Coincidence Points of a Sequence of Multivalued Mappings in Metric Space with a Graph

Abstract

In this article the coincidence points of a self map and a sequence of multivalued maps are found in the settings of complete metric space endowed with a graph. A novel result of Asrifa and Vetrivel is generalized and as an application we obtain an existence theorem for a special type of fractional integral equation. Moreover, we establish a result on the convergence of successive approximation of a system of Bernstein operators on a Banach space.

1. Introduction and Preliminaries

For the metric space $(X,d),$using the notions of Nadler [1] and Hu [2], denote $CB(X),C\left(X\right)$ and $2^X$ by the collection of nonempty closed and bounded, compact and all nonempty subsets of X respectively. Consider $A,B\in CB(X)$ the distance between sets A and B is defined by $d(A,B)={\displaystyle \underset{x\in A,y\in B}{inf}}d(x,y),$ which does not allow to enjoy the properties of metric on $CB\left(X\right)$ therefore a well known idea of Hausdorff–Pompeiu distance H on $CB(X)$induced by d is used to define a metric on $CB\left(X\right)$ as follows:

$$H(A,B)=inf\{ϵ>0:A\subseteq N(ϵ,B),B\subseteq N(ϵ,A)\},$$

where:

$$N(ϵ,A)=\{x\in X:d(x,a)<ϵ,\mathrm{for}\mathrm{some}a\in A\}.$$

In 1969, Nadler [1] proved fixed point results for multivalued mappings in complete metric spaces, using the Hausdorff distance H, which was the generalization of Banach contraction principle in the settings of set-valued mappings. Covitz and Nadler [3] extended the idea of multivalued mappings in the generalized metric spaces. Reich [4] in 1972 published a fixed point result for the multivalued maps on the compact subsets of a complete metric space and posed the question, “can $C\left(X\right)$ be replaced by $CB\left(X\right)$?”. In 1989, Mizoguchi and Takahashi answered this question in Theorem 5 of [5] and they also provide some Caristi type theorems for multivalued operators. Whereas Hu [2] in 1980 extended the multivalued fixed point results from complete metric space to complete $\epsilon$-chainable metric space. Azam and Arshad [6] have extended the Theorem 6 of [1] by finding the fixed points of a sequence of locally contractive multivalued maps in $\epsilon$-chainable metric space. Further Feng and Liu [7] used the concept of lower semi-continuity and a generalized contractive condition to extend the result of Nadler [1] and Caristi type theorems as defined in [5]. For more references the readers are referred to the work of Ciric [8], Klim and Wardowski [9,10] , Nicolae [11].

Jachymski [12] in 2007 unified and extended the work of Nieto [13] and Ran and Reuring [14] by defining a new class of contractions (G-contraction ) on metric space$\left(X,d\right)$ endowed with a graph. The connectivity of the graph brings more attractions regarding a necessary and sufficient condition for any G-contractive operator to be a Picard operator.

In the present article, fascinated by [6] the existence of coincidence points of a sequence of multivalued maps with a self map are taken into account with a generalized form of G-contraction. This provides a new way to generalize many existing results in the literature (see [1,6] and the references therein).

Let us recall some definitions from graph theory with the perspective of using them in our ideas and results. For a metric space $(X,d)$ let Δ be the diagonal of the Cartesian product $X\times X.$ Consider a directed graph$G$ such that $X=V(G),$ where $V(G)$ is the set of vertices of $G.$ The set $E(G)$ of edges of G contains all the loops. If G has no parallel edge then we can identify G with the pair $\left(V(G),E(G)\right).$ Further, the graph G can be dealt with as a weighted graph if each edge is assigned by the distance between its edges.

Consider a directed graph $G,$ then $G^{-1}$ denote the graph obtained from G by reversing the direction of edges and if we ignore the direction of edges in graph G we get an undirected graph $\tilde{G}$. The pair $\left(V^{\prime },E^{\prime }\right)$ is said to be a subgraph of G if $V^{\prime }\subseteq V\left(G\right)$ and $E^{\prime }\subseteq E\left(G\right)$ and for any edge $\left(a,b\right)\in E^{\prime }$ for all $a,b\in V^{\prime }.$

Recall some fundamental definitions regarding the connectivity of graphs, which can be found in [15].

Definition 1.

A path in G from the vertex p to q of length K, is a sequence $\left\{p_i\right\}$ of $K+1$ vertices such that $p_0=p,$...,$p_K=q$ and $(p_{j-1},p_j)\in E(G)$ for $j=1,2,...,K.$

Definition 2.

A graph G is called connected if there is a path between any two vertices. Graph G is weakly connected if $\tilde{G}$ is connected.

Definition 3.

For $a,b$ and c in $V\left(G\right),$${\left[a\right]}_G$ denote the equivalence class of the relation ∼ defined on $V\left(G\right)$ by the rule:

$$b\sim c\mathit{if}\mathit{there}\mathit{is}a\mathit{path}\mathit{in}G\mathit{from}b\mathit{to}c.$$

For $v\in V(G)$ and $K\in \mathbb{N}\cup \{0\}$ by ${[v]}_G^K$ we denote the set

$${[v]}_G^K:=\{u\in V(G):thereisapathoflengthKfromvtou\}.$$

Following is the definition of G-contraction by Jachymski [12].

Definition 4.

[12] Let $(X,d)$ be a metric space endowed with a graph G. We say that a mapping $T:X\to X$ is a G-contraction if T preserves edges of G i.e.,

$$\underset{x,y\in X}{\forall }(x,y)\in E(G)\Rightarrow (Tx,Ty)\in E(G),$$

and there exists some $\alpha \in [0,1)$ such that:

$$\underset{x,y\in X}{\forall }(x,y)\in E(G)\Rightarrow d(Tx,Ty)\le \alpha d(x,y).$$

Mizoguchi and Takahashi [5] had defined a $MT-$function as follows:

Definition 5.

[16] A function φ: $[0,+\infty )\to [0,1)$ is said to be a $MT-$function if it satisfies Mizoguchi and Takahashi’s condition (i.e., $lim{\displaystyle \underset{r\to t^{+}}{sup}}\phi (r)<1$ for all $t\in [0,+\infty )$). Clearly, if φ: $[0,+\infty )\to [0,1)$ is a nondecreasing function or a nonincreasing function, then it is a $MT-$function.

Now we state some results from the basic theory of multivalued mappings.

Lemma 1.

[17] Let $(X,d)$ be a metric space and $A,B\in CB(X),$ with $H(A,B)<ϵ$, then for each $a\in A$, there exists an element $b\in B$ such that:

$$d(a,b)<ϵ.$$

Lemma 2.

[18] Let $(X,d)$be a metric space and$A,B\in CB(X)$, then for each $a\in A$:

$$d(a,B)\le H(A,B).$$

Lemma 3.

[19] Let $\{A_n\}$ be a sequence in $CB(X)$ and there exists $A\in CB(X)$such that ${\displaystyle \underset{n\to \infty }{lim}}H(A_n,A)\to 0.$ If$x_n\in A_n(n=1,2,3,...)$ and there exists $x\in X$ such that ${\displaystyle \underset{n\to \infty }{lim}}d(x_n,x)\to 0$then $x\in A.$

2. Main Results

Definition 6.

[20] A multivalued mapping $F:X\to CB\left(X\right)$ is said to be Mizoguchi-Takahashi G-contraction if for all $x,$ y in $X,$$x\ne y$ with $\left(x,y\right)\in E\left(G\right):$

(i) 

$H\left(F\left(x\right),F\left(y\right)\right)\le \phi \left(d\left(x,y\right)\right)d\left(x,y\right);$

(ii) 

If $u\in F\left(x\right)$ and $v\in F\left(y\right)$ are such that $d\left(u,v\right)\le d\left(x,y\right)$, then $\left(u,v\right)\in E\left(G\right)$.

Motivated by the Definition 2.1 of [20], in a more general settings, we define the sequence of multivalued $G_f$-contraction as follows:

Definition 7.

Let $f:X\to X$ be a edge preserving surjection . A sequence of multivalued mappings ${\{T_q\}}_{q=1}^{\infty }$ from X into $CB(X)$ is said to be sequence of multivalued $G_f$-contraction if $(fu,fv)\in E(G),$implies:

$$H(T_q(u),T_r(v))\le \mu (d(fu,fv))d(fu,fv),\mathit{for}\mathit{all}q,r\in \mathbb{N}.$$

(1)

For $x\in T_q(u)$ and $y\in T_r(v)$ satisfying $d(fx,fy)\le d(fu,fv)$ implies $\left(fx,fy\right)\in E(G),$ where μ: $[0,\infty )\to [0,1)$ is a $MT$-function.

The next theorem provides the way to find the coincidence of a self map and a sequence of multivalued maps.

Theorem 1.

Let $(X,d)$ a complete metric space,${\{T_q\}}_{q=1}^{\infty }$ a sequence of multivalued $G_f$-contraction from X into $CB(X)$ and $f:X\to X$ a surjection. If there exist $m\in \mathbb{N}$ and $v_0\in X,$ such that:

(i) 

$T_1(v_0)\cap {[f{v}_0]}_G^m\ne \varphi ;$

(ii) 

For any sequence $\left\{v_n\right\}$ in X, if $v_n\to v$ and $v_n\in T_n(v_{n-1})\cap {\left[v_{n-1}\right]}_G^m$ for all $n\in \mathbb{N},$ then there exists a subsequence $\left\{v_{n_k}\right\}$ such that $\left(v_{n_k},v\right)\in E(G)$ for all $k\in \mathbb{N}.$

Then f and sequence of mappings ${\{T_q\}}_{q=1}^{\infty }$ have a coincidence point, i.e., there exists $v^{*}\in X$ such that $f{v}^{*}\in {\displaystyle \bigcap _{q\in \mathbb{N}}}{T}_q(v^{*})$.

Proof. 

Choose any $v_1\in X$ such that $f{v}_1\in T_1(v_0)\cap {\left[f{v}_0\right]}_G^m$ then there exists a path from $f{v}_0$ to $f{v}_1,$ i.e., $f{v}_0=f{u}_0^{\left(1\right)},...f{u}_m^{\left(1\right)}=f{v}_1\in T_1(v_0),\mathrm{and}\left(f{u}_i^{\left(1\right)},f{u}_{i+1}^{\left(1\right)}\right)\in E(G)\mathrm{for}\mathrm{all}i=0,1,2,...,m-1.$ ☐

Without any loss of generality, assume that $f{u}_k^{\left(1\right)}\ne f{u}_j^{\left(1\right)}$ for each $k,j\in \{0,1,2,...,m\}$ with $k\ne j.$ Since $(f{u}_0^{\left(1\right)},f{u}_1^{\left(1\right)})\in E(G),$ so:

$$\begin{array}{ccc}\hfill H(T_1(u_0^{\left(1\right)}),T_2(u_1^{\left(1\right)}))& \le & \mu (d(f{u}_0^{\left(1\right)},f{u}_1^{\left(1\right)}))d(f{u}_0^{\left(1\right)},f{u}_1^{\left(1\right)})\hfill \\ & <& \sqrt{\mu (d(f{u}_0^{\left(1\right)},f{u}_1^{\left(1\right)}))}d(f{u}_0^{\left(1\right)},f{u}_1^{\left(1\right)})\hfill \\ & <& d(f{u}_0^{\left(1\right)},f{u}_1^{\left(1\right)})\hfill \end{array}$$

Rename $f{v}_1$ as $f{u}_0^{\left(2\right)}$. As $f{u}_0^{\left(2\right)}\in T_1(u_0^{\left(1\right)}),$ and using Lemma 1 one can find some $f{u}_1^{\left(2\right)}\in T_2(u_1^{\left(1\right)})$ such that:

$$d(f{u}_0^{\left(2\right)},f{u}_1^{\left(2\right)})<d(f{u}_0^{\left(1\right)},f{u}_1^{\left(1\right)}).$$

Since $(f{u}_1^{\left(1\right)},f{u}_2^{\left(1\right)})\in E(G),$ so:

$$\begin{array}{ccc}\hfill H(T_2(u_1^{\left(1\right)}),T_2(u_2^{\left(1\right)}))& \le & \mu (d(f{u}_1^{\left(1\right)},f{u}_2^{\left(1\right)}))d(f{u}_1^{\left(1\right)},f{u}_2^{\left(1\right)})\hfill \\ & <& d(f{u}_1^{\left(1\right)},f{u}_2^{\left(1\right)}).\hfill \end{array}$$

Similarly since $f{u}_1^{\left(2\right)}\in T_2(u_1^{\left(1\right)}),$ again using Lemma 1 one can find some $f{u}_2^{\left(2\right)}\in T_2(u_2^{\left(1\right)})$ such that:

$$d(f{u}_1^{\left(2\right)},f{u}_2^{\left(2\right)})<d(f{u}_1^{\left(1\right)},f{u}_2^{\left(1\right)}).$$

Thus we obtain $\{f{u}_0^{\left(2\right)},f{u}_1^{\left(2\right)},f{u}_2^{\left(2\right)},\cdots ,f{u}_m^{\left(2\right)}\}$ of $m+1$ vertices of X such that $f{u}_0^{\left(2\right)}\in T_1(u_0^{\left(1\right)})$ and $f{u}_s^{\left(2\right)}\in T_2(u_s^{\left(1\right)})$ for $s=1,2,\dots .,m,$ with:

$$d(f{u}_s^{\left(2\right)},f{u}_{s+1}^{\left(2\right)})<d(f{u}_s^{\left(1\right)},f{u}_{s+1}^{\left(1\right)}),$$

for $s=0,1,2,\dots .,m-1.$ As $(f{u}_s^{\left(1\right)},f{u}_{s+1}^{\left(1\right)})\in E(G)$ for all $s=0,1,2,\dots .,m-1,$ thus $(f{u}_s^{\left(2\right)},f{u}_{s+1}^{\left(2\right)})\in E(G)$ for all $s=0,1,2,\dots .,m-1.$

Let $f{u}_m^{\left(2\right)}=f{v}_2.$ Thus the set of points $f{v}_1=f{u}_0^{\left(2\right)},f{u}_1^{\left(2\right)},f{u}_2^{\left(2\right)},\cdots ,f{u}_m^{\left(2\right)}=f{v}_2\in T_2(v_1)$ is a path from $f{v}_1$ to $f{v}_2.$Rename $f{v}_2$ as $f{u}_0^{\left(3\right)}$. Then by the same procedure we obtain a path:

$$f{v}_2=f{u}_0^{\left(3\right)},f{u}_1^{\left(3\right)},f{u}_2^{\left(3\right)},\cdots ,f{u}_m^{\left(3\right)}=f{v}_3\in T_3(v_2)$$

from $f{v}_2$ to $f{v}_3$. Inductively, obtained:

$$f{v}_h=f{u}_0^{\left(h+1\right)},f{u}_1^{\left(h+1\right)},f{u}_2^{\left(h+1\right)},\cdots ,f{u}_m^{\left(h+1\right)}=f{v}_{h+1}\in T_{h+1}(v_h)$$

with:

$$d(f{u}_t^{\left(h+1\right)},f{u}_{t+1}^{\left(h+1\right)})<d(f{u}_t^{\left(h\right)},f{u}_{t+1}^{\left(h\right)}),$$

(2)

hence $(f{u}_t^{\left(h+1\right)},f{u}_{t+1}^{\left(h+1\right)})\in E(G)$ for $t=0,1,2,\dots .,m-1.$

Consequently, construct a sequence ${\{f{v}_h\}}_{h=1}^{\infty }$ of points of X with:

$$\begin{array}{ccc}\hfill f{v}_1& =& f{u}_m^{\left(1\right)}=f{u}_0^{\left(2\right)}\in T_1(v_0),\hfill \\ \hfill f{v}_2& =& f{u}_m^{\left(2\right)}=f{u}_0^{\left(3\right)}\in T_2(v_1),\hfill \\ \hfill f{v}_3& =& f{u}_m^{\left(3\right)}=f{u}_0^{\left(4\right)}\in T_3(v_2),\hfill \\ & & ⋮\hfill \\ \hfill f{v}_{h+1}& =& f{u}_m^{\left(h+1\right)}=f{u}_0^{\left(h+2\right)}\in T_{h+1}(v_h),\hfill \end{array}$$

for all $h\in \mathbb{N}.$

For each $t\in \{0,1,2,...,m-1\},$ and from $\left(2\right)$, clearly ${\{d(f{u}_t^{\left(h\right)},f{u}_{t+1}^{\left(h\right)})\}}_{h=1}^{\infty }$ is a decreasing sequence of non-negative real numbers and so there exists $a_t\ge 0$ such that:

$$\underset{h\to \infty }{lim}d(f{u}_t^{\left(h\right)},f{u}_{t+1}^{\left(h\right)})=a_t.$$

By assumption, $lim{sup}_{t\to a_t^{+}}\mu (t)<1,$ so there exists $k_t\in \mathbb{N}$ such that $\mu (d(f{u}_t^{\left(h\right)},f{u}_{t+1}^{\left(h\right)}))<\omega (a_t)$ for all $h\ge k_t$ where $lim{sup}_{t\to a_t^{+}}\mu (t)<\omega (a_t)<1.$

Now put:

$${\Theta }_t=max\left\{\underset{r=1,...,k_t}{max}\sqrt{\mu (d(f{u}_t^{\left(r\right)},f{u}_{t+1}^{\left(r\right)}))},\sqrt{\omega (a_t)}\right\}.$$

Then, for every $h>k_t,$ consider:

$$\begin{array}{ccc}\hfill d(f{u}_t^{\left(h+1\right)},f{u}_{t+1}^{\left(h+1\right)})& <& \sqrt{\mu (d(f{u}_t^{\left(h\right)},f{u}_{t+1}^{\left(h\right)}))}d(f{u}_t^{\left(h\right)},f{u}_{t+1}^{\left(h\right)})\hfill \\ & <& \sqrt{\omega (a_t)}d(f{u}_t^{\left(h\right)},f{u}_{t+1}^{\left(h\right)})\hfill \\ & \le & {\Theta }_{t}d(f{u}_t^{\left(h\right)},f{u}_{t+1}^{\left(h\right)})\hfill \\ & \le & {({\Theta }_t)}^{2}d(f{u}_t^{\left(h-1\right)},f{u}_{t+1}^{\left(h-1\right)})\hfill \\ & \le & ...\hfill \\ & \le & {({\Theta }_t)}^{h}d(f{u}_t^{\left(1\right)},f{u}_{t+1}^{\left(1\right)}).\hfill \end{array}$$

Putting $q=max\{k_t:t=0,1,2,...,m-1\},$ gives:

$$\begin{array}{ccc}\hfill d(f{v}_h,f{v}_{h+1})& =& d(f{u}_0^{\left(h+1\right)},f{u}_m^{\left(h+1\right)})\hfill \\ & \le & {\displaystyle \sum }_{t=0}^{m-1}d(f{u}_t^{\left(h+1\right)},f{u}_{t+1}^{\left(h+1\right)})\hfill \\ & <& {\displaystyle \sum }_{t=0}^{m-1}{({\Theta }_t)}^{h}d(f{u}_t^{\left(1\right)},f{u}_{t+1}^{\left(1\right)}),\mathrm{for}\mathrm{all}h>q.\hfill \end{array}$$

Now for $p>h>q,$ consider:

$$\begin{array}{ccc}d(f{v}_h,f{v}_p)\hfill & \le \hfill & d(f{v}_h,f{v}_{h+1})+d(f{v}_{h+1},f{v}_{h+2})+\cdots +d(f{v}_{p-1},f{v}_p)\hfill \\ & <\hfill & {\displaystyle {\displaystyle \sum }_{t=0}^{m-1}}{\left({\Theta }_t\right)}^{h}d(f{u}_t^{\left(1\right)},f{u}_{t+1}^{\left(1\right)})+\cdots +{\displaystyle {\displaystyle \sum }_{t=0}^{m-1}}{\left({\Theta }_t\right)}^{p-1}d(f{u}_t^{\left(1\right)},f{u}_{t+1}^{\left(1\right)}).\hfill \end{array}$$

(3)

Since ${\Theta }_t<1$ for all $t\in \{0,1,2,...,m-1\},$ it follows that $\{f{v}_h=f{u}_m^{\left(h\right)}\}$ is a Cauchy sequence. Using completeness of X, find $v^{*}\in X$ such that $f{v}_h\to f{v}^{*}$. Now using the fact that $f{v}_n\in T\left(v_{n-1}\right)\cap {\left[f{v}_{n-1}\right]}_G^m$ for all $n\in \mathbb{N},$ find a subsequence $\left\{f{v}_{n_k}\right\}$ of $\left\{f{v}_h\right\}$such that $\left(f{v}_{n_k},f{v}^{*}\right)\in E(G)$ for all $k\in \mathbb{N}.$ Now for any $q\in \mathbb{N}:$

$$\begin{array}{ccc}\hfill d\left(f{v}^{*},T_q(v^{*})\right)& \le & d\left(f{v}^{*},f{v}_{h+1}\right)+d\left(f{v}_{h+1},T_q(v^{*})\right)\hfill \\ & \le & d\left(f{v}^{*},f{v}_{h+1}\right)+H\left(T_{h+1}(v_h),T_q(v^{*})\right)\hfill \\ & \le & d\left(f{v}^{*},f{v}_{h+1}\right)+\mu \left(d\left(f{v}_h,f{v}^{*}\right)\right)d\left(f{v}_h,f{v}^{*}\right).\hfill \end{array}$$

Letting $h\to \infty$ in the above inequality, gives $d\left(f{v}^{*},T_q(v^{*})\right)\to 0,$ which implies $f{v}^{*}\in T_q(v^{*})$ for all $q\in \mathbb{N}$. Hence, $f{v}^{*}\in {\displaystyle \bigcap _{q\in \mathbb{N}}}{T}_q(v^{*})$ as required.

Example 1.

Let $X=\left\{0\right\}\cup \left\{\frac{1}{q^n}:n\in \mathbb{N}\cup \left\{0\right\}\right\}$ for $q\in \mathbb{N}.$ Consider the graph G such that $V\left(G\right)=X$ and for all x and y in $X:$

$$E\left(G\right)=\left\{\left(x,y\right):x\ne y\right\}.$$

For $q\in \mathbb{N},$ let $T_q:X\to CB(X)$ be defined by:

$$T_q\left(x\right)=\left\{\begin{array}{cc}\left\{0,\frac{1}{q}+1,1\right\}& ifx=0,\\ \left\{\frac{1}{q^{n+1}}+1,1\right\}& ifx=\frac{1}{q^n},n\in \mathbb{N},\\ \left\{\frac{1}{q}+1\right\}& ifx=1.\end{array}\right.$$

If we assume $f:X\to X$ as an identity map then sequence of multivalued mappings ${\{T_q\}}_{q=1}^{\infty }$ from X into $CB(X)$ is a sequence of multivalued $G_f$-contraction.

It satisfies the conditions of Theorem 1 and $1\in X$ is the fixed point of sequence of multivalued maps $T_q$ for $q\in \mathbb{N}.$

The next theorem provides a way to find the coincidence point of a hybrid pair.

Theorem 2.

Let $(X,d)$ be a complete metric space, $T:X\to CB(X)$ and $f:X\to X$ a surjection. If $u,v\in X($with $u\ne v)$ such that $(fu,fv)\in E(G),$implies:

$$H(T(u),T(v))\le \mu (d(fu,fv))d(fu,fv),$$

(4)

where $\mu :[0,\infty )\to [0,1)$ is a MT-function, if there exist $m\in \mathbb{N}$ and $v_0\in X,$ such that:

(i) 

$T(v_0)\cap {[f{v}_0]}_G^m\ne \varphi ;$

(ii) 

For any sequence $\left\{v_n\right\}$ in X, if $v_n\to v$ and $v_n\in T(v_{n-1})\cap {\left[v_{n-1}\right]}_G^m$ for all $n\in \mathbb{N}$ and $j=1,2,...,$ then there exists a subsequence $\left\{v_{n_k}\right\}$ such that $\left(v_{n_k},v\right)\in E(G)$ for all $k\in \mathbb{N}.$

Then f and T have a coincidence point, i.e., there exists $v^{*}\in X$ such that $f{v}^{*}\in T(v^{*})$.

Proof. 

Take $T_q:=T$ for all $q\in \mathbb{N}$ in Theorem 1 and proof is following the same procedure. ☐

Corollary 1.

Let $(X,d)$ be a complete metric space,${\{T_q\}}_{q=1}^{\infty }$ a sequence of the self mappings on X and $f:X\to X$ a surjection. If $u,v\in X($with $u\ne v)$ such that $(fu,fv)\in E(G),$implies:

$$d(T_q(u),T_r(v))\le \mu (d(fu,fv))d(fu,fv),$$

(5)

for all $q,r\in \mathbb{N},$ where $\mu :[0,\infty )\to [0,1)$ is a MT function, if there exist $m\in \mathbb{N}$ and $v_0\in X,$ such that:

(i) 

$T_1(v_0)\cap {[f{v}_0]}_G^m\ne \varphi ;$

(ii) 

For any sequence $\left\{v_n\right\}$ in X, if $v_n\to v$ and $v_n=T_n(v_{n-1})\cap {\left[v_{n-1}\right]}_G^m$ for all $n\in \mathbb{N},$

then there exists a subsequence $\left\{v_{n_k}\right\}$ such that $\left(v_{n_k},v\right)\in E(G)$ for all $k\in \mathbb{N}.$

Then f and sequence of mappings ${\{T_q\}}_{q=1}^{\infty }$ have a coincidence point, i.e., there exists $v^{*}\in X$ such that $f{v}^{*}={\displaystyle \bigcap _{q\in \mathbb{N}}}{T}_q(v^{*})$.

Corollary 2.

Let $(X,d)$ be a complete metric space, $T:X\to CB(X)$ and if $u,v\in X($with $u\ne v)$ such that $(u,v)\in E(G),$implies:

$$H(T(u),T(v))\le \mu (d(u,v))d(u,v),$$

(6)

where μ: $[0,\infty )\to [0,1)$ is a MT-function, if there exist $m\in \mathbb{N}$ and $v_0\in X,$ such that:

(i) 

$T(v_0)\cap {\left[v_0\right]}_G^m\ne \varphi ;$

(ii) 

For any sequence $\left\{v_n\right\}$ in X, if $v_n\to v$ and $v_n\in T(v_{n-1})\cap {\left[v_{n-1}\right]}_G^m$ for all $n\in \mathbb{N}$ and $j=1,2,...,$ then there exists a subsequence $\left\{v_{n_k}\right\}$ such that $\left(v_{n_k},v\right)\in E(G)$ for all $k\in \mathbb{N}.$

Then T has a fixed point, i.e., $v^{*}=T(v^{*})$.

The following are the consequence of the Theorem 1 and Theorem 2 for the case of self mappings.

Corollary 3.

Let $(X,d)$ be a complete metric space, $T:X\to X$ and $f:X\to X$ a surjection. If $u,v\in X($with $u\ne v)$ such that $(fu,fv)\in E(G),$implies:

$$d(T(u),T(v))\le \mu (d(fu,fv))d(fu,fv),$$

(7)

where $\mu :[0,\infty )\to [0,1)$ is a MT function, if there exist $m\in \mathbb{N}$ and $v_0\in X,$ such that:

(i) 

$T(v_0)\cap {[f{v}_0]}_G^m\ne \varphi ;$

(ii) 

For any sequence $\left\{v_n\right\}$ in X, if $v_n\to v$ and $v_n=T(v_{n-1})\cap {\left[v_{n-1}\right]}_G^m$ for all $n\in \mathbb{N}$ and $j=1,2,...,$ then there exists a subsequence $\left\{v_{n_k}\right\}$ such that $\left(v_{n_k},v\right)\in E(G)$ for all $k\in \mathbb{N}.$

Then f and T have a coincidence point, i.e., there exists $v^{*}\in X$ such that $f{v}^{*}=T(v^{*})$.

Corollary 4.

Let $(X,d)$ be a complete metric space, $T:X\to X$ and if $u,v\in X($with $u\ne v)$ such that $(u,v)\in E(G),$implies:

$$d(T(u),T(v))\le \mu (d(u,v))d(u,v),$$

(8)

where $\mu :[0,\infty )\to [0,1)$ is a MT-function, if there exist $m\in \mathbb{N}$ and $v_0\in X,$ such that:

(i) 

$T(v_0)\cap {\left[v_0\right]}_G^m\ne \varphi ;$

(ii) 

For any sequence $\left\{v_n\right\}$ in X, if $v_n\to v$ and $v_n=T(v_{n-1})\cap {\left[v_{n-1}\right]}_G^m$ for all $n\in \mathbb{N}$ and $j=1,2,...,$ then there exists a subsequence $\left\{v_{n_k}\right\}$ such that $\left(v_{n_k},v\right)\in E(G)$ for all $k\in \mathbb{N}.$

Then T has a fixed point, i.e., $v^{*}=T(v^{*})$.

The next remark highlights the applications of all the above results in settings of complete metric spaces, complete metric spaces endowed with partial order and $\epsilon$-chainable complete metric spaces.

Remark 1.

Consider the following cases:

R1. 

Let $\left(X,d\right)$ be a complete metric space, consider the graph $G_0$ with:

$$E\left(G_0\right)=X\times X.$$

R2. 

Let $\left(X,d\right)$ be a complete metric space with partial order ⪯ on $X,$ consider the graphs $G_1$ and $G_2$ with:

$$E\left(G_1\right)=\left\{\left(x,y\right)\in X\times X:x⪯y\right\},$$

and:

$$E\left(G_2\right)=\left\{\left(x,y\right)\in X\times X:x⪯yory⪯x\right\}.$$

R3. 

Let $\epsilon >0$ and $\left(X,d\right)$ be a complete ε-chainable metric space, consider the graph:

$$G_3:=\left\{\left(x,y\right)\in X\times X:0<d\left(x,y\right)<\epsilon ,\mathrm{for}\epsilon >0\right\}.$$

We remark that all above results are valid under the above construction of remarks $\left(R1\right),\left(R2\right)$ and $\left(R3\right).$

Further, in an application of Theorem 1 we generalize the Theorem 6 of [20]. It establishes the convergence of successive approximations of operators on a Banach space, which consequently yields the Kelisky-Rivlin theorem on iterates of Bernstein operators on the space $C\left(I\right)$, where I is the closed unit interval.

Theorem 3.

Let X be a Banach space and $X_0$ be a closed subspace of X. Let $T,f:X\to X$ be maps such that f is surjection and:

$$∥Tx-Ty∥\le \phi \left(∥fx-fy∥\right)∥fx-fy∥\mathrm{whenever}fx-fy\in X_0,x\ne y.$$

(9)

If $\left(I-f\right)\left(X\right)\subseteq X_0$ and $\left(f-T\right)(X)\subseteq X_0,$ then for all $x\in X$, $\left\{T^{n}x\right\}$ converges to $Coin\left\{T,f\right\},$ where $Coin\left\{T,f\right\}=\left\{x\in X:Tx=fx\right\}.$

Proof. 

Consider the graph $G=\left(V\left(G\right),E\left(G\right)\right)$ where $V\left(G\right)=X$ and $E(G)=\left\{\left(x,y\right)\in X\times X:x-y\in X_0\right\}.$ Clearly, $△\subseteq E(G),$$\tilde{G}=G$ and G has no parallel edges. Consider $\left(x,y\right)\in E\left(G\right),$ then $fx-fy=\left(y-fy\right)-\left(x-fx\right)+(x-y)\in X_0,$ since $\left(I-f\right)\left(X\right)\subseteq X_0.$ Hence and by given contractive condition $\left(9\right)$, we see that $\forall \left(x,y\right)\in E\left(G\right)$ with $x\ne y,$$\left(6\right)$ holds. Also $Tx-Ty=\left(fy-Ty\right)-\left(fx-Tx\right)+(fx-fy)\in X_0,$ since $\left(f-T\right)\left(X\right)\subseteq X_0.$

The use of $\left(f-T\right)\left(X\right)\subseteq X_0,$implies that $\left(fx,Tx\right)\in E\left(G\right)$ for x in $X.$ Therefore condition $\left(i\right)$ of Corollary 4 holds with $x=v_0=x_0$ and $N=1.$ Thus we are able to generate a sequence such that $T{x}_{n-1}=f{x}_n$ for all $n\in \mathbb{N}.$ Assume that $T{x}_n\to v^{*}\in X$ but since f is surjection so there exists some v in X such that $v^{*}=fv.$ Here also $T{x}_n\in {\left[T{x}_{n-1}\right]}_G^1$ for all $n\in \mathbb{N},$ which implies that $\left(T{x}_n,T{x}_{n-1}\right)\in E\left(G\right)$ for all $n\in \mathbb{N}.$ Now using the outline of the proof of Theorem $4.1$ of [12], $\left(T{x}_n,fv\right)\in E\left(G\right)$ for all $n\in \mathbb{N}.$ Now assume:

$$\begin{array}{ccc}∥fv-Tv∥\hfill & \le \hfill & ∥fv-f{x}_{n+1}∥+∥f{x}_{n+1}-Tv∥\hfill \\ & =\hfill & ∥fv-f{x}_{n+1}∥+∥T{x}_n-Tv∥.\hfill \end{array}$$

(10)

Since$\left(T{x}_n,fv\right)\in E\left(G\right)$ for all $n\in \mathbb{N},$ thus from $\left(9\right)$ and $\left(10\right)$we have:

$$∥fv-Tv∥\le ∥fv-f{x}_{n+1}∥+\phi \left(∥f{x}_n-fv∥\right)∥f{x}_n-fv∥.$$

As $n\to \infty ,$ we get $fv=Tv.$ Thus v is the coincidence point of f and $T,$ by using Corollary 4. For the uniqueness of the coincidence point we let two coincidence points u, v of f and $T,$ then:

$$\begin{array}{ccc}\hfill ∥Tu-Tv∥& \le & \phi \left(∥fu-fv∥\right)∥fu-fv∥\hfill \\ \hfill \left(1-\phi \left(∥fu-fv∥\right)\right)∥Tu-Tv∥& \le & 0.\hfill \end{array}$$

This implies that $Tu=Tv.$ ☐

In the next result, we discussed the generalization of fractional differential equation described in [21]. For the closed interval $I=\left[0,1\right],$ assume function $g\in C\left(I,\mathbb{R}\right)$ and $f:I\times \mathbb{R}\to \mathbb{R}$ is a continuous function. The fractional differential equation is given as follows:

$$D^{\alpha }x\left(t\right)+f\left(t,g\left(x\left(t\right)\right)\right)=0\left(0\le t\le 1,\alpha >1\right)$$

(11)

with boundary conditions $x\left(0\right)=x\left(1\right)=0.$ It is to be noted that associated Green’s function with the problem $\left(11\right)$ is:

$$G\left(t,s\right)=\left\{\begin{array}{cc}{\left(t\left(1-s\right)\right)}^{\alpha -1}-{\left(t-s\right)}^{\alpha -1}\hfill & 0\le s\le t\le 1,\hfill \\ \frac{{\left(t\left(1-s\right)\right)}^{\alpha -1}}{\Gamma \left(\alpha \right)}\hfill & 0\le t\le s\le 1.\hfill \end{array}\right.,$$

where $\Gamma \left(.\right)$ represents the Gamma function.

Theorem 4.

Consider the surjective function $g\in C\left(I,\mathbb{R}\right)$and $f:I\times \mathbb{R}\to \mathbb{R}$ satisfies:

(i) 

$\left|\left(f\left(s,g\left(x\left(s\right)\right)\right)-f\left(s,g\left(y\left(s\right)\right)\right)\right)\right|\le \left|g\left(x\left(s\right)\right)-g\left(y\left(s\right)\right)\right|$for all $s\in I;$

(ii) 

${sup}_{t\in I}\underset{0}{\overset{1}{{\displaystyle \int }}}G\left(t,s\right)ds\le k<1.$

Then, problem $(11)$ has a unique solution.

Proof. 

Assume space $X=C\left(I,\mathbb{R}\right)$ , and we have $d\left(x,y\right)=\underset{t\in \left[0,1\right]}{max}\left|x\left(t\right)-y\left(t\right)\right|\mathrm{for}x\mathrm{and}y\mathrm{in}X.$ It is well known that $x\in X$ is a solution of $\left(11\right)$ if and only if it is a solution of the integral equation:

$$x\left(t\right)=\underset{0}{\overset{1}{{\displaystyle \int }}}G\left(t,s\right)f\left(s,\left(gx\right)\left(s\right)\right)ds\mathrm{for}\mathrm{all}t\in I.$$

Define the operator $F:X\to X$ by:

$$Fx\left(t\right)=\underset{0}{\overset{1}{{\displaystyle \int }}}G\left(t,s\right)f\left(s,\left(gx\right)\left(s\right)\right)ds\mathrm{for}\mathrm{all}t\in I,$$

and $S:X\to X$ by:

$$Sx=gx,\mathrm{with}\left(Sx\right)\left(t\right)=\left(gx\right)\left(t\right)\mathrm{for}t\in I.$$

Thus, for finding a solution of $\left(11\right)$, it is sufficient to show that F has a coincidence point with $g.$ Now let $x,y\in C\left(I\right)$ for all $s\in I.$ Here we have:

$$\begin{array}{ccc}\hfill \left|Fx\left(t\right)-Fy\left(t\right)\right|& =& \left|\underset{0}{\overset{1}{{\displaystyle \int }}}G\left(t,s\right)\left(f\left(s,\left(gx\right)\left(s\right)\right)-f\left(s,\left(gy\right)\left(s\right)\right)\right)ds\right|\hfill \\ & \le & \underset{0}{\overset{1}{{\displaystyle \int }}}G\left(t,s\right)\left|\left(f\left(s,\left(gx\right)\left(s\right)\right)-f\left(s,\left(gy\right)\left(s\right)\right)\right)\right|ds\hfill \\ & \le & \underset{0}{\overset{1}{{\displaystyle \int }}}G\left(t,s\right)\left|\left(gx\right)\left(s\right)-\left(gy\right)\left(s\right)\right|ds\hfill \\ & \le & \underset{0}{\overset{1}{{\displaystyle \int }}}G\left(t,s\right)\left|\left(Sx\right)\left(s\right)-\left(Sy\right)\left(s\right)\right|ds\hfill \\ & \le & \underset{0}{\overset{1}{{\displaystyle \int }}}G\left(t,s\right)d\left(Sx,Sy\right)ds\hfill \\ & \le & d\left(Sx,Sy\right)\underset{t\in I}{sup}\underset{0}{\overset{1}{{\displaystyle \int }}}G\left(t,s\right)ds\hfill \\ & \le & kd\left(Sx,Sy\right).\hfill \end{array}$$

This implies that for each $x,y\in X,$ we have:

$$d\left(Fx,Fy\right)\le kd\left(Sx,Sy\right).$$

Now the use of Corollary 3 with graph $G=G_0,$ we have $x^{*}\in X$ such that $F{x}^{*}=S{x}^{*}$ with $\left(S{x}^{*}\right)\left(t\right)=\left(g{x}^{*}\right)\left(t\right)$ for $t\in I.$ Thus $x^{*}$ is the required coincidence point of F and $g.$ ☐