</sup>*<sup>∗</sup> be continuous pseudomonotone and BRGN mappings, respectively with <sup>Ω</sup> <sup>¼</sup> *<sup>G</sup>*�<sup>1</sup> ð Þ 0 ∩ *Fg*ð Þ *T* 6¼ ∅. Then, the sequens f g *xn* generated by Algorithm 2 converge strongly to an element *<sup>x</sup>*<sup>∗</sup> <sup>¼</sup> *<sup>P</sup><sup>g</sup>* <sup>Ω</sup>ð Þ*v* .

If in Algorithm 2, we put *T* ¼ ∇ *g*, the identity mapping in *E*, then we get the following corollary for zero point of continuous pseudomonotone.

**Corollary 1.14** Suppose the Assumptions (A1) and (A2) hold. Let *<sup>G</sup>* : *<sup>E</sup>* ! *<sup>E</sup>*<sup>∗</sup> be a continuous pseudomonotone mapping with *G*�<sup>1</sup> ð Þ 0 6¼ ∅. Then, the sequens f g *xn* generated by Algorithm 2 converge strongly to an element *<sup>x</sup>*<sup>∗</sup> <sup>¼</sup> *<sup>P</sup><sup>g</sup> G*�<sup>1</sup> ð Þ <sup>0</sup> ð Þ*<sup>v</sup>* .

#### **2.1 Application to convex minimization problem**

In this section, we apply Corollary 1.14 to find the minimum point of the convex function in Banach Spaces.

Let *f* : *E* ! be a convex smooth function. We consider the problem of finding a point *z*∈*E* such that

$$f(\mathbf{z}) = \min\_{\mathbf{x} \in E} \{ f(\mathbf{x}) \}. \tag{99}$$

According to Fermat's rule, this problem is equivalent to the problem of finding *z*∈*E* such that

$$\nabla \ f \mathbf{\hat{z}} = \mathbf{0},\tag{100}$$

where ∇ *f* is a gradient of *f*. We note that ∇ *f* is monotone mapping (see, e.g., [36, 37]) and hence pseudomonotone mapping.

Now, if in Algorithm 2, we assume *G* ¼ ∇ *f*, then we obtain the following Algorithm 3 for the minimum point problem of convex functions in real reflexive Banach spaces.

#### **Algorithm 3:** For any *x*0,*v*∈*E*, define an algorithm by.

**Step 1.** Compute

$$\mathbf{y}\_n = \nabla \text{ g}^\* \left[ \nabla \text{ g} \mathbf{x}\_n - \beta \nabla \text{ f} \mathbf{x}\_n \right] \text{ and } \; d(\mathbf{y}\_n) = \mathbf{x}\_n - \mathbf{y}\_n. \tag{101}$$

If *d yn* � � <sup>¼</sup> 0, then stop and *xn* <sup>∈</sup> <sup>Ω</sup>. Otherwise,

**Step 2.** Compute *pn* ¼ *xn* � *τnd yn* � �,

where *τ<sup>n</sup>* ¼ *l j <sup>n</sup>* and *j <sup>n</sup>* is the smallest nonnegative integer *j* satisfying

$$
\left< \nabla \ f \mathbf{x}\_n - \nabla \ f p\_n, \ d \left( \mathbf{y}\_n \right) \right> \le \mu \mathcal{D}\_\mathbf{g} \left( \mathbf{y}\_n, \ \mathbf{x}\_n \right). \tag{102}
$$

**Step 3.** Compute

$$\begin{cases} \mathfrak{u}\_{n} = \operatorname{P}\_{\operatorname{P}\_{\operatorname{s}}}^{\operatorname{f}} \nabla \operatorname{ g}^{\ast} \left( \nabla \operatorname{g} \mathfrak{x}\_{n} - \beta \nabla \operatorname{f} \mathfrak{p}\_{n} \right), \\ \mathfrak{r}\_{n} = \nabla \operatorname{ g}^{\ast} \left( \eta\_{n} \nabla \operatorname{ g} \mathfrak{x}\_{n} + (\mathbbm{1} - \eta\_{n}) \nabla \operatorname{ g} \mathfrak{u}\_{n} \right), \\ \mathfrak{x}\_{n+1} = \nabla \operatorname{ g}^{\ast} \left( a\_{n} \nabla \operatorname{ g} \mathfrak{v} + (\mathbbm{1} - a\_{n}) \nabla \operatorname{ g} \mathfrak{r}\_{n} \right), \end{cases} \tag{103}$$

where *g* ∈Gð Þ *E* , *Pn* ¼ *p*∈*C* : ∇ *fpn*, *p* � *pn* � � <sup>¼</sup> <sup>0</sup> � �, and <sup>f</sup>*η<sup>n</sup>* <sup>⊂</sup>½ Þ <sup>ϵ</sup>,1 <sup>⊂</sup>ð Þ 0, 1 ,∀*n*≥0. **Step 4.** Set *n* ≔ *n* þ 1 and go to **Step 1**.

The method of proof Theorem 1.11 provides the proof of the following theorem of finding the minimum point of a convex function in reflexive Banach spaces.

**Theorem 1.15** Suppose the Assumptions (A1) and (A2) hold. Let *f* : *E* ! be a convex smooth function with ∇*f* is continuous and Ω ¼ f g *z* : *f z*ð Þ¼ min *<sup>x</sup>*∈*Ef x*ð Þ ¼6 ∅. Then, the sequens f g *xn* generated by Algorithm 3 converge strongly to an element *<sup>x</sup>*<sup>∗</sup> <sup>¼</sup> *<sup>P</sup><sup>g</sup>* <sup>Ω</sup>ð Þ*v* .

#### **2.2 Numerical example**

In this section, we provide a numerical example to explain the conclusion of our main result. The following example verifies the conclusion of Theorem 1.11.

**Example 1.16.** Let *E* ¼ be with the standard topology. Define *g* : ! , by *g x*ð Þ¼ *<sup>x</sup>*<sup>2</sup> <sup>2</sup> , then *<sup>g</sup>* <sup>∗</sup> *<sup>x</sup>*<sup>∗</sup> ð Þ¼ *<sup>x</sup>*<sup>∗</sup> <sup>2</sup> <sup>2</sup> and <sup>∇</sup> *g x*ð Þ¼ *<sup>x</sup>* <sup>¼</sup> <sup>∇</sup> *<sup>g</sup>* <sup>∗</sup> *<sup>x</sup>*<sup>∗</sup> ð Þ¼ *<sup>x</sup>*<sup>∗</sup> , where *<sup>x</sup>* <sup>¼</sup> ð Þ *x*1, *x*2, *x*<sup>3</sup> ∈ . Let *C* ¼ f g *x*∈ : k*x*k≤1 . Let *G*,*T* : *C* ! be defined by *G x*ð Þ¼ 1, *x*2, *x*<sup>3</sup> ð Þ *x*1, *x*2, *x*<sup>3</sup> 1*:*8 � ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi *x*2 <sup>1</sup> <sup>þ</sup> *<sup>x</sup>*<sup>2</sup> <sup>2</sup> <sup>þ</sup> *<sup>x</sup>*<sup>2</sup> 3 � � q and *T x*ð Þ¼ 1, *<sup>x</sup>*2, *<sup>x</sup>*<sup>3</sup> ð Þ *<sup>x</sup>*1, *<sup>x</sup>*2, *<sup>x</sup>*<sup>3</sup> <sup>5</sup> , then *G* is continuous pseudomonotone mapping and *T* is BRGN mapping with Ω ¼ *VI C*ð Þ¼ , *G* f g0 ¼ *Fg* ð Þ *T* ¼6 ∅. Now, if we assume *v* ¼ ð Þ¼ *v*1, *v*2, *v*<sup>3</sup> ð Þ 0,0*:*5,0*:*5 , *α<sup>n</sup>* ¼ 1 *<sup>n</sup>*þ10, *<sup>η</sup><sup>n</sup>*,1 <sup>¼</sup> *<sup>η</sup><sup>n</sup>*,2 <sup>¼</sup> <sup>0</sup>*:*<sup>001</sup> <sup>þ</sup> <sup>1</sup> *<sup>n</sup>*þ<sup>1000</sup> and *<sup>η</sup><sup>n</sup>*,3 <sup>¼</sup> <sup>0</sup>*:*<sup>998</sup> � <sup>2</sup> *<sup>n</sup>*þ1000, *<sup>l</sup>* <sup>¼</sup> <sup>0</sup>*:*8, *<sup>μ</sup>* <sup>¼</sup> <sup>0</sup>*:*9 and *<sup>λ</sup>* <sup>¼</sup> <sup>1</sup> for all *n* ≥0, and take different initial points *x*<sup>0</sup> ¼ ð Þ 0, 1, �1 , *x*<sup>0</sup> <sup>0</sup> ¼ ð Þ 1*:*2233,2, �1*:*4532

*Iterative Algorithms for Common Solutions of Nonlinear Problems in Banach Spaces DOI: http://dx.doi.org/10.5772/intechopen.106547*


#### **Table 1.**

*Convergence of the sequence x*f g*<sup>n</sup> generated by Algorithm 1 for different choices of x*0*.*

**Figure 1.** *The graph of* <sup>k</sup>*xn* � *<sup>x</sup>*<sup>∗</sup> <sup>k</sup> *versus number of iterations with different choices of x*0*.*

and *x*<sup>0</sup> 0 <sup>0</sup> ¼ ð Þ 1,2,3 , then in all cases the numerical example result using MATLAB provides that the sequence f g *xn* , generated by Algorithm 1 converges strongly to *<sup>x</sup>*<sup>∗</sup> <sup>¼</sup> ð Þ 0,0,0 (see, **Table 1**). In addition, we have sketched the error term <sup>k</sup>*xn* � *<sup>x</sup>*<sup>∗</sup> <sup>k</sup> for each initial point. From the sketch, we observe that <sup>k</sup>*xn* � *<sup>x</sup>*<sup>∗</sup> k ! 0 as *<sup>n</sup>* ! <sup>∞</sup> (see, **Figure 1**).

#### **3. Conclusions**

In this, manuscript, we introduced an iterative method for approximating a common solution of VIP of continuous pseudomonotone and GFPP of BRGN mappings and proved strong convergence of the sequence generated by the method to a

common solution in real reflexive Banach spaces. In addition, we gave an application of our main result to find a minimum point of convex functions in real reflexive Banach spaces. Finally, a numerical example that supports our main result is presented. Our results extend and generalize many results in the literature. In particular, Theorem 1.11 extends the results in [3, 4, 7, 13, 16, 17, 38] from real Hilbert spaces to real reflexive Banach spaces. Moreover, Theorem 1.11 extends the classes of mappings in Theorem 3.1 of Tufa and Zegeye [17] and Theorem 3.2 of Wega and Zegeye [18] from Lipschitz monotone mapping to continuous pseudomonotone mappings in reflexive real Banach spaces.

### **Conflict of interest**

The authors declare that they have no competing interests.

### **Author details**

Getahun Bekele Wega Jimma University, Jimma, Ethiopia

\*Address all correspondence to: getahunbekele2012@gmail.com

© 2022 The Author(s). Licensee IntechOpen. This chapter is distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

*Iterative Algorithms for Common Solutions of Nonlinear Problems in Banach Spaces DOI: http://dx.doi.org/10.5772/intechopen.106547*

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Section 3
