1. Introduction

Let H nð Þ denote the set of all n � n Hermitian matrices, P nð Þ the set of all n � n Hermitian positive definite matrices, S nð Þ the set of all n � n positive semidefinite matrices. Instead of X ∈P nð Þ we will write X . 0. Furthermore, X ≥0 means X ∈S nð Þ. Also we will use X ≥ Y Xð Þ ≤ Y instead of X � Y ≥0ð Þ Y � X ≥0 . The symbol k:k denotes the spectral norm, that is,

$$\|\mathbf{A}\| = \sqrt{\lambda^+(\mathbf{A}^\*\mathbf{A})},$$

where <sup>λ</sup><sup>þ</sup> <sup>A</sup><sup>∗</sup> ð Þ <sup>A</sup> is the largest eigenvalue of <sup>A</sup><sup>∗</sup> A. We denote by k k: <sup>1</sup> the Ky Fan norm defined by

$$\|\mathbf{A}\|\_1 = \sum\_{i=1}^n s\_i(\mathbf{A}),$$

where sið Þ A , i ¼ 1, …, n, are the singular values of A. Also,

$$\|\mathbf{A}\|\_1 = tr\Big(\left(\mathbf{A}^\*\mathbf{A}\right)^{1/2}\Big),$$

which is tr Að Þ for (Hermitian) nonnegative matrices. Then the set H nð Þ endowed with this norm is a complete metric space. Moreover, H nð Þ is a partially ordered set with partial order ≼, where X ≼Y ⇔ Y ≼X. In this section, denote d Xð Þ¼ ; Y k k Y � X <sup>1</sup> ¼ tr Yð Þ � X . Now, consider the non-linear matrix equation

$$X = Q + \sum\_{i=1}^{m} A\_i^\* \boldsymbol{\gamma}(X) A\_i. \tag{1}$$

d ið Þ ; j ≤d ið Þþ ; k d kð Þ ; j ,

where D xð Þ¼ ; B inff g d xð Þ ; y : y∈ B . Then H is called generalized Pompeiu Hausdorff distance on CLð Þ X . It is well known that H is a metric on CBð Þ X , which is

point <sup>x</sup> if <sup>T</sup> <sup>x</sup> <sup>¼</sup> <sup>x</sup> and if <sup>T</sup> : <sup>X</sup> ! <sup>2</sup><sup>X</sup> is multivalued mapping, then <sup>T</sup> is said to have a fixed point x if x∈T x. We denote by Fix Tf g, the set of all fixed points of

Definition 2.1 [22] Let <sup>T</sup> : <sup>X</sup> ! <sup>2</sup><sup>X</sup> be a multivalued map on a metric space ð Þ X; d , α, η : X � X ! IRþ be two functions where η is bounded, then T is an α∗-

αð Þ y; z ≥ηð Þ y; z implies that α∗ð Þ T y; T z ≥η∗ð Þ T y; T z , y, z∈ X,

Definition 2.2 [23] Let <sup>T</sup> : <sup>X</sup> ! <sup>2</sup><sup>X</sup> be a multivalued map on a metric space ð Þ X; d , α, η : X � X ! ½ Þ 0; ∞ be two functions. We say that T is generalized α∗-

αð Þ y; z ≥ηð Þ y; z implies that αð Þ u; v ≥ηð Þ u; v , for all u ∈Ty, v∈Tz:

Consistent with Du and Khojasteh [24], we denote by MandðIRÞ, the set of all

ð Þ ϑ<sup>2</sup> for any bounded sequence f g tn ⊂ð Þ 0; þ∞ and any nondecreasing sequence

tn þ ϑð Þ tn; sn sn

Example 3.1 [24] Let r∈½ Þ 0; 1 . Then ϑ<sup>r</sup> : IR � IR ! IR defined by ϑrð Þ¼ t; s rs � t is

manageable functions ϑ : IR � IR ! IR fulfilling the following conditions:

lim<sup>n</sup>!<sup>∞</sup> sup

If ηð Þ¼ y; z 1 for all y, z∈ X, then T is said to be generalized α∗-admissible

αð Þ y; z , η∗ð Þ¼ A; B sup

If T : X ! X is a single valued self-mapping on X, then T is said to have a fixed

Denote by 2<sup>X</sup> , the family of all nonempty subsets of <sup>X</sup>, CLð Þ <sup>X</sup> , the family of all nonempty and closed subsets of X, CBð Þ X , the family of all nonempty, closed, and bounded subsets of X and Kð Þ X , the family of all nonempty compact subsets of X.

> D xð Þ ; B ;sup y ∈B

( )

D yð Þ ; A

<sup>y</sup>∈ A, <sup>z</sup> ∈B

ηð Þ y; z :

, 1: (3)

,

for all i, j, k∈ X and ð Þ X; d is called metric space.

<sup>H</sup>ð Þ¼ <sup>A</sup>; <sup>B</sup> max sup <sup>x</sup><sup>∈</sup> <sup>A</sup>

It is clear that, <sup>K</sup>ð Þ <sup>X</sup> <sup>⊆</sup>CBð Þ <sup>X</sup> <sup>⊆</sup>CLð Þ <sup>X</sup> <sup>⊆</sup>2<sup>X</sup> , let

Contraction Mappings and Applications DOI: http://dx.doi.org/10.5772/intechopen.81571

called Pompeiu Hausdorff metric induced by d.

admissible mapping with respect to η, if

α∗ð Þ¼ A; B inf

admissible mapping with respect to η, if

3. Some fixed point results

f g sn ⊂ ð Þ 0; þ∞ , it holds that

a manageable function.

11

ð Þ ϑ<sup>1</sup> ϑð Þ t; s , s � t for all s, t . 0;

<sup>y</sup>∈ A, <sup>z</sup> ∈B

Further, Definition 2.1 is generalized in the following way.

mapping T.

where

mapping.

where Q is a positive definite matrix, Ai, i ¼ 1, …, m, are arbitrary n � n matrices and γ is a mapping from H nð Þ to H nð Þ which maps P nð Þ into P nð Þ. Assume that γ is an order-preserving mapping (γ is order preserving if A, B ∈ H nð Þ with A ≼B implies that γð Þ A ≼ γð Þ B ). There are various kinds of problems in control theory, dynamical programming, ladder networks, etc., where the matrix equations plays a crucial role. Matrix Eq. (1) have been studied by many authors see [1–3].

At the same time, integral equations have been developed to solve boundary value problems for both ordinary and partial differential equations and play a very important role in nonlinear analysis. Many problems of mathematical physics, theory of elasticity, viscodynamics fluid and mixed problems of mechanics of continuous media reduce to the Fredholm integral Eq. A rich literature on existence of solutions for nonlinear integral equations, which contain particular cases of important integral and functional equations can be found, for example, see [4–14]. An important technique to solve integral equations is to construct an iterative procedure to generate approximate solutions and find their limit, a host of attractive methods have been proposed for the approximate solutions of Fredholm integral equations of the second kind, see [15–19]. We consider a non-homogeneous Fredholm integral equation of second kind of the form

$$z(r) = \int\_{b}^{\epsilon} \mathcal{B}(r, s, z(s)) \mathrm{d}s + \mathcal{g}(r), \tag{2}$$

where <sup>t</sup><sup>∈</sup> ½ � <sup>b</sup>;<sup>c</sup> , <sup>B</sup> : ½ �� <sup>b</sup>;<sup>c</sup> ½ �� <sup>b</sup>;<sup>c</sup> IR<sup>n</sup> ! IR<sup>n</sup> and <sup>g</sup> : IR<sup>n</sup> ! IR<sup>n</sup>.

An advancement in this direction is to find the solution of such mathematical models by using fixed point theorems. In this technique, we generate a sequence by iterative procedure for some self-map T and then look for a fixed point of T, that is actually the solution of given mathematical model. The simplest case is when T is a contraction mapping, that is a self-mapping satisfying

$$d(Tx, Ty) \le kd(x, y),$$

where k ∈½ Þ 0; 1 . The contraction mapping principle [20] guarantees that a contraction mapping of a complete metric space to itself has a unique fixed point which may be obtained as the limit of an iteration scheme defined by repeated images under the mapping of an arbitrary starting point in the space. The multivalued version of contraction mapping principle can be found in [21]. In general, fixed point theorems allow us to obtain existence theorems concerning investigated functional-operator equations.

In this chapter, we prove the existence of solution for matrix Eq. (1) and integral Eq. (2) by using newly developed fixed point theorems.

## 2. Background material from fixed point theory

Let X be a set of points, a distance function on X is a map d : X � X ! ½ Þ 0; ∞ that is symmetric, and satisfies d ið Þ¼ ; i 0 for all i ∈ X. The distance is said to be a metric if the triangle inequality holds, i.e.,

Contraction Mappings and Applications DOI: http://dx.doi.org/10.5772/intechopen.81571

$$d(i,j) \le d(i,k) + d(k,j),$$

for all i, j, k∈ X and ð Þ X; d is called metric space.

Denote by 2<sup>X</sup> , the family of all nonempty subsets of <sup>X</sup>, CLð Þ <sup>X</sup> , the family of all nonempty and closed subsets of X, CBð Þ X , the family of all nonempty, closed, and bounded subsets of X and Kð Þ X , the family of all nonempty compact subsets of X. It is clear that, <sup>K</sup>ð Þ <sup>X</sup> <sup>⊆</sup>CBð Þ <sup>X</sup> <sup>⊆</sup>CLð Þ <sup>X</sup> <sup>⊆</sup>2<sup>X</sup> , let

$$H(\mathcal{A}, \mathcal{B}) = \max \left\{ \sup\_{\boldsymbol{\varkappa} \in \mathcal{A}} D(\boldsymbol{\varkappa}, \mathcal{B}), \sup\_{\boldsymbol{\jmath} \in \mathcal{B}} D(\boldsymbol{\jmath}, \mathcal{A}) \right\},$$

where D xð Þ¼ ; B inff g d xð Þ ; y : y∈ B . Then H is called generalized Pompeiu Hausdorff distance on CLð Þ X . It is well known that H is a metric on CBð Þ X , which is called Pompeiu Hausdorff metric induced by d.

If T : X ! X is a single valued self-mapping on X, then T is said to have a fixed point <sup>x</sup> if <sup>T</sup> <sup>x</sup> <sup>¼</sup> <sup>x</sup> and if <sup>T</sup> : <sup>X</sup> ! <sup>2</sup><sup>X</sup> is multivalued mapping, then <sup>T</sup> is said to have a fixed point x if x∈T x. We denote by Fix Tf g, the set of all fixed points of mapping T.

Definition 2.1 [22] Let <sup>T</sup> : <sup>X</sup> ! <sup>2</sup><sup>X</sup> be a multivalued map on a metric space ð Þ X; d , α, η : X � X ! IRþ be two functions where η is bounded, then T is an α∗ admissible mapping with respect to η, if

αð Þ y; z ≥ηð Þ y; z implies that α∗ð Þ T y; T z ≥η∗ð Þ T y; T z , y, z∈ X,

where

ordered set with partial order ≼, where X ≼Y ⇔ Y ≼X. In this section, denote d Xð Þ¼ ; Y k k Y � X <sup>1</sup> ¼ tr Yð Þ � X . Now, consider the non-linear matrix equation

> m i¼1 A∗

where Q is a positive definite matrix, Ai, i ¼ 1, …, m, are arbitrary n � n matrices and γ is a mapping from H nð Þ to H nð Þ which maps P nð Þ into P nð Þ. Assume that γ is an order-preserving mapping (γ is order preserving if A, B ∈ H nð Þ with A ≼B implies that γð Þ A ≼ γð Þ B ). There are various kinds of problems in control theory, dynamical programming, ladder networks, etc., where the matrix equations plays a

At the same time, integral equations have been developed to solve boundary value problems for both ordinary and partial differential equations and play a very important role in nonlinear analysis. Many problems of mathematical physics, theory of elasticity, viscodynamics fluid and mixed problems of mechanics of continuous media reduce to the Fredholm integral Eq. A rich literature on existence of solutions for nonlinear integral equations, which contain particular cases of important integral and functional equations can be found, for example, see [4–14]. An important technique to solve integral equations is to construct an iterative procedure to generate approximate solutions and find their limit, a host of attractive methods have been proposed for the approximate solutions of Fredholm integral equations of the second kind, see [15–19]. We consider a non-homogeneous

<sup>i</sup> γð Þ X Ai, (1)

Bð Þ r; s; z sð Þ ds þ g rð Þ, (2)

X ¼ Q þ ∑

Recent Advances in Integral Equations

crucial role. Matrix Eq. (1) have been studied by many authors see [1–3].

Fredholm integral equation of second kind of the form

contraction mapping, that is a self-mapping satisfying

Eq. (2) by using newly developed fixed point theorems.

2. Background material from fixed point theory

metric if the triangle inequality holds, i.e.,

functional-operator equations.

10

z rð Þ¼

ðc b

where <sup>t</sup><sup>∈</sup> ½ � <sup>b</sup>;<sup>c</sup> , <sup>B</sup> : ½ �� <sup>b</sup>;<sup>c</sup> ½ �� <sup>b</sup>;<sup>c</sup> IR<sup>n</sup> ! IR<sup>n</sup> and <sup>g</sup> : IR<sup>n</sup> ! IR<sup>n</sup>.

An advancement in this direction is to find the solution of such mathematical models by using fixed point theorems. In this technique, we generate a sequence by iterative procedure for some self-map T and then look for a fixed point of T, that is actually the solution of given mathematical model. The simplest case is when T is a

d Tx ð Þ ; Ty ≤kd xð Þ ; y ,

where k ∈½ Þ 0; 1 . The contraction mapping principle [20] guarantees that a contraction mapping of a complete metric space to itself has a unique fixed point which may be obtained as the limit of an iteration scheme defined by repeated images under the mapping of an arbitrary starting point in the space. The multivalued version of contraction mapping principle can be found in [21]. In general, fixed point theorems allow us to obtain existence theorems concerning investigated

In this chapter, we prove the existence of solution for matrix Eq. (1) and integral

Let X be a set of points, a distance function on X is a map d : X � X ! ½ Þ 0; ∞ that is symmetric, and satisfies d ið Þ¼ ; i 0 for all i ∈ X. The distance is said to be a

$$a\_\*(\mathcal{A}, \mathcal{B}) = \inf\_{\mathcal{y} \in \mathcal{A}, z \in \mathcal{B}} a(\mathcal{y}, z), \quad \eta\_\*(\mathcal{A}, \mathcal{B}) = \sup\_{\mathcal{y} \in \mathcal{A}, z \in \mathcal{B}} \eta(\mathcal{y}, z).$$

Further, Definition 2.1 is generalized in the following way.

Definition 2.2 [23] Let <sup>T</sup> : <sup>X</sup> ! <sup>2</sup><sup>X</sup> be a multivalued map on a metric space ð Þ X; d , α, η : X � X ! ½ Þ 0; ∞ be two functions. We say that T is generalized α∗ admissible mapping with respect to η, if

αð Þ y; z ≥ηð Þ y; z implies that αð Þ u; v ≥ηð Þ u; v , for all u ∈Ty, v∈Tz:

If ηð Þ¼ y; z 1 for all y, z∈ X, then T is said to be generalized α∗-admissible mapping.

## 3. Some fixed point results

Consistent with Du and Khojasteh [24], we denote by MandðIRÞ, the set of all manageable functions ϑ : IR � IR ! IR fulfilling the following conditions:

ð Þ ϑ<sup>1</sup> ϑð Þ t; s , s � t for all s, t . 0;

ð Þ ϑ<sup>2</sup> for any bounded sequence f g tn ⊂ð Þ 0; þ∞ and any nondecreasing sequence f g sn ⊂ ð Þ 0; þ∞ , it holds that

$$\lim\_{n \to \infty} \sup \frac{t\_n + \theta(t\_n, s\_n)}{s\_n} < 1. \tag{3}$$

Example 3.1 [24] Let r∈½ Þ 0; 1 . Then ϑ<sup>r</sup> : IR � IR ! IR defined by ϑrð Þ¼ t; s rs � t is a manageable function.

Example 3.2 Let ϑ : IR � IR ! IR defined by

$$\theta(t,s) = \begin{cases} \psi(s) - t & \text{if } (t,s) \in [0, +\infty) \times [0, +\infty), \\\ f(t,s) & \text{otherwise}, \end{cases}$$

0 , t≤sλð Þ t; s : (8)

d zð Þ <sup>1</sup>; T z<sup>1</sup> : (10)

d zð Þ <sup>1</sup>; <sup>T</sup> <sup>z</sup><sup>1</sup> : (11)

d zð Þ <sup>1</sup>; T z<sup>1</sup> : (12)

d zð Þ <sup>n</sup>; T zn , (15)

d zð Þ <sup>n</sup>; T zn : (16)

0 , λð Þ Hð Þ T z0; T z<sup>1</sup> ; d zð Þ <sup>0</sup>; z<sup>1</sup> , 1: (9)

0 , d xð Þ <sup>n</sup>; T xn ≤ Hð Þ T zn�<sup>1</sup>; T zn , (13) ϑð Þ Hð Þ T zn�<sup>1</sup>; T zn ; d zð Þ <sup>n</sup>�<sup>1</sup>; zn ≥0, (14)

So, from (5) and (7), we get

Contraction Mappings and Applications DOI: http://dx.doi.org/10.5772/intechopen.81571

<sup>ε</sup><sup>1</sup> <sup>¼</sup> <sup>1</sup> ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

Since d zð Þ <sup>1</sup>; T z<sup>1</sup> . 0. So, by using (9), we get ε<sup>1</sup> . 0 and

This implies that there exists z<sup>2</sup> ∈T z<sup>1</sup> such that

p

zn ∈ T zn�1, zn 6¼ zn�1, zn ∉T zn, α∗ð Þ zn�<sup>1</sup>; zn ≥η∗ð Þ zn�<sup>1</sup>; zn ,

p

d zð Þ <sup>1</sup>; z<sup>2</sup> ,

d zð Þ <sup>n</sup>; znþ<sup>1</sup> ,

(17), for each n ∈IN, we get

From ð Þ ϑ<sup>2</sup> , we get

13

λð Þ Hð Þ T z0; T z<sup>1</sup> ; d zð Þ <sup>0</sup>; z<sup>1</sup> <sup>p</sup> � <sup>1</sup> !

d zð Þ <sup>1</sup>; T z<sup>1</sup> , d zð Þþ <sup>1</sup>; T z<sup>1</sup> ε<sup>1</sup>

<sup>¼</sup> <sup>1</sup> ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi <sup>λ</sup>ð Þ <sup>H</sup>ð Þ <sup>T</sup> <sup>z</sup>0; <sup>T</sup> <sup>z</sup><sup>1</sup> ; <sup>d</sup>ðz0; <sup>z</sup>1<sup>Þ</sup> <sup>p</sup> !

> 1 ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi λð Þ Hð Þ T z0; T z<sup>1</sup> ; d zð Þ <sup>0</sup>; z<sup>1</sup>

By induction, we form a sequence f g zn in X satisfying for each n∈IN,

1 ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi λð Þ Hð Þ T zn�<sup>1</sup>; T zn ; d zð Þ <sup>n</sup>�<sup>1</sup>; zn

<sup>ε</sup><sup>n</sup> <sup>¼</sup> <sup>1</sup> ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

By using (7), (8), (13), and (15), we get for each n ∈IN

d zð Þ <sup>n</sup>; znþ<sup>1</sup> , ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

λð Þ Hð Þ T zn�<sup>1</sup>; T zn ; d zð Þ <sup>n</sup>�<sup>1</sup>; zn <sup>p</sup> � <sup>1</sup> !

d zð Þ <sup>n</sup>; T zn ≤d zð Þ <sup>n</sup>�<sup>1</sup>; zn λð Þ Hð Þ T zn�<sup>1</sup>; T zn ; dðzn�<sup>1</sup>; znÞ ≤d zð Þ <sup>n</sup>�<sup>1</sup>; zn , (17)

this implies that f g d zð Þ <sup>n</sup>; T zn <sup>n</sup>∈IN is a bounded sequence. By combining (15) and

Which means that f g d zð Þ <sup>n</sup>�<sup>1</sup>; zn <sup>n</sup>∈IN is a monotonically decreasing sequence of non-negative reals and so it must be convergent. So, let lim<sup>n</sup>!<sup>∞</sup>d zð Þ¼ <sup>n</sup>; znþ<sup>1</sup> c≥0.

<sup>λ</sup>ð Þ <sup>H</sup>ð Þ <sup>T</sup> zn�<sup>1</sup>; <sup>T</sup> zn ; <sup>d</sup>ðzn�<sup>1</sup>; zn<sup>Þ</sup> � � <sup>p</sup> d zð Þ <sup>n</sup>�<sup>1</sup>; zn : (18)

!

!

Let

and

by taking

where <sup>ψ</sup> : ½ Þ! <sup>0</sup>; <sup>þ</sup><sup>∞</sup> ½ Þ <sup>0</sup>; <sup>þ</sup><sup>∞</sup> satisfying <sup>∑</sup><sup>∞</sup> <sup>n</sup>¼1ψnð Þ<sup>t</sup> , <sup>þ</sup><sup>∞</sup> for all <sup>t</sup> . 0 and

f : IR � IR ! IR is any function. Then ϑð Þ t; s ∈ MandðIRÞ. Indeed, by using Lemma 1 of [25], we have for any s, t . 0, ϑð Þ¼ t; s ψðÞ�s t , s � t, so, ð Þ ϑ<sup>1</sup> holds. Let f g tn ⊂ ð Þ 0; þ∞ be a bounded sequence and let f g sn ⊂ð Þ 0; þ∞ be a nonincreasing sequence. Then limn!<sup>∞</sup>sn ¼ infn∈INsn ¼ a for some a∈½ Þ 0; þ∞ , we get

$$\lim\_{n \to \infty} \sup \frac{t\_n + \theta(t\_n, s\_n)}{s\_n} = \lim\_{n \to \infty} \sup \frac{\nu(s\_n)}{(s\_n)} < \lim\_{n \to \infty} \frac{(s\_n)}{(s\_n)} = \mathbf{1},$$

so, ð Þ ϑ<sup>2</sup> is also satisfied.

Definition 3.3 Let ð Þ <sup>X</sup>; <sup>d</sup> be a metric space and <sup>T</sup> : <sup>X</sup> ! <sup>2</sup><sup>X</sup> be a closed valued mapping. Let α, η : X � X ! IRþ be two functions and ϑ∈ MandðIRÞ. Then T is called a multivalued α<sup>∗</sup> � η∗-manageable contraction with respect to ϑ if for all y, z∈ X

$$a\_\*(\mathcal{T}\mathcal{Y}, \mathcal{T}z) \ge \eta\_\*(\mathcal{T}\mathcal{Y}, \mathcal{T}z) \quad \text{implies} \quad \theta(H(\mathcal{T}\mathcal{Y}, \mathcal{T}z), d(\mathcal{y}, z)) \ge 0. \tag{4}$$

Now we prove first result of this section.

Theorem 3.4 Let ð Þ <sup>X</sup>; <sup>d</sup> be a complete metric space and let <sup>T</sup> : <sup>X</sup> ! <sup>2</sup><sup>X</sup> be a closed valued map satisfying following conditions:

1. T is α∗-admissible map with respect to η;


Then Fixf g T 6¼ ∅.

Proof. Let z<sup>1</sup> ∈T z<sup>0</sup> be such that αð Þ z0; z<sup>1</sup> ≥ηð Þ z0; z<sup>1</sup> . Since T is α∗-admissible map with respect to η, then α∗ð Þ T z0; T z<sup>1</sup> ≥η∗ð Þ T z0; T z<sup>1</sup> . Therefore, from (4) we have

$$\vartheta(H(\mathcal{T}z\_0, \mathcal{T}z\_1), d(z\_0, z\_1)) \ge 0. \tag{5}$$

If z<sup>1</sup> ¼ z0, then z<sup>0</sup> ∈ Fixf g T , also if z<sup>1</sup> ∈T z1, then z<sup>1</sup> ∈Fixf g T . So, we adopt that z<sup>0</sup> 6¼ z<sup>1</sup> and z<sup>1</sup> ∉T z1. Thus 0 , d zð Þ <sup>1</sup>; T z<sup>1</sup> ≤ Hð Þ T z0; T z<sup>1</sup> . Define λ : IR � IR ! IR by

$$
\lambda(t,s) = \begin{cases}
\frac{t+\theta(t,s)}{s} & \text{if } t, s \ge 0 \\
& \mathbf{0} \\
& \text{otherwise}.
\end{cases}
\tag{6}
$$

By ð Þ ϑ<sup>1</sup> , we know that

$$0 \le \lambda(t, s) \le 1 \qquad\qquad\text{for all } t, s \ge 0.\tag{7}$$

Also note that if ϑð Þ t; s ≥0, then

Contraction Mappings and Applications DOI: http://dx.doi.org/10.5772/intechopen.81571

$$0 \le t \le \varsigma \lambda(t, \varsigma). \tag{8}$$

So, from (5) and (7), we get

$$0 \le \lambda(H(\mathcal{T}z\_0, \mathcal{T}z\_1), d(z\_0, z\_1)) \le 1. \tag{9}$$

Let

Example 3.2 Let ϑ : IR � IR ! IR defined by

�

where <sup>ψ</sup> : ½ Þ! <sup>0</sup>; <sup>þ</sup><sup>∞</sup> ½ Þ <sup>0</sup>; <sup>þ</sup><sup>∞</sup> satisfying <sup>∑</sup><sup>∞</sup>

ψðÞ�s t if ð Þ t; s ∈ ½ Þ� 0; þ∞ ½ Þ 0; þ∞ ,

ψð Þ sn ð Þ sn

, limn!<sup>∞</sup>

<sup>n</sup>¼1ψnð Þ<sup>t</sup> , <sup>þ</sup><sup>∞</sup> for all <sup>t</sup> . 0 and

ð Þ sn ð Þ sn

¼ 1,

f tð Þ ; s otherwise,

f : IR � IR ! IR is any function. Then ϑð Þ t; s ∈ MandðIRÞ. Indeed, by using Lemma 1 of

f g tn ⊂ ð Þ 0; þ∞ be a bounded sequence and let f g sn ⊂ð Þ 0; þ∞ be a nonincreasing

<sup>¼</sup> limn!<sup>∞</sup> sup

Definition 3.3 Let ð Þ <sup>X</sup>; <sup>d</sup> be a metric space and <sup>T</sup> : <sup>X</sup> ! <sup>2</sup><sup>X</sup> be a closed valued mapping. Let α, η : X � X ! IRþ be two functions and ϑ∈ MandðIRÞ. Then T is called a multivalued α<sup>∗</sup> � η∗-manageable contraction with respect to ϑ if for all y, z∈ X

Theorem 3.4 Let ð Þ <sup>X</sup>; <sup>d</sup> be a complete metric space and let <sup>T</sup> : <sup>X</sup> ! <sup>2</sup><sup>X</sup> be a closed

4. for a sequence zf g<sup>n</sup> ⊂ X, lim<sup>n</sup>!<sup>∞</sup>f g zn ¼ x and αð Þ zn; znþ<sup>1</sup> ≥ηð Þ zn; znþ<sup>1</sup> for all

Proof. Let z<sup>1</sup> ∈T z<sup>0</sup> be such that αð Þ z0; z<sup>1</sup> ≥ηð Þ z0; z<sup>1</sup> . Since T is α∗-admissible map with respect to η, then α∗ð Þ T z0; T z<sup>1</sup> ≥η∗ð Þ T z0; T z<sup>1</sup> . Therefore, from (4) we have

If z<sup>1</sup> ¼ z0, then z<sup>0</sup> ∈ Fixf g T , also if z<sup>1</sup> ∈T z1, then z<sup>1</sup> ∈Fixf g T . So, we adopt that z<sup>0</sup> 6¼ z<sup>1</sup> and z<sup>1</sup> ∉T z1. Thus 0 , d zð Þ <sup>1</sup>; T z<sup>1</sup> ≤ Hð Þ T z0; T z<sup>1</sup> . Define λ : IR � IR ! IR by

> t þ ϑð Þ t; s s

8 < :

ϑð Þ Hð Þ T z0; T z<sup>1</sup> ; d zð Þ <sup>0</sup>; z<sup>1</sup> ≥ 0: (5)

if t, s . 0

0 , λð Þ t; s , 1 for all t, s . 0: (7)

(6)

0 otherwise:

α∗ð Þ T y; T z ≥η∗ð Þ T y; T z implies ϑð Þ Hð Þ T y; T z ; d yð Þ ; z ≥0: (4)

[25], we have for any s, t . 0, ϑð Þ¼ t; s ψðÞ�s t , s � t, so, ð Þ ϑ<sup>1</sup> holds. Let

sequence. Then limn!<sup>∞</sup>sn ¼ infn∈INsn ¼ a for some a∈½ Þ 0; þ∞ , we get

tn þ ϑð Þ tn; sn sn

ϑð Þ¼ t; s

Recent Advances in Integral Equations

limn!<sup>∞</sup> sup

Now we prove first result of this section.

1. T is α∗-admissible map with respect to η;

2. T is α<sup>∗</sup> � η<sup>∗</sup> manageable contraction with respect to ϑ;

n ∈IN, implies αð Þ zn; x ≥ηð Þ zn; x for all n∈IN.

λð Þ¼ t; s

3.there exists z<sup>0</sup> ∈ X and z<sup>1</sup> ∈T z<sup>0</sup> such that αð Þ z0; z<sup>1</sup> ≥ ηð Þ z0; z<sup>1</sup> ;

valued map satisfying following conditions:

so, ð Þ ϑ<sup>2</sup> is also satisfied.

Then Fixf g T 6¼ ∅.

By ð Þ ϑ<sup>1</sup> , we know that

12

Also note that if ϑð Þ t; s ≥0, then

$$\varepsilon\_{1} = \left(\frac{1}{\sqrt{\lambda(H(Tz\_{0}, Tz\_{1}), d(z\_{0}, z\_{1}))}} - 1\right) d(z\_{1}, Tz\_{1}).\tag{10}$$

Since d zð Þ <sup>1</sup>; T z<sup>1</sup> . 0. So, by using (9), we get ε<sup>1</sup> . 0 and

$$d(z\_1, Tz\_1) \le d(z\_1, Tz\_1) + \varepsilon\_1$$

$$\varepsilon\_1 = \left(\frac{1}{\sqrt{\lambda(H(Tz\_0, Tz\_1), d(z\_0, z\_1))}}\right) d(z\_1, Tz\_1). \tag{11}$$

This implies that there exists z<sup>2</sup> ∈T z<sup>1</sup> such that

$$d(z\_1, z\_2) < \left(\frac{1}{\sqrt{\lambda(H(Tz\_0, Tz\_1), d(z\_0, z\_1))}}\right) d(z\_1, Tz\_1). \tag{12}$$

By induction, we form a sequence f g zn in X satisfying for each n∈IN, zn ∈ T zn�1, zn 6¼ zn�1, zn ∉T zn, α∗ð Þ zn�<sup>1</sup>; zn ≥η∗ð Þ zn�<sup>1</sup>; zn ,

$$0 \le d(\boldsymbol{x}\_n, \boldsymbol{T}\boldsymbol{x}\_n) \le H(\boldsymbol{T}\boldsymbol{z}\_{n-1}, \boldsymbol{T}\boldsymbol{z}\_n),\tag{13}$$

$$\mathfrak{gl}(H(\mathcal{T}z\_{n-1}, \mathcal{T}z\_n), d(z\_{n-1}, z\_n)) \ge 0,\tag{14}$$

and

$$d(\mathbf{z}\_n, \mathbf{z}\_{n+1}) \, < \left(\frac{1}{\sqrt{\lambda(H(\mathbf{T}z\_{n-1}, \mathbf{T}z\_n), d(\mathbf{z}\_{n-1}, \mathbf{z}\_n))}}\right) d(\mathbf{z}\_n, \mathbf{T}z\_n), \tag{15}$$

by taking

$$\varepsilon\_{n} = \left(\frac{1}{\sqrt{\lambda(H(\mathcal{T}z\_{n-1}, \mathcal{T}z\_{n}), d(z\_{n-1}, z\_{n}))}} - 1\right) d(z\_{n}, \mathcal{T}z\_{n}).\tag{16}$$

By using (7), (8), (13), and (15), we get for each n ∈IN

$$d(z\_n, Tz\_n) \le d(z\_{n-1}, z\_n) \lambda(H(Tz\_{n-1}, Tz\_n), d(z\_{n-1}, z\_n)) \le d(z\_{n-1}, z\_n),\tag{17}$$

this implies that f g d zð Þ <sup>n</sup>; T zn <sup>n</sup>∈IN is a bounded sequence. By combining (15) and (17), for each n ∈IN, we get

$$d(z\_n, z\_{n+1}) \le \left(\sqrt{\lambda(H(Tz\_{n-1}, Tz\_n), d(z\_{n-1}, z\_n))}\right) d(z\_{n-1}, z\_n). \tag{18}$$

Which means that f g d zð Þ <sup>n</sup>�<sup>1</sup>; zn <sup>n</sup>∈IN is a monotonically decreasing sequence of non-negative reals and so it must be convergent. So, let lim<sup>n</sup>!<sup>∞</sup>d zð Þ¼ <sup>n</sup>; znþ<sup>1</sup> c≥0. From ð Þ ϑ<sup>2</sup> , we get

$$\lim\_{n \to \infty} \sup \lambda(H(\mathcal{T}z\_n, \mathcal{T}z\_n), d(z\_{n-1}, z\_n)) < 1. \tag{19}$$

dð Þ T x; T y . 0 implies τ þ Fð Þ dð Þ T x; T y ≤ Fð Þ d xð Þ ; y :

Theorem 3.6 [26] Let ð Þ X; d be a complete metric space and let T : X ! X be an F-contraction. Then T has a unique fixed point x<sup>∗</sup> ∈ X and for every x<sup>0</sup> ∈ X a sequence

Definition 3.7 ([27]). Let ð Þ X; d be a metric space and T : X ! CBð Þ X be a mapping. Then T is a multivalued F-contraction, if F ∈ Δð Þ F and there exists τ . 0

Hð Þ T x; T y . 0 ) τ þ Fð Þ Hð Þ T x; T y ≤ Fð Þ d xð Þ ; y :

multivalued F-contraction, then T has a fixed point in X.

<sup>σ</sup> with D zð Þ ; T z . 0 satisfying

M zð Þ¼ ; y max d zð Þ ; y ; Dðz; T zÞ; Dðy; T yÞ;

D yð Þ ; T y ½ � 1 þ D zð Þ ; T z <sup>1</sup> <sup>þ</sup> d zð Þ ; <sup>y</sup> ;

<sup>σ</sup> with D zð Þ ; T z . 0 satisfying

1. T is multivalued α-orbital admissible mapping;

2. the map z ! D zð Þ ; T z is lower semi-continuous;

4. τ satisfies lim<sup>t</sup>!s<sup>þ</sup> infτð Þt . σ for all s ≥0.

3.there exists z<sup>0</sup> ∈ X and z<sup>1</sup> ∈T z<sup>0</sup> such that αð Þ z0; z<sup>1</sup> ≥ 1;

Fz

Theorem 3.8 ([27]). Let ð Þ X; d be a complete metric space and T : X ! Kð Þ X be a

Theorem 3.9 ([27]). Let ð Þ X; d be a complete metric space and T : X ! Cð Þ X be a

Definition 3.10 Let <sup>T</sup> : <sup>X</sup> ! <sup>2</sup><sup>X</sup> be a multivalued mapping on a metric space ð Þ X; d , then T is said to be an multivalued α-F-weak-contraction on X, if there exists σ . 0, τ : ð Þ! 0; ∞ ð Þ σ; ∞ , F ∈ Δð Þ F and α : X � X ! ½ Þ 0; þ∞ such that for

<sup>σ</sup> ¼ f g y∈T z : Fð Þ d zð Þ ; y ≤ Fð Þþ D zð Þ ; T z σ :

<sup>σ</sup> 6¼ ∅ in both cases when F ∈ Δð Þ F and F ∈ Δð Þ F<sup>∗</sup> [32]. Definition 3.11 Let T : X ! Pð Þ X be a multivalued mapping on a metric space ð Þ X; d , then T is said to be an multivalued α-F-contraction on X, if there exists σ . 0, τ : ð Þ! 0; ∞ ð Þ σ; ∞ , F ∈ Δð Þ F and α : X � X ! ½ Þ 0; þ∞ such that for all

Theorem 3.12 Let ð Þ X; d be a complete metric space and T : X ! Kð Þ X be an

multivalued α-F-weak-contraction satisfying the following assertions:

τð Þþ d zð Þ ; y Fð Þ αð Þ z; y D yð Þ ; T y ≤ Fð Þ M zð Þ ; y , (28)

τð Þþ d zð Þ ; y Fð Þ αð Þ z; y D yð Þ ; T y ≤ Fð Þ d xð Þ ; y , (30)

D yð Þ ; T z ½ � 1 þ D zð Þ ; T y 1 þ d zð Þ ; y

D yð Þþ ; T z D zð Þ ; T y

<sup>2</sup> ;

: (29)

multivalued F-contraction. Suppose F ∈ Δð Þ F<sup>∗</sup> , then T has a fixed point in X. For more in this direction, see, [28–31]. Here, we give the concept of multivalued α-F-weak-contractions and prove some fixed point results.

Tnx0n ∈IN is convergent to x<sup>∗</sup>.

Contraction Mappings and Applications DOI: http://dx.doi.org/10.5772/intechopen.81571

such that for all x, y∈ X,

all z∈ X, y∈F<sup>z</sup>

where,

and

z∈ X, y∈F<sup>z</sup>

15

Note that F<sup>z</sup>

Now, if c . 0, then by taking the limn!<sup>∞</sup> sup in (18) and using (19), we have

$$c \le \sqrt{\lim\_{n \to \infty} \sup \lambda(H(\mathcal{T}z\_{n-1}, \mathcal{T}z\_n), d(z\_{n-1}, z\_n))}c \le c. \tag{20}$$

This contradiction shows that c ¼ 0. Hence, limn!<sup>∞</sup> d zð Þ¼ <sup>n</sup>; znþ<sup>1</sup> 0. Next, we prove that f g zn <sup>n</sup>∈IN is a Cauchy sequence in X. Let, for each n∈ IN,

$$
\sigma\_n = \sqrt{\lambda(H(Tz\_{n-1}, Tz\_n), d(z\_{n-1}, z\_n))},\tag{21}
$$

then from Eq. (9), we have σ<sup>n</sup> ∈ð Þ 0; 1 . By (18), we obtain

$$d(z\_n, z\_{n+1}) < \sigma\_n d(z\_{n-1}, z\_n). \tag{22}$$

(19) implies that lim<sup>n</sup>!<sup>∞</sup> σ<sup>n</sup> , 1, so there exists γ ∈ ½ Þ 0; 1 and n<sup>0</sup> ∈IN, such that

$$
\sigma\_n \le \gamma \qquad \text{for all } n \in \mathbb{N}, n \ge n\_0. \tag{23}
$$

For any n≥n0, since σ<sup>n</sup> ∈ð Þ 0; 1 for all n∈ IN and γ ∈½ Þ 0; 1 , (22, 23) implies that

$$d(z\_n, z\_{n+1}) \le \sigma\_n d(z\_{n-1}, z\_n) \le \sigma\_n \sigma\_{n-1} d(z\_{n-2}, z\_{n-1}) \cdots \le \gamma^{n-n\_0+1} d(z\_0, z\_1). \tag{24}$$

Put <sup>β</sup><sup>n</sup> <sup>¼</sup> <sup>γ</sup>n�n0þ<sup>1</sup> 1�γ � �d zð Þ <sup>0</sup>; <sup>z</sup><sup>1</sup> , <sup>n</sup><sup>∈</sup> IN. For m, n<sup>∈</sup> IN with <sup>m</sup> . <sup>n</sup>≥n0, we have from (24) that

$$d(z\_n, z\_m) \le \sum\_{j=n}^{m-1} d\left(z\_j, z\_{j+1}\right) < \beta\_n. \tag{25}$$

Since γ ∈ ½ Þ 0; 1 , lim<sup>n</sup>!∞β<sup>n</sup> ¼ 0. Hence lim<sup>n</sup>!<sup>∞</sup>supf g d zð Þ <sup>n</sup>; zm : m . n ¼ 0. This shows that f g zn is a Cauchy sequence in X. Completeness of X ensures the existence of z∈ X such that zn ! z as n ! ∞. Now, since αð Þ zn; z ≥ηð Þ zn; z for all n ∈IN, α∗ð Þ T zn; T z ≥ η∗ð Þ T zn; T z , and so from (4), we have ϑð Þ Hð Þ T zn; T z ; d zð Þ <sup>n</sup>; z ≥ 0. Then from (7, 8), we have

$$H(\mathcal{T}z\_n, \mathcal{T}z) \le \lambda(H(\mathcal{T}z\_n, \mathcal{T}z), d(z\_n, z))d(z\_n, z) \le d(z\_n, z). \tag{26}$$

Since 0 , d zð Þ ; T z ≤ Hð Þþ T zn; T z d zð Þ <sup>n</sup>; z , so by using (26), we get

$$0 \le d(z, \mathcal{T}z) \le 2d(z\_n, z). \tag{27}$$

Letting limit n ! ∞ in above inequality, we get d zð Þ¼ ; T z 0. Hence z∈Fixf g T .

□ Let Δð Þ F be the set of all functions F : IRþ ! IR satisfying following conditions: ð Þ F<sup>1</sup> F is strictly increasing;

ð Þ F<sup>2</sup> for all sequence f g α<sup>n</sup> ⊆Rþ, lim<sup>n</sup>!<sup>∞</sup> α<sup>n</sup> ¼ 0 if and only if lim<sup>n</sup>!<sup>∞</sup>Fð Þ¼� α<sup>n</sup> ∞;

ð Þ <sup>F</sup><sup>3</sup> there exist 0 , <sup>k</sup> , 1 such that lim<sup>n</sup>!0<sup>þ</sup> <sup>α</sup><sup>k</sup>Fð Þ¼ <sup>α</sup> <sup>0</sup>,

Δð Þ F<sup>∗</sup> , if F also satisfies the following:

ð Þ F<sup>4</sup> Fð Þ¼ infA infFð Þ A for all A ⊂ð Þ 0; ∞ with inf A . 0,

Definition 3.5 [27] Let ð Þ X; d be a metric space. A mapping T : X ! X is said to be F-contraction of there exists τ . 0 such that

Contraction Mappings and Applications DOI: http://dx.doi.org/10.5772/intechopen.81571

dð Þ T x; T y . 0 implies τ þ Fð Þ dð Þ T x; T y ≤ Fð Þ d xð Þ ; y :

Theorem 3.6 [26] Let ð Þ X; d be a complete metric space and let T : X ! X be an F-contraction. Then T has a unique fixed point x<sup>∗</sup> ∈ X and for every x<sup>0</sup> ∈ X a sequence Tnx0n ∈IN is convergent to x<sup>∗</sup>.

Definition 3.7 ([27]). Let ð Þ X; d be a metric space and T : X ! CBð Þ X be a mapping. Then T is a multivalued F-contraction, if F ∈ Δð Þ F and there exists τ . 0 such that for all x, y∈ X,

$$H(\mathcal{T}\mathfrak{x}, \mathcal{T}\mathfrak{y}) \succeq \mathbf{0} \Rightarrow \mathfrak{r} + \mathcal{F}(H(\mathcal{T}\mathfrak{x}, \mathcal{T}\mathfrak{y})) \leq \mathcal{F}(d(\mathfrak{x}, \mathfrak{y})).$$

Theorem 3.8 ([27]). Let ð Þ X; d be a complete metric space and T : X ! Kð Þ X be a multivalued F-contraction, then T has a fixed point in X.

Theorem 3.9 ([27]). Let ð Þ X; d be a complete metric space and T : X ! Cð Þ X be a multivalued F-contraction. Suppose F ∈ Δð Þ F<sup>∗</sup> , then T has a fixed point in X.

For more in this direction, see, [28–31]. Here, we give the concept of multivalued α-F-weak-contractions and prove some fixed point results.

Definition 3.10 Let <sup>T</sup> : <sup>X</sup> ! <sup>2</sup><sup>X</sup> be a multivalued mapping on a metric space ð Þ X; d , then T is said to be an multivalued α-F-weak-contraction on X, if there exists σ . 0, τ : ð Þ! 0; ∞ ð Þ σ; ∞ , F ∈ Δð Þ F and α : X � X ! ½ Þ 0; þ∞ such that for all z∈ X, y∈F<sup>z</sup> <sup>σ</sup> with D zð Þ ; T z . 0 satisfying

$$
\pi(d(z, y)) + \mathcal{F}(a(z, y)D(y, \mathcal{T}y)) \le \mathcal{F}(M(z, y)),
\tag{28}
$$

where,

limn!<sup>∞</sup> supλð Þ <sup>H</sup>ð Þ <sup>T</sup> zn; <sup>T</sup> zn ; d zð Þ <sup>n</sup>�1; zn , <sup>1</sup>: (19)

q c , c: (20)

p , (21)

d zð Þ <sup>n</sup>; znþ<sup>1</sup> , σnd zð Þ <sup>n</sup>�1; zn : (22)

σ<sup>n</sup> ≤γ for all n∈ IN, n≥ n0: (23)

d zð Þ <sup>0</sup>; z<sup>1</sup> , n∈ IN. For m, n∈ IN with m . n≥n0, we have from

� � , βn: (25)

d zð Þ <sup>0</sup>; z<sup>1</sup> : (24)

□

Now, if c . 0, then by taking the limn!<sup>∞</sup> sup in (18) and using (19), we have

This contradiction shows that c ¼ 0. Hence, limn!<sup>∞</sup> d zð Þ¼ <sup>n</sup>; znþ<sup>1</sup> 0. Next, we

<sup>σ</sup><sup>n</sup> <sup>¼</sup> ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi λð Þ Hð Þ T zn�1; T zn ; d zð Þ <sup>n</sup>�1; zn

(19) implies that lim<sup>n</sup>!<sup>∞</sup> σ<sup>n</sup> , 1, so there exists γ ∈ ½ Þ 0; 1 and n<sup>0</sup> ∈IN, such that

For any n≥n0, since σ<sup>n</sup> ∈ð Þ 0; 1 for all n∈ IN and γ ∈½ Þ 0; 1 , (22, 23) implies that

<sup>c</sup><sup>≤</sup> ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi limn!<sup>∞</sup> sup <sup>λ</sup>ð Þ <sup>H</sup>ð Þ <sup>T</sup> zn�1; <sup>T</sup> zn ; d zð Þ <sup>n</sup>�1; zn

prove that f g zn <sup>n</sup>∈IN is a Cauchy sequence in X. Let, for each n∈ IN,

then from Eq. (9), we have σ<sup>n</sup> ∈ð Þ 0; 1 . By (18), we obtain

d zð Þ <sup>n</sup>; znþ<sup>1</sup> , <sup>σ</sup>nd zð Þ <sup>n</sup>�<sup>1</sup>; zn , <sup>σ</sup>nσ<sup>n</sup>�<sup>1</sup>d zð Þ <sup>n</sup>�<sup>2</sup>; zn�<sup>1</sup> ⋯ ≤γ<sup>n</sup>�n0þ<sup>1</sup>

d zð Þ <sup>n</sup>; zm ≤ ∑<sup>m</sup>�<sup>1</sup>

j¼n

Since 0 , d zð Þ ; T z ≤ Hð Þþ T zn; T z d zð Þ <sup>n</sup>; z , so by using (26), we get

ð Þ F<sup>2</sup> for all sequence f g α<sup>n</sup> ⊆Rþ, lim<sup>n</sup>!<sup>∞</sup> α<sup>n</sup> ¼ 0 if and only if

ð Þ <sup>F</sup><sup>3</sup> there exist 0 , <sup>k</sup> , 1 such that lim<sup>n</sup>!0<sup>þ</sup> <sup>α</sup><sup>k</sup>Fð Þ¼ <sup>α</sup> <sup>0</sup>,

ð Þ F<sup>4</sup> Fð Þ¼ infA infFð Þ A for all A ⊂ð Þ 0; ∞ with inf A . 0,

Since γ ∈ ½ Þ 0; 1 , lim<sup>n</sup>!∞β<sup>n</sup> ¼ 0. Hence lim<sup>n</sup>!<sup>∞</sup>supf g d zð Þ <sup>n</sup>; zm : m . n ¼ 0. This shows that f g zn is a Cauchy sequence in X. Completeness of X ensures the existence of z∈ X such that zn ! z as n ! ∞. Now, since αð Þ zn; z ≥ηð Þ zn; z for all n ∈IN, α∗ð Þ T zn; T z ≥ η∗ð Þ T zn; T z , and so from (4), we have ϑð Þ Hð Þ T zn; T z ; d zð Þ <sup>n</sup>; z ≥ 0.

d zj; zjþ<sup>1</sup>

Hð Þ T zn; T z ≤λð Þ Hð Þ T zn; T z ; dðzn; zÞ d zð Þ <sup>n</sup>; z , d zð Þ <sup>n</sup>; z : (26)

Letting limit n ! ∞ in above inequality, we get d zð Þ¼ ; T z 0. Hence z∈Fixf g T .

Let Δð Þ F be the set of all functions F : IRþ ! IR satisfying following conditions:

Definition 3.5 [27] Let ð Þ X; d be a metric space. A mapping T : X ! X is said to

0 , d zð Þ ; T z , 2d zð Þ <sup>n</sup>; z : (27)

Put <sup>β</sup><sup>n</sup> <sup>¼</sup> <sup>γ</sup>n�n0þ<sup>1</sup>

Then from (7, 8), we have

ð Þ F<sup>1</sup> F is strictly increasing;

Δð Þ F<sup>∗</sup> , if F also satisfies the following:

be F-contraction of there exists τ . 0 such that

lim<sup>n</sup>!<sup>∞</sup>Fð Þ¼� α<sup>n</sup> ∞;

14

(24) that

1�γ � �

Recent Advances in Integral Equations

$$M(\mathbf{z}, \mathbf{y}) = \max\left\{ d(\mathbf{z}, \mathbf{y}), D(\mathbf{z}, \mathbf{T}\mathbf{z}), D(\mathbf{y}, \mathbf{T}\mathbf{y}), \frac{D(\mathbf{y}, \mathbf{T}\mathbf{z}) + D(\mathbf{z}, \mathbf{T}\mathbf{y})}{2},$$

$$\frac{D(\mathbf{y}, \mathbf{T}\mathbf{y})[\mathbf{1} + D(\mathbf{z}, \mathbf{T}\mathbf{z})]}{\mathbf{1} + d(\mathbf{z}, \mathbf{y})}, \frac{D(\mathbf{y}, \mathbf{T}\mathbf{z})[\mathbf{1} + D(\mathbf{z}, \mathbf{T}\mathbf{y})]}{\mathbf{1} + d(\mathbf{z}, \mathbf{y})} \right\}.\tag{29}$$

and

$$\mathcal{F}\_{\sigma}^{\sharp} = \{ \mathcal{y} \in \mathcal{T}z : \mathcal{F}(d(z, \mathcal{y})) \le \mathcal{F}(D(z, \mathcal{T}z)) + \sigma \}.$$

Note that F<sup>z</sup> <sup>σ</sup> 6¼ ∅ in both cases when F ∈ Δð Þ F and F ∈ Δð Þ F<sup>∗</sup> [32].

Definition 3.11 Let T : X ! Pð Þ X be a multivalued mapping on a metric space ð Þ X; d , then T is said to be an multivalued α-F-contraction on X, if there exists σ . 0, τ : ð Þ! 0; ∞ ð Þ σ; ∞ , F ∈ Δð Þ F and α : X � X ! ½ Þ 0; þ∞ such that for all z∈ X, y∈F<sup>z</sup> <sup>σ</sup> with D zð Þ ; T z . 0 satisfying

$$
\pi(d(x, y)) + \mathcal{F}(a(x, y)D(y, \mathcal{T}y)) \le \mathcal{F}(d(x, y)),
\tag{30}
$$

Theorem 3.12 Let ð Þ X; d be a complete metric space and T : X ! Kð Þ X be an multivalued α-F-weak-contraction satisfying the following assertions:

1. T is multivalued α-orbital admissible mapping;

2. the map z ! D zð Þ ; T z is lower semi-continuous;

3.there exists z<sup>0</sup> ∈ X and z<sup>1</sup> ∈T z<sup>0</sup> such that αð Þ z0; z<sup>1</sup> ≥ 1;

4. τ satisfies lim<sup>t</sup>!s<sup>þ</sup> infτð Þt . σ for all s ≥0.

Then T has a fixed point in X.

Proof. Let <sup>z</sup><sup>0</sup> <sup>∈</sup> <sup>X</sup>, since <sup>T</sup> <sup>z</sup>∈Kð Þ <sup>X</sup> for every <sup>z</sup><sup>∈</sup> <sup>X</sup>, the set <sup>F</sup><sup>z</sup> <sup>σ</sup> is non-empty for any σ . 0, then there exists z<sup>1</sup> ∈ Fz<sup>0</sup> <sup>σ</sup> and by hypothesis αð Þ z0; z<sup>1</sup> ≥1. Assume that z<sup>1</sup> ∉T z1, otherwise z<sup>1</sup> is the fixed point of T . Then, since T z<sup>1</sup> is closed, D zð Þ <sup>1</sup>; T z<sup>1</sup> . 0, so, from (28), we have

$$\sigma(d(z\_0, z\_1)) + \mathcal{F}(a(z\_0, z\_1)D(z\_1, Tz\_1)) \le \mathcal{F}(\mathbf{M}(z\_0, z\_1)),\tag{31}$$

Fð Þ d zð Þ <sup>n</sup>; znþ<sup>1</sup> ≤ Fð Þþ d zð Þ <sup>n</sup>; znþ<sup>1</sup> σ: (40)

<sup>≤</sup> <sup>F</sup>ð Þþ <sup>d</sup><sup>0</sup> <sup>n</sup><sup>σ</sup> � <sup>τ</sup>ð Þ� dn <sup>τ</sup>ð Þ� dn�<sup>1</sup> <sup>⋯</sup> � <sup>τ</sup>ð Þ <sup>d</sup><sup>0</sup> : (42)

� � � � : (43)

� � � � : (44)

� �. Then we obtain a

� � � � for all

Fð Þ¼ dn 0. By (43), we

� � � � ≤ 0: (45)

! δþ, we

Þ ¼ 0,

Fð Þ d zð Þ <sup>n</sup>þ1; znþ<sup>2</sup> ≤ Fð Þþ d zð Þ <sup>n</sup>; znþ<sup>1</sup> σ � τð Þ d zð Þ <sup>n</sup>; znþ<sup>1</sup> (41)

� � � � . We distinguish two cases.

� � ¼ �<sup>∞</sup> and by ð Þ <sup>F</sup><sup>2</sup> , we obtain lim<sup>k</sup>!<sup>∞</sup>dpnk

� � � � for all <sup>m</sup> . <sup>n</sup>0. Hence lim<sup>m</sup>!∞Fð Þ¼� dm <sup>∞</sup>, so

n σ � τ d � pn

i¼n

<sup>1</sup>=<sup>k</sup> converges. Therefore, d zð Þ! <sup>n</sup>; zm 0 as m, n ! ∞. Thus,

<sup>n</sup>1=<sup>r</sup>, for all n . n1. Next, for m . n ≥n<sup>1</sup> we have

1 i <sup>1</sup>=<sup>k</sup> ,

� � . <sup>τ</sup> dpm

� � for all <sup>k</sup>. Since dpnk

� �<sup>≤</sup> <sup>F</sup>ð Þþ <sup>d</sup><sup>0</sup> <sup>n</sup> <sup>σ</sup> � <sup>τ</sup> dpnk

� � for all m . n0. Then

Let dn ¼ d zð Þ <sup>n</sup>; znþ<sup>1</sup> for n ∈IN, then dn . 0 and from (41) f g dn is decreasing. Therefore, there exists δ≥0 such that limn!<sup>∞</sup>dn ¼ δ. Now let δ . 0. From (41), we get

� � <sup>¼</sup> minf g <sup>τ</sup>ð Þ <sup>d</sup><sup>0</sup> ; <sup>τ</sup>ð Þ <sup>d</sup><sup>1</sup> ; <sup>⋯</sup>; <sup>τ</sup>ð Þ dn for all <sup>n</sup><sup>∈</sup> IN. From (42), we get

Fð Þ dnþ<sup>1</sup> ≤ Fð Þþ d<sup>0</sup> n σ � τ dpn

Fð Þ D zð Þ <sup>n</sup>þ<sup>1</sup>; T znþ<sup>1</sup> ≤ Fð Þþ D zð Þ <sup>0</sup>; T z<sup>0</sup> n σ � τ dpn

� � . <sup>τ</sup> dpnkþ<sup>1</sup>

� � . <sup>τ</sup> dpm

lim<sup>n</sup>!<sup>∞</sup> dn <sup>¼</sup> <sup>0</sup>:

<sup>F</sup>ð Þ <sup>d</sup><sup>0</sup> <sup>≤</sup>ð Þ dn <sup>r</sup>

Letting <sup>n</sup> ! <sup>∞</sup> in (45), we obtain lim<sup>n</sup>!<sup>∞</sup>n dð Þ<sup>n</sup> <sup>r</sup> <sup>¼</sup> 0. This implies that there exists

f g zn is a Cauchy sequence. Since <sup>X</sup> is complete, there exists <sup>z</sup><sup>∗</sup> <sup>∈</sup> <sup>X</sup> such that zn ! <sup>z</sup><sup>∗</sup>

<sup>0</sup><sup>≤</sup> D zð Þ ; Tz <sup>≤</sup> lim<sup>n</sup>!<sup>∞</sup> inf D zð Þ¼ <sup>n</sup>; <sup>T</sup> zn <sup>0</sup>:

d zi ð Þ ; ziþ<sup>1</sup> ≤ ∑<sup>m</sup>�<sup>1</sup>

Fð Þ dnþ<sup>1</sup> ≤ Fð Þþ dn σ � τð Þ dn ≤ Fð Þþ dn�<sup>1</sup> 2σ � τð Þ� dn τð Þ dn�<sup>1</sup> ⋯

Combining (39) and (40) gives

Contraction Mappings and Applications DOI: http://dx.doi.org/10.5772/intechopen.81571

Let τ dpn

subsequence dpnk

get for all n ∈IN

17

From (38), we also get

deduce that lim<sup>k</sup>!<sup>∞</sup>inf<sup>τ</sup> dpnk

k. Consequently, lim<sup>k</sup>!∞F dnk

<sup>F</sup>ð Þ dm <sup>≤</sup> <sup>F</sup>ð Þþ <sup>d</sup><sup>0</sup> <sup>m</sup> <sup>σ</sup> � <sup>τ</sup> dpn<sup>0</sup>

ð Þ dn <sup>r</sup>

<sup>n</sup><sup>1</sup> <sup>∈</sup>IN such that n dð Þ<sup>n</sup> <sup>r</sup> <sup>≤</sup> 1, or, dn <sup>≤</sup> <sup>1</sup>

since 0 , k , 1, ∑<sup>m</sup>�<sup>1</sup>

Now consider the sequence τ dpn

n o of dpn

which contradicts that lim<sup>n</sup>!<sup>∞</sup>dn . 0.

Case 2. There is <sup>n</sup><sup>0</sup> <sup>∈</sup> IN such that <sup>τ</sup> dpn<sup>0</sup>

Case 1. For each n∈IN, there is m . n such that τ dpn

� � with <sup>τ</sup> dpnk

lim<sup>m</sup>!<sup>∞</sup>dm ¼ 0, which contradicts that lim<sup>m</sup>!<sup>∞</sup>dm . 0. Thus,

From ð Þ <sup>F</sup><sup>3</sup> , there exists 0 , <sup>r</sup> , 1 such that lim<sup>n</sup>!<sup>∞</sup>ð Þ dn <sup>r</sup>

d zð Þ <sup>n</sup>; zm ≤ ∑<sup>m</sup>�<sup>1</sup>

i¼n

as n ! ∞. From Eqs. (44) and ð Þ F<sup>2</sup> , we have lim<sup>n</sup>!<sup>∞</sup>D zð Þ¼ <sup>n</sup>; T zn 0. Since

<sup>F</sup>ð Þ� dn ð Þ dn <sup>r</sup>

<sup>i</sup>¼<sup>n</sup> <sup>1</sup> i

z ! D zð Þ ; T z is lower semi-continuous, then

� � . <sup>σ</sup>. Hence <sup>F</sup> dnk

where

$$M(\mathbf{z}\_0, \mathbf{z}\_1) = \max\left\{ d(\mathbf{z}\_0, \mathbf{z}\_1), D(\mathbf{z}\_0, \mathbf{T}\mathbf{z}\_0), D(\mathbf{z}\_1, \mathbf{T}\mathbf{z}\_1), \frac{D(\mathbf{z}\_1, \mathbf{T}\mathbf{z}\_0) + D(\mathbf{z}\_0, \mathbf{T}\mathbf{z}\_1)}{2},$$

$$\frac{D(\mathbf{z}\_1, \mathbf{T}\mathbf{z}\_1)[\mathbf{1} + D(\mathbf{z}\_0, \mathbf{T}\mathbf{z}\_0)]}{\mathbf{1} + d(\mathbf{z}\_0, \mathbf{z}\_1)}, \frac{D(\mathbf{z}\_1, \mathbf{T}\mathbf{z}\_0)[\mathbf{1} + D(\mathbf{z}\_0, \mathbf{T}\mathbf{z}\_1)]}{\mathbf{1} + d(\mathbf{z}\_0, \mathbf{z}\_1)} \right\}.\tag{32}$$

Since T z<sup>0</sup> and T z<sup>1</sup> are compact, so we have

$$M(z\_0, z\_1) = \max\left\{d(z\_0, z\_1), d(z\_0, z\_1), d(z\_1, z\_2), \frac{d(z\_1, z\_1) + d(z\_0, z\_2)}{2}, \\\frac{d(z\_1, z\_2)[1 + d(z\_0, z\_1)]}{1 + d(z\_0, z\_1)}, \frac{d(z\_1, z\_1)[1 + d(z\_0, z\_2)]}{1 + d(z\_0, z\_1)}\right\}$$

$$= \max\left\{d(z\_0, z\_1), d(z\_1, z\_2), \frac{d(z\_0, z\_2)}{2}\right\}.\tag{33}$$

Since d zð Þ <sup>0</sup>;z<sup>2</sup> <sup>2</sup> <sup>≤</sup> d zð Þþ <sup>0</sup>;z<sup>1</sup> d zð Þ <sup>1</sup>;z<sup>2</sup> <sup>2</sup> ≤ maxf g d zð Þ <sup>0</sup>; z<sup>1</sup> ; d zð Þ <sup>1</sup>; z<sup>2</sup> , it follows that

$$M(z\_0, z\_1) \le \max\{d(z\_0, z\_1), d(z\_1, z\_2)\}.\tag{34}$$

Suppose that d zð Þ <sup>0</sup>; z<sup>1</sup> , d zð Þ <sup>1</sup>; z<sup>2</sup> , then (31) implies that

$$\begin{split} \pi(d(z\_0, z\_1)) + \mathcal{F}(\mathcal{D}(z\_1, \mathcal{T}z\_1)) &\leq \pi(d(z\_0, z\_1)) + \mathcal{F}(a(z\_0, z\_1)\mathcal{D}(z\_1, \mathcal{T}z\_1)) \\ &\leq \mathcal{F}(d(z\_1, z\_2)), \end{split} \tag{35}$$

consequently,

$$
\pi(d(z\_0, z\_1)) + \mathcal{F}(d(z\_1, z\_2)) \le \mathcal{F}(d(z\_1, z\_2)),
\tag{36}
$$

or, Fð Þ d zð Þ <sup>1</sup>; z<sup>2</sup> ≤ Fð Þ� d zð Þ <sup>1</sup>; z<sup>2</sup> τð Þ d zð Þ <sup>0</sup>; z<sup>1</sup> , which is a contradiction. Hence Mdz ð Þ ð Þ <sup>0</sup>; z<sup>1</sup> ≤d zð Þ <sup>0</sup>; z<sup>1</sup> , therefore by using ð Þ F<sup>1</sup> , (31) implies that

$$
\sigma(d(z\_0, z\_1)) + \mathcal{F}(a(z\_0, z\_1)d(z\_1, z\_2)) \le \mathcal{F}(d(z\_0, z\_1)).\tag{37}
$$

On continuing recursively, we get a sequence f g zn <sup>n</sup>∈IN in <sup>X</sup>, where znþ<sup>1</sup> <sup>∈</sup>Fzn σ , znþ<sup>1</sup> ∉T znþ1, αð Þ zn; znþ<sup>1</sup> ≥ 1, M zð Þ <sup>n</sup>; znþ<sup>1</sup> ≤d zð Þ <sup>n</sup>; znþ<sup>1</sup> and

$$
\sigma(d(z\_n, z\_{n+1})) + \mathcal{F}(D(z\_{n+1}, Tz\_{n+1})) \le \mathcal{F}(d(z\_n, z\_{n+1})).\tag{38}
$$

Since znþ<sup>1</sup> <sup>∈</sup>Fzn <sup>σ</sup> and T zn and T znþ<sup>1</sup> are compact, we have

$$\left(\pi(d(z\_n, z\_{n+1})) + \mathcal{F}(d(z\_{n+1}, z\_{n+2})) \leq \mathcal{F}(d(z\_n, z\_{n+1}))\right) \tag{39}$$

and

Contraction Mappings and Applications DOI: http://dx.doi.org/10.5772/intechopen.81571

$$\mathcal{F}(d(z\_n, z\_{n+1})) \le \mathcal{F}(d(z\_n, z\_{n+1})) + \sigma. \tag{40}$$

Combining (39) and (40) gives

Then T has a fixed point in X.

Recent Advances in Integral Equations

any σ . 0, then there exists z<sup>1</sup> ∈ Fz<sup>0</sup>

where

Since d zð Þ <sup>0</sup>;z<sup>2</sup>

consequently,

Since znþ<sup>1</sup> <sup>∈</sup>Fzn

and

16

D zð Þ <sup>1</sup>; T z<sup>1</sup> . 0, so, from (28), we have

<sup>2</sup> <sup>≤</sup> d zð Þþ <sup>0</sup>;z<sup>1</sup> d zð Þ <sup>1</sup>;z<sup>2</sup>

Proof. Let <sup>z</sup><sup>0</sup> <sup>∈</sup> <sup>X</sup>, since <sup>T</sup> <sup>z</sup>∈Kð Þ <sup>X</sup> for every <sup>z</sup><sup>∈</sup> <sup>X</sup>, the set <sup>F</sup><sup>z</sup>

M zð Þ¼ <sup>0</sup>; z<sup>1</sup> max d zð Þ <sup>0</sup>; z<sup>1</sup> ; Dðz0; T z0Þ; Dðz1; T z1Þ;

D zð Þ <sup>1</sup>; T z<sup>1</sup> ½ � 1 þ D zð Þ <sup>0</sup>; T z<sup>0</sup> 1 þ d zð Þ <sup>0</sup>; z<sup>1</sup>

M zð Þ¼ <sup>0</sup>; z<sup>1</sup> max d zð Þ <sup>0</sup>; z<sup>1</sup> ; dðz0; z1Þ; dðz1; z2Þ;

¼ max d zð Þ <sup>0</sup>; z<sup>1</sup> ; dðz1; z2Þ;

Suppose that d zð Þ <sup>0</sup>; z<sup>1</sup> , d zð Þ <sup>1</sup>; z<sup>2</sup> , then (31) implies that

Mdz ð Þ ð Þ <sup>0</sup>; z<sup>1</sup> ≤d zð Þ <sup>0</sup>; z<sup>1</sup> , therefore by using ð Þ F<sup>1</sup> , (31) implies that

znþ<sup>1</sup> ∉T znþ1, αð Þ zn; znþ<sup>1</sup> ≥ 1, M zð Þ <sup>n</sup>; znþ<sup>1</sup> ≤d zð Þ <sup>n</sup>; znþ<sup>1</sup> and

d zð Þ <sup>1</sup>; z<sup>2</sup> ½ � 1 þ d zð Þ <sup>0</sup>; z<sup>1</sup> 1 þ d zð Þ <sup>0</sup>; z<sup>1</sup>

Since T z<sup>0</sup> and T z<sup>1</sup> are compact, so we have

z<sup>1</sup> ∉T z1, otherwise z<sup>1</sup> is the fixed point of T . Then, since T z<sup>1</sup> is closed,

<sup>σ</sup> is non-empty for

<sup>2</sup> ;

: (32)

σ ,

D zð Þþ <sup>1</sup>; T z<sup>0</sup> D zð Þ <sup>0</sup>; T z<sup>1</sup>

d zð Þþ <sup>1</sup>; z<sup>1</sup> d zð Þ <sup>0</sup>; z<sup>2</sup>

<sup>2</sup> ;

: (33)

<sup>σ</sup> and by hypothesis αð Þ z0; z<sup>1</sup> ≥1. Assume that

D zð Þ <sup>1</sup>; T z<sup>0</sup> ½ � 1 þ D zð Þ <sup>0</sup>; T z<sup>1</sup> 1 þ d zð Þ <sup>0</sup>; z<sup>1</sup>

d zð Þ <sup>1</sup>; z<sup>1</sup> ½ � 1 þ d zð Þ <sup>0</sup>; z<sup>2</sup> 1 þ d zð Þ <sup>0</sup>; z<sup>1</sup>

M zð Þ <sup>0</sup>; z<sup>1</sup> ≤ maxf g d zð Þ <sup>0</sup>; z<sup>1</sup> ; d zð Þ <sup>1</sup>; z<sup>2</sup> : (34)

d zð Þ <sup>0</sup>; z<sup>2</sup> 2

<sup>2</sup> ≤ maxf g d zð Þ <sup>0</sup>; z<sup>1</sup> ; d zð Þ <sup>1</sup>; z<sup>2</sup> , it follows that

<sup>≤</sup> <sup>F</sup>ð Þ d zð Þ <sup>1</sup>; <sup>z</sup><sup>2</sup> , (35)

τð Þþ d zð Þ <sup>0</sup>; z<sup>1</sup> Fð Þ d zð Þ <sup>1</sup>; z<sup>2</sup> ≤ Fð Þ d zð Þ <sup>1</sup>; z<sup>2</sup> , (36)

τð Þþ d zð Þ <sup>0</sup>; z<sup>1</sup> Fð Þ αð Þ z0; z<sup>1</sup> d zð Þ <sup>1</sup>; z<sup>2</sup> ≤ Fð Þ d zð Þ <sup>0</sup>; z<sup>1</sup> : (37)

τð Þþ d zð Þ <sup>n</sup>; znþ<sup>1</sup> Fð Þ D zð Þ <sup>n</sup>þ<sup>1</sup>; T znþ<sup>1</sup> ≤ Fð Þ d zð Þ <sup>n</sup>; znþ<sup>1</sup> : (38)

τð Þþ d zð Þ <sup>n</sup>; znþ<sup>1</sup> Fð Þ d zð Þ <sup>n</sup>þ<sup>1</sup>; znþ<sup>2</sup> ≤ Fð Þ d zð Þ <sup>n</sup>; znþ<sup>1</sup> (39)

τð Þþ d zð Þ <sup>0</sup>; z<sup>1</sup> Fð Þ αð Þ z0; z<sup>1</sup> D zð Þ <sup>1</sup>; T z<sup>1</sup> ≤ Fð Þ M zð Þ <sup>0</sup>; z<sup>1</sup> , (31)

;

;

τð Þþ d zð Þ <sup>0</sup>; z<sup>1</sup> Fð Þ D zð Þ <sup>1</sup>; T z<sup>1</sup> ≤τð Þþ d zð Þ <sup>0</sup>; z<sup>1</sup> Fð Þ αð Þ z0; z<sup>1</sup> D zð Þ <sup>1</sup>; T z<sup>1</sup>

or, Fð Þ d zð Þ <sup>1</sup>; z<sup>2</sup> ≤ Fð Þ� d zð Þ <sup>1</sup>; z<sup>2</sup> τð Þ d zð Þ <sup>0</sup>; z<sup>1</sup> , which is a contradiction. Hence

On continuing recursively, we get a sequence f g zn <sup>n</sup>∈IN in <sup>X</sup>, where znþ<sup>1</sup> <sup>∈</sup>Fzn

<sup>σ</sup> and T zn and T znþ<sup>1</sup> are compact, we have

$$\mathcal{F}(d(z\_{n+1}, z\_{n+2})) \le \mathcal{F}(d(z\_n, z\_{n+1})) + \sigma - \tau(d(z\_n, z\_{n+1})) \tag{41}$$

Let dn ¼ d zð Þ <sup>n</sup>; znþ<sup>1</sup> for n ∈IN, then dn . 0 and from (41) f g dn is decreasing. Therefore, there exists δ≥0 such that limn!<sup>∞</sup>dn ¼ δ. Now let δ . 0. From (41), we get

$$\begin{split} \mathcal{F}(d\_{n+1}) \leq & \mathcal{F}(d\_n) + \sigma - \mathfrak{r}(d\_n) \leq \mathcal{F}(d\_{n-1}) + 2\sigma - \mathfrak{r}(d\_n) - \mathfrak{r}(d\_{n-1}) \cdots \\ \leq & \mathcal{F}(d\_0) + n\sigma - \mathfrak{r}(d\_n) - \mathfrak{r}(d\_{n-1}) - \cdots - \mathfrak{r}(d\_0). \end{split} \tag{42}$$

Let τ dpn � � <sup>¼</sup> minf g <sup>τ</sup>ð Þ <sup>d</sup><sup>0</sup> ; <sup>τ</sup>ð Þ <sup>d</sup><sup>1</sup> ; <sup>⋯</sup>; <sup>τ</sup>ð Þ dn for all <sup>n</sup><sup>∈</sup> IN. From (42), we get

$$\mathcal{F}(d\_{n+1}) \le \mathcal{F}(d\_0) + n\left(\sigma - \tau(d\_{p\_n})\right). \tag{43}$$

From (38), we also get

$$\mathcal{F}(\mathbf{D}(\mathbf{z}\_{n+1}, T\mathbf{z}\_{n+1})) \le \mathcal{F}(\mathbf{D}(\mathbf{z}\_0, T\mathbf{z}\_0)) + n\left(\sigma - \mathfrak{r}(d\_{p\_n})\right). \tag{44}$$

Now consider the sequence τ dpn � � � � . We distinguish two cases.

Case 1. For each n∈IN, there is m . n such that τ dpn � � . <sup>τ</sup> dpm � �. Then we obtain a subsequence dpnk n o of dpn � � with <sup>τ</sup> dpnk � � . <sup>τ</sup> dpnkþ<sup>1</sup> � � for all <sup>k</sup>. Since dpnk ! δþ, we deduce that lim<sup>k</sup>!<sup>∞</sup>inf<sup>τ</sup> dpnk � � . <sup>σ</sup>. Hence <sup>F</sup> dnk � �<sup>≤</sup> <sup>F</sup>ð Þþ <sup>d</sup><sup>0</sup> <sup>n</sup> <sup>σ</sup> � <sup>τ</sup> dpnk � � � � for all k. Consequently, lim<sup>k</sup>!∞F dnk � � ¼ �<sup>∞</sup> and by ð Þ <sup>F</sup><sup>2</sup> , we obtain lim<sup>k</sup>!<sup>∞</sup>dpnk Þ ¼ 0, which contradicts that lim<sup>n</sup>!<sup>∞</sup>dn . 0.

Case 2. There is <sup>n</sup><sup>0</sup> <sup>∈</sup> IN such that <sup>τ</sup> dpn<sup>0</sup> � � . <sup>τ</sup> dpm � � for all m . n0. Then <sup>F</sup>ð Þ dm <sup>≤</sup> <sup>F</sup>ð Þþ <sup>d</sup><sup>0</sup> <sup>m</sup> <sup>σ</sup> � <sup>τ</sup> dpn<sup>0</sup> � � � � for all <sup>m</sup> . <sup>n</sup>0. Hence lim<sup>m</sup>!∞Fð Þ¼� dm <sup>∞</sup>, so lim<sup>m</sup>!<sup>∞</sup>dm ¼ 0, which contradicts that lim<sup>m</sup>!<sup>∞</sup>dm . 0. Thus,

$$\lim\_{n \to \infty} d\_n = \mathbf{0}.$$

From ð Þ <sup>F</sup><sup>3</sup> , there exists 0 , <sup>r</sup> , 1 such that lim<sup>n</sup>!<sup>∞</sup>ð Þ dn <sup>r</sup> Fð Þ¼ dn 0. By (43), we get for all n ∈IN

$$n(d\_n)^r \mathcal{F}(d\_n) - (d\_n)^r \mathcal{F}(d\_0) \le (d\_n)^r n\left(\sigma - \tau(d - p\_n)\right) \le 0. \tag{45}$$

Letting <sup>n</sup> ! <sup>∞</sup> in (45), we obtain lim<sup>n</sup>!<sup>∞</sup>n dð Þ<sup>n</sup> <sup>r</sup> <sup>¼</sup> 0. This implies that there exists <sup>n</sup><sup>1</sup> <sup>∈</sup>IN such that n dð Þ<sup>n</sup> <sup>r</sup> <sup>≤</sup> 1, or, dn <sup>≤</sup> <sup>1</sup> <sup>n</sup>1=<sup>r</sup>, for all n . n1. Next, for m . n ≥n<sup>1</sup> we have

$$d(z\_n, z\_m) \le \sum\_{i=n}^{m-1} d(z\_i, z\_{i+1}) \le \sum\_{i=n}^{m-1} \frac{1}{i^{1/k}},$$

since 0 , k , 1, ∑<sup>m</sup>�<sup>1</sup> <sup>i</sup>¼<sup>n</sup> <sup>1</sup> i <sup>1</sup>=<sup>k</sup> converges. Therefore, d zð Þ! <sup>n</sup>; zm 0 as m, n ! ∞. Thus, f g zn is a Cauchy sequence. Since <sup>X</sup> is complete, there exists <sup>z</sup><sup>∗</sup> <sup>∈</sup> <sup>X</sup> such that zn ! <sup>z</sup><sup>∗</sup> as n ! ∞. From Eqs. (44) and ð Þ F<sup>2</sup> , we have lim<sup>n</sup>!<sup>∞</sup>D zð Þ¼ <sup>n</sup>; T zn 0. Since z ! D zð Þ ; T z is lower semi-continuous, then

$$0 \le D(\boldsymbol{z}, T\boldsymbol{z}) \le \lim\_{n \to \infty} \inf D(\boldsymbol{z}\_n, T\boldsymbol{z}\_n) = \mathbf{0}.$$

Thus, <sup>T</sup> has a fixed point. □ In the following theorem we take Cð Þ X instead of Kð Þ X , then we need to take F ∈ Δð Þ F<sup>∗</sup> in Definition 3.10.

Theorem 3.13 Let ð Þ X; d be a complete metric space and T : X ! Cð Þ X be an multivalued α-F-weak-contraction with F ∈ Δð Þ F<sup>∗</sup> satisfying all the assertions of Theorem 3.12. Then T has a fixed point in X.

Proof. Let <sup>z</sup><sup>0</sup> <sup>∈</sup> <sup>X</sup>, since <sup>T</sup> <sup>z</sup>∈Cð Þ <sup>X</sup> for every <sup>z</sup><sup>∈</sup> <sup>X</sup> and <sup>F</sup> <sup>∈</sup> <sup>Δ</sup>ð Þ <sup>F</sup><sup>∗</sup> , the set <sup>F</sup><sup>z</sup> <sup>σ</sup> is non-empty for any σ . 0, then there exists z<sup>1</sup> ∈Fz<sup>0</sup> <sup>σ</sup> and by hypothesis αð Þ z0; z<sup>1</sup> ≥1. Assume that z<sup>1</sup> ∉T z1, otherwise z<sup>1</sup> is the fixed point of T . Then, since T z<sup>1</sup> is closed, D zð Þ <sup>1</sup>; T z<sup>1</sup> . 0, so, from (28), we have

$$
\pi(d(z\_0, z\_1)) + a(z\_0, z\_1)\mathcal{F}(D(z\_1, \mathcal{T}z\_1)) \le \mathcal{F}(\mathcal{M}(z\_0, z\_1)),\tag{46}
$$

2. there exists z0, z<sup>1</sup> ∈ X such that αð Þ z0; z<sup>1</sup> ≥1;

lim

�

and for all z∈ X with dð Þ T z; T y . 0, there exist a function

m zð Þ¼ ; y max d zð Þ ; y ; dðz; T zÞ; dðy; T yÞ;

ordered metric spaces (see [34–36] and references therein).

3.there exists z<sup>0</sup> ∈X and z<sup>1</sup> ∈T z<sup>0</sup> such that z<sup>0</sup> ≼z1;

1 y≼ z

then for y, z∈ X with y ≼z, αð Þ y; z ≥ηð Þ y; z implies

T : X ! X be a self-map fulfilling the following assertions:

Proof. Define α, η : X � X ! ½ Þ 0; ∞ by

8 < :

d yð Þ ; T y ½ � 1 þ d zð Þ ; T z <sup>1</sup> <sup>þ</sup> d zð Þ ; <sup>y</sup> ;

<sup>t</sup>!s<sup>þ</sup> inf <sup>τ</sup>ð Þ<sup>t</sup> . 0 for all <sup>s</sup><sup>≥</sup> <sup>0</sup>

Now, let ð Þ <sup>X</sup>; <sup>d</sup>; <sup>≼</sup> be a partially ordered metric space. Recall that <sup>T</sup> : <sup>X</sup> ! <sup>2</sup><sup>X</sup> is monotone increasing if T y ≼T z for all y, z∈ X, for which y≼z (see [33]). There are many applications in differential and integral equations of monotone mappings in

Theorem 3.17 Let ð Þ X; d; ≼ be a complete partially ordered metric space and let <sup>T</sup> : <sup>X</sup> ! <sup>2</sup><sup>X</sup> be a closed valued mapping satisfying the following assertions for all y, z<sup>∈</sup> <sup>X</sup>

4.for a sequence zf g<sup>n</sup> ⊂ X, lim<sup>n</sup>!<sup>∞</sup>f g zn ¼ z and zn ≼ znþ<sup>1</sup> for all n ∈IN, we have

0 otherwise <sup>η</sup>ð Þ¼ <sup>y</sup>; <sup>z</sup>

all the conditions of Theorem 3.4 are satisfied and hence <sup>T</sup> has a fixed point. □ In case of single valued mapping Theorem 3.17 reduced to the following: Theorem 3.18 Let ð Þ X; d; ≼ be a complete partially ordered metric space and let

τð Þþ d zð Þ ; y αð Þ z; y Fð Þ dð Þ T z; T y ≤ Fð Þ m zð Þ ; y , (49)

d yð Þþ ; T z d zð Þ ; T y

d yð Þ ; T z ½ � 1 þ d zð Þ ; T y 1 þ d zð Þ ; y

1

8 < :

<sup>2</sup> ¼ η∗ð Þ T y; T z and α∗ð Þ¼ T y; T z η∗ð Þ¼ T y; T z 0 otherwise. Thus,

<sup>2</sup> <sup>y</sup>≼<sup>z</sup> 0 otherwise,

<sup>2</sup> ;

�

: (50)

3.there exists τ : ð Þ! 0; ∞ ð Þ 0; ∞ such that

Contraction Mappings and Applications DOI: http://dx.doi.org/10.5772/intechopen.81571

α : X � X ! �f g ∞ ∪ð Þ 0; þ∞ satisfying

Then T has a fixed point in X.

1. T is monotone increasing;

2. ϑð Þ Hð Þ T y; T z ; d yð Þ ; z ≥0;

zn ≼ z for all n∈IN.

αð Þ¼ y; z

1. T is monotone increasing;

Then Fixf g T 6¼ ∅.

<sup>α</sup>∗ð Þ¼ <sup>T</sup> <sup>y</sup>; <sup>T</sup> <sup>z</sup> <sup>1</sup> . <sup>1</sup>

19

where

with y ≼z:

where

$$M(\mathbf{z}\_0, \mathbf{z}\_1) = \max\left\{ d(\mathbf{z}\_0, \mathbf{z}\_1), D(\mathbf{z}\_0, \mathbf{T}\mathbf{z}\_0), D(\mathbf{z}\_1, \mathbf{T}\mathbf{z}\_1), \frac{D(\mathbf{z}\_1, \mathbf{T}\mathbf{z}\_0) + D(\mathbf{z}\_0, \mathbf{T}\mathbf{z}\_1)}{2},$$

$$\frac{D(\mathbf{z}\_1, \mathbf{T}\mathbf{z}\_1)[\mathbf{1} + D(\mathbf{z}\_0, \mathbf{T}\mathbf{z}\_0)]}{\mathbf{1} + d(\mathbf{z}\_0, \mathbf{z}\_1)}, \frac{D(\mathbf{z}\_1, \mathbf{T}\mathbf{z}\_0)[\mathbf{1} + D(\mathbf{z}\_0, \mathbf{T}\mathbf{z}\_1)]}{\mathbf{1} + d(\mathbf{z}\_0, \mathbf{z}\_1)} \right\}. \tag{47}$$

The rest of the proof can be completed as in the proof of Theorem 3.12 by considering the closedness of <sup>T</sup> <sup>z</sup>, for all <sup>z</sup><sup>∈</sup> <sup>X</sup>. □

Theorem 3.14 Let ð Þ X; d be a complete metric space, T : X ! Kð Þ X be a continuous mapping and F ∈ Δð Þ F . Assume that the following assertions hold:

1. T is generalized α∗-admissible mapping;

2. there exists z<sup>0</sup> ∈ X and z<sup>1</sup> ∈T z<sup>0</sup> such that αð Þ z0; z<sup>1</sup> ≥ 1;

3.there exists τ : ð Þ! 0; ∞ ð Þ 0; ∞ such that

$$\lim\_{t \to s^{+}} \inf \,\,\tau(t) \ge 0 \qquad\qquad \text{for all}\quad s \ge 0$$

and for all z∈ X with Hð Þ T z; T y . 0, there exist a function α : X � X ! �f g ∞ ∪ð Þ 0; þ∞ satisfying

$$
\sigma(d(z,\boldsymbol{y})) + a(z,\boldsymbol{y})\mathcal{F}(H(\mathcal{T}z,\mathcal{T}\boldsymbol{y})) \le \mathcal{F}(M(z,\boldsymbol{y})),\tag{48}
$$

where M zð Þ ; y is defined in (29).

Then T has a fixed point in X.

Proof. By following the steps in the proof of Theorem 3.12, we get the required result. □

Note that Theorem 3.14 cannot be obtained from Theorem 3.12, because in Theorem 3.12, σ cannot be equal to zero.

Theorem 3.15 Let ð Þ X; d be a complete metric space, T : X ! Cð Þ X be a continuous mapping and F ∈ Δð Þ F<sup>∗</sup> satisfying all assertions of Theorem 3.14. Then T has a fixed point in X.

From Theorems 3.14 and 3.15, we get the following fixed point result for single valued mappings:

Theorem 3.16 Let ð Þ X; d be a complete metric space, T : X ! X be a continuous mapping and F ∈ Δð Þ F . Assume that the following assertions hold:

1. T is α-admissible mapping;

Contraction Mappings and Applications DOI: http://dx.doi.org/10.5772/intechopen.81571

2. there exists z0, z<sup>1</sup> ∈ X such that αð Þ z0; z<sup>1</sup> ≥1;

3.there exists τ : ð Þ! 0; ∞ ð Þ 0; ∞ such that

$$\lim\_{t \to s^{+}} \inf \,\,\tau(t) \ge 0 \qquad\qquad \text{for all}\quad s \ge 0$$

and for all z∈ X with dð Þ T z; T y . 0, there exist a function α : X � X ! �f g ∞ ∪ð Þ 0; þ∞ satisfying

$$
\sigma(d(z,\boldsymbol{y})) + a(z,\boldsymbol{y})\mathcal{F}(d(\mathcal{T}z,\mathcal{T}\boldsymbol{y})) \le \mathcal{F}(m(z,\boldsymbol{y})),\tag{49}
$$

where

Thus, <sup>T</sup> has a fixed point. □ In the following theorem we take Cð Þ X instead of Kð Þ X , then we need to take

Theorem 3.13 Let ð Þ X; d be a complete metric space and T : X ! Cð Þ X be an multivalued α-F-weak-contraction with F ∈ Δð Þ F<sup>∗</sup> satisfying all the assertions of Theo-

Proof. Let <sup>z</sup><sup>0</sup> <sup>∈</sup> <sup>X</sup>, since <sup>T</sup> <sup>z</sup>∈Cð Þ <sup>X</sup> for every <sup>z</sup><sup>∈</sup> <sup>X</sup> and <sup>F</sup> <sup>∈</sup> <sup>Δ</sup>ð Þ <sup>F</sup><sup>∗</sup> , the set <sup>F</sup><sup>z</sup>

Assume that z<sup>1</sup> ∉T z1, otherwise z<sup>1</sup> is the fixed point of T . Then, since T z<sup>1</sup> is closed,

;

The rest of the proof can be completed as in the proof of Theorem 3.12 by considering the closedness of <sup>T</sup> <sup>z</sup>, for all <sup>z</sup><sup>∈</sup> <sup>X</sup>. □ Theorem 3.14 Let ð Þ X; d be a complete metric space, T : X ! Kð Þ X be a continu-

<sup>t</sup>!s<sup>þ</sup> inf <sup>τ</sup>ð Þ<sup>t</sup> . 0 for all <sup>s</sup><sup>≥</sup> <sup>0</sup>

Proof. By following the steps in the proof of Theorem 3.12, we get the required result. □ Note that Theorem 3.14 cannot be obtained from Theorem 3.12, because in

Theorem 3.15 Let ð Þ X; d be a complete metric space, T : X ! Cð Þ X be a continuous mapping and F ∈ Δð Þ F<sup>∗</sup> satisfying all assertions of Theorem 3.14. Then T has a fixed

From Theorems 3.14 and 3.15, we get the following fixed point result for single

Theorem 3.16 Let ð Þ X; d be a complete metric space, T : X ! X be a continuous

τð Þþ d zð Þ ; y αð Þ z; y Fð Þ Hð Þ T z; T y ≤ Fð Þ M zð Þ ; y , (48)

τð Þþ d zð Þ <sup>0</sup>; z<sup>1</sup> αð Þ z0; z<sup>1</sup> Fð Þ D zð Þ <sup>1</sup>; T z<sup>1</sup> ≤ Fð Þ M zð Þ <sup>0</sup>; z<sup>1</sup> , (46)

<sup>σ</sup> is

<sup>σ</sup> and by hypothesis αð Þ z0; z<sup>1</sup> ≥1.

D zð Þþ <sup>1</sup>; T z<sup>0</sup> D zð Þ <sup>0</sup>; T z<sup>1</sup>

D zð Þ <sup>1</sup>; T z<sup>0</sup> ½ � 1 þ D zð Þ <sup>0</sup>; T z<sup>1</sup> 1 þ d zð Þ <sup>0</sup>; z<sup>1</sup>

<sup>2</sup> ;

: (47)

F ∈ Δð Þ F<sup>∗</sup> in Definition 3.10.

Recent Advances in Integral Equations

where

rem 3.12. Then T has a fixed point in X.

D zð Þ <sup>1</sup>; T z<sup>1</sup> . 0, so, from (28), we have

1. T is generalized α∗-admissible mapping;

3.there exists τ : ð Þ! 0; ∞ ð Þ 0; ∞ such that

α : X � X ! �f g ∞ ∪ð Þ 0; þ∞ satisfying

where M zð Þ ; y is defined in (29). Then T has a fixed point in X.

Theorem 3.12, σ cannot be equal to zero.

1. T is α-admissible mapping;

point in X.

18

valued mappings:

lim

non-empty for any σ . 0, then there exists z<sup>1</sup> ∈Fz<sup>0</sup>

M zð Þ¼ <sup>0</sup>; z<sup>1</sup> max d zð Þ <sup>0</sup>; z<sup>1</sup> ; Dðz0; T z0Þ; Dðz1; T z1Þ;

D zð Þ <sup>1</sup>; T z<sup>1</sup> ½ � 1 þ D zð Þ <sup>0</sup>; T z<sup>0</sup> 1 þ d zð Þ <sup>0</sup>; z<sup>1</sup>

ous mapping and F ∈ Δð Þ F . Assume that the following assertions hold:

and for all z∈ X with Hð Þ T z; T y . 0, there exist a function

mapping and F ∈ Δð Þ F . Assume that the following assertions hold:

2. there exists z<sup>0</sup> ∈ X and z<sup>1</sup> ∈T z<sup>0</sup> such that αð Þ z0; z<sup>1</sup> ≥ 1;

$$m(z, \mathbf{y}) = \max\left\{ d(z, \mathbf{y}), d(z, \mathbf{T}z), d(\mathbf{y}, \mathbf{T}\mathbf{y}), \frac{d(\mathbf{y}, \mathbf{T}z) + d(\mathbf{z}, \mathbf{T}\mathbf{y})}{2},$$

$$\frac{d(\mathbf{y}, \mathbf{T}\mathbf{y})[\mathbf{1} + d(\mathbf{z}, \mathbf{T}z)]}{\mathbf{1} + d(\mathbf{z}, \mathbf{y})}, \frac{d(\mathbf{y}, \mathbf{T}z)[\mathbf{1} + d(\mathbf{z}, \mathbf{T}\mathbf{y})]}{\mathbf{1} + d(\mathbf{z}, \mathbf{y})}\right\}.\tag{50}$$

Then T has a fixed point in X.

Now, let ð Þ <sup>X</sup>; <sup>d</sup>; <sup>≼</sup> be a partially ordered metric space. Recall that <sup>T</sup> : <sup>X</sup> ! <sup>2</sup><sup>X</sup> is monotone increasing if T y ≼T z for all y, z∈ X, for which y≼z (see [33]). There are many applications in differential and integral equations of monotone mappings in ordered metric spaces (see [34–36] and references therein).

Theorem 3.17 Let ð Þ X; d; ≼ be a complete partially ordered metric space and let <sup>T</sup> : <sup>X</sup> ! <sup>2</sup><sup>X</sup> be a closed valued mapping satisfying the following assertions for all y, z<sup>∈</sup> <sup>X</sup> with y ≼z:

1. T is monotone increasing;


Then Fixf g T 6¼ ∅. Proof. Define α, η : X � X ! ½ Þ 0; ∞ by

$$a(\mathbf{y}, \mathbf{z}) = \begin{cases} 1 & \mathbf{y} \not\approx \mathbf{z} \\ 0 & \text{otherwise} \end{cases} \qquad \eta(\mathbf{y}, \mathbf{z}) = \begin{cases} 1 & \mathbf{y} \not\approx \mathbf{z} \\ 2 & \text{otherwise} \end{cases}$$

then for y, z∈ X with y ≼z, αð Þ y; z ≥ηð Þ y; z implies

<sup>α</sup>∗ð Þ¼ <sup>T</sup> <sup>y</sup>; <sup>T</sup> <sup>z</sup> <sup>1</sup> . <sup>1</sup> <sup>2</sup> ¼ η∗ð Þ T y; T z and α∗ð Þ¼ T y; T z η∗ð Þ¼ T y; T z 0 otherwise. Thus, all the conditions of Theorem 3.4 are satisfied and hence <sup>T</sup> has a fixed point. □

In case of single valued mapping Theorem 3.17 reduced to the following: Theorem 3.18 Let ð Þ X; d; ≼ be a complete partially ordered metric space and let

T : X ! X be a self-map fulfilling the following assertions:

1. T is monotone increasing;


for all y, z∈ X with y≼z and ϑ∈ MandðIRÞ. Then Fixf g T 6¼ ∅.

Definition 3.19 Let <sup>T</sup> : <sup>X</sup> ! <sup>2</sup><sup>X</sup> be a multivalued mapping on a partially ordered metric space ð Þ X; d; ≼ , then T is said to be an ordered F-τ-contraction on X, if there exists <sup>σ</sup> . 0 and <sup>τ</sup> : ð Þ! <sup>0</sup>; <sup>∞</sup> ð Þ <sup>σ</sup>; <sup>∞</sup> , <sup>F</sup> <sup>∈</sup> <sup>Δ</sup>ð Þ <sup>F</sup> such that for all <sup>z</sup><sup>∈</sup> <sup>X</sup>, <sup>y</sup><sup>∈</sup> <sup>F</sup><sup>z</sup> σ with z≼y and D zð Þ ; T z . 0 satisfying

$$
\pi(d(z,\boldsymbol{y})) + \mathcal{F}(D(\boldsymbol{y},T\boldsymbol{y})) \le \mathcal{F}(\mathsf{M}(z,\boldsymbol{y})),\tag{51}
$$

and for all z, y∈ X with z≼ y and Hð Þ T z; T y . 0 satisfying

where M zð Þ ; y is defined in (52). Then T has a fixed point in X.

Contraction Mappings and Applications DOI: http://dx.doi.org/10.5772/intechopen.81571

Theorem 3.22. Then T has a fixed point in X.

2. there exists z0, z<sup>1</sup> ∈ X such that z<sup>0</sup> ≼z1;

3.there exists τ : ð Þ! 0; ∞ ð Þ 0; ∞ such that

lim

�

Then T has a fixed point in X.

4. Existence of solution

and for all z, y∈ X with z≼ y and dð Þ T z; T y . 0 satisfying

m zð Þ¼ ; y max d zð Þ ; y ; dðz; T zÞ; dðy; T yÞ;

d yð Þ ; T y ½ � 1 þ d zð Þ ; T z <sup>1</sup> <sup>þ</sup> d zð Þ ; <sup>y</sup> ;

4.1 Solution of Fredholm integral equation of second kind

b

<sup>T</sup> z rð Þ¼ <sup>ð</sup><sup>c</sup>

Let <sup>≪</sup> be a partial order relation on IR<sup>n</sup>. Define <sup>T</sup> : <sup>X</sup> ! <sup>X</sup> by

1. <sup>B</sup> : ½ �� <sup>b</sup>;<sup>c</sup> ½ �� <sup>b</sup>;<sup>c</sup> IR<sup>n</sup> ! IR<sup>n</sup> and <sup>g</sup> : IR<sup>n</sup> ! IR<sup>n</sup> are continuous;

1. T is monotone increasing;

valued mapping.

tions hold:

where

21

τð Þþ d zð Þ ; y Fð Þ Hð Þ T z; T y ≤ Fð Þ M zð Þ ; y , (53)

Proof. By defining α : X � X ! ½ Þ 0; ∞ as in the proof of Theorem 3.17 and by using Theorem (3.14), we get the required result. □ Theorem 3.23 Let ð Þ X; d; ≼ be a complete partially ordered metric space, T : X ! Cð Þ X be a continuous mapping and F ∈ Δð Þ F<sup>∗</sup> satisfying all assertions of

From Theorems 3.22 and 3.23, we get the following fixed point result for single

<sup>t</sup>!s<sup>þ</sup> inf <sup>τ</sup>ð Þ<sup>t</sup> . <sup>0</sup> for all s <sup>≥</sup><sup>0</sup>

In this section, by using the fixed point results proved in the previous section,

Theorem 4.1 Let <sup>X</sup> <sup>¼</sup> C b½ � ;<sup>c</sup> ;IR<sup>n</sup> ð Þ with the usual spermium norm. Suppose that

we obtain the existence of the solution of integral Eq. (2) and matrix Eq. (1).

τð Þþ d zð Þ ; y Fð Þ dð Þ T z; T y ≤ Fð Þ m zð Þ ; y , (54)

d yð Þ ; T z ½ � 1 þ d zð Þ ; T y 1 þ d zð Þ ; y

Bð Þ r; s; z sð Þ ds þ g rð Þ, r∈½ � a; b : (56)

d yð Þþ ; T z d zð Þ ; T y

<sup>2</sup> ;

: (55)

�

Theorem 3.24 Let ð Þ X; d; ≼ be a complete partially ordered metric space, T : X ! X be a continuous mapping and F ∈ Δð Þ F . Assume that the following asser-

where,

$$M(\mathbf{z},\mathbf{y}) = \max\left\{d(\mathbf{z},\mathbf{y}), D(\mathbf{z},\mathbf{T}\mathbf{z}), D(\mathbf{y},\mathbf{T}\mathbf{y}), \frac{D(\mathbf{y},\mathbf{T}\mathbf{z}) + D(\mathbf{z},\mathbf{T}\mathbf{y})}{2},$$

$$\frac{D(\mathbf{y},\mathbf{T}\mathbf{y})[\mathbf{1} + D(\mathbf{z},\mathbf{T}\mathbf{z})]}{\mathbf{1} + d(\mathbf{z},\mathbf{y})}, \frac{D(\mathbf{y},\mathbf{T}\mathbf{z})[\mathbf{1} + D(\mathbf{z},\mathbf{T}\mathbf{y})]}{\mathbf{1} + d(\mathbf{z},\mathbf{y})}\right\}.\tag{52}$$

Theorem 3.20 Let ð Þ X; d; ≼ be a complete partially ordered metric space and T : X ! Kð Þ X be an ordered F-τ-contraction satisfying the following assertions:


4. τ satisfies

$$\lim\_{t \to s^{+}} \inf \,\,\tau(t) \ge \sigma \qquad\text{ for all}\quad s \ge 0$$

Then T has a fixed point in X.

Proof. By using the similar arguments as in the proof of Theorem 3.17 and using Theorem 3.12, we get the result. □

Theorem 3.21 Let ð Þ X; d; ≼ be a complete partially ordered metric space and T : X ! Cð Þ X be an ordered F-τ-contraction with F ∈ Δð Þ F<sup>∗</sup> satisfying all the assertions of Theorem 3.20. Then T has a fixed point in X.

Theorem 3.22 Let ð Þ X; d; ≼ be a complete partially ordered metric space, T : X ! Kð Þ X be a continuous mapping and F ∈ Δð Þ F . Assume that the following assertions hold:

1. T is monotone increasing;

2. there exists z<sup>0</sup> ∈ X and z<sup>1</sup> ∈T z<sup>0</sup> such that z<sup>0</sup> ≼z1;

3.there exists τ : ð Þ! 0; ∞ ð Þ 0; ∞ such that

$$\lim\_{t \to s^{+}} \inf \,\,\tau(t) \succeq 0 \qquad\qquad \text{for all}\quad s \ge 0$$

and for all z, y∈ X with z≼ y and Hð Þ T z; T y . 0 satisfying

$$\tau(d(z,\boldsymbol{y})) + \mathcal{F}(H(Tz,\mathcal{T}\boldsymbol{y})) \le \mathcal{F}(M(z,\boldsymbol{y})),\tag{53}$$

where M zð Þ ; y is defined in (52).

Then T has a fixed point in X.

2. ϑð Þ dð Þ T y; T z ; d yð Þ ; z ≥0;

Recent Advances in Integral Equations

zn ≼ z for all n∈IN.

where,

4. τ satisfies

assertions hold:

20

with z≼y and D zð Þ ; T z . 0 satisfying

1. T is monotone increasing;

Then T has a fixed point in X.

1. T is monotone increasing;

3.there exists z<sup>0</sup> ∈ X and z<sup>1</sup> ¼ T z<sup>0</sup> such that z<sup>0</sup> ≼z1;

for all y, z∈ X with y≼z and ϑ∈ MandðIRÞ. Then Fixf g T 6¼ ∅.

M zð Þ¼ ; y max d zð Þ ; y ; Dðz; T zÞ; Dðy; T yÞ;

D yð Þ ; T y ½ � 1 þ D zð Þ ; T z <sup>1</sup> <sup>þ</sup> d zð Þ ; <sup>y</sup> ;

�

2. the map z ! D zð Þ ; T z is lower semi-continuous;

3.there exists z<sup>0</sup> ∈ X and z<sup>1</sup> ∈T z<sup>0</sup> such that z<sup>0</sup> ≼z1;

lim

tions of Theorem 3.20. Then T has a fixed point in X.

2. there exists z<sup>0</sup> ∈ X and z<sup>1</sup> ∈T z<sup>0</sup> such that z<sup>0</sup> ≼z1;

3.there exists τ : ð Þ! 0; ∞ ð Þ 0; ∞ such that

lim

4. for a sequence zf g<sup>n</sup> ⊂ X, limn!<sup>∞</sup>f g zn ¼ z and zn ≼znþ<sup>1</sup> for all n∈ IN, we have

Definition 3.19 Let <sup>T</sup> : <sup>X</sup> ! <sup>2</sup><sup>X</sup> be a multivalued mapping on a partially ordered metric space ð Þ X; d; ≼ , then T is said to be an ordered F-τ-contraction on X, if there exists <sup>σ</sup> . 0 and <sup>τ</sup> : ð Þ! <sup>0</sup>; <sup>∞</sup> ð Þ <sup>σ</sup>; <sup>∞</sup> , <sup>F</sup> <sup>∈</sup> <sup>Δ</sup>ð Þ <sup>F</sup> such that for all <sup>z</sup><sup>∈</sup> <sup>X</sup>, <sup>y</sup><sup>∈</sup> <sup>F</sup><sup>z</sup>

Theorem 3.20 Let ð Þ X; d; ≼ be a complete partially ordered metric space and T : X ! Kð Þ X be an ordered F-τ-contraction satisfying the following assertions:

<sup>t</sup>!s<sup>þ</sup> inf <sup>τ</sup>ð Þ<sup>t</sup> . <sup>σ</sup> for all <sup>s</sup>≥<sup>0</sup>

Proof. By using the similar arguments as in the proof of Theorem 3.17 and using Theorem 3.12, we get the result. □ Theorem 3.21 Let ð Þ X; d; ≼ be a complete partially ordered metric space and T : X ! Cð Þ X be an ordered F-τ-contraction with F ∈ Δð Þ F<sup>∗</sup> satisfying all the asser-

Theorem 3.22 Let ð Þ X; d; ≼ be a complete partially ordered metric space, T : X ! Kð Þ X be a continuous mapping and F ∈ Δð Þ F . Assume that the following

<sup>t</sup>!s<sup>þ</sup> inf <sup>τ</sup>ð Þ<sup>t</sup> . 0 for all <sup>s</sup><sup>≥</sup> <sup>0</sup>

τð Þþ d zð Þ ; y Fð Þ D yð Þ ; T y ≤ Fð Þ M zð Þ ; y , (51)

D yð Þ ; T z ½ � 1 þ D zð Þ ; T y 1 þ d zð Þ ; y

D yð Þþ ; T z D zð Þ ; T y

<sup>2</sup> ;

: (52)

�

σ

Proof. By defining α : X � X ! ½ Þ 0; ∞ as in the proof of Theorem 3.17 and by using Theorem (3.14), we get the required result. □

Theorem 3.23 Let ð Þ X; d; ≼ be a complete partially ordered metric space, T : X ! Cð Þ X be a continuous mapping and F ∈ Δð Þ F<sup>∗</sup> satisfying all assertions of Theorem 3.22. Then T has a fixed point in X.

From Theorems 3.22 and 3.23, we get the following fixed point result for single valued mapping.

Theorem 3.24 Let ð Þ X; d; ≼ be a complete partially ordered metric space, T : X ! X be a continuous mapping and F ∈ Δð Þ F . Assume that the following assertions hold:

1. T is monotone increasing;

2. there exists z0, z<sup>1</sup> ∈ X such that z<sup>0</sup> ≼z1;

3.there exists τ : ð Þ! 0; ∞ ð Þ 0; ∞ such that

lim <sup>t</sup>!s<sup>þ</sup> inf <sup>τ</sup>ð Þ<sup>t</sup> . <sup>0</sup> for all s <sup>≥</sup><sup>0</sup>

and for all z, y∈ X with z≼ y and dð Þ T z; T y . 0 satisfying

$$
\pi(d(z, y)) + \mathcal{F}(d(Tz, Ty)) \le \mathcal{F}(m(z, y)),
\tag{54}
$$

where

$$m(\mathbf{z}, \mathbf{y}) = \max\left\{ d(\mathbf{z}, \mathbf{y}), d(\mathbf{z}, \mathbf{T}\mathbf{z}), d(\mathbf{y}, \mathbf{T}\mathbf{y}), \frac{d(\mathbf{y}, \mathbf{T}\mathbf{z}) + d(\mathbf{z}, \mathbf{T}\mathbf{y})}{2},$$

$$\frac{d(\mathbf{y}, \mathbf{T}\mathbf{y})[\mathbf{1} + d(\mathbf{z}, \mathbf{T}\mathbf{z})]}{\mathbf{1} + d(\mathbf{z}, \mathbf{y})}, \frac{d(\mathbf{y}, \mathbf{T}\mathbf{z})[\mathbf{1} + d(\mathbf{z}, \mathbf{T}\mathbf{y})]}{\mathbf{1} + d(\mathbf{z}, \mathbf{y})} \right\}.\tag{55}$$

Then T has a fixed point in X.
