A New Approach to Study Fixed Point of Multivalued Mappings in Modular Metric Spaces and Applications

Abstract

The purpose of this paper is to present a new approach to study the existence of fixed points for multivalued F-contraction in the setting of modular metric spaces. In establishing this connection, we introduce the notion of multivalued F-contraction and prove corresponding fixed point theorems in complete modular metric space with some specific assumption on the modular. Then we apply our results to establish the existence of solutions for a certain type of non-linear integral equations.

1. Introduction

The Banach contraction principle [1] is one of the most important analytical results and considered as the main source of metric fixed point theory. It is the most widely applied fixed point result in many branches of mathematics. This result has been generalized in many different directions. Subsequently, in 2012, Wordowski [2] introduced the concept of F-contraction which generalized the Banach contraction principal in many ways. Further, Sgroi et al. [3] obtained a multivalued version of Wordowski’s result.

On other hand, Chistyakov [4] introduced the concept of modular metric spaces and gave some fundamental results on this topic. The fixed point property in this space has been defined and investigated by many authors [5,6,7,8,9]. It is important to note that in the classical Banach contraction, the contractive condition of the mapping implies that any orbit is bounded (see [10]). In case of modular metric space, due to failure of triangle inequality, it is not always true that the contractive condition of the mapping implies the boundedness of the orbit. Therefore, it is very important to handle this obstacle when dealing with a fixed point in modular metric space. Keeping the above facts in mind, in this paper, we define multivalued F-contraction in the setting of modular metric spaces with specific modular situations. Our result is a partial extension of Nadler [11], Wardowski [2] and Sgroi [3] to modular metric spaces. We also give an application of our main results to establish the existence of the solution of a non-linear integral equation.

2. Preliminaries

Throughout the article $\mathbb{N},{\mathbb{R}}^{+}$ and $\mathbb{R}$ will denote the set of natural numbers, positive real numbers and real numbers respectively.

Let X be a nonempty set. Throughout this paper, for a function $w:(0,\infty )\times X\times X\to [0,\infty ],$ we write

$$w_{\lambda }(x,y)=w(\lambda ,x,y)$$

for all $\lambda >0$ and $x,y\in X.$

Definition 1. 

[4,5] Let X be a nonempty set. A function $w:(0,\infty )\times X\times X\to [0,\infty ]$ is said to be a metric modular on $X,$ it satisfies, for all $x,y,z\in X$ the following conditions:

(i)

$w_{\lambda }(x,y)=0$ for all $\lambda >0$ if and only if $x=y$;

(ii)

$w_{\lambda }(x,y)=w_{\lambda }(y,x)$ for all $\lambda >0$;

(iii)

$w_{\lambda +\mu }(x,y)\le w_{\lambda }(x,z)+w_{\mu }(z,y)$ for all $\lambda ,\mu >0$.

If instead of $\left(i\right)$ we have only the condition $\left(i^{\prime }\right)$

$$w_{\lambda }(x,x)=0\text{for}\text{all}\lambda >0,x\in X$$

then w is said to be a pseudomodular (metric) on $X.$ A modular metric w on X is said to be regular if the following weaker version of $\left(i\right)$ is satisfied:

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

This condition play a significant role to insure the existence of fixed point for contractive type mapping in the setting of modular metric.

Example 2. 

Let $X=\mathbb{R}$ and w is defined by $w_{\lambda }(x,y)=\infty$ if $\lambda <1,$ and $w_{\lambda }(x,y)=\frac{1}{\lambda }|x-y|$ if $\lambda \ge 1,$ it is easy to verify that w is regular but not metric modular on X.

Finally, w is said to be convex if, for $\lambda ,\mu >0$ and $x,y,z\in X,$ it satisfies the inequality

$$w_{\lambda +\mu }(x,y)\le {\displaystyle \frac{\lambda }{\lambda +\mu }}{w}_{\lambda }(x,z)+{\displaystyle \frac{\mu }{\lambda +\mu }}{w}_{\mu }(z,y)$$

Note that for a metric pseudomodular w on a set $X,$ and any $x,y\in X,$ the function $\lambda ⟶w_{\lambda }(x,y)$ is nonincreasing on $(0,\infty )$. Indeed, if $0<\mu <\lambda ,$ then

$$\begin{array}{ccc}\hfill w_{\lambda }(x,y)& \le w_{\lambda -\mu }(x,x)+w_{\mu }(x,y)& =w_{\mu }(x,y)\hfill \end{array}$$

Definition 3. 

[4,5] Let w be a pseudomodular on $X.$ Fix $x_0\in X.$ The set

$$X_w=X_w\left(x_0\right)=\{x\in X:w_{\lambda }(x,x_0)⟶0as\lambda ⟶\infty \}$$

is said to be modular spaces (around $x_0$).

Definition 4. 

[5] Let $X_w$ be a modular metric space.

(i)

The sequence ${\left(x_n\right)}_{n\in \mathbb{N}}$ in $X_w$ is said to be w-convergent to $x\in X_w$ if and only if $w_1(x_n,x)\to 0,$ as $n\to \infty$.

(ii)

The sequence ${\left(x_n\right)}_{n\in \mathbb{N}}$ in $X_w$ is said to be w-Cauchy if $w_1(x_m,x_n)\to 0,$ as $m,n\to \infty$.

(iii)

A subset D of $X_w$ is said to be w-complete if any w-Cauchy sequence in D is a convergent sequence and its limit is in D.

(iv)

A subset D of $X_w$ is said to be w-closed if the w-limit of a w-convergent sequence of D always belongs to $D.$

(v)

A subset D of $X_w$ is said to be w-bounded if for some $\lambda >0,$ we have

$${\delta }_w\left(D\right)=sup\{w_1(x,y);x,y\in D\}<\infty$$

(vi)

A subset D of $X_w$ is said to be w-compact if for any $\left\{x_n\right\}$ in D there exists a subset sequence $\left\{x_{n_k}\right\}$ and $x\in D$ such that $w_1(x_{n_k},x)⟶0.$

In general, if $\underset{n⟶\infty }{lim}{w}_{\lambda }(x_n,x)=0,$ for some $\lambda >0,$ then we may not have $\underset{n⟶\infty }{lim}{w}_{\lambda }(x_n,x)=0,$ for all $\lambda >0.$ Therefore, as is done in modular function spaces, we will say that w satisfies the ${\Delta }_2$-condition (see page 4 in [5] ) if this the case, i.e., $\underset{n⟶\infty }{lim}{w}_{\lambda }(x_n,x)=0,$ for some $\lambda >0$ implies $\underset{n⟶\infty }{lim}{w}_{\lambda }(x_n,x)=0,$ for all $\lambda >0$.

The motivation of the following definition can easily be predicted from the last step of proof of Cauchy sequence in Theorems 13 and 15 (given below).

Definition 5. 

[12] Let $X_w$ be a modular metric space and ${\left\{x_n\right\}}_{n\in \mathbb{N}}$ be sequence in $X_w.$ We will say that w satisfies the ${\Delta }_M$-condition if this the case, i.e., $\underset{m,n⟶\infty }{lim}{w}_{m-n}(x_n,x_m)=0$ for $(m,n\in \mathbb{N},m>n)$ implies $\underset{m,n⟶\infty }{lim}{w}_{\lambda }(x_n,x_m)=0$ for some $\lambda >0.$

Let $CB\left(D\right):=\{C:C$ is nonempty w-closed and w-bounded subsets of $D\},$ $K\left(D\right):=\{C:C$ is nonempty w-compact subsets of $D\}$ and the Hausdorff metric modular defined on $CB\left(D\right)$ by

$$H_w(A,B):=max\{\underset{x\in A}{sup}{w}_1(x,B),\underset{y\in B}{sup}{w}_1(A,y)\}$$

where $w_1(x,B)=\underset{y\in B}{inf}{w}_1(x,y).$

Lemma 6. 

[5] Let $(X,w)$ be a modular metric space. Assume that w satisfies ${\Delta }_2$-condition. Let D be a nonempty subset of $X_w.$ Let $A_n$ be a sequence of sets in $CB\left(D\right),$ and suppose $\underset{n⟶\infty }{lim}{H}_w(A_n,A_0)=0$ where $A_0\in CB\left(D\right).$ Then if $x_n\in A_n$ and $\underset{n⟶\infty }{lim}{x}_n=x_0,$ it follows that $x_0\in A_0.$

3. Fixed Point Results for Multivalued F-Contractions

Definition 7. 

[2] Let $F:{\mathbb{R}}^{+}⟶\mathbb{R}$ satisfying the following condition:

$(F1$)

F is strictly increasing on ${\mathbb{R}}^{+},$

$(F2$)

for every sequence $\left\{s_n\right\}$ in ${\mathbb{R}}^{+},$ we have $\underset{n⟶\infty }{lim}{s}_n=0$ if and only if $\underset{n⟶\infty }{lim}F\left(s_n\right)=-\infty ,$

$\left(F3\right)$

there exists a number $k\in (0,1)$ such that $\underset{s⟶0^{+}}{lim}{s}^{k}F\left(s\right)=0.$

We denote by $\mathcal{F}$ the family of all function that satisfy the conditions (F1)–(F3).

Example 8. 

The following functions $F:{\mathbb{R}}^{+}\to \mathbb{R}$ belong to $\mathcal{F}$:

(i)

$F\left(s\right)=lns,$ with $s>0,$

(ii)

$F\left(s\right)=-\frac{1}{\sqrt{s}},s>0$

Definition 9. 

Let $(X,w)$ be a modular metric space. Let D be non empty bounded subset of X. A multivalued mapping $T:D⟶CB\left(D\right)$ is called F-contraction on X if $F\in \mathcal{F},$ and $\tau \in {\mathbb{R}}^{+}$, for all $x,y\in D$ with $y\in Tx$ there exists $z\in Ty$ such that $w_1(y,z)>0,$ the following inequality holds:

$$\begin{array}{c}\hfill \tau +F\left(w_1(y,z)\right)\le F\left(\mathcal{M}(x,y)\right)\end{array}$$

(3.1)

where $\mathcal{M}(x,y)=max\left\{w_1(x,y),w_1(x,Tx),w_1(y,Ty),w_1(y,Tx)\right\}.$

Definition 10. 

Let $(X,w)$ be a modular metric space. Let D be a nonempty subset of $X_w$. A multivalued mapping $T:D⟶CB\left(D\right)$ is said to be F-contraction of Hardy-Rogers-type if $F\in \mathcal{F}$ and $\tau \in {\mathbb{R}}^{+}$ such that,

$$2\tau +F\left(H_w(Tx,Ty)\right)\le F(\alpha w_1(x,y)+\beta w_1(x,Tx)+\gamma w_1(y,Ty)+L{w}_1(y,Tx))$$

(3.2)

for all $x,y\in D$ with $H_w(Tx,Ty)>0,$ where $\alpha ,\beta ,\gamma ,L\ge 0,\alpha +\beta +\gamma =1$ and $\gamma \ne 1.$

Example 11. 

Let $F:{\mathbb{R}}^{+}⟶\mathbb{R}$ be given by $F\left(s\right)=lns.$ For each multivalued mapping $T:D⟶CB\left(D\right)$ satisfying Equation (3.1) we have

$$w_1(y,z)\le e^{-\tau }\mathcal{M}(x,y),\mathit{\text{for all}}x,y\in D,y\ne z$$

It is clear that for $z,y\in D$ such that $y=z$ the previous inequality also holds.

Example 12. 

Let $F:{\mathbb{R}}^{+}⟶\mathbb{R}$ be given by $F\left(s\right)=lns.$ It is clear that F satisfies $\left(F1\right)-\left(F3\right)$ for any $k\in (0,1).$ Each mapping $T:D⟶CB\left(D\right)$ satisfying Equation (3.2) is an F-contraction such that

$$H_w(Tx,Ty)\le e^{-\tau }{w}_1(x,y),\mathit{\text{for all}}x,y\in D,Tx\ne Ty$$

It is clear that for $x,y\in D$ such that $Tx=Ty$ the previous inequality also holds and hence T is a contraction.

Theorem 13. 

Let $(X,w)$ be a modular metric space. Assume that w is a regular modular satisfying ${\Delta }_M$-condition and ${\Delta }_2$-condition. Let D be a nonempty w-bounded and w-complete subset of $X_w.$ Let $T:D⟶CB\left(D\right)$ be a continuous F-contraction. Then T has a fixed point.

Proof. 

Let $x_0\in D$ be an arbitrary point of D and choose $x_1\in T{x}_0$. If $x_1=x_0,$ then $x_1$ is a fixed point of T and the proof is completed. Suppose that $x_1\ne x_0.$ Since T is an F-contraction, then there exists $x_2\in T{x}_1$ such that

$$\begin{array}{c}\hfill \tau +F(w_1(x_1,x_2)\le F\left(\mathcal{M}(x_0,x_1)\right)\text{and}{x}_1\ne x_2\end{array}$$

Therefore, we have that there exists $x_3\in T{x}_2$ such that

$$\begin{array}{c}\hfill \tau +F(w_1(x_2,x_3)\le F\left(\mathcal{M}(x_1,x_2)\right)\text{and}{x}_2\ne x_3\end{array}$$

Repeating this process, we find that there exists a sequence $\left\{x_n\right\}$ with initial point $x_0$ such that $x_{n+1}\in T{x}_n,x_{n+1}\ne x_n$ and

$$\tau +F\left(w_1(x_n,x_{n+1})\right)\le F\left(\mathcal{M}(x_{n-1},x_n)\right)\text{for}\text{all}n\in \mathbb{N}$$

This implies

$$F\left(w_1(x_n,x_{n+1})\right)<F\left(\mathcal{M}(x_{n-1},x_n)\right)\text{for}\text{all}n\in \mathbb{N}$$

Consequently,

$$\begin{array}{ccc}\hfill w_1(x_n,x_{n+1})& <& \mathcal{M}(x_{n-1},x_n)(\text{Since}\mathrm{F}\text{is}\text{strictly}\text{increasing}.)\hfill \\ & =& max\{w_1(x_{n-1},x_n),w_1(x_{n-1},T{x}_{n-1}),w_1(x_n,T{x}_n),w_1(x_n,T{x}_{n-1})\}\hfill \\ & =& max\{w_1(x_{n-1},x_n),w_1(x_n,T{x}_n)\}\hfill \\ & \le & max\{w_1(x_{n-1},x_n),w_1(x_n,x_{n+1})\}\hfill \end{array}$$

Obviously, if $max\{w_1(x_{n-1},x_n),w_1(x_n,x_{n+1})\}=w_1(x_n,x_{n+1}),$ we have a contradiction and so $max\{w_1(x_{n-1},x_n),w_1(x_n,x_{n+1})\}=w_1(x_{n-1},x_n).$

Consequently, By $\left(F1\right)$ we have

$$\tau +F(w_1(x_n,x_{n+1})\le F\left(w_1(x_{n-1},x_n)\right)\text{for}\text{all}n\in \mathbb{N}$$

(3.3)

By Equation (3.3), we have

$$F\left(w_1(x_n,x_{n+1})\right)\le F\left(w_1(x_{n-1},x_n)\right)-\tau \le \cdots \le F\left(w_1(x_0,x_1)\right)-n\tau ,\text{for}\text{all}n\in \mathbb{N}$$

(3.4)

and hence $\underset{n\to \infty }{lim}F(w_1(x_n,x_{n+1})=-\infty .$ By $\left(F2\right)$ we have that $w_1(x_n,x_{n+1})\to 0$ as $n\to \infty .$ Now, let $k\in (0,1)$ such that $\underset{n\to \infty }{lim}({\left(w_1(x_n,x_{n+1})\right)}^{k}F\left(w_1(x_n,x_{n+1})\right).$ By Equation (3.4), the following holds for all $n\in \mathbb{N}:$

$${\left(w_1(x_{n+1},x_n)\right)}^k\left(F\left(w_1(x_{n+1},x_n)\right)-F\left(w_1(x_0,x_1)\right)\right)\le -{\left(w_1(x_{n+1},x_n)\right)}^{k}n\tau \le 0$$

(3.5)

Taking $n\to \infty$ in Equation (3.5), we deduce

$$\underset{n\to \infty }{lim}\left(n{\left(w_1(x_{n+1},x_n)\right)}^k\right)=0$$

Then there exists $n_1\in \mathbb{N}$ such that $n{\left(w_1(x_{n+1},x_n)\right)}^k\le 1$ for all $n\ge n_1,$ that is,

$$w_1(x_n,x_{n+1})\le \frac{1}{n^{1/k}}\text{for}\text{all}n\ge n_1$$

Now, For all $m,n\ge n_1$ with $m>n,$ we have

$$\begin{array}{ccc}\hfill w_{m-n}(x_n,x_m)& \le & w_1(x_n,x_{n+1})+w_1(x_{n+1},x_{n+2})+\cdots +w_1(x_{m-1},x_m)\hfill \\ & \le & \frac{1}{n^{1/k}}+\frac{1}{{(n+1)}^{1/k}}+\cdots +\frac{1}{m^{1/k}}\hfill \\ & <& {\Sigma }_{i=n}^{\infty }\frac{1}{i^{1/k}}\hfill \end{array}$$

Since the series ${\Sigma }_{i=n}^{\infty }\frac{1}{i^{1/k}}$ is convergent, this implies

$$\underset{m,n\to \infty }{lim}\left(w_{m-n}(x_n,x_m)\right)=0$$

Since w satisfies ${\Delta }_M$-condition. Hence, we have

$$\underset{m,n⟶\infty }{lim}{w}_1(x_n,x_m)=0$$

This shows that $\left\{x_n\right\}$ is a w-Cauchy sequence. D is w-complete, there exists $v\in D$ such that $x_n⟶v$ as $n⟶\infty .$ Now, we prove that v is a fixed point of T.

Let $T{x}_n$ be a sequence in CB(D). Since T is continuous then we have $T{x}_n⟶Tv$ so $\underset{n⟶\infty }{lim}{H}_w(T{x}_n,Tv)=0$, where $Tv\in CB\left(D\right).$ Then if $x_{n+1}\in T{x}_n$ and $\underset{n⟶\infty }{lim}{x}_{n+1}=v,$ it follows from Lemma 6 that $v\in Tv.$ Hence v is a fixed point of T. ☐

Example 14. 

Let $X_w=D=\{x_n=\frac{n(n+1)}{2}:n\in \mathbb{N}\}$ and $w_1(x,y)=\frac{1}{\lambda }|x-y|,x,y\in D.$ Then $(X,w)$ is a w-complete modular metric space. Define the mapping $T:D\to CB\left(D\right)$ by the:

$$T\left(x\right)=\left\{\begin{array}{c}\begin{array}{ccc}\left\{x_1\right\}\hfill & ,& x=x_1\hfill \\ \{x_1,x_2,\cdots ,x_{n-1}\}\hfill & ,& x=x_n\hfill \end{array}\hfill \end{array}\right.$$

Then, as shown in Example 3 of [13], T is a multivalued F-contraction with respect to $F\left(s\right)=lns+s$ and $\tau =1.$ Therefore, Theorem 13 are satisfied and so T has a fixed point in $X_w.$

On the other hand, since

$$\underset{n\to \infty }{lim}{\displaystyle \frac{H_w(T{x}_n,T{x}_1)}{\mathcal{M}(x_n,x_1)}}=\underset{n\to \infty }{lim}{\displaystyle \frac{x_{n-1}-1}{x_n-1}}=1$$

then T is not multivalued contraction.

Next, we give a fixed point result for multivalued F-contractions of Hardy-Rogers-type in modular metric space.

Theorem 15. 

Let $(X,w)$ be a modular metric space. Assume that w is a regular modular satisfying ${\Delta }_M$-condition and ${\Delta }_2$-condition. Let D be a nonempty w-bounded and w-complete subset of $X_w$ and $T:X⟶K\left(D\right)$ be an F-contractions of Hardy-Rogers-type. Then T has a fixed point.

Proof. 

Let $x_0$ be an arbitrary point in D. As $Tx$ is nonempty for all $x\in X$, we can choose $x_1\in T{x}_0.$ If $x_1\in T{x}_1$, then $x_1$ is a fixed point of T and so the proof is complete. Assume $x_1\notin T{x}_1.$ Then, since $T{x}_1$ is closed, $w(x,T{x}_1)>0.$ On the other hand, from $w(x_1,T{x}_1)\le H_w(T{x}_0,T{x}_1)$ and (F1)

$$F\left(w(x_1,T{x}_1)\right)\le F\left(H_w(T{x}_0,T{x}_1)\right)$$

From Equation (3.2), we can write that

$$\begin{array}{ccc}\hfill F(w(x_1,T{x}_1)\le F\left(H_w(T{x}_0,T{x}_1)\right)& \le & F(\alpha w_1(x_0,x_1)+\beta w_1(x_0,T{x}_0)+\gamma w_1(x_1,T{x}_1)\hfill \\ & & +L{w}_1(x_1,T{x}_0))-2\tau \hfill \end{array}$$

Since $T{x}_1$ is compact, there exists $x_2\in T{x}_1$ such that

$$w_1(x_1,x_2)=w_1(x_1,T{x}_1)$$

Then,

$$\begin{array}{ccc}\hfill F\left(w_1(x_1,x_2)\right)=F(w(x_1,T{x}_1)& \le & F\left(H_w(T{x}_0,T{x}_1)\right)\hfill \\ & \le & F(\alpha w_1(x_0,x_1)+\beta w_1(x_0,T{x}_0)+\gamma w_1(x_1,T{x}_1)\hfill \\ & & +L{w}_1(x_1,T{x}_0))-2\tau \hfill \end{array}$$

Thus,

$$\begin{array}{ccc}\hfill F\left(w_1(x_1,x_2)\right)& \le & F\left(H_w(T{x}_0,T{x}_1)\right)\hfill \\ & \le & F(\alpha w_1(x_0,x_1)+\beta w_1(x_0,T{x}_0)+\gamma w_1(x_1,T{x}_1)\hfill \\ & & +L{w}_1(x_1,T{x}_0))-2\tau \hfill \\ & \le & F(\alpha w_1(x_0,x_1)+\beta w_1(x_0,x_1)+\gamma w_1(x_1,x_2))-2\tau \hfill \\ & \le & F((\alpha +\beta )w_1(x_0,x_1)+\gamma w_1(x_1,x_2))\hfill \end{array}$$

Thus,

$$F\left(w_1(x_1,x_2)\right)\le F((\alpha +\beta )w_1(x_0,x_1)+\gamma w_1(x_1,x_2))$$

Since F is strictly increasing, we deduce that

$$w_1(x_1,x_2)\le (\alpha +\beta )w_1(x_0,x_1)+\gamma w_1(x_1,x_2)$$

and hence

$$(1-\gamma )w_1(x_1,x_2)<(\alpha +\beta )w_1(x_0,x_1)$$

From $\alpha +\beta +\gamma =1$ and $\gamma \ne 1,$ we deduce that $1-\gamma >0$ and so

$$w_1(x_1,x_2)<{\displaystyle \frac{\alpha +\beta }{1-\gamma }}{w}_1(x_0,x_1)=w_1(x_0,x_1)$$

Consequently,

$$\tau +F\left(w_1(x_1,x_2)\right)\le F\left(w_1(x_0,x_1)\right)$$

Continuing in this manner, we can define a sequence $\left\{x_n\right\}\subset D$ such that $x_n\notin T{x}_n,x_{n+1}\in T{x}_n$ and

$$\tau +F\left(w_1(x_{n+1},x_{n+2})\right)\le F\left(w_1(x_n,x_{n+1})\right)\text{for}\text{all}n\in \mathbb{N}\cup \left\{0\right\}$$

Proceeding as in the proof of Theorem 13, we obtain that $\left\{x_n\right\}$ is a w-Cauchy sequence. Since D is a w-complete modular metric space, there exists $v\in D$ such that $x_n⟶v$ as $n⟶\infty .$ Now, we prove that v is a fixed point of $T.$ If there exists an increasing sequence $\left\{n_k\right\}\subset \mathbb{N}$ such that $x_{n_k}\in Tv$ for all $k\in \mathbb{N},$ since $Tv$ is w-closed and $x_{n_k}⟶v,$ we have $v\in Tv$ and the proof is completed. So we can assume that there exists $n_0\in \mathbb{N}$ such that $x_n\notin Tv$ for all $n\ge n_0.$ This implies that $T{x}_{n-1}\ne Tv$ for all $n\ge n_0.$ Now, using Equation (3.2) with $x=x_n$ and $y=v,$ we obtain

$$\begin{array}{ccc}2\tau +F\left(H_w(T{x}_n,Tv)\right)\hfill & \le & F(\alpha w_1(x_n,v)+\beta w_1(x_n,T{x}_n)+\gamma w_1(v,Tv)\hfill \\ & & +L{w}_1(v,T{x}_n))\hfill \end{array}$$

which implies

$$\begin{array}{ccc}2\tau +F\left(w_1(x_{n+1},Tv)\right)\hfill & \le & 2\tau +F\left(H_w(T{x}_n,Tv)\right)\hfill \\ & \le & F(\alpha w_1(x_n,v)+\beta w_1(x_n,T{x}_n)+\gamma w_1(v,Tv)\hfill \\ & & +L{w}_1(v,T{x}_n))\hfill \\ & \le & F(\alpha w_1(x_n,v)+\beta w_1(x_n,x_{n+1})+\gamma w_1(v,Tv)\hfill \\ & & +L{w}_1(v,x_{n+1}))\hfill \end{array}$$

Since F is strictly increasing, we have

$$\begin{array}{ccc}\hfill w_1(x_{n+1},Tv)& <& \alpha w_1(x_n,v)+\beta w_1(x_n,x_{n+1})+\gamma w_1(v,Tv)\hfill \\ & & +L{w}_1(v,x_{n+1}).\hfill \end{array}$$

Letting $n⟶\infty$ in the previous inequality, as $\gamma <1$ we have $w_1(v,Tv)\le \gamma w_1(v,Tv)<w_1(v,Tv),$ which implies $w_1(v,Tv)=0.$ Since $Tv$ is w-closed, we obtain that $v\in Tv,$ that is, v is a fixed point of T. ☐

Remark 16. 

If we consider $T:X⟶CB\left(T\right)$ in Theorem 15 i.e., we are relaxing compactness of co-domain of mapping T but then we have to assume T be continuous. In this case, we can write proof as Theorem 15 upto Cauchy. Further, by the completeness of D, we have $v\in D$ such that $x_n\to v.$ Since T is continuous, we have $\underset{n⟶\infty }{lim}{H}_w(T{x}_n,T_v)=0$ and as $x_{n+1}\in T{x}_{n}with{x}_{n+1}\to v$ then by Lemma 6 we obtain $v\in Tv.$ Hence v is fixed point of $T.$

4. Application to Integral Equations

Integral equations arise in many scientific and engineering problems. A large class of initial and boundary value problem can be converted to Volterra or Fredholm integral equation (see for instant [14]).

In this section we consider the following integral equation:

$$u\left(t\right)=\beta A\left(u\left(t\right)\right)+\gamma B\left(u\left(t\right)\right)+g\left(t\right),t\in [0,\mathit{T}],\mathit{T}>0$$

(4.1)

where

$$A\left(u\left(t\right)\right)={\int }_0^t{K}_1\left(t,s,u\left(s\right)\right)ds,B\left(u\left(t\right)\right)={\int }_0^t{K}_2\left(t,s,u\left(s\right)\right)ds\text{and}\beta ,\gamma \ge 0$$

Let $C(I,\mathbb{R})$ be the space of all continuous functions on $I,$ where $I=[0,\mathit{T}]$ with the norm $\parallel u\parallel ={sup}_{t\in I}\left|u\left(t\right)\right|$ and the metric $w_{\lambda }(u,v):=\frac{1}{\lambda }\parallel u-v\parallel =\frac{1}{\lambda }d(u,v)$ for all $u,v\in C(I,\mathbb{R}).$ For $r>0$ and $u\in C(I,\mathbb{R})$ we denote by $B_{\lambda }(u,r)=\{v\in C(I,\mathbb{R}):w_{\lambda }(u,v)\le r\}$ the closed ball concerned at u and of radius $r.$

Theorem 17. 

Let $r>0$ be a fixed real number and the following conditions are satisfied:

(i) 

$K:I\times I\times \mathbb{R}\to \mathbb{R}$ and $g:I\to \mathbb{R}$ are continuous;

(ii) 

there exists $u_0\in C(I,\mathbb{R})$ such that $\beta A\left(u_0\left(t\right)\right)+\gamma B\left(u_0\left(t\right)\right)+g\left(t\right)\in B(u_0,r);$

(iii) 

if $v\in B_{\lambda }(u,r)$, $\lambda >0,$ then

$$|K_i\left(t,s,u\left(s\right)\right)-K_i\left(t,s,v\left(s\right)\right)|\le L_i(t,s,u\left(s\right),v\left(s\right))\frac{\left|u\right(s)-v(s\left)\right|}{{\left(1+\tau \sqrt{\frac{\left|u\right(s)-v(s\left)\right|}{\lambda }}\right)}^2},i=1,2$$

for all $t,s\in I,u,v\in \mathbb{R}$ and for some continuous functions $L_1,L_2:I\times I\times \mathbb{R}\times \mathbb{R}\to {\mathbb{R}}^{+}.$

such that $L_i\left(t,s,u\left(s\right),v\left(s\right)\right)(\beta +\gamma )\mathit{T}\le 1,i=1,2$ for all $s,t\in I$, then the integral Equation (4.1) admit a solution.

Proof. 

Note that $\left(C(I,\mathbb{R}),w_{\lambda }\right)$ is a complete modular metric space. Define $T:C(I,\mathbb{R})\to C(I,\mathbb{R})$ by

$$T\left(u\left(t\right)\right)=\beta A\left(u\left(t\right)\right)+\gamma B\left(u\left(t\right)\right)+g\left(t\right),t\in I$$

Since $v\in B_{\lambda }(u,r),$ then by the definition of T and (iii) we have

$$\begin{array}{ccc}\hfill w_{\lambda }(Tu,Tv)& =& \frac{1}{\lambda }\underset{t\in I}{sup}|\beta A\left(u\left(t\right)\right)+\gamma B\left(u\left(t\right)\right)-\beta A\left(v\left(t\right)\right)-\gamma B\left(v\left(t\right)\right)|\hfill \\ & =& \frac{1}{\lambda }\underset{t\in I}{sup}|\beta {\int }_0^t[K_1\left(t,s,u\left(s\right)\right)-K_1\left(t,s,v\left(s\right)\right)]ds\hfill \\ & & +\gamma {\int }_0^t[K_2\left(t,s,u\left(s\right)\right)-K_2\left(t,s,v\left(s\right)\right)]ds|\hfill \\ & \le & \frac{1}{\lambda }\underset{t\in I}{sup}\{\beta {\int }_0^t|K_1\left(t,s,u\left(s\right)\right)-K_1\left(t,s,v\left(s\right)\right)|ds\hfill \\ & & +\gamma {\int }_0^t|K_2\left(t,s,u\left(s\right)\right)-K_2\left(t,s,v\left(s\right)\right)|ds\}\hfill \\ & \le & \frac{1}{\lambda }\underset{t\in I}{sup}\{\beta {\int }_0^t|L_1(t,s,u\left(s\right),v\left(s\right))\frac{\left|u\right(s)-v(s\left)\right|}{{\left(1+\tau \sqrt{\frac{\left|u\right(s)-v(s\left)\right|}{\lambda }}\right)}^2}ds\hfill \\ & & +\gamma {\int }_0^t|L_2(t,s,u\left(s\right),v\left(s\right))\frac{\left|u\right(s)-v(s\left)\right|}{{\left(1+\tau \sqrt{\frac{\left|u\right(s)-v(s\left)\right|}{\lambda }}\right)}^2}ds\}\hfill \\ & \le & \frac{\left|u\right(s)-v(s\left)\right|}{\lambda }\frac{1}{{\left(1+\tau \sqrt{\frac{\left|u\right(s)-v(s\left)\right|}{\lambda }}\right)}^2}\underset{t\in I}{sup}\left\{\beta {\int }_0^t\frac{1}{(\beta +\gamma )T}ds+\gamma {\int }_0^t\frac{1}{(\beta +\gamma )T}ds\right\}\hfill \\ & \le & \frac{w_{\lambda }(u,v)}{{\left(1+\tau \sqrt{w_{\lambda }(u,v)}\right)}^2}\underset{t\in I}{sup}\left\{\frac{t}{\mathit{T}}\right\}\hfill \end{array}$$

This implies

$$w_{\lambda }(Tu,Tv)\le \frac{w_{\lambda }(u,v)}{{\left(1+\tau \sqrt{w_{\lambda }(u,v)}\right)}^2}$$

☐

Now, we observe that the function $F:{\mathbb{R}}^{+}\to \mathbb{R}$ defined by $F\left(\alpha \right)=-\frac{1}{\sqrt{\alpha }},\alpha >0$ is in $\mathcal{F}$ and so we deduce that the mapping T satisfies all condition of Theorem 13 with $\mathcal{M}(u,v)=w_{\lambda }(u,v)\text{for}\lambda =1.$ Hence there exists a solution of the integral Equation (4.1).

Remark 18. 

Our above Theorem 4.1 is an abstract application of F- contraction mapping which can not be covered by Banach contraction principle.