Fixed point theorems for Reich type contractions on metric spaces with a graph

Abstract

Let $(X,d)$ be a metric space endowed with a graph $G$ such that the set $V\left(G\right)$ of vertices of $G$ coincides with $X$. We define the notion of $G$-Reich type maps and obtain a fixed point theorem for such mappings. This extends and subsumes many recent results which were obtained for other contractive type mappings on ordered metric spaces and for cyclic operators.

Introduction

Let $T$ be a selfmap of a metric space $(X,d)$. Following Petruşel and Rus [1], we say that $T$ is a Picard operator (abbr., PO) if $T$ has a unique fixed point $x^{\ast }$ and ${lim}_{n\to \infty }{T}^{n}x=x^{\ast }$ for all $x\in X$ and is called a weakly Picard operator (abbr. WPO) if the sequence ${\left(T^{n}x\right)}_{n\in \mathbb{N}}$ converges, for all $x\in X$ and the limit (which may depends on $x$) is a fixed point of $T$.

Let $(X,d)$ be a metric space. Let $\Delta$ denote the diagonal of the Cartesian product $X\times X$. Consider a directed graph $G$ such that the set $V\left(G\right)$ of its vertices coincides with $X$, and the set $E\left(G\right)$ of its edges contains all loops, i.e., $E\left(G\right)\supseteq \Delta$. We assume $G$ has no parallel edges, so we can identify $G$ with the pair $(V\left(G\right),E\left(G\right))$. Moreover, we may treat $G$ as a weighted graph (see [2, p. 309]) by assigning to each edge the distance between its vertices. By $G^{-1}$ we denote the conversion of a graph $G$, i.e., the graph obtained from $G$ by reversing the direction of edges. Thus we have

$$E\left(G^{-1}\right)=\{(x,y)\mid (y,x)\in G\}\text{.}$$

The letter $\tilde{G}$ denotes the undirected graph obtained from $G$ by ignoring the direction of edges. Actually, it will be more convenient for us to treat $\tilde{G}$ as a directed graph for which the set of its edges is symmetric. Under this convention,

$$E\left(\tilde{G}\right)=E\left(G\right)\cup E\left(G^{-1}\right)\text{.}$$

We call $(V^{\prime },E^{\prime })$ a subgraph of $G$ if $V^{\prime }\subseteq V\left(G\right),E^{\prime }\subseteq E\left(G\right)$ and, for any edge $(x,y)\in E^{\prime },x,y\in V^{\prime }$.

Now we recall a few basic notions concerning the connectivity of graphs. All of them can be found, e.g., in [2]. If $x$ and $y$ are vertices in a graph $G$, then a path in $G$ from $x$ to $y$ of length $N(N\in \mathbb{N})$ is a sequence ${\left(x_i\right)}_{i=0}^N$ of $N+1$ vertices such that $x_0=x,x_N=y$ and $(x_{n-1},x_n)\in E\left(G\right)$ for $i=1,\dots ,N$. A graph $G$ is connected if there is a path between any two vertices. $G$ is weakly connected if $\tilde{G}$ is connected. If $G$ is such that $E\left(G\right)$ is symmetric and $x$ is a vertex in $G$, then the subgraph $G_x$ consisting of all edges and vertices which are contained in some path beginning at $x$ is called the component of $G$ containing $x$. In this case $V\left(G_x\right)={\left[x\right]}_G$, where ${\left[x\right]}_G$ is the equivalence class of the following relation $R$ defined on $V\left(G\right)$ by the rule:

$$yRz\text{ if there is a path in }G\text{ from }y\text{ to }z\text{.}$$

Clearly, $G_x$ is connected.

Now we discuss some types of continuity of mappings. The first of them is well known and often used in the metric fixed point theory.

Definition 1

A mapping $T:X\to X$ is called orbitally continuous if for all $x\in X$ and any sequence ${\left(k_n\right)}_{n\in \mathbb{N}}$ of positive integers, $T^{k_n}x\to y\in X$ implies $T\left(T^{k_n}x\right)\to Ty$ as $n\to \infty$.

Definition 2

A mapping $T:X\to X$ is called orbitally G-continuous if given $x\in X$ and a sequence ${\left(x_n\right)}_{n\in \mathbb{N}}$,

$$x_n\to x\text{and}(x_n,x_{n+1})\in E\left(G\right)\text{for }n\in \mathbb{N}\text{ imply }T{x}_n\to Tx\text{.}$$

Recently, some results have appeared giving sufficient conditions for a mapping to be a PO if $(X,d)$ is endowed with a graph. The first result in this direction was given by Jachymski [3] who also presented its applications to the Kelisky–Rivlin theorem on iterates of the Bernstein operators on the space $C[0,1]$.

Definition 3 [3, Definition 2.1]

We say that a mapping $f:X\to X$ is a Banach $G$-contraction or simply a $G$-contraction if $f$ preserves edges of $G$, i.e.,

$$\forall x,y\in X((x,y)\in E\left(G\right)\Rightarrow (f\left(x\right),f\left(y\right))\in E\left(G\right))$$

and $f$ decreases weights of edges of $G$ in the following way:

$$\exists \alpha \in (0,1),\forall x,y\in X((x,y)\in E\left(G\right)\Rightarrow d(f\left(x\right),f\left(y\right))⩽\alpha d(x,y))\text{.}$$

The main result in [3] is giving by the next theorem.

Theorem 1 [3, Theorem 3.2]

Let $(X,d)$ be complete, and let the triple $(X,d,G)$ have the following property.

For any ${\left(x_n\right)}_{n\in \mathbb{N}}$ in $X$, if $x_n\to x$ and $(x_n,x_{n+1})\in E\left(G\right)$ for $n\in \mathbb{N}$then there is a subsequence .

Let $f:X\to X$ be a Banach $G$-contraction, and $X_f=\{x\in X|(x,fx)\in E\left(G\right)\}$ . Then the following statements hold.

• $\text{<mi mathvariant="italic" is="true">card Fix</mi>}f=\text{<mi mathvariant="italic" is="true">card</mi>}\left\{{\left[x\right]}_{\tilde{G}}\right|x\in X_f\}$.

• $\text{<mi mathvariant="italic" is="true">Fix</mi>}f\ne 0̸\text{<mi mathvariant="italic" is="true">iff</mi>}{X}_f\ne 0̸$.

• $f$ has a unique fixed point iff there exists $x_0\in X_f$ such that $X_f\subseteq {\left[x_0\right]}_{\tilde{G}}$.

• For any $x\in X_f,f{|}_{{\left[x\right]}_{\tilde{G}}}$ is a PO.

• If $X_f\ne 0̸$ and $G$ is weakly connected, then $f$ is a PO.

• If $X^{\prime }≔\cup \left\{{\left[x\right]}_{\tilde{G}}\right|x\in G\}$then $f{|}_{X^{\prime }}$ is a WPO.

• If $f\subseteq E\left(G\right)$, then $f$ is a WPO.

Subsequently, Bega et al. [4] extended Theorem 1 for set valued mappings.

Definition 4 [4, Definition 2.6]

Let $F:X\rightsquigarrow X$ be a set valued mapping with nonempty closed and bounded values. The mapping $F$ is said to be a $G$-contraction if there exists a $k\in (0,1)$ such that

$$D(Fx,Fy)⩽kd(x,y)\text{for all }x,y\in E\left(G\right)$$

and if $u\in Fx$ and $v\in Fy$ are such that

$$d(u,v)⩽kd(x,y)+\alpha \text{,}\text{for each }\alpha >0$$

then $(u,v)\in E\left(G\right)$.

Theorem 2 [4, Theorem 3.1]

Let $(X,d)$ be a complete metric space and suppose that the triple $(X,d,G)$ has the property.

For any ${\left(x_n\right)}_{n\in \mathbb{N}}$ in $X$, if $x_n\to x$ and $(x_n,x_{n+1})\in E\left(G\right)$ for $n\in \mathbb{N}$then there is a subsequence ${\left(x_{k_n}\right)}_{n\in \mathbb{N}}$ with $(x_{k_n},x)\in E\left(G\right)$ for $n\in \mathbb{N}$.

Let $F:X\rightsquigarrow X$ be a $G$-contraction and

Then the following statements hold.

• For any $x\in X_F,F{|}_{{\left[x\right]}_{\tilde{G}}}$ has a fixed point.

• If $X_F\ne 0̸$ and $G$ is weakly connected, then $F$ has a fixed point in $X$.

• If $X^{\prime }≔\cup \{{\left[x\right]}_{\tilde{G}}:x\in X_F\}$, then $F{|}_{X^{\prime }}$ has a fixed point.

• If $F\subseteq E\left(G\right)$then $F$ has a fixed point.

• $\mathrm{Fix}F\ne 0̸$ if and only if $X_F\ne 0̸$.

In [5], the author considered the problem of existence of a fixed point for $\phi$-contractions in metric spaces endowed with a graph.

Definition 5

Let $(X,d)$ be a metric space and $G$ a graph. The mapping $T:X\to X$ is said to be a $(G,\phi )-\mathrm{contraction}$ if:

• $\forall x,y\in X((x,y)\in E\left(G\right)\Rightarrow (Tx,Ty)\in E\left(G\right))$;

• there exists a comparison function $\phi :{\mathbb{R}}_{+}\to {\mathbb{R}}_{+}$ such that:

  $$d(Tx,Ty)⩽\phi \left(d(x,y)\right)$$

  for all $(x,y)\in E\left(G\right)$.

Using this concept the author proved in [5] the following theorem.

Theorem 3

Let $(X,d)$ be a metric space endowed with a graph $G$ and $T:X\to X$ be an operator. We suppose that:

• $G$ is weakly connected;

• for any sequence ${\left(x_n\right)}_{n\in \mathbb{N}}\subset X$ with $d(x_n,x_{n+1})\to 0$there exists $k,n_0\in \mathbb{N}$ such that $(x_{kn},x_{km})\in E\left(G\right)$ for all $m,n\in \mathbb{N}m,n⩾n_0$ ;

• $T$ is orbitally continuous

• or

• $T$ is orbitally $G$-continuous and there exists a subsequence ${\left(T^{n_k}{x}_0\right)}_{k\in \mathbb{N}}$ of ${\left(T^n{x}_0\right)}_{n\in \mathbb{N}}$ such that $(T^{n_k}{x}_0,x^{\ast })\in E\left(G\right)$ for each $k\in N$ ;

• there exists a comparison function $\phi :{\mathbb{R}}_{+}\to {\mathbb{R}}_{+}$ such that $T$ is a $(G,\phi )-\mathrm{contraction}$ ;

• the metric $d$ is complete.

Then $T$ is a PO.

We recall the following concept.

Definition 6

Let $(X,d)$ be a metric space. The operator $T:X\to X$ is called a C̀iric̀–Reich–Rus operator if there exist nonnegative numbers $a,b,c$ with $a+b+c<1$, such that:

$$d(Tx,Ty)⩽ad(x,y)+bd(x,Tx)+cd(y,Ty)\text{,}$$

for all $x,y\in X$.

Remark 1

Using the symmetry of distance, the above condition is equivalent with: there exist nonnegative numbers $\alpha$ and $\beta$ with $\alpha +2\beta <1$ such that:

$$d(Tx,Ty)⩽\alpha d(x,y)+\beta [d(x,Tx)+d(y,Ty)]\text{,}$$

for all $x,y\in X$.

Reich [6], Ćirić [7] and Rus [8] proved that if $(X,d)$ is complete, then every C̀iric̀–Reich–Rus operator has a unique fixed point. The aim of this paper is to study the existence of fixed points for C̀iric̀–Reich–Rus operator in metric spaces endowed with a graph $G$ by introducing the concept of $G$-C̀iric̀–Reich–Rus operators.