Relaxing the Φ-Family Auxiliary Functions and Related Results

Abstract

This article establishes the existence of fixed points and common fixed points for set-valued mappings satisfying an implicit-type contraction inequality involving a new auxiliary function in a complete metric space equipped with a binary relation. Through a novel family of functions referred to as the $\Delta$-family, which simplifies the axioms in comparison to the previously defined $\Phi$-family, the study unifies a few classical fixed-point theorems. The practical relevance of the theoretical findings is demonstrated by applying the results to investigate the existence of solutions for a system of integral equations.

1. Introduction and Preliminaries

One of the most well-known and practical results for functional analysis is the traditional Banach fixed point theorem [1]. This theorem establishes the existence and uniqueness of the fixed point of a self-map that meets the contraction condition defined on a complete metric space. In addition to that, it offers a methodical approach to determining the fixed point of a self-map via an iterative process. This iterative process guarantees the convergence of the iterative sequence to the fixed point of a self-map in the underlying space. Consequently, the Banach fixed point theorem has far-reaching implications in numerous fields, including differential equations, optimization, and even computer science. In the context of the application, a recent article [2] added a significant contribution by discussing the existence of the solution of the fractional-order Chua’s attractor model.

The Kannan fixed point result [3] and the Chatterjea [4] fixed point result have significant contributions to the literature, particularly related to the study of fixed points for discontinuous mappings under certain conditions in metric space. These results are considered the most basic generalizations of the Banach fixed point result. Some other classical generalizations are presented by Reich [5], Bianchini [6], Zamftrescu [7], Hardy and Rogers [8], and Ciric [9].

The conventional Banach contraction principle was extended by Alam and Imdad [10] to a complete metric space endowed with a binary relation. In this study, the validation of the contraction condition is required only for the components that are related under the underlying relation, rather than the entire space. This technique makes the contraction condition weaker and more applicable in problem scenarios. The work of Alam and Imdad [10] reduces to the Banach contraction principle under the universal relation. Further, they explained how the several well-known fixed point results become particular cases of their study. For instance, the results of Nieto and Rodríguez-López [11] are followed by defining the relation as a partial order. This approach inspired a large number of researchers, and we see much enhancement to the available literature. For instance, Shukla et al. [12] introduced the Prešić-Ćirić-type results in metric space using a binary relation, and Almarri et al. [13] investigated various fixed point results in M-metric space with a binary relation, while Almalki [14] and Din et al. [15] investigated a few generalizations of Perov-type fixed point results in vector metric space with a binary relation.

This approach of a binary relation is not restricted to contraction-type inequalities; with the passage of time, many researchers incorporated this idea along with the structure of metric and metric-like spaces. As a result, we have orthogonal metric spaces [16], graphical metric spaces [17], Czerwik vector-valued R-metric spaces [18], etc.

We now go through some basic ideas related to binary relations.

Definition 1

([19]). Let W be a set that is nonempty. The set $W\times W=\{(w,q):w,q\in W\}$ defines the Cartesian product on W. A binary relation on W, denoted by $R_W$, is defined by a subset of $W\times W$.

The following two situations should be known while discussing binary relations.

(1)

If $(w,q)\notin R_W$, then w does not relate to q under the relation $R_W$. Another way to denote $(w,q)\notin R_W$ is $w /R_{W}q$.

(2)

If $(w,q)\in R_W$, then w relates to q under the relation $R_W$. Another way to denote $(w,q)\in R_W$ is $w{R}_{W}q$.

A binary relation can be categorized into multiple categories. A few categories are mentioned below.

Definition 2

([19]). A binary relation $R_W$ on a nonempty set W is known as

• reflexive if $(w,w)\in R_W\forall w\in W$,
• non-reflexive if $(w,w)\notin R_W$ for some $w\in W$,
• irreflexive if $(w,w)\notin R_W$$\forall w\in W$,
• non-irreflexive if $(w,w)\in R_W$ for some $w\in W$,
• symmetric if $(w,q)\in R_W$ implies $(q,w)\in W$$\forall w,q\in W$,
• anti-symmetric if $(w,q)\in R_W$ and $(q,w)\in R_W$ implies $w=q$$\forall w, q\in W$,
• transitive if $(w,q)\in R_W$ and $(q,t)\in R_W$ implies $(w,t)\in R_W$$\forall w,q, t\in W$,
• a sharp order or strict order if $R_W$ is irreflexive and transitive,
• a near order if $R_W$ is anti-symmetric and transitive,
• a pre-order or quasi order if $R_W$ is reflexive and transitive,
• a partial order, if $R_W$ is reflexive, anti-symmetric, and transitive,
• a pseudo order if $R_W$ is reflexive and anti-symmetric,
• an equivalence order if $R_W$ is reflexive, symmetric and transitive,
• a universal relation (full relation) if $R_W=W\times W$, and
• an empty relation if $R_W$ is empty set.

For convenience, we just write a “binary relation” rather than a “non-empty binary relation” even though we designate $R_W$ for a non-empty binary relation throughout the article.

Alam and Imdad [10] introduced the concept of $R_W$-comparative elements in the following manner.

Definition 3

([10]). Consider a binary relation $R_W$ on a nonempty set W. Any two elements $w,q\in W$ are called $R_W$-comparative if either $(w,q)\in R_W$ or $(q,w)\in R_W$. The notion $[w,q]\in R_W$ means either $(w,q)\in R_W$ or $(q,w)\in R_W$.

The concept of an $R_W$-preserving sequence was listed by Alam and Imdad [10] as follows.

Definition 4

([10]). Consider a binary relation $R_W$ on a nonempty set W. A sequence $\left\{w_n\right\}\subseteq W$ is called an $R_W$-preserving sequence if $(w_n,w_{n+1})\in R_W\forall n\in \mathbb{N}.$

Almalki et al. extended the notion of $\Psi$-closed given by Alam and Imdad [10] as follows.

Definition 5

([14]). Consider a binary relation $R_W$ on a nonempty set W and a mapping $\Psi :W\to CL(W)$. A relation $R_W$ is called Ψ-closed if for each $w,q\in W$ with $(w,q)\in R_W$ we obtain $(g,h)\in R_W$$\forall g\in \Psi w$ and $h\in \Psi q$.

The concepts of $\alpha$-$\psi$-contractive and $\alpha$-admissible self-mappings were proposed by Samet et al. [20], they also established some fixed point results for such mappings in metric spaces. Karapinar and Samet [21] expanded these ideas and came up with a more generalized fixed point result. Asl et al. [22] expanded the ideas of Samet et al. [20] to set-valued mappings by providing the concepts of ${\alpha }_{\ast }$-$\psi$-contractive and ${\alpha }_{\ast }$-admissible mappings, and proposed associated results. A few more works in this direction are available in [23,24,25,26,27].

In this article, $CL(W)$ is the set of all nonempty closed subsets of $(W,d_w)$. For $A\in CL(W)$ and $p\in W$, $d_w(p,A)=inf\{d_w(p,a):a\in A\}$. From the definition of $d_w(p,A)$, it is clear that, for each $ϵ>0$, there is a $q\in A$ such that

$$d_w(p,q)\le d_w(p,A)+ϵ.$$

For each $A,B\in CL(W)$, the generalized Hausdorff metric induced by $d_w$ is defined as

$$H_w(A,B)=\left\{\begin{array}{cc}max\{\underset{a\in A}{sup}{d}_w(a,B),\underset{b\in B}{sup}{d}_w(b,A)\},\hfill & \mathrm{if} \mathrm{the} \mathrm{maximum} \mathrm{exists};\hfill \\ \infty ,\hfill & \mathrm{otherwise}.\hfill \end{array}\right.$$

In the literature, we see that the Hausdorff distance is defined on the collection of all nonempty closed and bounded subsets of $(W,d_w)$, while the generalized Hausdorff is defined on the collection of all nonempty closed subsets of $(W,d_w)$ with a scenario that $H_w(A,B)=\infty$ if the ${sup}_{a\in A}{d}_w(a,B)$ or ${sup}_{b\in B}{d}_w(A,b)$ fail to exist. This raises the question of why $H_w(\Psi x,\Psi y)=\infty$ does not appear in the proof of those fixed point theorems that are based on contraction-type multi-valued mappings $\Psi :W\to CL(W)$ involving generalized Hausdorff distance. The existence of the contraction-type inequality itself is a straightforward justification for not employing $H_w(\Psi x,\Psi y)=\infty$. This means that, if $H_w(\Psi x,\Psi y)=\infty$ appears in some case on the left side of the inequality, then a contraction-type inequality does not exist. To stay consistent with the mathematical concept of distance measure in $CL(W)$, we see the use of generalized Hausdorff distance in theorems rather than Hausdorff distance.

To integrate a few well-known fixed point results, Ali and Vetro [26] introduced the following family of functions.

The collection of functions $\varphi :{[0,\infty )}^4\to [0,\infty )$ that meet the following criteria is called the $\Phi$-family.

(i)

$\varphi$ is nondecreasing and continuous in every coordinate;

(ii)

let $g_1,g_2,f\in [0,\infty )$, if $g_1<g_2$, $g_1<qf$, for any $q>1$, and $f\le \varphi (g_2,g_2,g_1,g_2)$, then there exists $\xi \in \Xi$ with $f\le \xi (g_2)$;

(iii)

let $g_1,g_2,f\in [0,\infty )$, if $g_1\ge g_2$, $g_1\le qf$, for any $q>1$, and $f\le \varphi (g_1,g_2,g_1,g_1)$, then $f=0$;

(iv)

let $g\in [0,\infty )$, if $g\le \varphi (0,0,g,{\displaystyle \frac{1}{2}}g)$ or $g\le \varphi (0,g,0,{\displaystyle \frac{1}{2}}g)$, then $g=0$.

Note that, throughout the article, $\Xi$ represents the collection of strictly increasing functions $\xi :[0,\infty )\to [0,\infty )$ with ${\sum }_{n=1}^{\infty }{\xi }^n(t)<\infty$$\forall t\ge 0$. Further, $\xi (t)<t$ for all $t>0$.

To integrate some of the most well-known results from the literature, Ali and Vetro [26] proposed the following implicit-type fixed point result.

Theorem 1.

Let $\Psi :W\to CL(W)$ be an implicit-type ζ-ϕ-contractive mapping on complete metric space $(W,d_w)$. That is, there is a $\zeta :W\times W\to [0,\infty )$ and $\varphi \in \Phi$ such that

$$H_w(\Psi g,\Psi h)\le \varphi (d_w(g,h),D_w(g,\Psi g),D_w(h,\Psi h),{\displaystyle \frac{1}{2}}(D_w(h,\Psi g)+D_w(g,\Psi h))),$$

(1)

for all $g,h\in W$ with $\zeta (g,h)\ge 1$. Additionally, assume the following axioms are vaild:

(i) 

Ψ is ζ-admissible, that is, for each $g\in W$ and $h\in \Psi g$ with $\zeta (g,h)\ge 1$, we have $\zeta (h,z)\ge 1$ for all $z\in \Psi h$;

(ii) 

there is a $w_0\in W$ and $w_1\in \Psi w_0$ with $\zeta (w_0,w_1)\ge 1$;

(iii) 

a. Ψ is continuous;

or

b. if $\left\{w_n\right\}$ is a sequence in W such that $w_n\to w$ as $n\to \infty$ and $\zeta (w_n,w_{n+1})\ge 1$ for each $n\in \mathbb{N}$, then $\zeta (w_n,w)\ge 1$ for each $n\in \mathbb{N}$.

Then Ψ has a fixed point.

Regarding the notion of the $\Phi$-family, the following observations and queries might be brought up.

(1)

The axioms (ii) and (iii) of the $\Phi$-family are very complicated axioms. Is it possible that the role of these axioms in the above result can be achieved by some simple axioms?

(2)

As $\varphi :{[0,\infty )}^4\to [0,\infty )$, Theorem 1 does not imply quasi-contraction-type results by using this family. Is it possible to extend the domain of this family to incorporate a few more fixed point results?

(3)

$\Phi$-family functions are useless in establishing the existence of common fixed points for two mappings. Hence, Theorem 1 cannot be extended using the $\Phi$-family to ensure the existence of common fixed points for two mappings.

In view of the above observations, this article presents new classes of auxiliary functions called the $\Delta$-family and the ${\Delta }_C$-family. We may consider the $\Delta$-family as an improved form of the $\Phi$-family because it will lead to the positive answer of the first two above-listed observations. That is, the $\Delta$-family involves an easy axiom that replaces the role played by the complex axioms (ii) and (iii) of the $\Phi$-family to establish a result. Moreover, in this new class, the domain of the functions is extended from ${[0,\infty )}^4$ to ${[0,\infty )}^5$, so that a few more types of contractions can be discussed under this family. The ${\Delta }_C$-family is presented by keeping the third observation in mind to ensure the existence of common fixed points for set-valued mappings.

2. Main Result

This section presents the new classes of auxiliary functions and studies the existence of fixed points and common fixed points for set-valued mappings satisfying implicit contractive inequalities involving these auxiliary functions.

2.1. Fixed Point Result Involving Auxiliary Function

The $\Delta$-family is the collection of all functions $\delta :{[0,\infty )}^5\to [0,\infty )$ that meet the following three criteria:

(i)

$\delta$ is nondecreasing and continuous in every coordinate.

(ii)

If $h\le \delta (g,g,h,0,g+h)$, then either $h=0$ or $h<\xi (g)$, where $\xi \in \Xi$ and $\xi$ is dependent on $\delta$ but not on the elements of $\delta$.

(iii)

If $h\le \delta (0,0,h,0,h)$, then $h=0$.

The $\Delta$-family provides the conceptual improvement in the $\Phi$-family and uses the axioms that are easy to understand as compared to the $\Phi$-family. The following are a few examples of functions that are part of the $\Delta$-family.

• ${\delta }_1(g_1,g_2,g_3,g_4,g_5)=kmax\{g_1,g_2,g_3,g_4,g_5\}$, where $k\in [0,1/2)$.
• ${\delta }_2(g_1,g_2,g_3,g_4,g_5)=k{g}_1$, where $k\in [0,1)$.
• ${\delta }_3(g_1,g_2,g_3,g_4,g_5)=a{g}_1+b{g}_2+c{g}_3+d{g}_4+e{g}_5$, where $a,b,c,d,e\in [0,1)$ with $a+b+c+d+2e<1.$
• ${\delta }_4(g_1,g_2,g_3,g_4,g_5)=k(g_2+g_3)$, where $k\in [0,1/2)$.
• ${\delta }_5(g_1,g_2,g_3,g_4,g_5)=k(g_4+g_5)$, where $k\in [0,1/2)$.
• ${\delta }_6(g_1,g_2,g_3,g_4,g_5)=kmax\{g_1,g_2,g_3\}$, where $k\in [0,1)$.
• ${\delta }_7(g_1,g_2,g_3,g_4,g_5)=a{g}_1+b{g}_2+c{g}_3$, where $a,b,c\in [0,1)$ with $a+b+c<1.$
• ${\delta }_8(g_1,g_2,g_3,g_4,g_5)=a{g}_1+b{\displaystyle \frac{g_2+g_3}{2}}+c{\displaystyle \frac{g_4+g_5}{2}}$, where $a,b,c\in [0,1)$ with $a+b+c<1.$

The following definition presents the concept of implicit-type $\delta$-contractive mapping.

Definition 6.

Let $(W,d_w)$ be a metric space equipped with a binary relation $R_W$. A mapping $\Psi :W\to CL(W)$ is called an implicit-type δ-contractive mapping if

$$H_w(\Psi g,\Psi h)\le \delta \left(d_w(g,h),d_w(g,\Psi g),d_w(h,\Psi h),d_w(h,\Psi g),d_w(g,\Psi h)\right)$$

(2)

for all $g,h\in W$ with $(g,h)\in R_W$, where $\delta \in \Delta$.

The following theorem helps us to study the existence of a fixed point of implicit-type $\delta$-contractive mapping.

Theorem 2.

Let $\Psi :W\to CL(W)$ be an implicit-type δ-contractive mapping on a complete metric space $(W,d_w)$ equipped with a binary relation $R_W$, and let the following axioms exist:

(i) 

There is a $w_0\in W$ and a $w_1\in \Psi w_0$ with $(w_0,w_1)\in R_W$.

(ii) 

Ψ is relation-admissible, that is, for each $g\in W$ and $h\in \Psi g$ with $(g,h)\in R_W$, we have $(h,z)\in R_W$ for all $z\in \Psi h$.

(iii) 

If $\left\{w_n\right\}$ is a sequence in W that converges to $w\in W$ and $(w_n,w_{n+1})\in R_W$$\forall n\in \mathbb{N}$, then $(w_n,w)\in R_W$$\forall n\in \mathbb{N}$.

Then, Ψ has a fixed point in W, that is, there is a $w^{\ast }\in W$ with $w^{\ast }\in \Psi w^{\ast }$.

Proof. 

The hypothesis states that, for certain $w_0\in W$ and $w_1\in \Psi w_0$, we have $(w_0,w_1)\in R_W$. The proof proceeds by considering $w_0\ne w_1$. From (2), we have

$$\begin{array}{c}\hfill H_w(\Psi w_0,\Psi w_1)\le \delta \left(d_w(w_0,w_1),d_w(w_0,\Psi w_0),d_w(w_1,\Psi w_1),d_w(w_1,\Psi w_0),d_w(w_0,\Psi w_1)\right).\end{array}$$

(3)

It was chosen that $w_1\in \Psi w_0$ and $\delta$ has a nondecreasing characteristic; thus, by (3), we obtain

$$\begin{array}{c}\hfill d_w(w_1,\Psi w_1)\le \delta \left(d_w(w_0,w_1),d_w(w_0,w_1),d_w(w_1,\Psi w_1),0,d_w(w_0,\Psi w_1)\right).\end{array}$$

This further implies the following inequality by incorporating the fact that $d_w(w_0,\Psi w_1)\le d_w(w_0,w_1)+d_w(w_1,\Psi w_1)$. Hence,

$$\begin{array}{c}\hfill d_w(w_1,\Psi w_1)\le \delta \left(d_w(w_0,w_1),d_w(w_0,w_1),d_w(w_1,\Psi w_1),0,d_w(w_0,w_1)+d_w(w_1,\Psi w_1)\right).\end{array}$$

(4)

We obtain either $d_w(w_1,\Psi w_1)=0$ or $d_w(w_1,\Psi w_1)<\xi (d_w(w_0,w_1))$ by taking into account (4) and axiom (ii) of $\Delta$. If $d_w(w_1,\Psi w_1)=0$, then $w_1\in \Psi w_1$, and the outcome of the result is accomplished. Hence, assuming that $d_w(w_1,\Psi w_1)<\xi (d_w(w_0,w_1))$, we can move forward. As $d_w(w_1,\Psi w_1)<\xi (d_w(w_0,w_1))$, there is an ${ϵ}_1>0$ such that $d_w(w_1,\Psi w_1)+{ϵ}_1\le \xi (d_w(w_0,w_1))$. Hence, we obtain $w_2\in \Psi w_1$ such that

$$\begin{array}{c}\hfill d_w(w_1,w_2)\le d_w(w_1,\Psi w_1)+{ϵ}_1\le \xi (d_w(w_0,w_1)).\end{array}$$

(5)

As $(w_0,w_1)\in R_W$ for $w_0\in W$ and $w_1\in \Psi w_0$, by hypothesis (ii) of the theorem, we obtain $(w_1,w_2)\in R_W$. Again, from (2), we obtain

$$\begin{array}{c}\hfill H_w(\Psi w_1,\Psi w_2)\le \delta \left(d_w(w_1,w_2),d_w(w_1,\Psi w_1),d_w(w_2,\Psi w_2),d_w(w_2,\Psi w_1),d_w(w_1,\Psi w_2)\right).\end{array}$$

(6)

$w_2\in \Psi w_1$ and $\delta$ is a nondecreasing in every coordinate; thus, by (6), we obtain

$$\begin{array}{c}\hfill d_w(w_2,\Psi w_2)\le \delta \left(d_w(w_1,w_2),d_w(w_1,w_2),d_w(w_2,\Psi w_2),0,d_w(w_1,\Psi w_2)\right).\end{array}$$

This yields that

$$\begin{array}{c}\hfill d_w(w_2,\Psi w_2)\le \delta \left(d_w(w_1,w_2),d_w(w_1,w_2),d_w(w_2,\Psi w_2),0,d_w(w_1,w_2)+d_w(w_2,\Psi w_2)\right).\end{array}$$

(7)

We obtain either $d_w(w_2,\Psi w_2)=0$ or $d_w(w_2,\Psi w_2)<\xi (d_w(w_1,w_2))$ by taking into account (7) and axiom (ii) of $\Delta$. If $d_w(w_2,\Psi w_2)=0$, then $w_2\in \Psi w_2$, and the outcome of the result is achieved. Hence, we move forward with proof by assuming that $d_w(w_2,\Psi w_2)<\xi (d_w(w_1,w_2))$. As $d_w(w_2,\Psi w_2)<\xi (d_w(w_1,w_2))$, there is an ${ϵ}_2>0$ such that $d_w(w_2,\Psi w_2)+{ϵ}_2\le \xi (d_w(w_1,w_2))$. Hence, we obtain $w_3\in \Psi w_2$ such that

$$\begin{array}{c}\hfill d_w(w_2,w_3)\le d_w(w_2,\Psi w_2)+{ϵ}_2\le \xi (d_w(w_1,w_2)).\end{array}$$

(8)

Additionally, by hypothesis (ii), we have $(w_2,w_3)\in R_W$. As $\xi$ is an increasing function, by (5) and (8), we obtain

$$\begin{array}{c}\hfill d_w(w_2,w_3)\le {\xi }^2(d_w(w_0,w_1)).\end{array}$$

(9)

The sequence $\left\{w_n\right\}$ in W that results from repeating the above process satisfies the following requirements:

• $w_n\in \Psi w_{n-1}\forall n\in \mathbb{N}$;
• $d_w(w_n,w_{n+1})\le {\xi }^n(d_w(w_0,w_1))\forall n\in \mathbb{N}$;
• $(w_{n-1},w_n)\in R_W\forall n\in \mathbb{N}$.

We next demonstrate the Cauchy status of $\left\{w_n\right\}$. Given any $n,m\in \mathbb{N}$ where $n>m$, the triangle inequality provides the following:

$$\begin{array}{ccc}\hfill d_w(w_m,w_n)& \le & d_w(w_m,w_{m+1})+d_w(w_{m+1},w_{m+2})+\cdots +d_w(w_{n-1},w_n)\hfill \\ & \le & \sum _{i=m}^{n-1}{\xi }^i(d_w(w_0,w_1)).\hfill \end{array}$$

Hence, we obtain ${lim}_{n,m\to \infty }{d}_w(w_m,w_n)=0$, since ${\sum }_{i=1}^{\infty }{\xi }^i(d_w(w_0,w_1))$ is convergent. This confirms the Cauchy status of $\left\{w_n\right\}$ in complete space W. Consequently, there is some $w_{\ast }\in W$ with $w_n\to w_{\ast }$. As $(w_{n-1},w_n)\in R_W\forall n\in \mathbb{N}$ and $w_n\to w_{\ast }$, by hypothesis (iii), we obtain $(w_{n-1},w_{\ast })\in R_W\forall n\in \mathbb{N}$. By (2), we obtain

$$\begin{array}{ccc}\hfill H_w(\Psi w_n,\Psi w_{\ast })& \le & \delta (d_w(w_n,w_{\ast }),d_w(w_n,\Psi w_n),d_w(w_{\ast },\Psi w_{\ast }),\hfill \\ & & d_w(w_{\ast },\Psi w_n),d_w(w_n,\Psi w_{\ast }))\forall n\in \mathbb{N}.\hfill \end{array}$$

This provides

$$\begin{array}{ccc}\hfill d_w(w_{n+1},\Psi w_{\ast })& \le & \delta (d_w(w_n,w_{\ast }),d_w(w_n,w_{n+1}),d_w(w_{\ast },\Psi w_{\ast }),\hfill \\ & & d_w(w_{\ast },w_{n+1}),d_w(w_n,\Psi w_{\ast }))\forall n\in \mathbb{N}.\hfill \end{array}$$

It is given that $\delta$ is nondecreasing and continuous in every coordinate. Additionally, we have $w_n\to w_{\ast }$ as $n\to \infty$. Thus, by applying the limit as $n\to \infty$ in the above inequality, we obtain the following:

$$d_w(w_{\ast },\Psi w_{\ast })\le \delta \left(0,0,d_w(w_{\ast },\Psi w_{\ast }),0,d_w(w_{\ast },\Psi w_{\ast })\right).$$

By axiom (iii) of $\Delta$ and the above inequality, we obtain $d_w(w_{\ast },\Psi w_{\ast })=0$. Hence, $w_{\ast }\in \Psi w_{\ast }$. □

Remark 1.

The aforementioned theorem’s result remains valid even if we replace the hypothesis (iii) with the condition that Ψ is a continuous map. It is worth mentioning that proceeding with the continuity of Ψ, the characteristic (iii) of the delta mapping and the continuity of the delta mapping are additional properties, and we can accomplish the conclusion of the theorem without these restrictions.

Example 1.

Let $W=C([0,1],\mathbb{R})$ be the collection of all continuous real-valued functions defined on $[0,1]$ and $d_w(g,h)={sup}_{j\in [0,1]}|g(j)-h(j)|=\parallel g-h\parallel$. Define a binary relation $R_W$ on W as $(g,h)\in R_W$ if and only if $g(j)\ge 0$, $h(j)\ge 0$$\forall j\in [0,1]$. Define $\Psi :W\to CL(W)$ by

$$\Psi (g)(j)=\left\{\begin{array}{c}\left\{{\displaystyle \frac{1}{2}}(g(j)+j)+{\displaystyle \frac{1}{4}}\left|{\int }_0^j(g(t)-t)dt\right|\right\},g(j)\ge 0\forall j\in [0,1]\hfill \\ \left\{g(j),{(g(j))}^2\right\}, otherwise.\hfill \end{array}\right.$$

First, we show that Ψ is an implicit-type δ-contractive mapping with $\delta (g_1,g_2,g_3,g_4,g_5)={\displaystyle \frac{4}{5}}{g}_1$. For $g,h\in W$ with $(g,h)\in R_W$, we have

$$\begin{array}{ccc}\hfill H_w(\Psi g,\Psi h)& =& \underset{j\in [0,1]}{sup}|\left({\displaystyle \frac{1}{2}}(g(j)+j)+{\displaystyle \frac{1}{4}}\left|{\int }_0^j(g(t)-t)dt\right|\right)\hfill \\ & & -\left({\displaystyle \frac{1}{2}}(h(j)+j)+{\displaystyle \frac{1}{4}}\left|{\int }_0^j(h(t)-t)dt\right|\right)|\hfill \\ & =& \underset{j\in [0,1]}{sup}|\left({\displaystyle \frac{1}{2}}(g(j)+j)-{\displaystyle \frac{1}{2}}(h(j)+j)\right)\hfill \\ & & +\left({\displaystyle \frac{1}{4}}\left|{\int }_0^j(g(t)-t)dt\right|-{\displaystyle \frac{1}{4}}\left|{\int }_0^j(h(t)-t)dt\right|\right)|\hfill \\ & \le & {\displaystyle \frac{1}{2}}\underset{j\in [0,1]}{sup}|g(j)-h(j)|+{\displaystyle \frac{1}{4}}\underset{j\in [0,1]}{sup}\left|\left(\left|{\int }_0^j(g(t)-t)dt\right|-\left|{\int }_0^j(h(t)-t)dt\right|\right)\right|\hfill \\ & \le & {\displaystyle \frac{1}{2}}\underset{j\in [0,1]}{sup}|g(j)-h(j)|+{\displaystyle \frac{1}{4}}\underset{j\in [0,1]}{sup}\left(|{\int }_0^j(g(t)-t)dt-{\int }_0^j(h(t)-t)dt|\right)\hfill \\ & \le & {\displaystyle \frac{1}{2}}\underset{j\in [0,1]}{sup}|g(j)-h(j)|+{\displaystyle \frac{1}{4}}\underset{j\in [0,1]}{sup}\left({\int }_0^j|g(t)-h(t)|dt\right)\hfill \\ & \le & {\displaystyle \frac{1}{2}}\parallel g-h\parallel +{\displaystyle \frac{1}{4}}\parallel g-h\parallel \hfill \\ & \le & \delta \left(d_w(g,h),d_w(g,\Psi g),d_w(h,\Psi h),d_w(h,\Psi g),d_w(g,\Psi h)\right).\hfill \end{array}$$

To assure axiom (i), consider $w_0=e^j$ and $w_1={\displaystyle \frac{1}{2}}(e^j+j)+{\displaystyle \frac{1}{4}}\left|{\int }_0^j(e^t-t)dt\right|\in \Psi w_0$, then $(w_0,w_1)\in R_W$. Clearly, Ψ is relation-admissible, since, for each $g\ge 0$ in W, that is, $g(j)\ge 0$$\forall j\in [0,1]$, and $h\ge 0$ in $\Psi g$, we have $z\ge 0$ for all $z\in \Psi h$. Furthermore, if $\left\{w_n\right\}$ is a sequence in W with $w_n\ge 0$$\forall n\in \mathbb{N}$ and converges to any $w\in W$, then $w\ge 0$; that is, if $w_n\to w$ and $(w_n,w_{n+1})\in R_W$$\forall n\in \mathbb{N}$, then $(w_n,w)\in R_W$$\forall n\in \mathbb{N}$. Hence, Theorem 2 implies that Ψ has a fixed point in W.

2.2. Common Fixed Point Result Involving an Auxiliary Function

This subsection presents the ${\Delta }_C$ subclass of the $\Delta$-family that helps to extend Theorem 2 to study the existence of common fixed points for two mappings.

The ${\Delta }_C$-family is a collection of functions $\delta :{[0,\infty )}^5\to [0,\infty )$ that meet the following three criteria:

(i)

$\delta$ is nondecreasing and continuous in every coordinate.

(ii)

(a): If $h\le \delta (g,g,h,0,g+h)$, then either $h=0$ or $h<\xi (g)$;

(b): If $h\le \delta (g,h,g,g+h,0)$, then either $h=0$ or $h<\xi (g)$;

where $\xi \in \Xi$ and $\xi$ is dependent on $\delta$ but not on the elements of $\delta$.

(iii)

(a): If $h\le \delta (0,0,h,0,h)$, then $h=0$;

(b): If $h\le \delta (0,h,0,h,0)$, then $h=0$.

The following definition presents the concept of implicit-type $\delta$-contractive mappings.

Definition 7.

Let $(W,d_w)$ be a metric space equipped with a binary relation $R_W$. Two mappings $\Psi ,\Lambda :W\to CL(W)$ are called implicit-type δ-contractive mappings if

$$H_w(\Psi g,\Lambda h)\le \delta \left(d_w(g,h),d_w(g,\Psi g),d_w(h,\Lambda h),d_w(h,\Psi g),d_w(g,\Lambda h)\right)$$

(10)

for all $g,h\in W$ with $(g,h)\in R_W$ or $(h,g)\in R_W$, where $\delta \in {\Delta }_C$.

In order to ensure that the aforementioned concept has a common fixed point, we derived the following result.

Theorem 3.

Let $\Psi ,\Lambda :W\to CL(W)$ be implicit-type δ-contractive mappings on a complete metric space $(W,d_w)$ equipped with a binary relation $R_W$, and let the following axioms exist:

(i) 

There are $w_0\in W$ and $w_1\in \Psi w_0$ or $w_1\in \Lambda w_0$ with $(w_0,w_1)\in R_W$.

(ii) 

$(\Psi ,\Lambda )$ is relation-admissible, which means that, for each $g\in W$ and $h\in \Psi g$ with $(g,h)\in R_W$, we have $(h,z)\in R_W$ for all $z\in \Lambda h$; moreover, for each $g\in W$ and $h\in \Lambda g$ with $(g,h)\in R_W$, we have $(h,z)\in R_W$ for all $z\in \Psi h$.

(iii) 

If $\left\{w_n\right\}$ is a sequence in W that converges to $w\in W$ and $(w_n,w_{n+1})\in R_W$$\forall n\in \mathbb{N}$, then $(w_n,w)\in R_W$$\forall n\in \mathbb{N}$.

Then, Ψ and Λ have a common fixed point in W; that is, there is a $w^{\ast }\in W$ with $w^{\ast }\in \Psi w^{\ast }$ and $w^{\ast }\in \Lambda w^{\ast }$.

Proof. 

Without a loss of generality, we say that the hypothesis (i) guarantees that there is a $w_0\in W$ and a $w_1\in \Psi w_0$ with $(w_0,w_1)\in R_W$. From (10), we obtain the following:

$$\begin{array}{c}\hfill H_w(\Psi w_0,\Lambda w_1)\le \delta \left(d_w(w_0,w_1),d_w(w_0,\Psi w_0),d_w(w_1,\Lambda w_1),d_w(w_1,\Psi w_0),d_w(w_0,\Lambda w_1)\right).\end{array}$$

(11)

$w_1\in \Psi w_0$ and $\delta$ has a nondecreasing characteristic; thus, by (11), we obtain

$$\begin{array}{c}\hfill d_w(w_1,\Lambda w_1)\le \delta \left(d_w(w_0,w_1),d_w(w_0,w_1),d_w(w_1,\Lambda w_1),0,d_w(w_0,\Lambda w_1)\right).\end{array}$$

As $d_w(w_0,\Lambda w_1)\le d_w(w_0,w_1)+d_w(w_1,\Lambda w_1)$, by incorporating this fact in the above inequality, we obtain

$$\begin{array}{c}\hfill d_w(w_1,\Lambda w_1)\le \delta \left(d_w(w_0,w_1),d_w(w_0,w_1),d_w(w_1,\Lambda w_1),0,d_w(w_0,w_1)+d_w(w_1,\Lambda w_1)\right).\end{array}$$

(12)

Here, we discuss the two possibilities of $w_0$ and $w_1$.

Case 1: If $w_0=w_1$, then by (12), we obtain

$$\begin{array}{c}\hfill d_w(w_1,\Lambda w_1)\le \delta \left(0,0,d_w(w_1,\Lambda w_1),0,0+d_w(w_1,\Lambda w_1)\right).\end{array}$$

Thus, using axiom (iii) of ${\Delta }_C$, we obtain $d_w(w_1,\Lambda w_1)=0$. Hence, $w_1\in \Lambda w_1$ and $w_1\in \Psi w_1$.

Case 2: If $w_0\ne w_1$, then by (12) and axiom (ii) of ${\Delta }_C$, we obtain $d_w(w_1,\Lambda w_1)<\xi (d_w(w_0,w_1))$. As $d_w(w_1,\Lambda w_1)<\xi (d_w(w_0,w_1))$, there is an ${ϵ}_1>0$ with $d_w(w_1,\Lambda w_1)+{ϵ}_1\le \xi (d_w(w_0,w_1))$. Hence, we obtain $w_2\in \Lambda w_1$ such that

$$\begin{array}{c}\hfill d_w(w_1,w_2)\le d_w(w_1,\Lambda w_1)+{ϵ}_1\le \xi (d_w(w_0,w_1)).\end{array}$$

(13)

It is mentioned above that $w_0\in W$ and $w_1\in \Psi w_0$ with $(w_0,w_1)\in R_W$, by hypothesis (ii) of the theorem, we obtain $(w_1,w_2)\in R_W$, since $w_2\in \Lambda w_1$. As $(w_1,w_2)\in R_W$, by using (10) with $h=w_1$ and $g=w_2$, we obtain

$$\begin{array}{c}\hfill H_w(\Psi w_2,\Lambda w_1)\le \delta \left(d_w(w_2,w_1),d_w(w_2,\Psi w_2),d_w(w_1,\Lambda w_1),d_w(w_1,\Psi w_2),d_w(w_2,\Lambda w_1)\right).\end{array}$$

(14)

$w_2\in \Lambda w_1$ and $\delta$ has a nondecreasing characteristic; thus, by (14), we obtain

$$\begin{array}{c}\hfill d_w(w_2,\Psi w_2)\le \delta \left(d_w(w_1,w_2),d_w(w_2,\Psi w_2),d_w(w_1,w_2),d_w(w_1,\Psi w_2),0\right).\end{array}$$

As $d_w(w_1,\Psi w_2)\le d_w(w_1,w_2)+d_w(w_2,\Psi w_2)$, by incorporating this fact in the above inequality, we obtain

$$\begin{array}{c}\hfill d_w(w_2,\Psi w_2)\le \delta \left(d_w(w_1,w_2),d_w(w_2,\Psi w_2),d_w(w_1,w_2),d_w(w_1,w_2)+d_w(w_2,\Psi w_2),0\right).\end{array}$$

(15)

We now again discuss the two possibilities of $w_1$ and $w_2$

Case 1: If $w_1=w_2$, then by (15), we obtain

$$\begin{array}{c}\hfill d_w(w_2,\Psi w_2)\le \delta \left(0,d_w(w_2,\Psi w_2),0,d_w(w_2,\Psi w_2),0\right).\end{array}$$

Thus, using axiom (iii) of ${\Delta }_C$, we obtain $d_w(w_2,\Psi w_2)=0$. Hence, $w_2\in \Psi w_2$ and $w_2\in \Lambda w_2.$

Case 2: If $w_1\ne w_2$, then by (15) and axiom (ii) of ${\Delta }_C$, we obtain $d_w(w_2,\Psi w_2)<\xi (d_w(w_1,w_2))$. As $d_w(w_2,\Psi w_2)<\xi (d_w(w_1,w_2))$, there is an ${ϵ}_2>0$ with $d_w(w_2,\Psi w_2)+{ϵ}_2\le \xi (d_w(w_1,w_2))$. Hence, we obtain $w_3\in \Psi w_2$ such that

$$\begin{array}{c}\hfill d_w(w_2,w_3)\le d_w(w_2,\Psi w_2)+{ϵ}_2\le \xi (d_w(w_1,w_2)).\end{array}$$

(16)

By using (16) and (13), we obtain

$$\begin{array}{c}\hfill d_w(w_2,w_3)\le {\xi }^2(d_w(w_0,w_1)).\end{array}$$

As $w_1\in W$ and $w_2\in \Lambda w_1$ with $(w_1,w_2)\in R_W$, by the hypothesis (ii) of the theorem, we obtain $(w_2,w_3)\in R_W$, since $w_3\in \Psi w_2$. By proceeding with the proof in this way, we obtain a sequence $\left\{w_n\right\}$ in W satisfying the following conditions:

• $w_{2n+1}\in \Psi w_{2n}$ and $w_{2n+2}\in \Lambda w_{2n+1}$ for each $n\in \mathbb{N}\cup \left\{0\right\}$;
• $d_w(w_n,w_{n+1})\le {\xi }^n(d_w(w_0,w_1))\forall n\in \mathbb{N}$;
• $(w_{n-1},w_n)\in R_W\forall n\in \mathbb{N}$.

For any $n,m\in \mathbb{N}$ with $n>m$, we obtain the following with the help of the triangle inequality:

$$\begin{array}{ccc}\hfill d_w(w_m,w_n)& \le & d_w(w_m,w_{m+1})+d_w(w_{m+1},w_{m+2})+\cdots +d_w(w_{n-1},w_n)\hfill \\ & \le & \sum _{i=m}^{n-1}{\xi }^i(d_w(w_0,w_1)).\hfill \end{array}$$

The inequality mentioned above leads to ${lim}_{n,m\to \infty }{d}_w(w_m,w_n)=0$. Hence, the Cauchy status of $\left\{w_n\right\}$ is confirmed in W. The completeness of W now provides $w_{\ast }\in W$ with $w_n\to w_{\ast }$. As $(w_{n-1},w_n)\in R_W\forall n\in \mathbb{N}$ and $w_n\to w_{\ast }$, by the hypothesis (iii), we obtain $(w_{n-1},w_{\ast })\in R_W\forall n\in \mathbb{N}$. As $(w_{2n},w_{\ast })\in R_W\forall n\in \mathbb{N}\cup \left\{0\right\}$, by (10), we obtain

$$\begin{array}{ccc}\hfill H_w(\Psi w_{2n},\Lambda w_{\ast })& \le & \delta (d_w(w_{2n},w_{\ast }),d_w(w_{2n},\Psi w_{2n}),d_w(w_{\ast },\Lambda w_{\ast }),\hfill \\ & & d_w(w_{\ast },\Psi w_{2n}),d_w(w_{2n},\Lambda w_{\ast }))\hfill \end{array}$$

for all $n\in \mathbb{N}\cup \left\{0\right\}$. This provides

$$\begin{array}{ccc}\hfill d_w(w_{2n+1},\Lambda w_{\ast })& \le & \delta (d_w(w_{2n},w_{\ast }),d_w(w_{2n},w_{2n+1}),d_w(w_{\ast },\Lambda w_{\ast }),\hfill \\ & & d_w(w_{\ast },w_{2n+1}),d_w(w_{2n},\Lambda w_{\ast }))\forall n\in \mathbb{N}.\hfill \end{array}$$

Given that $\delta$ is nondecreasing and continuous in every coordinate. We also have $w_n\to w_{\ast }$. Thus, applying the limit as $n\to \infty$ in the above inequality, we obtain the following:

$$d_w(w_{\ast },\Lambda w_{\ast })\le \delta \left(0,0,d_w(w_{\ast },\Lambda w_{\ast }),0,d_w(w_{\ast },\Lambda w_{\ast })\right).$$

By the axiom (iii) of ${\Delta }_C$ and the above inequality, we obtain $d_w(w_{\ast },\Lambda w_{\ast })=0$. Hence, $w_{\ast }\in \Lambda w_{\ast }.$

As $(w_{2n+1},w_{\ast })\in R_W\forall n\in \mathbb{N}\cup \left\{0\right\}$, again by (10) with $h=w_{2n+1}$ and $g=w_{\ast }$, we obtain

$$\begin{array}{ccc}\hfill H_w(\Psi w_{\ast },\Lambda w_{2n+1})& \le & \delta (d_w(w_{\ast },w_{2n+1}),d_w(w_{\ast },\Psi w_{\ast }),d_w(w_{2n+1},\Lambda w_{2n+1}),\hfill \\ & & d_w(w_{2n+1},\Psi w_{\ast }),d_w(w_{\ast },\Lambda w_{2n+1}))\hfill \end{array}$$

for all $n\in \mathbb{N}\cup \left\{0\right\}$. This provides

$$\begin{array}{ccc}\hfill d_w(w_{2n+2},\Psi w_{\ast })& \le & \delta (d_w(w_{2n+1},w_{\ast }),d_w(w_{\ast },\Psi w_{\ast }),d_w(w_{2n+1},w_{2n+2}),\hfill \\ & & d_w(w_{2n+1},\Psi w_{\ast }),d_w(w_{\ast },w_{2n+2}))\forall n\in \mathbb{N}.\hfill \end{array}$$

The following inequality results from letting $n\to \infty$ in the previous inequality. This is because $\delta$ is continuous and nondecreasing in each coordinate, and $w_n\to w_{\ast }$.

$$d_w(w_{\ast },\Psi w_{\ast })\le \delta \left(0,d_w(w_{\ast },\Psi w_{\ast }),0,d_w(w_{\ast },\Psi w_{\ast }),0\right).$$

By the axiom (iii) of ${\Delta }_C$ and the above inequality, we obtain $d_w(w_{\ast },\Psi w_{\ast })=0$. Hence, $w_{\ast }\in \Psi w_{\ast }$. Therefore, we say that $w_{\ast }$ is a common fixed point of $\Psi$ and $\Lambda$; that is, $w_{\ast }\in \Psi w_{\ast }$ and $w_{\ast }\in \Lambda w_{\ast }$. □

Example 2.

Let $W=C([0,1],\mathbb{R})$ be the collection of all continuous real-valued functions defined on $[0,1]$ and $d_w(g,h)={sup}_{j\in [0,1]}|g(j)-h(j)|=\parallel g-h\parallel$. Define a binary relation $R_W$ on W as $(g,h)\in R_W$ if and only if $g(j)\ge 0$, $h(j)\ge 0$$\forall j\in [0,1]$. Define $\Psi :W\to CL(W)$ by

$$\Psi (g)(j)=\left\{\begin{array}{c}\left\{{\displaystyle \frac{g(j)}{4}}\right\},g(j)\ge 0\forall j\in [0,1]\hfill \\ \left\{-\parallel g\parallel \right\}, otherwise.\hfill \end{array}\right.$$

Define $\Lambda :W\to CL(W)$ by

$$\Lambda (h)(j)=\left\{\begin{array}{c}\left\{{\displaystyle \frac{h(j)}{6}}\right\},h(j)\ge 0\forall j\in [0,1]\hfill \\ \left\{{\int }_0^{j}h(t)dt,{(h(j))}^2\right\}, otherwise.\hfill \end{array}\right.$$

First, we prove that $\Psi ,\Lambda$ are implicit-type δ-contractive mappings with $\delta (g_1,g_2,g_3,g_4,g_5)={\displaystyle \frac{1}{3}}{g}_1+{\displaystyle \frac{1}{6}}{g}_2+{\displaystyle \frac{1}{6}}{g}_3$. For $g,h\in W$ with $(g,h)\in R_W$, we have

$$\begin{array}{ccc}\hfill H_w(\Psi g,\Lambda h)& =& \underset{j\in [0,1]}{sup}|{\displaystyle \frac{g(j)}{4}}-{\displaystyle \frac{h(j)}{6}}|\hfill \\ & =& \underset{j\in [0,1]}{sup}|{\displaystyle \frac{g(j)}{4}}-{\displaystyle \frac{h(j)}{4}}+{\displaystyle \frac{h(j)}{4}}-{\displaystyle \frac{h(j)}{6}}|\hfill \\ & \le & {\displaystyle \frac{1}{4}}\underset{j\in [0,1]}{sup}|g(j)-h(j)|+\underset{j\in [0,1]}{sup}|{\displaystyle \frac{h(j)}{4}}-{\displaystyle \frac{h(j)}{6}}|\hfill \\ & =& {\displaystyle \frac{1}{4}}\underset{j\in [0,1]}{sup}|g(j)-h(j)|+{\displaystyle \frac{1}{10}}\underset{j\in [0,1]}{sup}\left|{\displaystyle \frac{5}{6}}h(j)\right|\hfill \\ & =& {\displaystyle \frac{1}{4}}\underset{j\in [0,1]}{sup}|g(j)-h(j)|+{\displaystyle \frac{1}{10}}\underset{j\in [0,1]}{sup}|h(j)-{\displaystyle \frac{1}{6}}h(j)|\hfill \\ & \le & \delta \left(d_w(g,h),d_w(g,\Psi g),d_w(h,\Lambda h),d_w(h,\Psi g),d_w(g,\Lambda h)\right).\hfill \end{array}$$

It is now obvious that the remaining conditions of Theorem 3 also exist. Hence, Theorem 3 ensures that Ψ and Λ have at least one common fixed point in W.

If $\Psi$ and $\Lambda$ are single-valued mappings, then it is easy to present the following theorem, followed by Theorem 3.

Theorem 4.

Let $\Psi ,\Lambda :W\to W$ be two mappings on a complete metric space $(W,d_w)$ equipped with a binary relation $R_W$ such that

$$d_w(\Psi g,\Lambda h)\le \delta \left(d_w(g,h),d_w(g,\Psi g),d_w(h,\Lambda h),d_w(h,\Psi g),d_w(g,\Lambda h)\right)$$

(17)

for all $g,h\in W$ with $(g,h)\in R_W$ or $(h,g)\in R_W$, where $\delta \in {\Delta }_C$. Further, consider the following axioms exist:

(i) 

There is a $w_0\in W$ with $(w_0,\Psi w_0)\in R_W$, or $(w_0,\Lambda w_0)\in R_W$.

(ii) 

$(\Psi ,\Lambda )$ is relation-admissible, which means that, for each $g\in W$ with $(g,\Psi g)\in R_W$, we have $(\Psi g,\Lambda (\Psi g))\in R_W$; similarly, for each $g\in W$ with $(g,\Lambda g)\in R_W$, we have $(\Lambda g,\Psi (\Lambda g))\in R_W$.

(iii) 

If $\left\{w_n\right\}$ is a sequence in W that converges to $w\in W$ and $(w_n,w_{n+1})\in R_W$$\forall n\in \mathbb{N}$, then $(w_n,w)\in R_W$$\forall n\in \mathbb{N}$.

Then, Ψ and Λ have a common fixed point in W; that is, there is a $w^{\ast }\in W$ with $w^{\ast }=\Psi w^{\ast }$ and $w^{\ast }=\Lambda w^{\ast }$.

Remark 2.

Theorem 4 provides an extension of the result of Alam and Imdad [10] and ensures the existence of common fixed points for two single-valued mappings, whereas Theorem 2 and Theorem 3 provide the extensions of the result of Alam and Imdad [10] in the case of multivalued mappings.

2.3. Consequences

The following theorem is a direct consequence of Theorem 2 and can be viewed as an extended version of Theorem 1 along with the results presented in [28,29].

Theorem 5.

Let $\Psi :W\to CL(W)$ be a mapping on a complete metric space $(W,d_w)$ and let $\beta ,\gamma :W\times W\to [0,\infty )$ be two functions such that

$$H_w(\Psi g,\Psi h)\le \delta \left(d_w(g,h),d_w(g,\Psi g),d_w(h,\Psi h),d_w(h,\Psi g),d_w(g,\Psi h)\right)$$

for all $g,h\in W$ with $\beta (g,h)\ge \gamma (g,h)$, where $\delta \in \Delta$. Further, consider the following axioms exist:

(i) 

There is a $w_0\in W$ and a $w_1\in \Psi w_0$ with $\beta (w_0,w_1)\ge \gamma (w_0,w_1)$.

(ii) 

Ψ is $(\beta ,\gamma )$-admissible; that is, for each $g\in W$ and $h\in \Psi g$ with $\beta (g,h)\ge \gamma (g,h)$, we have $\beta (h,z)\ge \gamma (h,z)$ for all $z\in \Psi h$.

(iii) 

If $\left\{w_n\right\}$ is a sequence in W that converges to $w\in W$ and $\beta (w_n,w_{n+1})\ge \gamma (w_n,w_{n+1})$$\forall n\in \mathbb{N}$, then $\beta (w_n,w)\ge \gamma (w_n,w)$$\forall n\in \mathbb{N}$.

Then, Ψ has a fixed point in W; that is, there is a $w^{\ast }\in W$ with $w^{\ast }\in \Psi w^{\ast }$.

Proof. 

It is given that $\beta ,\gamma :W\times W\to [0,\infty )$ are two functions. By using these functions, we define a binary relation $R_W$ on W such that

$$R_W=\left\{(p,q)\right|\beta (p,q)\ge \gamma (p,q)\}.$$

The axiom (i) of the theorem ensures that $R_W$ is a nonempty binary relation. It is easy to check that the conditions of Theorem 2 become true by considering the above-defined binary relation and the axioms of the given theorem. Hence, T has a fixed point in W. □

Following the above demonstration, it is trivial to mention that the following result is a consequence of Theorem 3.

Theorem 6.

Let $\Psi ,\Lambda :W\to CL(W)$ be two mappings on a complete metric space $(W,d_w)$ and let $\beta ,\gamma :W\times W\to [0,\infty )$ be two functions such that

$$H_w(\Psi g,\Lambda h)\le \delta \left(d_w(g,h),d_w(g,\Psi g),d_w(h,\Lambda h),d_w(h,\Psi g),d_w(g,\Lambda h)\right)$$

for all $g,h\in W$ with $\beta (g,h)\ge \gamma (g,h)$ or $\beta (h,g)\ge \gamma (h,g)$, where $\delta \in {\Delta }_C$. Further, consider the following axioms exist:

(i)

There are $w_0\in W$ and $w_1\in \Psi w_0$ or $w_1\in \Lambda w_0$ with $\beta (w_0,w_1)\ge \gamma (w_0,w_1)$;

(ii)

$(\Psi ,\Lambda )$ is $(\beta ,\gamma )$-admissible, that is, for each $g\in W$ and $h\in \Psi g$ with $\beta (g,h)\ge \gamma (g,h)$, we have $\beta (h,z)\ge \gamma (h,z)$ for all $z\in \Lambda h$, similarly, for each $g\in W$ and $h\in \Lambda g$ with $\beta (g,h)\ge \gamma (g,h)$, we have $\beta (h,z)\ge \gamma (h,z)$ for all $z\in \Psi h$;

(iii)

If $\left\{w_n\right\}$ is a sequence in W that converges to $w\in W$ and $\beta (w_n,w_{n+1})\ge \gamma (w_n,w_{n+1})$$\forall n\in \mathbb{N}$, then $\beta (w_n,w)\ge \gamma (w_n,w)$$\forall n\in \mathbb{N}$.

Then, Ψ and Λ have a common fixed point in W, that is, there exists $w^{\ast }\in W$ with $w^{\ast }\in \Psi w^{\ast }$ and $w^{\ast }\in \Lambda w^{\ast }$.

The following result discusses the existence of a fixed point for multivalued mappings on a complete metric space endowed with a directed graph G. Note that $G=(V;E)$ is a directed graph such that vertex set $V=W$, edge set $E\subseteq W\times W$, and $\{(w,w):w\in W\}\subseteq E$. Furthermore, no parallel edge exists in G. This approach was introduced by Jachymski [30].

Theorem 7.

Let $\Psi :W\to CL(W)$ be a mapping on a complete metric space $(W,d_w)$ endowed with the graph G such that for all $g,h\in W$ with $(g,h)\in E$, we have

$$H_w(\Psi g,\Psi h)\le \delta \left(d_w(g,h),d_w(g,\Psi g),d_w(h,\Psi h),d_w(h,\Psi g),d_w(g,\Psi h)\right)$$

where $\delta \in \Delta$. Further, consider the following axioms exist:

(i) 

There is a $w_0\in W$ and a $w_1\in \Psi w_0$ with $(w_0,w_1)\in E$.

(ii) 

Ψ is E-admissible; that is, for each $g\in W$ and $h\in \Psi g$ with $(g,h)\in E$, we have $(h,z)\in E$ for all $z\in \Psi h$.

(iii) 

If $\left\{w_n\right\}$ is a sequence in W that converges to $w\in W$ and $(w_n,w_{n+1})\in E$$\forall n\in \mathbb{N}$, then $(w_n,w)\in E$$\forall n\in \mathbb{N}$.

Then, Ψ has a fixed point in W; that is, there is a $w^{\ast }\in W$ with $w^{\ast }\in \Psi w^{\ast }$.

This result immediately follows from Theorem 2 by using the edge set to define a binary relation $R_W$ on W such that

$$R_W=\left\{(p,q)\right|(p,q)\in E\}.$$

Remark 3.

The above result extends the result of [31] to a metric space endowed with the G. Similarly, Theorem 2 extends the result of [31] to a metric space endowed with a binary relation, and Theorem 3 provides an extension for common fixed points. Note that Theorem 1 does not extend the result presented in [31] that involves a set-valued quasi-type contraction.

3. Application to a System of Integral Equations

This section focuses on proving the existence of a solution for a system of integral equations via the obtained common fixed point result. We start the discussion by providing the required information and notations. Let $W=C([a,b],\mathbb{R})$ be the set of all continuous real-valued functions on $[a,b]$, and $d_w:W\times W\to [0,\infty )$ be defined by

$$d_w(p,q)={∥p-q∥}_{\infty }\forall p,q\in W.$$

It is obvious that $(W,d_w)$ is a complete metric space.

Consider the following system of integral equations:

$$\begin{array}{c}\hfill \left\{\begin{array}{c}p(t)=h(t)+{\int }_a^{b}M(t,j)f(j,p(j))dj\hfill \\ p(t)=h(t)+{\int }_a^{b}M(t,j)g(j,p(j))dj\hfill \end{array}\right.\end{array}$$

(18)

where $f,g:[a,b]\times \mathbb{R}\to \mathbb{R}$ and $h:[a,b]\to \mathbb{R}$ are continuous functions and $M:[a,b]\times [a,b]\to [0,\infty )$ is a function such that $M(t,·)\in L^1([a,b])$ for each $t\in [a,b]$.

We now present the following existence theorem for the system of the integral Equation (18).

Theorem 8.

Consider $W=C([a,b],\mathbb{R})$ and consider $\Psi ,\Lambda :W\to W$ be the following defined operators:

$$\begin{array}{c}\hfill \Psi (p)(t)=h(t)+{\int }_a^{b}M(t,j)f(j,p(j))dj\end{array}$$

(19)

$$\begin{array}{c}\hfill \Lambda (p)(t)=h(t)+{\int }_a^{b}M(t,j)g(j,p(j))dj\end{array}$$

(20)

where $f,g:[a,b]\times \mathbb{R}\to \mathbb{R}$ and $h:[a,b]\to \mathbb{R}$ are continuous functions and $M:[a,b]\times [a,b]\to [0,\infty )$ is a function such that $M(t,·)\in L^1([a,b])$ for each $t\in [a,b]$. Further, consider that the following axioms hold:

(i) 

$R_W$ is a nonempty binary relation on W.

(ii) 

There is a $\beta :W\times W\to [0,\infty )$ and $l\in (0,1)$ such that, for every $j\in [a,b]$, we have

$$|f(j,p(j))-g(j,q(j))|\le \beta (p,q)|p(j)-q(j)|\forall p,q\in Wwith(p,q)\in R_W$$

and

$${∥{\int }_a^{b}M(t,j)\beta (p,q)dj∥}_{\infty }<l.$$

(iii) 

For each $p\in W$ with $(p,\Psi p)\in R_W$, we have $(\Psi p,\Lambda (\Psi p))\in R_W$; similarly, for each $p\in W$ with $(p,\Lambda p)\in R_W$, we have $(\Lambda p,\Psi (\Lambda p))\in R_W$.

(iv) 

There is a $p_0\in W$ with $(p_0,\Psi (p_0))\in R_W$, or $(p_0,\Lambda (p_0))\in R_W$.

(v) 

If $\left\{w_n\right\}$ is a sequence in W that converges to $w\in W$ and $(w_n,w_{n+1})\in R_W$$\forall n\in \mathbb{N}$, then $(w_n,w)\in R_W$$\forall n\in \mathbb{N}$.

Then the system of integral Equation (18) has a solution in W.

Proof. 

It is understood that a common fixed point of the integral operators (19) and (20) is a solution of (18). By axiom $(ii)$, for each $p,q\in W$ with $(p,q)\in R_W$, we obtain

$$\begin{array}{ccc}\hfill |\Psi (p)(t)-\Lambda (q)(t)|& =& \left|{\int }_a^{b}M(t,j)[f(j,p(j))-g(j,q(j))]dj\right|\hfill \\ & \le & {\int }_a^{b}M(t,j)|f(j,p(j))-g(j,q(j))|dj\hfill \\ & \le & {\int }_a^{b}M(t,j)\beta (p,q)|p(j)-q(j)|dj\hfill \\ & \le & {∥p-q∥}_{\infty }{\int }_a^{b}M(t,j)\beta (p,q)dj.\hfill \end{array}$$

This gives

$$\begin{array}{c}\hfill {∥\Psi (p)-\Lambda (q)∥}_{\infty }\le {∥p-q∥}_{\infty }{∥{\int }_a^{b}M(t,j)\beta (p,q)dj∥}_{\infty }.\end{array}$$

Hence, we write

$$\begin{array}{c}\hfill d_w(\Psi (p),\Lambda (q))\le l{d}_w(p,q)\forall p,q\in W\mathrm{with}(p,q)\in R_W.\end{array}$$

This ensures that (17) of Theorem 4 holds with $\delta (g_1,g_2,g_3,g_4,g_5)=l{g}_1$. The remaining conditions of Theorem 4 are immediately followed by the given assumptions of the result. Hence, the operators $\Psi$ and $\Lambda$ have a common fixed point; that is, there is a solution of the system of integral Equation (18) in W. □

4. Conclusions

The article presents a class of auxiliary functions, called the $\Delta$-family, which simplifies the complexity of the previously defined $\Phi$-family. Through the newly defined auxiliary functions, the authors presented implicit-type contraction inequalities for set-valued mappings on a metric space equipped with a binary relation and studied the existence of fixed points and common fixed points for such mappings. By defining a particular binary relation, another type of result is also achieved as a consequence of the results. The practical application related to one of the derived theoretical results is also demonstrated in a system of integral equations.