2.1. Fixed point

Definition 2.1. Let <sup>X</sup> be a nonempty set and <sup>T</sup> : <sup>X</sup> ! <sup>X</sup> be a mapping. A point <sup>x</sup><sup>∗</sup> <sup>∈</sup> <sup>X</sup> is said to be a fixed point of <sup>T</sup> if T x<sup>∗</sup> ð Þ¼ <sup>x</sup><sup>∗</sup>:

Definition 2.2. Let ð Þ X; d be a metric space. The mapping T : X ! X is said to be Lipschitzian if there exists a constant α > 0 (called Lipschitz constant) such that

$$d(T\mathfrak{x}, T\mathfrak{y}) \le \alpha d(\mathfrak{x}, \mathfrak{y}) \quad \text{for all} \ \mathfrak{x}, \mathfrak{y} \in X.$$

A mapping T with a Lipschitz constant α < 1 is called contraction.

Definition 2.3. Let F and X be normed spaces over the field K, T : F ! X an operator and c∈F. We say that T is continuous at c if for every ε > 0 there exists δ > 0 such that ∥T xð Þ� T cð Þ∥ < e whenever ∥x � c∥ < δ and x∈ F. If T is continuous at each x ∈F, then T is said to be continuous on T.

Definition 2.4. Let X and Y be normed spaces. The mapping T : X ! Y is said to be completely continuous if T Cð Þ is a compact subset of Y for every bounded subset C of X.

Definition 2.5. Compact operator is a linear operator L form a Banach space X to another Banach space Y, such that the image under L of any bounded subset of X is a relatively compact subset (has compact closure) of Y such an operator is necessarily a bounded operator, and so continuous.

or in other words, <sup>∥</sup>f ux ð Þ� ð Þ f un ð Þ0 <sup>∥</sup><sup>L</sup>

i. continuous as map from L<sup>2</sup>

ii. compact as map form L<sup>2</sup>

follows as �Δ þ μId

�Δ þ μId �<sup>1</sup> : L<sup>2</sup>

2.2. Fuzzy

quence this convergence u holds for un. □

Proof. The first part is due to the fact that L<sup>2</sup>

ð Þ! <sup>Ω</sup> <sup>H</sup><sup>1</sup>

�<sup>1</sup> : L<sup>2</sup>

set of A, denoted by A<sup>α</sup> is defined by

H Að Þ ; B the Huasdorff distance.

Let I ¼ ½ � 0; 1 and W Xð Þ⊂ I

Wαð Þ X is given by A ∈ I

p

ð Þ <sup>Ω</sup> to <sup>H</sup><sup>1</sup>

∥v∥H<sup>1</sup>

ð Þ Ω .

sition of a compact linear operator a continuous linear operator is again compact.

∥u∥Lpð Þ <sup>Ω</sup> ≤ C∥∇u∥Lp <sup>Ω</sup>:R<sup>n</sup> ð Þ. A key tool to obtain the compactness of the fixed point maps.

<sup>0</sup>ð Þ <sup>Ω</sup> and the compact embedding <sup>H</sup><sup>1</sup>

Theorem 2.11. (Poincare) For p <sup>∈</sup>½ Þ <sup>1</sup>; <sup>∞</sup> , there exists a constant C <sup>¼</sup> <sup>C</sup>ð Þ <sup>Ω</sup>; <sup>p</sup> such that ∀ ∈ <sup>W</sup><sup>1</sup>,p

A fuzzy set in X is a function with domain X and values in 0½ � ; 1 . If A is a fuzzy set on X and x∈ X, then the functional value Ax is called the grade of membership of x in A. The α� level

A<sup>α</sup> ¼ f g x : Ax ≥ α if α∈ ð � 0; 1 , A<sup>0</sup> ¼ f g x : Ax > 0 ,

where denotes by A the closure of the set A. For any A and B are subset of X we denote by

Definition 2.12. A fuzzy set A in a metric linear space is called an approximate quantity if and

For a metric space ð Þ X; d we denoted by V Xð Þ the collection of fuzzy sets A in X for which A<sup>α</sup> is compact and supAx ¼ 1 for all α∈ ½ � 0; 1 . Clearly, when X is a metric linear space W Xð Þ⊂V Xð Þ:

<sup>X</sup> be the collection of all approximate in <sup>X</sup>. For <sup>α</sup> <sup>∈</sup>½ � <sup>0</sup>; <sup>1</sup> , the family

d xð Þ ; y , Dαð Þ¼ A; B H Að Þ <sup>α</sup>; B<sup>α</sup>

only if A<sup>α</sup> is convex and compact in X for each α ∈½ � 0; 1 and sup<sup>x</sup><sup>∈</sup> <sup>X</sup>Ax ¼ 1:

is nonempty and compact}.

<sup>X</sup> : A<sup>α</sup>

<sup>p</sup>αð Þ¼ <sup>A</sup>; <sup>B</sup> inf <sup>x</sup><sup>∈</sup> <sup>A</sup>α, <sup>y</sup><sup>∈</sup> <sup>B</sup><sup>α</sup>

Definition 2.13. Let A, B∈V Xð Þ, α∈ ½ � 0; 1 : Then

where H is the Hausdorff distance.

ð Þ <sup>Ω</sup> to <sup>L</sup><sup>2</sup>

ð Þ! <sup>Ω</sup> <sup>L</sup><sup>2</sup>

Corollary 2.10. ref. [5] Let μ ≥ 0. Then the map g ↦ �Δ þ μId

<sup>r</sup>ð Þ <sup>Ω</sup> ! <sup>0</sup>: Since the limit does not depend on the subse-

http://dx.doi.org/10.5772/intechopen.74560

Fixed Point Theory Approach to Existence of Solutions with Differential Equations

g is

ð Þ Ω can be viewed as composition of the continuous map

<sup>0</sup>ð Þ <sup>Ω</sup> <sup>↣</sup>L<sup>2</sup>

ð Þ Ω . The second part

ð Þ Ω and as the compo-

<sup>0</sup> ð Þ Ω ;

7

�<sup>1</sup>

ð Þ <sup>Ω</sup> is continuously in <sup>H</sup>�<sup>1</sup>

<sup>0</sup>ð Þ Ω in other words

<sup>0</sup>ð Þ <sup>Ω</sup> <sup>≤</sup> <sup>C</sup>ð Þ <sup>Ω</sup> <sup>∥</sup>g∥L2ð Þ <sup>Ω</sup> :

Definition 2.6. A subset C of a normed linear space X is said to be convex subset in X if λx þ ð Þ 1 � λ y∈ C for each x, y ∈C and for each scalar λ∈ ½ � 0; 1 .

Definition 2.7. v is called the αth weak derivative of u

$$D^a u = v$$

if

$$\int\_{\Omega} \mathfrak{u} D^{\alpha} \psi d\mathfrak{x} = (-1)^{|\alpha|} \int\_{\Omega} v \psi d\mathfrak{x}$$

for all test function ψ∈ C<sup>∞</sup> <sup>c</sup> ð Þ Ω .

Theorem 2.8. (Schauder's Fixed Point Theorem) Let X be a Banach space, M ⊂ X be nonempty, convex, bounded, closed and T : M ⊂ X ! M be a compact operator. Then T has a fixed point.

Lemma 2.9. ref. [5] Given <sup>f</sup> <sup>∈</sup>Cð Þ <sup>R</sup> such that <sup>∣</sup>f tð Þ<sup>∣</sup> <sup>≤</sup> <sup>a</sup> <sup>¼</sup> b tj j<sup>r</sup> where <sup>a</sup> <sup>&</sup>gt; <sup>0</sup>, b <sup>&</sup>gt; 0 and <sup>r</sup> <sup>&</sup>gt; 0 are positive constants. Then the map <sup>u</sup> <sup>↦</sup> f uð Þ is continuous for <sup>L</sup><sup>p</sup> ð Þ Ω to L p <sup>r</sup>ð Þ Ω for p ≥ max 1ð Þ ;r and maps bounded subset of L<sup>p</sup> ð Þ Ω to bounded subset of L p <sup>r</sup>ð Þ Ω .

Proof. Form to Jensen's inequality

$$(a+b|t|^r)^{\left(\frac{p}{r}\right)} \leq 2^{\frac{p}{r}-1}a^{\frac{p}{r}} + 2^{\frac{p}{r}-1}b^{\frac{p}{r}}|t|^p \leq \mathbb{C}(1+|t|^p)$$

where C is a positive constant depending on a, b, p and r only, since u∈Lp ð Þ Ω , we have

$$\int\_{\Omega} |f(u)|^{\frac{p}{r}} d\mathfrak{x} \le \mathsf{C}(a, b, p, r) \left( |\Omega| + \int\_{\Omega} u^p d\mathfrak{x} \right) < \infty$$

therefore f uð Þ∈ L p <sup>r</sup>ð Þ <sup>Ω</sup> . Let un be a sequence converging to <sup>u</sup> in <sup>L</sup><sup>p</sup> ð Þ Ω . There exists a subsequence un, and a function g ∈Lp ð Þ Ω such that set, un<sup>0</sup> ! u xð Þ, and ∣un0ð Þx ∣ ≤ g xð Þ, almost everywhere. This is sometimes called the generalized DCT, or the partial converse of the DCT, or the Riesz-Fisher Theorem. From the continuity of f , ∣f ux ð Þ� ð Þ f un ð Þ0 ∣ ! 0 on Ω\ℕ, and

$$|f(\mu(\mathbf{x})) - f(\mu\_{n'})|^\frac{p}{r} \le C(1 + \mathcal{g}(\mathbf{x})^p + |f(\mu)|^p),$$

where C is another positive constant depending on a, b, p and r only, the left-hand-side is independent of n 0 and is in L<sup>1</sup> ð Þ Ω . We can apply the Dominated Convergence Theorem to conclude the

$$\int\_{\Omega} |f(u(\mathfrak{x})) - f(u\_{n'})|^\sharp d\mathfrak{x} \to 0$$

or in other words, <sup>∥</sup>f ux ð Þ� ð Þ f un ð Þ0 <sup>∥</sup><sup>L</sup> p <sup>r</sup>ð Þ <sup>Ω</sup> ! <sup>0</sup>: Since the limit does not depend on the subsequence this convergence u holds for un. □

Corollary 2.10. ref. [5] Let μ ≥ 0. Then the map g ↦ �Δ þ μId �<sup>1</sup> g is

i. continuous as map from L<sup>2</sup> ð Þ <sup>Ω</sup> to <sup>H</sup><sup>1</sup> <sup>0</sup>ð Þ Ω in other words

$$\|\|\boldsymbol{\upsilon}\|\|\_{H^{1}\_{0}(\Omega)} \leq \mathsf{C}(\Omega) \|\|\boldsymbol{g}\|\|\_{L^{2}(\Omega)}.$$

ii. compact as map form L<sup>2</sup> ð Þ <sup>Ω</sup> to <sup>L</sup><sup>2</sup> ð Þ Ω .

Proof. The first part is due to the fact that L<sup>2</sup> ð Þ <sup>Ω</sup> is continuously in <sup>H</sup>�<sup>1</sup> ð Þ Ω . The second part follows as �Δ þ μId �<sup>1</sup> : L<sup>2</sup> ð Þ! <sup>Ω</sup> <sup>L</sup><sup>2</sup> ð Þ Ω can be viewed as composition of the continuous map �Δ þ μId �<sup>1</sup> : L<sup>2</sup> ð Þ! <sup>Ω</sup> <sup>H</sup><sup>1</sup> <sup>0</sup>ð Þ <sup>Ω</sup> and the compact embedding <sup>H</sup><sup>1</sup> <sup>0</sup>ð Þ <sup>Ω</sup> <sup>↣</sup>L<sup>2</sup> ð Þ Ω and as the composition of a compact linear operator a continuous linear operator is again compact.

Theorem 2.11. (Poincare) For p <sup>∈</sup>½ Þ <sup>1</sup>; <sup>∞</sup> , there exists a constant C <sup>¼</sup> <sup>C</sup>ð Þ <sup>Ω</sup>; <sup>p</sup> such that ∀ ∈ <sup>W</sup><sup>1</sup>,p <sup>0</sup> ð Þ Ω ; ∥u∥Lpð Þ <sup>Ω</sup> ≤ C∥∇u∥Lp <sup>Ω</sup>:R<sup>n</sup> ð Þ. A key tool to obtain the compactness of the fixed point maps.

### 2.2. Fuzzy

Definition 2.5. Compact operator is a linear operator L form a Banach space X to another Banach space Y, such that the image under L of any bounded subset of X is a relatively compact subset (has compact closure) of Y such an operator is necessarily a bounded operator,

Definition 2.6. A subset C of a normed linear space X is said to be convex subset in X if

<sup>D</sup><sup>α</sup><sup>u</sup> <sup>¼</sup> <sup>v</sup>

Theorem 2.8. (Schauder's Fixed Point Theorem) Let X be a Banach space, M ⊂ X be nonempty,

Lemma 2.9. ref. [5] Given <sup>f</sup> <sup>∈</sup>Cð Þ <sup>R</sup> such that <sup>∣</sup>f tð Þ<sup>∣</sup> <sup>≤</sup> <sup>a</sup> <sup>¼</sup> b tj j<sup>r</sup> where <sup>a</sup> <sup>&</sup>gt; <sup>0</sup>, b <sup>&</sup>gt; 0 and <sup>r</sup> <sup>&</sup>gt; 0 are

ð Ω vψdx

> p <sup>r</sup>ð Þ Ω .

> > ð Ω up dx

<sup>r</sup> <sup>≤</sup><sup>C</sup> <sup>1</sup> <sup>þ</sup> g xð Þ<sup>p</sup> <sup>þ</sup> j j f uð Þ <sup>p</sup> ð Þ

p rdx ! 0

� �

ð Þ Ω to L p

< ∞

ð Þ Ω such that set, un<sup>0</sup> ! u xð Þ, and ∣un0ð Þx ∣ ≤ g xð Þ, almost every-

ð Þ Ω . We can apply the Dominated Convergence Theorem to

<sup>r</sup>j j<sup>t</sup> <sup>p</sup> <sup>≤</sup><sup>C</sup> <sup>1</sup> <sup>þ</sup> j j<sup>t</sup> <sup>p</sup> ð Þ

<sup>r</sup>ð Þ Ω for p ≥ max 1ð Þ ;r and

ð Þ Ω , we have

ð Þ Ω . There exists a subse-

uDαψdx ¼ �ð Þ<sup>1</sup> <sup>∣</sup>α<sup>∣</sup>

convex, bounded, closed and T : M ⊂ X ! M be a compact operator. Then T has a fixed point.

ð Þ Ω to bounded subset of L

rdx ≤C að Þj ; b; p;r Ωj þ

<sup>r</sup>ð Þ <sup>Ω</sup> . Let un be a sequence converging to <sup>u</sup> in <sup>L</sup><sup>p</sup>

Riesz-Fisher Theorem. From the continuity of f , ∣f ux ð Þ� ð Þ f un ð Þ0 ∣ ! 0 on Ω\ℕ, and

p

f ux ð Þ� ð Þ f un j j ð Þ0

f ux ð Þ� ð Þ f un j j ð Þ0

ð Ω

where. This is sometimes called the generalized DCT, or the partial converse of the DCT, or the

where C is another positive constant depending on a, b, p and r only, the left-hand-side is

ð Þ<sup>r</sup> ≤ 2 p r �1 a p <sup>r</sup> þ 2 p r �1 b p

where C is a positive constant depending on a, b, p and r only, since u∈Lp

λx þ ð Þ 1 � λ y∈ C for each x, y ∈C and for each scalar λ∈ ½ � 0; 1 .

ð Ω

Definition 2.7. v is called the αth weak derivative of u

<sup>c</sup> ð Þ Ω .

positive constants. Then the map <sup>u</sup> <sup>↦</sup> f uð Þ is continuous for <sup>L</sup><sup>p</sup>

<sup>a</sup> <sup>þ</sup> b tj j<sup>r</sup> ð Þ <sup>p</sup>

ð Ω j j f uð Þ <sup>p</sup>

and is in L<sup>1</sup>

and so continuous.

6 Differential Equations - Theory and Current Research

for all test function ψ∈ C<sup>∞</sup>

maps bounded subset of L<sup>p</sup>

therefore f uð Þ∈ L

independent of n

conclude the

Proof. Form to Jensen's inequality

p

0

quence un, and a function g ∈Lp

if

A fuzzy set in X is a function with domain X and values in 0½ � ; 1 . If A is a fuzzy set on X and x∈ X, then the functional value Ax is called the grade of membership of x in A. The α� level set of A, denoted by A<sup>α</sup> is defined by

$$A\_{\alpha} = \{ \mathbf{x} : A \mathbf{x} \succeq \alpha \} \quad \text{if} \ \alpha \in (0, 1], \quad A\_0 = \overline{\{ \mathbf{x} : A \mathbf{x} > 0 \}} \dots$$

where denotes by A the closure of the set A. For any A and B are subset of X we denote by H Að Þ ; B the Huasdorff distance.

Definition 2.12. A fuzzy set A in a metric linear space is called an approximate quantity if and only if A<sup>α</sup> is convex and compact in X for each α ∈½ � 0; 1 and sup<sup>x</sup><sup>∈</sup> <sup>X</sup>Ax ¼ 1:

Let I ¼ ½ � 0; 1 and W Xð Þ⊂ I <sup>X</sup> be the collection of all approximate in <sup>X</sup>. For <sup>α</sup> <sup>∈</sup>½ � <sup>0</sup>; <sup>1</sup> , the family Wαð Þ X is given by A ∈ I <sup>X</sup> : A<sup>α</sup> is nonempty and compact}.

For a metric space ð Þ X; d we denoted by V Xð Þ the collection of fuzzy sets A in X for which A<sup>α</sup> is compact and supAx ¼ 1 for all α∈ ½ � 0; 1 . Clearly, when X is a metric linear space W Xð Þ⊂V Xð Þ:

Definition 2.13. Let A, B∈V Xð Þ, α∈ ½ � 0; 1 : Then

$$p\_{\boldsymbol{\alpha}}(A,B) = \inf\_{\boldsymbol{x} \in A\_{\boldsymbol{\alpha}}, \boldsymbol{y} \in B\_{\boldsymbol{\alpha}}} d(\boldsymbol{x}, \boldsymbol{y}), \quad D\_{\boldsymbol{\alpha}}(A,B) = H(A\_{\boldsymbol{\alpha}}, B\_{\boldsymbol{\alpha}}).$$

where H is the Hausdorff distance.

Definition 2.14. Let A, B∈ V Xð Þ: Then A is said to be more accurate than B (or B includes A), denoted by A ⊂B, if and only if Ax ≤ Bx for each x∈ X:

p1

p2

p3

p4

x, y, z ∈ X, the following conditions hold:

Then a pair ð Þ X; σ is called a metric-like space.

examples:

space X; dp , p xð Þ ; <sup>x</sup> <sup>≤</sup> p xð Þ ; <sup>y</sup> ,

p xð Þ¼ ; <sup>x</sup> p yð Þ ; <sup>x</sup> ,

Example 2.19. [8] Let X ¼ f g 0; 1 and σ : X � X ! R<sup>þ</sup> be defined by

Lemma 2.20. ref. [9] Let ð Þ X; p be a partial metric space. Then

Definition 2.21. [8, 10] Let ð Þ X; σ be a metric-like space. Then:

converges (in τσ) to a point x∈ X such that σð Þ¼ x; x 0:

ii. X is complete if and only if the metric space X; dp

σð Þ¼ x; y

Then ð Þ X; σ is a metric-like space, but it is not a partial metric space, cause σð Þ 0; 0 ≰σð Þ 0; 1 :

i. f g xn is a Cauchy sequence in ð Þ X; p if and only if it is a Cauchy sequence in the metric

i. A sequence f g xn in X converges to a point x ∈ X if lim<sup>n</sup>!<sup>∞</sup>σð Þ¼ xn; x σð Þ x; x : The sequence f g xn is said to be σ� Cauchy if limn,m!<sup>∞</sup>σð Þ xn; xm exists and is finite. The space ð Þ X; σ is called complete if for every σ� Cauchy sequence in f g xn , there exists x ∈ X such that

lim<sup>n</sup>!<sup>∞</sup> <sup>σ</sup>ð Þ¼ xn; <sup>x</sup> <sup>σ</sup>ð Þ¼ <sup>x</sup>; <sup>x</sup> lim n, <sup>m</sup>!<sup>∞</sup> <sup>σ</sup>ð Þ xn; xm :

ii. A sequence f g xn in ð Þ X; σ is said to be a 0 � σ� Cauchy sequence if limn,m!<sup>∞</sup>σð Þ¼ xn; xm 0: The space ð Þ X; σ is said to be 0 � σ� complete if every 0 � σ� Cauchy sequence in X

p xð Þ¼ ; <sup>x</sup> p xð Þ¼ ; <sup>y</sup> p yð Þ ; <sup>y</sup> if and only if <sup>x</sup> <sup>¼</sup> y,

Fixed Point Theory Approach to Existence of Solutions with Differential Equations

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9

p xð Þ¼ ; <sup>y</sup> p xð Þþ ; <sup>z</sup> p zð Þ� ; <sup>y</sup> p zð Þ ; <sup>z</sup> :

Then p is called a partial metric on X, so a pair ð Þ X; p is said to be a partial metric space.

ð Þ σ<sup>1</sup> σð Þ¼ x; y 0 ) x ¼ y;

ð Þ σ<sup>2</sup> σð Þ¼ x; y σð Þ y; x ;

Definition 2.18. [8] A metric-like on nonempty set X is a function σ : X � X ! R<sup>þ</sup>. If for all

ð Þ σ<sup>3</sup> σð Þ¼ x; y σð Þþ x; z σð Þ z; y :

It is easy to see that a metric space is a partial metric space and each partial metric space is a metric-like space, but the converse is not true but the converse is not true as in the following

> 2, if x ¼ y ¼ 0, 1, otherwise:

> > is complete.

Denote with Φ, the family of nondecreasing function ϕ : ½ Þ! 0; þ∞ ½ Þ 0; þ∞ such that P<sup>∞</sup> <sup>n</sup>¼<sup>1</sup> <sup>ϕ</sup><sup>n</sup>ð Þ<sup>t</sup> <sup>&</sup>lt; <sup>∞</sup> for all <sup>t</sup> <sup>&</sup>gt; <sup>0</sup>:

Theorem 2.15. ref. [6] Let ð Þ X; d;≼ be a complete ordered metric space and T1, T<sup>2</sup> : X ! Wαð Þ X be two fuzzy mapping satisfying

<sup>D</sup>αð Þ <sup>T</sup>1x; <sup>T</sup>2<sup>y</sup> <sup>≤</sup> <sup>ϕ</sup>ð Þþ M xð Þ ; <sup>y</sup> <sup>L</sup> min <sup>p</sup>αð Þ <sup>x</sup>; <sup>T</sup>1<sup>x</sup> ; <sup>p</sup>αð Þ <sup>y</sup>; <sup>T</sup>2<sup>y</sup> ; <sup>p</sup>αð Þ <sup>x</sup>; <sup>T</sup>2<sup>y</sup> ; <sup>p</sup>αð Þ <sup>y</sup>; <sup>T</sup>1<sup>x</sup> � �

for all comparable element x, y ∈ X, where L ≥ 0 and

$$M(\mathbf{x}, y) = \max\left\{ d(\mathbf{x}, y), p\_a(\mathbf{x}, T\_1 \mathbf{x}), p\_a(y, T\_2 y), \frac{1}{2} \left[ p\_a(\mathbf{x}, T\_2 y) + p\_a(y, T\_1 \mathbf{x}) \right] \right\}.$$

Also suppose that


Then there exists a point x ∈ X such that x<sup>α</sup> ⊂ T1x and x<sup>α</sup> ⊂ T2x:

Proof. See in [6].

Corollary 2.16. ref. [6] Let ð Þ X; d;≼ be a complete ordered metric space and T1, T<sup>2</sup> : X ! Wαð Þ X be two fuzzy mappings satisfying

$$D\_a(T\_1\mathbf{x}, T\_2y) \le q \max\left\{ d(\mathbf{x}, y), p\_a(\mathbf{x}, T\_1\mathbf{x}), p\_a(y, T\_2y), \frac{1}{2} \left[ p\_a(\mathbf{x}, T\_2y) + p\_a(y, T\_1\mathbf{x}) \right] \right\}$$

for all comparable elements x, y ∈ X. Also suppose that


Then there exists a point x ∈ X such that x<sup>α</sup> ⊂ T1x and x<sup>α</sup> ⊂ T2x:

#### 2.3. Metric-like space

Definition 2.17. [7] Let X be nonempty set and function p : X � X ! R<sup>þ</sup> be a function satisfying the following condition: for all x, y, z ∈ X,

$$\begin{aligned} \left(p\_1\right)\left(p(\mathbf{x},\mathbf{x})=p(\mathbf{x},y)=p(y,y)\text{ if and only if }\mathbf{x}=y,\\ \left(p\_2\right)p(\mathbf{x},\mathbf{x})\le p(\mathbf{x},y),\\ \left(p\_3\right)p(\mathbf{x},\mathbf{x})=p(y,\mathbf{x}),\\ \left(p\_4\right)p(\mathbf{x},y)=p(\mathbf{x},z)+p(z,y)-p(z,z).\end{aligned}$$

Then p is called a partial metric on X, so a pair ð Þ X; p is said to be a partial metric space.

Definition 2.18. [8] A metric-like on nonempty set X is a function σ : X � X ! R<sup>þ</sup>. If for all x, y, z ∈ X, the following conditions hold:

$$\begin{aligned} \left(\sigma\_1\right)\sigma(\mathbf{x},y) &= \mathbf{0} \Rightarrow \mathbf{x} = y; \\ \left(\sigma\_2\right)\sigma(\mathbf{x},y) &= \sigma(y,\mathbf{x}); \\ \left(\sigma\_3\right)\sigma(\mathbf{x},y) &= \sigma(\mathbf{x},z) + \sigma(z,y). \end{aligned}$$

Then a pair ð Þ X; σ is called a metric-like space.

Definition 2.14. Let A, B∈ V Xð Þ: Then A is said to be more accurate than B (or B includes A),

Denote with Φ, the family of nondecreasing function ϕ : ½ Þ! 0; þ∞ ½ Þ 0; þ∞ such that

Theorem 2.15. ref. [6] Let ð Þ X; d;≼ be a complete ordered metric space and T1, T<sup>2</sup> : X ! Wαð Þ X be

<sup>D</sup>αð Þ <sup>T</sup>1x; <sup>T</sup>2<sup>y</sup> <sup>≤</sup> <sup>ϕ</sup>ð Þþ M xð Þ ; <sup>y</sup> <sup>L</sup> min <sup>p</sup>αð Þ <sup>x</sup>; <sup>T</sup>1<sup>x</sup> ; <sup>p</sup>αð Þ <sup>y</sup>; <sup>T</sup>2<sup>y</sup> ; <sup>p</sup>αð Þ <sup>x</sup>; <sup>T</sup>2<sup>y</sup> ; <sup>p</sup>αð Þ <sup>y</sup>; <sup>T</sup>1<sup>x</sup> � �

ii. if x, y∈ X are comparable, then every u∈ð Þ T1x <sup>α</sup> and every v∈ ð Þ T2y <sup>α</sup> are comparable,

iii. if a sequence f g xn in X converges to x ∈ X and its consecutive terms are comparable, then

Corollary 2.16. ref. [6] Let ð Þ X; d;≼ be a complete ordered metric space and T1, T<sup>2</sup> : X ! Wαð Þ X be

ii. if x, y∈ X are comparable, then every u∈ð Þ T1x <sup>α</sup> and every v∈ ð Þ T2y <sup>α</sup> are comparable,

iii. if a sequence f g xn in X converges to x ∈ X and its consecutive terms are comparable, then

Definition 2.17. [7] Let X be nonempty set and function p : X � X ! R<sup>þ</sup> be a function satisfy-

1 <sup>2</sup> <sup>p</sup>αð Þþ <sup>x</sup>; <sup>T</sup>2<sup>y</sup> <sup>p</sup>αð Þ <sup>y</sup>; <sup>T</sup>1<sup>x</sup> � � � �

1 <sup>2</sup> <sup>p</sup>αð Þþ <sup>x</sup>; <sup>T</sup>2<sup>y</sup> <sup>p</sup>αð Þ <sup>y</sup>; <sup>T</sup>1<sup>x</sup> � � � �

:

denoted by A ⊂B, if and only if Ax ≤ Bx for each x∈ X:

for all comparable element x, y ∈ X, where L ≥ 0 and

i. if y∈ ð Þ T1x<sup>0</sup> <sup>α</sup>, then y, x<sup>0</sup> ∈ X are comparable,

Then there exists a point x ∈ X such that x<sup>α</sup> ⊂ T1x and x<sup>α</sup> ⊂ T2x:

Dαð Þ T1x; T2y ≤ q max d xð Þ ; y ; pαð Þ x; T1x ; pαð Þ y; T2y ;

for all comparable elements x, y ∈ X. Also suppose that

Then there exists a point x ∈ X such that x<sup>α</sup> ⊂ T1x and x<sup>α</sup> ⊂ T2x:

i. if y∈ ð Þ T1x<sup>0</sup> <sup>α</sup>, then y, x<sup>0</sup> ∈ X are comparable,

xn and x are comparable for all n.

ing the following condition: for all x, y, z ∈ X,

xn and x are comparable for all n.

M xð Þ¼ ; y max d xð Þ ; y ; pαð Þ x; T1x ; pαð Þ y; T2y ;

<sup>n</sup>¼<sup>1</sup> <sup>ϕ</sup><sup>n</sup>ð Þ<sup>t</sup> <sup>&</sup>lt; <sup>∞</sup> for all <sup>t</sup> <sup>&</sup>gt; <sup>0</sup>:

8 Differential Equations - Theory and Current Research

two fuzzy mapping satisfying

Also suppose that

Proof. See in [6].

two fuzzy mappings satisfying

2.3. Metric-like space

P<sup>∞</sup>

It is easy to see that a metric space is a partial metric space and each partial metric space is a metric-like space, but the converse is not true but the converse is not true as in the following examples:

Example 2.19. [8] Let X ¼ f g 0; 1 and σ : X � X ! R<sup>þ</sup> be defined by

$$
\sigma(\mathbf{x}, y) = \begin{cases} \mathbf{2}, & \text{if } \mathbf{x} = y = \mathbf{0}, \\ \mathbf{1}, & \text{otherwise.} \end{cases}
$$

Then ð Þ X; σ is a metric-like space, but it is not a partial metric space, cause σð Þ 0; 0 ≰σð Þ 0; 1 :

Lemma 2.20. ref. [9] Let ð Þ X; p be a partial metric space. Then


Definition 2.21. [8, 10] Let ð Þ X; σ be a metric-like space. Then:

i. A sequence f g xn in X converges to a point x ∈ X if lim<sup>n</sup>!<sup>∞</sup>σð Þ¼ xn; x σð Þ x; x : The sequence f g xn is said to be σ� Cauchy if limn,m!<sup>∞</sup>σð Þ xn; xm exists and is finite. The space ð Þ X; σ is called complete if for every σ� Cauchy sequence in f g xn , there exists x ∈ X such that

$$\lim\_{n \to \infty} \sigma(\mathbf{x}\_n, \mathbf{x}) = \sigma(\mathbf{x}, \mathbf{x}) = \lim\_{n\_\nu \le n \to \infty} \sigma(\mathbf{x}\_n, \mathbf{x}\_m).$$

ii. A sequence f g xn in ð Þ X; σ is said to be a 0 � σ� Cauchy sequence if limn,m!<sup>∞</sup>σð Þ¼ xn; xm 0: The space ð Þ X; σ is said to be 0 � σ� complete if every 0 � σ� Cauchy sequence in X converges (in τσ) to a point x∈ X such that σð Þ¼ x; x 0:

iii. A mapping T : X ! X is continuous, if the following limits exist (finite) and

$$\lim\_{n \to \infty} \sigma(\mathfrak{x}\_n, \mathfrak{x}) = \sigma(T\mathfrak{x}, \mathfrak{x}).$$

Following Wardowski [11], we denote by F the family of all function, F : R<sup>þ</sup> ! R satisfying the following conditions:

(F1) F is strictly increasing on R<sup>þ</sup>,

(F2) for every sequence f g sn in <sup>R</sup><sup>þ</sup>, we have lim<sup>n</sup>!<sup>∞</sup> sn <sup>¼</sup> 0 if and only if lim<sup>n</sup>!<sup>∞</sup> F sð Þ¼� <sup>n</sup> <sup>∞</sup>,

(F3) there exists a number <sup>k</sup><sup>∈</sup> ð Þ <sup>0</sup>; <sup>1</sup> such that lim<sup>s</sup>!0<sup>þ</sup> <sup>s</sup> k F sð Þ¼ 0:

Example 2.22. The following function F : R<sup>þ</sup> ! R belongs to F: i. F sð Þ¼ ln s, with s > 0,

ii. F sð Þ¼ ln s þ s, with s > 0:

Definition 2.23. [11] Let ð Þ X; d be a metric space. A self-mapping T on X is called an Fcontraction mapping if there exist F∈ F and τ∈ R<sup>þ</sup> such that

$$\forall \mathbf{x}, y \in \mathcal{X}, \quad [d(\mathbf{Tx}, Ty) > 0 \Rightarrow \pi + F(d(\mathbf{Tx}, Ty)) \leq F(d(\mathbf{x}, y))].\tag{2.1}$$

Definition 2.24. [12] Let ð Þ X; σ be a metric-like space. A mapping T : X ! X is called a generalized Roger Hardy type F� contraction mapping, if there exist F∈ F and τ∈ R<sup>þ</sup> such that

$$\begin{aligned} \sigma(\text{Tx}, \text{Ty}) > 0 &\Rightarrow \text{\raisebox{0.1pt}{ $\pi$ }} + F(\sigma(\text{Tx}, \text{Ty})) &\leq & F(a\sigma(\text{x}, \text{y}) + \beta \sigma(\text{x}, \text{Tx}) + \gamma \sigma(\text{y}, \text{Ty}) \\ &+ \eta \sigma(\text{x}, \text{Ty}) + \delta \sigma(\text{y}, \text{Tx})) \end{aligned} \tag{2.2}$$

for all x, y∈ X and α, β, γ, η, δ ≥ 0 with α þ β þ γ þ 2η þ 2δ < 1.

Theorem 2.25. ref. [12] Let ð Þ X; σ be 0 � σ� complete metric-like spaces and T : X ! X be a generalized Roger Hardy type F� contraction. Then T has a unique fixed point in X, either T or F is continuous.

Proof. See in [12]. □

#### 2.4. Modular metric space

Let X be a nonempty set. Throughout this paper, for a function ω : ð Þ� 0; ∞ X � X ! ½ � 0; ∞ , we write

$$
\omega\_\lambda(\mathbf{x}, \mathbf{y}) = \omega(\lambda, \mathbf{x}, \mathbf{y}),
$$

for all λ > 0 and x, y ∈ X:

Definition 2.26 [13, 14] Let X be a nonempty set. A function ω : ð Þ� 0; ∞ X � X ! ½ � 0; ∞ is called a metric modular on X if satisfying, for all x, y, z ∈ X the following conditions hold:

i. ωλð Þ¼ x; y 0 for all λ > 0 if and only if x ¼ y,

iii. ωλþ<sup>μ</sup>ð Þ x; y ≤ ωλð Þþ x; z ωμð Þ z; y for all λ, μ > 0.

regular if the following weaker version of (i) is satisfied:

λ ↦ ωλð Þ x; y is nonincreasing on 0ð Þ ; ∞ : Indeed, if 0 < μ < λ, then

ωλð Þ¼ x; x 0 for all λ > 0, x ∈ X,

Fixed Point Theory Approach to Existence of Solutions with Differential Equations

http://dx.doi.org/10.5772/intechopen.74560

11

then ω is said to be a pseudomodular (metric) on X: A modular metric ω on X is said to be

Note that for a metric (pseudo)modular ω on a set X, and any x, y ∈ X, the function

ωλð Þ x; y ≤ ωλ�<sup>μ</sup>ð Þþ x; x ωμð Þ¼ x; y ωμð Þ x; y :

Example 2.27. Let X ¼ R and ω is defined by ωλð Þ¼ x; y ∞ if λ < 1, and ωλð Þ¼ x; y ∣x � y∣ if

X<sup>ω</sup> ¼ Xωð Þ¼ x<sup>0</sup> f g x ∈ X : ωλð Þ! x; x<sup>0</sup> 0 as λ ! ∞

Throughout this section we assume that ð Þ X; ω is a modular metric space, D be a nonempty

Definition 2.29. [15, 16] The pair ð Þ D; G<sup>ω</sup> has Property (A) if for any sequence f g xn <sup>n</sup> <sup>∈</sup><sup>ℕ</sup> in D,

Definition 2.30. ref. [17] Let F∈ F and G<sup>ω</sup> ∈ G: A mapping T : D ! D is said to be F-Gω-

ω1ð Þ Tx; Ty > 0 ) τ þ Fð Þ ω1ð Þ Tx; Ty ≤ Fð Þ ω1ð Þ Rx;Ry

Example 2.31. ref. [17] Let F∈ F be arbitrary. Then every F-contractive mapping w.r.t. R is an

F-Gω-contraction w.r.t. R for the graph G<sup>ω</sup> given by V Gð Þ¼ <sup>ω</sup> D and E Gð Þ¼ <sup>ω</sup> D � D.

i. ð Þ Rx;Ry ∈E Gð Þ) <sup>ω</sup> ð Þ Tx; Ty ∈ E Gð Þ <sup>ω</sup> for all x, y ∈ D, i.e. T preserves edges w.r.t. R,

subset of X<sup>ω</sup> and G≔fG<sup>ω</sup> is a directed graph with V Gð Þ¼ <sup>ω</sup> D and Δ⊆E Gð Þg <sup>ω</sup> :

with xn ! x as n ! ∞ and ð Þ xn; xnþ<sup>1</sup> ∈ E Gð Þ <sup>ω</sup> , then ð Þ xn; x ∈ E Gð Þ <sup>ω</sup> , for all n:

<sup>ω</sup>ð Þ¼ x<sup>0</sup> f g x ∈ X : ∃λ ¼ λð Þx > 0 such that ωλð Þ x; x<sup>0</sup> < ∞

Note that every modular metric is regular but converse may not necessarily be true.

λ ≥ 1, it is easy to verify that ω is regular modular metric but not modular metric.

Definition 2.28. [13, 14] Let X<sup>ω</sup> be a (pseudo)modular on X: Fix x<sup>0</sup> ∈ X: The two sets

If instead of (i) we have only the condition (i')

x ¼ y if and only if ωλð Þ¼ x; y 0 for some λ > 0:

and

X∗ <sup>ω</sup> <sup>¼</sup> <sup>X</sup><sup>∗</sup>

are said to be modular spaces (around x0).

contraction with respect to R : D ! D if

ii. there exists a number τ > 0 such that

for all x, y∈ D with ð Þ Rx; Ry ∈ E Gð Þ <sup>ω</sup> :

ii. ωλð Þ¼ x; y ωλð Þ y; x for all λ > 0,


$$\text{iii.}\quad \omega\_{\lambda+\mu}(\mathbf{x},y) \le \omega\_{\lambda}(\mathbf{x},z) + \omega\_{\mu}(z,y) \text{ for all } \lambda, \mu > 0.$$

If instead of (i) we have only the condition (i')

ωλð Þ¼ x; x 0 for all λ > 0, x ∈ X,

then ω is said to be a pseudomodular (metric) on X: A modular metric ω on X is said to be regular if the following weaker version of (i) is satisfied:

$$\alpha = y \text{ if and only if } w\_{\lambda}(\mathbf{x}, y) = 0 \text{ for some } \lambda > 0.$$

Note that for a metric (pseudo)modular ω on a set X, and any x, y ∈ X, the function λ ↦ ωλð Þ x; y is nonincreasing on 0ð Þ ; ∞ : Indeed, if 0 < μ < λ, then

ωλð Þ x; y ≤ ωλ�<sup>μ</sup>ð Þþ x; x ωμð Þ¼ x; y ωμð Þ x; y :

Note that every modular metric is regular but converse may not necessarily be true.

Example 2.27. Let X ¼ R and ω is defined by ωλð Þ¼ x; y ∞ if λ < 1, and ωλð Þ¼ x; y ∣x � y∣ if λ ≥ 1, it is easy to verify that ω is regular modular metric but not modular metric.

Definition 2.28. [13, 14] Let X<sup>ω</sup> be a (pseudo)modular on X: Fix x<sup>0</sup> ∈ X: The two sets

$$X\_{\omega} = X\_{\omega}(\mathfrak{x}\_0) = \{ \mathfrak{x} \in X : \omega\_{\lambda}(\mathfrak{x}, \mathfrak{x}\_0) \to 0 \text{ as } \lambda \to \infty \},$$

and

iii. A mapping T : X ! X is continuous, if the following limits exist (finite) and

(F2) for every sequence f g sn in <sup>R</sup><sup>þ</sup>, we have lim<sup>n</sup>!<sup>∞</sup> sn <sup>¼</sup> 0 if and only if lim<sup>n</sup>!<sup>∞</sup> F sð Þ¼� <sup>n</sup> <sup>∞</sup>,

the following conditions:

i. F sð Þ¼ ln s, with s > 0,

that

F is continuous.

write

2.4. Modular metric space

for all λ > 0 and x, y ∈ X:

ii. F sð Þ¼ ln s þ s, with s > 0:

(F1) F is strictly increasing on R<sup>þ</sup>,

10 Differential Equations - Theory and Current Research

(F3) there exists a number <sup>k</sup><sup>∈</sup> ð Þ <sup>0</sup>; <sup>1</sup> such that lim<sup>s</sup>!0<sup>þ</sup> <sup>s</sup>

Example 2.22. The following function F : R<sup>þ</sup> ! R belongs to F:

contraction mapping if there exist F∈ F and τ∈ R<sup>þ</sup> such that

for all x, y∈ X and α, β, γ, η, δ ≥ 0 with α þ β þ γ þ 2η þ 2δ < 1.

lim<sup>n</sup>!<sup>∞</sup> <sup>σ</sup>ð Þ¼ xn; <sup>x</sup> <sup>σ</sup>ð Þ Tx; <sup>x</sup> :

Following Wardowski [11], we denote by F the family of all function, F : R<sup>þ</sup> ! R satisfying

Definition 2.23. [11] Let ð Þ X; d be a metric space. A self-mapping T on X is called an F-

Definition 2.24. [12] Let ð Þ X; σ be a metric-like space. A mapping T : X ! X is called a generalized Roger Hardy type F� contraction mapping, if there exist F∈ F and τ∈ R<sup>þ</sup> such

σð Þ Tx; Ty > 0 ) τ þ Fð Þ σð Þ Tx; Ty ≤ Fðασð Þþ x; y βσð Þþ x; Tx γσð Þ y; Ty

Theorem 2.25. ref. [12] Let ð Þ X; σ be 0 � σ� complete metric-like spaces and T : X ! X be a generalized Roger Hardy type F� contraction. Then T has a unique fixed point in X, either T or

Proof. See in [12]. □

Let X be a nonempty set. Throughout this paper, for a function ω : ð Þ� 0; ∞ X � X ! ½ � 0; ∞ , we

ωλð Þ¼ x; y ω λð Þ ; x; y

Definition 2.26 [13, 14] Let X be a nonempty set. A function ω : ð Þ� 0; ∞ X � X ! ½ � 0; ∞ is called a metric modular on X if satisfying, for all x, y, z ∈ X the following conditions hold:

k

F sð Þ¼ 0:

∀x, y∈ X, d Tx ½ � ð Þ ; Ty > 0 ) τ þ F d Tx ð Þ ð Þ ; Ty ≤ Fdx ð Þ ð Þ ; y : (2.1)

<sup>þ</sup>ησð Þþ <sup>x</sup>; Ty δσð ÞÞ <sup>y</sup>; Tx (2.2)

$$X^\*\_{\omega} = X^\*\_{\omega}(\mathbf{x}\_0) = \{ \mathbf{x} \in X : \exists \lambda = \lambda(\mathbf{x}) > 0 \text{ such that } \omega\_{\lambda}(\mathbf{x}, \mathbf{x}\_0) < \infty \}.$$

are said to be modular spaces (around x0).

Throughout this section we assume that ð Þ X; ω is a modular metric space, D be a nonempty subset of X<sup>ω</sup> and G≔fG<sup>ω</sup> is a directed graph with V Gð Þ¼ <sup>ω</sup> D and Δ⊆E Gð Þg <sup>ω</sup> :

Definition 2.29. [15, 16] The pair ð Þ D; G<sup>ω</sup> has Property (A) if for any sequence f g xn <sup>n</sup> <sup>∈</sup><sup>ℕ</sup> in D, with xn ! x as n ! ∞ and ð Þ xn; xnþ<sup>1</sup> ∈ E Gð Þ <sup>ω</sup> , then ð Þ xn; x ∈ E Gð Þ <sup>ω</sup> , for all n:

Definition 2.30. ref. [17] Let F∈ F and G<sup>ω</sup> ∈ G: A mapping T : D ! D is said to be F-Gωcontraction with respect to R : D ! D if

$$\text{i.i.} \quad (Rx, Ry) \in E(\mathbb{G}\_w) \Rightarrow (Tx, Ty) \in E(\mathbb{G}\_w) \text{ for all } x, y \in D, \text{ i.e. } T \text{ preserves edges w.r.t. } R, \text{ i.e. } R \text{ and } R \text{ are elements of } R \text{ and } R \text{ are elements of } R \text{ and } R \text{ are elements of } R \text{ and } R \text{ are elements of } R \text{ and } R \text{ are elements of } R \text{ and } R \text{ are elements of } R \text{ and } R \text{ are elements of } R \text{ and } R \text{ are elements of } R \text{ and } R \text{ are elements of } R \text{ and } R \text{ are elements of } R \text{ and } R \text{ are elements of } R \text{ and } R \text{ are elements of } R \text{ and } R \text{ are elements of } R \text{ and } R \text{ are elements of } R \text{ and } R \text{ are elements of } R \text{ and } R \text{ are elements of } R \text{ and } R \text{ are elements of } R \text{ and } R \text{ are elements of } R \text{ and } R \text{ are elements of } R \text{ and } R \text{ are elements of } R \text{ and } R \text{ are elements of } R \text{ and } R \text{ are elements of } R \text{ and } R \text{ are elements of } R \text{ and } R \text{ are elements of } R \text{ and } R \text{ are elements of } R \text{ and } R \text{ are elements of } R \text{ and } R \text{ are elements of } R \text{ and } R \text{ are elements of } R \text{ and } R \text{ are elements of } R \text{ and } R \text{ are elements of } R \text{ and } R \text{ are elements of } R \text{ and } R \text{ are elements of } R \text{ and } R \text{ are elements of } R \text{ and } R \text{ are elements of } R \text{ and } R \text{ are elements of } R \text{ and } R \text{ are$$

ii. there exists a number τ > 0 such that

$$
\omega\_1(T\mathfrak{x}, T\mathfrak{y}) > 0 \\
\Rightarrow \pi + F(\omega\_1(T\mathfrak{x}, T\mathfrak{y})) \leq F(\omega\_1(R\mathfrak{x}, R\mathfrak{y})) 
$$

for all x, y∈ D with ð Þ Rx; Ry ∈ E Gð Þ <sup>ω</sup> :

Example 2.31. ref. [17] Let F∈ F be arbitrary. Then every F-contractive mapping w.r.t. R is an F-Gω-contraction w.r.t. R for the graph G<sup>ω</sup> given by V Gð Þ¼ <sup>ω</sup> D and E Gð Þ¼ <sup>ω</sup> D � D.

We denote C Tð Þ ;R ≔f g x ∈ D : Tx ¼ Rx the set of all coincidence points of two self-mappings T and R, defined on D.

Theorem 2.32. ref. [17] Let ð Þ X; ω be a regular modular metric space with a graph Gω: Assume that D ¼ V Gð Þ <sup>ω</sup> is a nonempty ω-bounded subset of X<sup>ω</sup> and the pair ð Þ D; G<sup>ω</sup> has property (A) and also satisfy ΔM-condition. Let R, T : D ! D be two self mappings satisfying the following conditions:


Then C Rð Þ ; T 6¼ Ø.

Proof. See in [17]. □
