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# Fixed points and strict fixed points for multivalued contractions of Reich type on metric spaces endowed with a graph

Fixed Point Theory and Applications20132013:203

https://doi.org/10.1186/1687-1812-2013-203

• Received: 8 May 2013
• Accepted: 10 July 2013
• Published:

## Abstract

The purpose of this paper is to present some strict fixed point theorems for multivalued operators satisfying a Reich-type condition on a metric space endowed with a graph. The well-posedness of the fixed point problem is also studied.

MSC:47H10, 54H25.

## Keywords

• fixed point
• strict fixed point
• metric space endowed with a graph
• well-posed problem

## 1 Preliminaries

A new approach in the theory of fixed points was recently given by Jachymski  and Gwóźdź-Lukawska and Jachymski  by using the context of metric spaces endowed with a graph. Other recent results for single-valued and multivalued operators in such metric spaces are given by Nicolae, O’Regan and Petruşel in  and by Beg, Butt and Radojevic in .

Let $\left(X,d\right)$ be a metric space and let Δ be the diagonal of $X×X$. Let G be a directed graph such that the set $V\left(G\right)$ of its vertices coincides with X and $\mathrm{\Delta }\subseteq E\left(G\right)$, where $E\left(G\right)$ is the set of the edges of the graph. Assume also that G has no parallel edges and, thus, one can identify G with the pair $\left(V\left(G\right),E\left(G\right)\right)$.

If x and y are vertices of G, then a path in G from x to y of length $k\in \mathbb{N}$ is a finite sequence ${\left({x}_{n}\right)}_{n\in \left\{0,1,2,\dots ,k\right\}}$ of vertices such that ${x}_{0}=x$, ${x}_{k}=y$ and $\left({x}_{i-1},{x}_{i}\right)\in E\left(G\right)$ for $i\in \left\{1,2,\dots ,k\right\}$. Notice that a graph G is connected if there is a path between any two vertices and it is weakly connected if $\stackrel{˜}{G}$ is connected, where $\stackrel{˜}{G}$ denotes the undirected graph obtained from G by ignoring the direction of edges.

Denote by ${G}^{-1}$ the graph obtained from G by reversing the direction of edges. Thus,
Since it is more convenient to treat $\stackrel{˜}{G}$ as a directed graph for which the set of its edges is symmetric, under this convention, we have that
If G is such that $E\left(G\right)$ is symmetric, then for $x\in V\left(G\right)$, the symbol ${\left[x\right]}_{G}$ denotes the equivalence class of the relation defined on $V\left(G\right)$ by the rule:
Let us consider the following families of subsets of a metric space $\left(X,d\right)$:
The gap functional between the sets A and B in the metric space $\left(X,d\right)$ is given by
$D:P\left(X\right)×P\left(X\right)\to {\mathbb{R}}_{+}\cup \left\{+\mathrm{\infty }\right\},\phantom{\rule{2em}{0ex}}D\left(A,B\right)=inf\left\{d\left(a,b\right)\mid a\in A,b\in B\right\}.$

In particular, if ${x}_{0}\in X$ then $D\left({x}_{0},B\right):=D\left(\left\{{x}_{0}\right\},B\right)$.

The Pompeiu-Hausdorff functional is defined by
$\begin{array}{c}H:P\left(X\right)×P\left(X\right)\to {\mathbb{R}}_{+}\cup \left\{+\mathrm{\infty }\right\},\hfill \\ H\left(A,B\right)=max\left\{\underset{a\in A}{sup}D\left(a,B\right),\underset{b\in B}{sup}D\left(A,b\right)\right\}.\hfill \end{array}$
The diameter generalized functional generated by d is given by
$\begin{array}{c}\delta :P\left(X\right)×P\left(X\right)\to {\mathbb{R}}_{+}\cup \left\{+\mathrm{\infty }\right\},\hfill \\ \delta \left(A,B\right)=sup\left\{d\left(a,b\right)\mid a\in A,b\in B\right\}.\hfill \end{array}$

In particular, we denote by $\delta \left(A\right):=\delta \left(A,A\right)$ the diameter of the set A.

Let $\left(X,d\right)$ be a metric space. If $T:X\to P\left(X\right)$ is a multivalued operator, then $x\in X$ is called a fixed point for T if and only if $x\in T\left(x\right)$. The set $Fix\left(T\right):=\left\{x\in X\mid x\in T\left(x\right)\right\}$ is called the fixed point set of T, while $SFix\left(T\right)=\left\{x\in X\mid \left\{x\right\}=Tx\right\}$ is called the strict fixed point set of T. $Graph\left(T\right):=\left\{\left(x,y\right)\mid y\in T\left(x\right)\right\}$ denotes the graph of T.

Definition 1.1 Let $\phi :{\mathbb{R}}_{+}\to {\mathbb{R}}_{+}$ be a mapping. Then φ is called a strong comparison function if the following assertions hold:
1. (i)

φ is increasing;

2. (ii)

${\phi }^{n}\left(t\right)\to 0$ as $n\to \mathrm{\infty }$ for all $t\in {\mathbb{R}}_{+}$;

3. (iii)

${\sum }_{n=1}^{\mathrm{\infty }}{\phi }^{n}\left(t\right)<\mathrm{\infty }$ for all $t\in {\mathbb{R}}_{+}$.

Definition 1.2 Let $\left(X,d\right)$ be a complete metric space, let G be a directed graph, and let $T:X\to {P}_{\mathrm{b}}\left(X\right)$ be a multivalued operator. By definition, T is called a $\left(\delta ,\phi \right)$-G-contraction if there exists $\phi :{\mathbb{R}}_{+}\to {\mathbb{R}}_{+}$, a strong comparison function, such that

In this paper, we present some fixed point and strict fixed point theorems for multivalued operators satisfying a contractive condition of Reich type involving the functional δ (see [5, 6]). The equality between $Fix\left(T\right)$ and $SFix\left(T\right)$ and the well-posedness of the fixed point problem are also studied.

Our results also generalize and extend some fixed point theorems in partially ordered complete metric spaces given in Harjani and Sadarangani , Nicolae et al. , Nieto and Rodríguez-López  and , Nieto et al. , O’Regan and Petruşel , Petruşel and Rus , and Ran and Reurings .

## 2 Fixed point and strict fixed point theorems

We begin this section by presenting a strict fixed point theorem for a Reich type contraction with respect to the functional δ.

Theorem 2.1 Let $\left(X,d\right)$ be a complete metric space and let G be a directed graph such that the triple $\left(X,d,G\right)$ satisfies the following property:
$\begin{array}{rl}\left(P\right)& \begin{array}{l}\mathit{\text{for any sequence}}\phantom{\rule{0.25em}{0ex}}{\left({x}_{n}\right)}_{n\in \mathbb{N}}\subset X\phantom{\rule{0.25em}{0ex}}\mathit{\text{with}}\phantom{\rule{0.25em}{0ex}}{x}_{n}\to x\phantom{\rule{0.25em}{0ex}}\mathit{\text{as}}\phantom{\rule{0.25em}{0ex}}n\to \mathrm{\infty },\\ \mathit{\text{there exists a subsequence}}\phantom{\rule{0.25em}{0ex}}{\left({x}_{{k}_{n}}\right)}_{n\in \mathbb{N}}\phantom{\rule{0.25em}{0ex}}\mathit{\text{of}}\phantom{\rule{0.25em}{0ex}}{\left({x}_{n}\right)}_{n\in \mathbb{N}}\phantom{\rule{0.25em}{0ex}}\mathit{\text{such that}}\phantom{\rule{0.25em}{0ex}}\left({x}_{{k}_{n}},x\right)\in E\left(G\right).\end{array}\end{array}$
Let $T:X\to {P}_{\mathrm{b}}\left(X\right)$ be a multivalued operator. Suppose that the following assertions hold:
1. (i)
There exists $a,b,c\in {\mathbb{R}}_{+}$ with $b\ne 0$ and $a+b+c<1$ such that
$\delta \left(T\left(x\right),T\left(y\right)\right)\le ad\left(x,y\right)+b\delta \left(x,T\left(x\right)\right)+c\delta \left(y,T\left(y\right)\right)$

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

2. (ii)
For each $x\in X$, the set
$\begin{array}{rcl}{\stackrel{˜}{X}}_{T}\left(x\right)& :=& \left\{y\in T\left(x\right):\left(x,y\right)\in E\left(G\right)\phantom{\rule{0.25em}{0ex}}\mathit{\text{and}}\phantom{\rule{0.25em}{0ex}}\delta \left(x,T\left(x\right)\right)\le qd\left(x,y\right)\\ \mathit{\text{for some}}\phantom{\rule{0.25em}{0ex}}q\in \phantom{\rule{0.2em}{0ex}}\right]1,\frac{1-a-c}{b}\left[\right\}\end{array}$

is nonempty.

Then we have:
1. (a)

$Fix\left(T\right)=SFix\left(T\right)\ne \mathrm{\varnothing }$;

2. (b)
If we additionally suppose that
${x}^{\ast },{y}^{\ast }\in Fix\left(T\right)\phantom{\rule{1em}{0ex}}⇒\phantom{\rule{1em}{0ex}}\left({x}^{\ast },{y}^{\ast }\right)\in E\left(G\right),$

then $Fix\left(T\right)=SFix\left(T\right)=\left\{{x}^{\ast }\right\}$.

Proof (a) Let ${x}_{0}\in X$. Since ${\stackrel{˜}{X}}_{T}\left({x}_{0}\right)\ne \mathrm{\varnothing }$, there exists ${x}_{1}\in T\left({x}_{0}\right)$ and $1 such that $\left({x}_{0},{x}_{1}\right)\in E\left(G\right)$ and
$\delta \left({x}_{0},T\left({x}_{0}\right)\right)\le qd\left({x}_{0},{x}_{1}\right).$
By (i) we have that
$\begin{array}{rl}\delta \left({x}_{1},T\left({x}_{1}\right)\right)& \le \delta \left(T\left({x}_{0}\right),T\left({x}_{1}\right)\right)\le ad\left({x}_{0},{x}_{1}\right)+b\delta \left({x}_{0},T\left({x}_{0}\right)\right)+c\delta \left({x}_{1},T\left({x}_{1}\right)\right)\\ \le ad\left({x}_{0},{x}_{1}\right)+bqd\left({x}_{0},{x}_{1}\right)+c\delta \left({x}_{1},T\left({x}_{1}\right)\right).\end{array}$
Hence,
$\delta \left({x}_{1},T\left({x}_{1}\right)\right)\le \frac{a+bq}{1-c}d\left({x}_{0},{x}_{1}\right).$
(2.1)
For ${x}_{1}\in X$, since ${\stackrel{˜}{X}}_{T}\left({x}_{1}\right)\ne \mathrm{\varnothing }$, we get again that there exists ${x}_{2}\in T\left({x}_{1}\right)$ such that $\delta \left({x}_{1},T\left({x}_{1}\right)\right)\le qd\left({x}_{1},{x}_{2}\right)$ and $\left({x}_{1},{x}_{2}\right)\in E\left(G\right)$. Then
$d\left({x}_{1},{x}_{2}\right)\le \delta \left({x}_{1},T\left({x}_{1}\right)\right)\le \frac{a+bq}{1-c}d\left({x}_{0},{x}_{1}\right).$
(2.2)
On the other hand, by (i), we have that
$\begin{array}{rcl}\delta \left({x}_{2},T\left({x}_{2}\right)\right)& \le & \delta \left(T\left({x}_{1}\right),T\left({x}_{2}\right)\right)\le ad\left({x}_{1},{x}_{2}\right)+b\delta \left({x}_{1},T\left({x}_{1}\right)\right)+c\delta \left({x}_{2},T\left({x}_{2}\right)\right)\\ \le & ad\left({x}_{1},{x}_{2}\right)+bqd\left({x}_{1},{x}_{2}\right)+c\delta \left({x}_{2},T\left({x}_{2}\right)\right).\end{array}$
Using (2.2) we obtain
$\delta \left({x}_{2},T\left({x}_{2}\right)\right)\le \frac{a+bq}{1-c}d\left({x}_{1},{x}_{2}\right)\le {\left(\frac{a+bq}{1-c}\right)}^{2}d\left({x}_{0},{x}_{1}\right).$
(2.3)

For ${x}_{2}\in X$, we have ${\stackrel{˜}{X}}_{T}\left({x}_{2}\right)\ne \mathrm{\varnothing }$, and so there exists ${x}_{3}\in T\left({x}_{2}\right)$ such that $\delta \left({x}_{2},T\left({x}_{2}\right)\right)\le qd\left({x}_{2},{x}_{3}\right)$ and $\left({x}_{2},{x}_{3}\right)\in E\left(G\right)$.

Then
$d\left({x}_{2},{x}_{3}\right)\le \delta \left({x}_{2},T\left({x}_{2}\right)\right)\le {\left(\frac{a+bq}{1-c}\right)}^{2}d\left({x}_{0},{x}_{1}\right).$
(2.4)
By these procedures, we obtain a sequence ${\left({x}_{n}\right)}_{n\in \mathbb{N}}$ with the following properties:
1. (1)

$\left({x}_{n},{x}_{n+1}\right)\in E\left(G\right)$ for each $n\in \mathbb{N}$;

2. (2)

$d\left({x}_{n},{x}_{n+1}\right)\le {\left(\frac{a+bq}{1-c}\right)}^{n}d\left({x}_{0},{x}_{1}\right)$ for each $n\in \mathbb{N}$;

3. (3)

$\delta \left({x}_{n},T\left({x}_{n}\right)\right)\le {\left(\frac{a+bq}{1-c}\right)}^{n}d\left({x}_{0},{x}_{1}\right)$ for each $n\in \mathbb{N}$.

From (2) we obtain that the sequence ${\left({x}_{n}\right)}_{n\in \mathbb{N}}$ is Cauchy. Since the metric space X is complete, we get that the sequence is convergent, i.e., ${x}_{n}\to {x}^{\ast }$ as $n\to \mathrm{\infty }$. By the property (P), there exists a subsequence ${\left({x}_{{k}_{n}}\right)}_{n\in \mathbb{N}}$ of ${\left({x}_{n}\right)}_{n\in \mathbb{N}}$ such that $\left({x}_{{k}_{n}},{x}^{\ast }\right)\in E\left(G\right)$ for each $n\in \mathbb{N}$.

We have
$\begin{array}{l}\delta \left({x}^{\ast },T\left({x}^{\ast }\right)\right)\le d\left({x}^{\ast },{x}_{{k}_{n+1}}\right)+\delta \left({x}_{{k}_{n+1}},T\left({x}^{\ast }\right)\right)\le d\left({x}^{\ast },{x}_{{k}_{n+1}}\right)+\delta \left(T\left({x}_{{k}_{n}}\right),T\left({x}^{\ast }\right)\right)\\ \phantom{\delta \left({x}^{\ast },T\left({x}^{\ast }\right)\right)}\le d\left({x}^{\ast },{x}_{{k}_{n+1}}\right)+ad\left({x}_{{k}_{n}},{x}^{\ast }\right)+b\delta \left({x}_{{k}_{n}},T\left({x}_{{k}_{n}}\right)\right)+c\delta \left({x}^{\ast },T\left({x}^{\ast }\right)\right)\\ \phantom{\delta \left({x}^{\ast },T\left({x}^{\ast }\right)\right)}\le d\left({x}^{\ast },{x}_{{k}_{n+1}}\right)+ad\left({x}_{{k}_{n}},{x}^{\ast }\right)+b{\left(\frac{a+bq}{1-c}\right)}^{{k}_{n}}d\left({x}_{0},{x}_{1}\right)+c\delta \left({x}^{\ast },T\left({x}^{\ast }\right)\right),\\ \delta \left({x}^{\ast },T\left({x}^{\ast }\right)\right)\le \frac{1}{1-c}d\left({x}^{\ast },{x}_{{k}_{n+1}}\right)+\frac{a}{1-c}d\left({x}_{{k}_{n}},{x}^{\ast }\right)+\frac{b}{1-c}{\left(\frac{a+bq}{1-c}\right)}^{{k}_{n}}d\left({x}_{0},{x}_{1}\right).\end{array}$
(2.5)

But $d\left({x}^{\ast },{x}_{{k}_{n+1}}\right)\to 0$ as $n\to \mathrm{\infty }$ and $d\left({x}_{{k}_{n}},{x}^{\ast }\right)\to 0$ as $n\to \mathrm{\infty }$. Hence, $\delta \left({x}^{\ast },T\left({x}^{\ast }\right)\right)=0$, which implies that ${x}^{\ast }\in SFix\left(T\right)$. Thus $SFix\left(T\right)\ne \mathrm{\varnothing }$.

We shall prove now that $Fix\left(T\right)=SFix\left(T\right)$.

Because $SFix\left(T\right)\subset Fix\left(T\right)$, we need to show that $Fix\left(T\right)\subset SFix\left(T\right)$.

Let ${x}^{\ast }\in Fix\left(T\right)⇒{x}^{\ast }\in T\left({x}^{\ast }\right)$. Because $\mathrm{\Delta }\subset E\left(G\right)$, we have that $\left({x}^{\ast },{x}^{\ast }\right)\in E\left(G\right)$. Using (ii) with $x=y={x}^{\ast }$, we obtain
$\delta \left(T\left({x}^{\ast }\right)\right)\le ad\left({x}^{\ast },{x}^{\ast }\right)+b\delta \left({x}^{\ast },T\left({x}^{\ast }\right)\right)+c\delta \left({x}^{\ast },T\left({x}^{\ast }\right)\right).$
So, $\delta \left(T\left({x}^{\ast }\right)\right)\le \left(b+c\right)\delta \left({x}^{\ast },T\left({x}^{\ast }\right)\right)$. Because ${x}^{\ast }\in T\left({x}^{\ast }\right)$, we get that $\delta \left({x}^{\ast },T\left({x}^{\ast }\right)\right)\le \delta \left(T\left({x}^{\ast }\right)\right)$. Hence, we have
$\delta \left(T\left({x}^{\ast }\right)\right)\le \left(b+c\right)\delta \left(T\left({x}^{\ast }\right)\right).$
(2.6)

Suppose that $card\left(T\left({x}^{\ast }\right)\right)>1$. This implies that $\delta \left(T\left({x}^{\ast }\right)\right)>0$. Thus from (2.6) we obtain that $b+c>1$, which contradicts the hypothesis $a+b+c<1$.

Thus $\delta \left(T\left({x}^{\ast }\right)\right)=0⇒T\left({x}^{\ast }\right)=\left\{{x}^{\ast }\right\}$, i.e., ${x}^{\ast }\in SFix\left(T\right)$ and $Fix\left(T\right)\subset SFix\left(T\right)$.

Hence, $Fix\left(T\right)=SFix\left(T\right)\ne \mathrm{\varnothing }$.

(b) Suppose that there exist ${x}^{\ast },{y}^{\ast }\in Fix\left(T\right)=SFix\left(T\right)$ with ${x}^{\ast }\ne {y}^{\ast }$. We have that

• ${x}^{\ast }\in SFix\left(T\right)⇒\delta \left({x}^{\ast },T\left({x}^{\ast }\right)\right)=0$;

• ${y}^{\ast }\in SFix\left(T\right)⇒\delta \left({y}^{\ast },T\left({y}^{\ast }\right)\right)=0$;

• $\left({x}^{\ast },{y}^{\ast }\right)\in E\left(G\right)$.

Using (i) we obtain
$d\left({x}^{\ast },{y}^{\ast }\right)=\delta \left(T\left({x}^{\ast }\right),T\left({y}^{\ast }\right)\right)\le ad\left({x}^{\ast },{y}^{\ast }\right)+b\delta \left({x}^{\ast },T\left({x}^{\ast }\right)\right)+c\delta \left({y}^{\ast },T\left({y}^{\ast }\right)\right).$

Thus, $d\left({x}^{\ast },{y}^{\ast }\right)\le ad\left({x}^{\ast },{y}^{\ast }\right)$, which implies that $a\ge 1$, which is a contradiction.

Hence, $Fix\left(T\right)=SFix\left(T\right)=\left\{{x}^{\ast }\right\}$. □

Next we present some examples and counterexamples of multivalued operators which satisfy the hypothesis in Theorem 2.1.

Example 2.1 Let $X:=\left\{\left(0,0\right),\left(0,1\right),\left(1,0\right),\left(1,1\right)\right\}$ and $T:X\to {P}_{\mathrm{cl}}\left(X\right)$ be given by
$T\left(x\right)=\left\{\begin{array}{cc}\left\{\left(0,0\right)\right\},\hfill & x=\left(0,0\right),\hfill \\ \left\{\left(0,0\right)\right\},\hfill & x=\left(0,1\right),\hfill \\ \left\{\left(0,0\right),\left(0,1\right)\right\},\hfill & x=\left(1,0\right),\hfill \\ \left\{\left(0,0\right),\left(0,1\right)\right\},\hfill & x=\left(1,1\right).\hfill \end{array}$
(2.7)

Let $E\left(G\right):=\left\{\left(\left(0,1\right),\left(0,0\right)\right),\left(\left(1,0\right),\left(0,1\right)\right),\left(\left(1,1\right),\left(0,0\right)\right)\right\}\cup \mathrm{\Delta }$.

Notice that all the hypotheses in Theorem 2.1 are satisfied (the condition (i) is verified for $a=c=0.01$, $b=0.97$ and so $Fix\left(T\right)=SFix\left(T\right)=\left\{\left(0,0\right)\right\}$.

The following remarks show that it is not possible to have elements in ${F}_{T}\setminus S{F}_{T}$.

Remark 2.1 If we suppose that there exists $x\in {F}_{T}\setminus S{F}_{T}$, then, since $\left(x,x\right)\in \mathrm{\Delta }$, we get (using the condition (i) in the above theorem with $y=x$) that $\delta \left(T\left(x\right)\right)\le \left(b+c\right)\delta \left(T\left(x\right)\right)$, which is a contradiction with $a+b+c<1$.

Remark 2.2 If, in the previous theorem, instead of the property (P), we suppose that T has a closed graph, then we obtain again the conclusion $Fix\left(T\right)=SFix\left(T\right)\ne \mathrm{\varnothing }$.

Remark 2.3 If, in the above remark, we additionally suppose that
${x}^{\ast },{y}^{\ast }\in Fix\left(T\right)\phantom{\rule{1em}{0ex}}⇒\phantom{\rule{1em}{0ex}}\left({x}^{\ast },{y}^{\ast }\right)\in E\left(G\right),$

then $Fix\left(T\right)=SFix\left(T\right)=\left\{{x}^{\ast }\right\}$.

The next result presents a strict fixed point theorem where the operator T satisfies a $\left(\delta ,\phi \right)$-G-contractive condition on $E\left(G\right)$.

Theorem 2.2 Let $\left(X,d\right)$ be a complete metric space and let G be a directed graph such that the triple $\left(X,d,G\right)$ satisfies the following property:
$\begin{array}{rl}\left(P\right)& \begin{array}{l}\mathit{\text{for any sequence}}\phantom{\rule{0.25em}{0ex}}{\left({x}_{n}\right)}_{n\in \mathbb{N}}\subset X\phantom{\rule{0.25em}{0ex}}\mathit{\text{with}}\phantom{\rule{0.25em}{0ex}}{x}_{n}\to x\phantom{\rule{0.25em}{0ex}}\mathit{\text{as}}\phantom{\rule{0.25em}{0ex}}n\to \mathrm{\infty },\\ \mathit{\text{there exists a subsequence}}\phantom{\rule{0.25em}{0ex}}{\left({x}_{{k}_{n}}\right)}_{n\in \mathbb{N}}\phantom{\rule{0.25em}{0ex}}\mathit{\text{of}}\phantom{\rule{0.25em}{0ex}}{\left({x}_{n}\right)}_{n\in \mathbb{N}}\phantom{\rule{0.25em}{0ex}}\mathit{\text{such that}}\phantom{\rule{0.25em}{0ex}}\left({x}_{{k}_{n}},x\right)\in E\left(G\right).\end{array}\end{array}$
Let $T:X\to {P}_{\mathrm{b}}\left(X\right)$ be a multivalued operator. Suppose that the following assertions hold:
1. (i)

T is a $\left(\delta ,\phi \right)$-G-contraction.

2. (ii)
For each $x\in X$, the set
${\stackrel{˜}{X}}_{T}:=\left\{y\in T\left(x\right):\left(x,y\right)\in E\left(G\right)\phantom{\rule{0.25em}{0ex}}\mathit{\text{and}}\phantom{\rule{0.25em}{0ex}}\delta \left(x,T\left(x\right)\right)\le qd\left(x,y\right)\phantom{\rule{0.25em}{0ex}}\mathit{\text{for some}}\phantom{\rule{0.25em}{0ex}}q\in \phantom{\rule{0.2em}{0ex}}\right]1,\frac{1-a-c}{b}\left[\right\}$

is nonempty.

Then we have:
1. (a)

$Fix\left(T\right)=SFix\left(T\right)\ne \mathrm{\varnothing }$;

2. (b)
If, in addition, the following implication holds:
${x}^{\ast },{y}^{\ast }\in Fix\left(T\right)\phantom{\rule{1em}{0ex}}⇒\phantom{\rule{1em}{0ex}}\left({x}^{\ast },{y}^{\ast }\right)\in E\left(G\right),$

then $Fix\left(T\right)=SFix\left(T\right)=\left\{{x}^{\ast }\right\}$.

Proof (a) Let ${x}_{0}\in X$. Then, since ${\stackrel{˜}{X}}_{T}\left({x}_{0}\right)$ is nonempty, there exist ${x}_{1}\in T\left({x}_{0}\right)$ and $q\in \phantom{\rule{0.2em}{0ex}}\right]1,\frac{1-a-c}{b}\left[$ such that $\left({x}_{0},{x}_{1}\right)\in E\left(G\right)$ and
$\delta \left({x}_{0},T\left({x}_{0}\right)\right)\le qd\left({x}_{0},{x}_{1}\right).$
By (i) we have that
$\delta \left({x}_{1},T\left({x}_{1}\right)\right)\le \delta \left(T\left({x}_{0}\right),T\left({x}_{1}\right)\right)\le \phi \left(d\left({x}_{0},{x}_{1}\right)\right).$

For ${x}_{1}\in X$, by the same approach as before, there exists ${x}_{2}\in T\left({x}_{1}\right)$ such that $\delta \left({x}_{1},T\left({x}_{1}\right)\right)\le qd\left({x}_{1},{x}_{2}\right)$ and $\left({x}_{1},{x}_{2}\right)\in E\left(G\right)$.

We have
$d\left({x}_{1},{x}_{2}\right)\le \delta \left({x}_{1},T\left({x}_{1}\right)\right)\le \delta \left(T\left({x}_{0}\right),T\left({x}_{1}\right)\right)\le \phi \left(d\left({x}_{0},{x}_{1}\right)\right).$
(2.8)
On the other hand, by (i) we have that
$\delta \left({x}_{2},T\left({x}_{2}\right)\right)\le \delta \left(T\left({x}_{1}\right),T\left({x}_{2}\right)\right)\le \phi \left(d\left({x}_{1},{x}_{2}\right)\right)\le {\phi }^{2}\left(d\left({x}_{0},{x}_{1}\right)\right).$
By the same procedure, for ${x}_{2}\in X$ there exists ${x}_{3}\in T\left({x}_{2}\right)$ such that $\delta \left({x}_{2},T\left({x}_{2}\right)\right)\le qd\left({x}_{2},{x}_{3}\right)$ and $\left({x}_{2},{x}_{3}\right)\in E\left(G\right)$. Thus
$d\left({x}_{2},{x}_{3}\right)\le \delta \left({x}_{2},T\left({x}_{2}\right)\right)\le {\phi }^{2}\left(d\left({x}_{0},{x}_{1}\right)\right).$
(2.9)
We have
$\delta \left({x}_{3},T\left({x}_{3}\right)\right)\le \delta \left(T\left({x}_{2}\right),T\left({x}_{3}\right)\right)\le \phi \left(d\left({x}_{2},{x}_{3}\right)\right)\le {\phi }^{3}\left(d\left({x}_{0},{x}_{1}\right)\right).$
By these procedures, we obtain a sequence ${\left({x}_{n}\right)}_{n\in \mathbb{N}}$ with the following properties:
1. (1)

$\left({x}_{n},{x}_{n+1}\right)\in E\left(G\right)$ for each $n\in \mathbb{N}$;

2. (2)

$d\left({x}_{n},{x}_{n+1}\right)\le {\phi }^{n}\left(d\left({x}_{0},{x}_{1}\right)\right)$ for each $n\in \mathbb{N}$;

3. (3)

$\delta \left({x}_{n},T\left({x}_{n}\right)\right)\le {\phi }^{n}\left(d\left({x}_{0},{x}_{1}\right)\right)$ for each $n\in \mathbb{N}$.

By (2), using the properties of φ, we obtain that the sequence ${\left({x}_{n}\right)}_{n\in \mathbb{N}}$ is Cauchy. Since the metric space is complete, we have that the sequence is convergent, i.e., ${x}_{n}\to {x}^{\ast }$ as $n\to \mathrm{\infty }$. By the property (P), we get that there exists a subsequence ${\left({x}_{{k}_{n}}\right)}_{n\in \mathbb{N}}$ of ${\left({x}_{n}\right)}_{n\in \mathbb{N}}$ such that $\left({x}_{{k}_{n}},{x}^{\ast }\right)\in E\left(G\right)$ for each $n\in \mathbb{N}$.

We shall prove now that ${x}^{\ast }\in SFix\left(T\right)$. We have
$\begin{array}{rcl}\delta \left({x}^{\ast },T\left({x}^{\ast }\right)\right)& \le & d\left({x}^{\ast },{x}_{{k}_{n+1}}\right)+\delta \left({x}_{{k}_{n+1}},T\left({x}^{\ast }\right)\right)\le d\left({x}^{\ast },{x}_{{k}_{n+1}}\right)+\delta \left(T\left({x}_{{k}_{n}}\right),T\left({x}^{\ast }\right)\right)\\ \le & d\left({x}^{\ast },{x}_{{k}_{n+1}}\right)+\phi \left(d\left({x}_{{k}_{n}},{x}^{\ast }\right)\right).\end{array}$

Since $d\left({x}^{\ast },{x}_{{k}_{n+1}}\right)\to 0$ as $n\to \mathrm{\infty }$ and φ is continuous in 0 with $\phi \left(0\right)=0$, we get that $\delta \left({x}^{\ast },T\left({x}^{\ast }\right)\right)=0$.

Hence, ${x}^{\ast }\in SFix\left(T\right)⇒SFix\left(T\right)\ne \mathrm{\varnothing }$.

We shall prove now that $Fix\left(T\right)=SFix\left(T\right)$.

Because $SFix\left(T\right)\subset Fix\left(T\right)$, we need to show that $Fix\left(T\right)\subset SFix\left(T\right)$.

Let ${x}^{\ast }\in Fix\left(T\right)$. Because $\mathrm{\Delta }\subset E\left(G\right)$, we have that $\left({x}^{\ast },{x}^{\ast }\right)\in E\left(G\right)$. Using (i) with $x=y={x}^{\ast }$, we obtain
$\delta \left(T\left({x}^{\ast }\right)\right)\le \phi \left(d\left({x}^{\ast },{x}^{\ast }\right)\right)=0.$

Hence, ${x}^{\ast }\in SFix\left(T\right)$ and the proof of this conclusion is complete.

(b) Suppose that there exist ${x}^{\ast },{y}^{\ast }\in Fix\left(T\right)=SFix\left(T\right)$ with ${x}^{\ast }\ne {y}^{\ast }$. We have that

• ${x}^{\ast }\in SFix\left(T\right)⇒\delta \left({x}^{\ast },T\left({x}^{\ast }\right)\right)=0$;

• ${y}^{\ast }\in SFix\left(T\right)⇒\delta \left({y}^{\ast },T\left({y}^{\ast }\right)\right)=0$;

• $\left({x}^{\ast },{y}^{\ast }\right)\in E\left(G\right)$.

Using (i) we obtain
$d\left({x}^{\ast },{y}^{\ast }\right)=\delta \left(T\left({x}^{\ast }\right),T\left({y}^{\ast }\right)\right)\le \phi \left(d\left({x}^{\ast },{y}^{\ast }\right)\right)

This is a contradiction. Hence, $Fix\left(T\right)=SFix\left(T\right)=\left\{{x}^{\ast }\right\}$. □

In the next result, the operator T satisfies another contractive condition with respect to δ on $E\left(G\right)\cap Graph\left(T\right)$.

Theorem 2.3 Let $\left(X,d\right)$ be a complete metric space, let G be a directed graph, and let $T:X\to {P}_{\mathrm{b}}\left(X\right)$ be a multivalued operator. Suppose that $f:X\to {\mathbb{R}}_{+}$ defined $f\left(x\right):=\delta \left(x,T\left(x\right)\right)$ is a lower semicontinuous mapping. Suppose that the following assertions hold:
1. (i)
There exist $a,b\in {\mathbb{R}}_{+}$, with $b\ne 0$ and $a+b<1$, such that
$\delta \left(y,T\left(y\right)\right)\le ad\left(x,y\right)+b\delta \left(x,T\left(x\right)\right)\phantom{\rule{1em}{0ex}}\mathit{\text{for all}}\phantom{\rule{0.25em}{0ex}}\left(x,y\right)\in E\left(G\right)\cap Graph\left(T\right).$

2. (ii)
For each $x\in X$, the set
${\stackrel{˜}{X}}_{T}\left(x\right):=\left\{y\in T\left(x\right):\left(x,y\right)\in E\left(G\right)\phantom{\rule{0.25em}{0ex}}\mathit{\text{and}}\phantom{\rule{0.25em}{0ex}}\delta \left(x,T\left(x\right)\right)\le qd\left(x,y\right)\phantom{\rule{0.25em}{0ex}}\mathit{\text{for some}}\phantom{\rule{0.25em}{0ex}}q\in \phantom{\rule{0.2em}{0ex}}\right]1,\frac{1-a}{b}\left[\right\}$

is nonempty.

Then $Fix\left(T\right)=SFix\left(T\right)\ne \mathrm{\varnothing }$.

Proof Let ${x}_{0}\in X$. Then, since ${\stackrel{˜}{X}}_{T}\left({x}_{0}\right)$ is nonempty, there exist ${x}_{1}\in T\left({x}_{0}\right)$ and $1 such that
$\delta \left({x}_{0},T\left({x}_{0}\right)\right)\le qd\left({x}_{0},{x}_{1}\right)$

and $\left({x}_{0},{x}_{1}\right)\in E\left(G\right)$. Since ${x}_{1}\in T\left({x}_{0}\right)$, we get that $\left({x}_{0},{x}_{1}\right)\in E\left(G\right)\cap Graph\left(T\right)$.

By (i), taking $y={x}_{1}$ and $x={x}_{0}$, we have that
$\begin{array}{rcl}\delta \left({x}_{1},T\left({x}_{1}\right)\right)& \le & ad\left({x}_{0},{x}_{1}\right)+b\delta \left({x}_{0},T\left({x}_{0}\right)\right)\\ \le & ad\left({x}_{0},{x}_{1}\right)+bqd\left({x}_{0},{x}_{1}\right).\end{array}$
Hence,
$\delta \left({x}_{1},T\left({x}_{1}\right)\right)\le \left(a+bq\right)d\left({x}_{0},{x}_{1}\right).$
(2.10)

For ${x}_{1}\in X$ (since ${\stackrel{˜}{X}}_{T}\left({x}_{1}\right)\ne \mathrm{\varnothing }$), there exists ${x}_{2}\in T\left({x}_{1}\right)$ such that $\delta \left({x}_{1},T\left({x}_{1}\right)\right)\le qd\left({x}_{1},{x}_{2}\right)$ and $\left({x}_{1},{x}_{2}\right)\in E\left(G\right)$. But ${x}_{2}\in T\left({x}_{1}\right)$ and so $\left({x}_{1},{x}_{2}\right)\in E\left(G\right)\cap Graph\left(T\right)$.

Then
$d\left({x}_{1},{x}_{2}\right)\le \delta \left({x}_{1},T\left({x}_{1}\right)\right)\le \left(a+bq\right)d\left({x}_{0},{x}_{1}\right).$
(2.11)
By (i), taking $y={x}_{2}$ and $x={x}_{1}$, we have that
$\begin{array}{rcl}\delta \left({x}_{2},T\left({x}_{2}\right)\right)& \le & ad\left({x}_{1},{x}_{2}\right)+b\delta \left({x}_{1},T\left({x}_{1}\right)\right)\\ \le & ad\left({x}_{1},{x}_{2}\right)+bqd\left({x}_{1},{x}_{2}\right)=\left(a+bq\right)d\left({x}_{1},{x}_{2}\right)\le {\left(a+bq\right)}^{2}d\left({x}_{0},{x}_{1}\right).\end{array}$
By these procedures, we obtain a sequence ${\left({x}_{n}\right)}_{n\in \mathbb{N}}$ with the following properties:
1. (1)

$\left({x}_{n},{x}_{n+1}\right)\in E\left(G\right)\cap Graph\left(T\right)$ for each $n\in \mathbb{N}$;

2. (2)

$d\left({x}_{n},{x}_{n+1}\right)\le {\left(a+bq\right)}^{n}d\left({x}_{0},{x}_{1}\right)$ for each $n\in \mathbb{N}$;

3. (3)

$\delta \left({x}_{n},T\left({x}_{n}\right)\right)\le {\left(a+bq\right)}^{n}d\left({x}_{0},{x}_{1}\right)$ for each $n\in \mathbb{N}$.

From (2) we obtain that the sequence ${\left({x}_{n}\right)}_{n\in \mathbb{N}}$ is Cauchy. Since the metric space X is complete, we have that the sequence is convergent, i.e., ${x}_{n}\to {x}^{\ast }$ as $n\to \mathrm{\infty }$. Now, by the lower semicontinuity of the function f, we have
$0\le f\left({x}^{\ast }\right)\le \underset{n\to \mathrm{\infty }}{lim inf}f\left({x}_{n}\right)=0.$

Thus $f\left({x}^{\ast }\right)=0$, which means that $\delta \left({x}^{\ast },T\left({x}^{\ast }\right)\right)=0$. Thus ${x}^{\ast }\in SFix\left(T\right)$.

Let ${x}^{\ast }\in Fix\left(T\right)$. Then $\left({x}^{\ast },{x}^{\ast }\right)\in Graph\left(T\right)$ and hence $\left({x}^{\ast },{x}^{\ast }\right)\in E\left(G\right)\cap Graph\left(T\right)$.

Using (i) with $x=y={x}^{\ast }$, we obtain
$\delta \left(T\left({x}^{\ast }\right)\right)=\delta \left({x}^{\ast },T\left({x}^{\ast }\right)\right)\le ad\left({x}^{\ast },{x}^{\ast }\right)+b\delta \left({x}^{\ast },T\left({x}^{\ast }\right)\right).$

So, $\delta \left(T\left({x}^{\ast }\right)\right)\le b\delta \left(T\left({x}^{\ast }\right)\right)$. If we suppose that $cardT\left({x}^{\ast }\right)>1$, then $\delta \left(T\left({x}^{\ast }\right)\right)>0$. Thus, $b\ge 1$, which contradicts the hypothesis.

Thus $\delta \left(T\left({x}^{\ast }\right)\right)=0$ and so $T\left({x}^{\ast }\right)=\left\{{x}^{\ast }\right\}$. The proof is now complete. □

Remark 2.4 Example 2.1 satisfies the conditions from Theorem 2.3 for $a=0.01$ and $b=0.97$.

## 3 Well-posedness of the fixed point problem

In this section we present some well-posedness results for the fixed point problem. We consider both the well-posedness and the well-posedness in the generalized sense for a multivalued operator T.

We begin by recalling the definition of these notions from  and .

Definition 3.1 Let $\left(X,d\right)$ be a metric space and let $T:X\to P\left(X\right)$ be a multivalued operator. By definition, the fixed point problem is well posed for T with respect to H if:
1. (i)

$SFix\left(T\right)=\left\{{x}^{\ast }\right\}$;

2. (ii)

If ${\left({x}_{n}\right)}_{n\in \mathbb{N}}$ is a sequence in X such that $H\left({x}_{n},T\left({x}_{n}\right)\right)\to 0$ as $n\to \mathrm{\infty }$, then ${x}_{n}\stackrel{d}{\to }{x}^{\ast }$ as $n\to \mathrm{\infty }$.

Definition 3.2 Let $\left(X,d\right)$ be a metric space and let $T:X\to P\left(X\right)$ be a multivalued operator. By definition, the fixed point problem is well posed in the generalized sense for T with respect to H if:
1. (i)

$SFixT\ne \mathrm{\varnothing }$;

2. (ii)

If ${\left({x}_{n}\right)}_{n\in \mathbb{N}}$ is a sequence in X such that $H\left({x}_{n},T\left({x}_{n}\right)\right)\to 0$ as $n\to \mathrm{\infty }$, then there exists a subsequence ${\left({x}_{{k}_{n}}\right)}_{n\in \mathbb{N}}$ of ${\left({x}_{n}\right)}_{n\in \mathbb{N}}$ such that ${x}_{{k}_{n}}\stackrel{d}{\to }{x}^{\ast }$ as $n\to \mathrm{\infty }$.

In our first result we will establish the well-posedness of the fixed point problem for the operator T, where T is a Reich-type δ-contraction.

Theorem 3.1 Let $\left(X,d\right)$ be a complete metric space and let G be a directed graph such that the triple $\left(X,d,G\right)$ satisfies the property (P).

Let $T:X\to {P}_{\mathrm{b}}\left(X\right)$ be a multivalued operator. Suppose that
1. (i)

conditions (i) and (ii) in Theorem  2.1 hold;

2. (ii)

if ${x}^{\ast },{y}^{\ast }\in Fix\left(T\right)$, then $\left({x}^{\ast },{y}^{\ast }\right)\in E\left(G\right)$;

3. (iii)

for any sequence ${\left({x}_{n}\right)}_{n\in \mathbb{N}}$, ${x}_{n}\in X$ with $H\left({x}_{n},T\left({x}_{n}\right)\right)\to 0$ as $n\to \mathrm{\infty }$, we have $\left({x}_{n},{x}^{\ast }\right)\in E\left(G\right)$.

In these conditions the fixed point problem is well posed for T with respect to H.

Proof From (i) and (ii) we obtain that $SFix\left(T\right)=\left\{{x}^{\ast }\right\}$. Let ${\left({x}_{n}\right)}_{n\in \mathbb{N}}\subset X$ be a sequence which satisfies (iii). It is obvious that $H\left({x}_{n},T\left({x}_{n}\right)\right)=\delta \left({x}_{n},T\left({x}_{n}\right)\right)$,
$\begin{array}{rcl}d\left({x}_{n},{x}^{\ast }\right)& \le & \delta \left({x}_{n},T\left({x}^{\ast }\right)\right)\le \delta \left({x}_{n},T\left({x}_{n}\right)\right)+\delta \left(T\left({x}_{n}\right),T\left({x}^{\ast }\right)\right)\\ \le & \delta \left({x}_{n},T\left({x}_{n}\right)\right)+ad\left({x}_{n},{x}^{\ast }\right)+b\delta \left({x}_{n},T\left({x}_{n}\right)\right)+c\delta \left({x}^{\ast },T\left({x}^{\ast }\right)\right).\end{array}$
Thus

Hence, ${x}_{n}\to {x}^{\ast }$ as $n\to \mathrm{\infty }$. □

Remark 3.1 If we replace the property (P) with the condition that T has a closed graph, we reach the same conclusion.

The next result deals with the well-posedness of the fixed point problem in the generalized sense.

Theorem 3.2 Let $\left(X,d\right)$ be a complete metric space and let G be a directed graph such that the triple $\left(X,d,G\right)$ satisfies the property (P).

Let $T:X\to {P}_{\mathrm{b}}\left(X\right)$ be a multivalued operator. Suppose that
1. (i)

conditions (i) and (ii) in Theorem  2.1 hold;

2. (ii)

for any sequence ${\left({x}_{n}\right)}_{n\in \mathbb{N}}$, ${x}_{n}\in X$ with $H\left({x}_{n},T\left({x}_{n}\right)\right)\to 0$ as $n\to \mathrm{\infty }$, there exists a subsequence ${\left({x}_{{k}_{n}}\right)}_{n\in \mathbb{N}}$ such that $\left({x}_{{k}_{n}},{x}^{\ast }\right)\in E\left(G\right)$ and $H\left({x}_{{k}_{n}},T\left({x}_{{k}_{n}}\right)\right)\to 0$.

In these conditions the fixed point problem is well posed in the generalized sense for T with respect to H.

Proof From (i) we have that $SFix\left(T\right)\ne \mathrm{\varnothing }$. Let ${\left({x}_{n}\right)}_{n\in \mathbb{N}}\subset X$ be a sequence which satisfies (ii). Then there exists a subsequence ${\left({x}_{{k}_{n}}\right)}_{n\in \mathbb{N}}$ such that $\left({x}_{{k}_{n}},{x}^{\ast }\right)\in E\left(G\right)$.

We have $H\left({x}_{{k}_{n}},T\left({x}_{{k}_{n}}\right)\right)=\delta \left({x}_{{k}_{n}},T\left({x}_{{k}_{n}}\right)\right)$,
$\begin{array}{rcl}d\left({x}_{{k}_{n}},{x}^{\ast }\right)& \le & \delta \left({x}_{{k}_{n}},T\left({x}^{\ast }\right)\right)\le \delta \left({x}_{{k}_{n}},T\left({x}_{{k}_{n}}\right)\right)+\delta \left(T\left({x}_{{k}_{n}}\right),T\left({x}^{\ast }\right)\right)\\ \le & \delta \left({x}_{{k}_{n}},T\left({x}_{{k}_{n}}\right)\right)+ad\left({x}_{{k}_{n}},{x}^{\ast }\right)+b\delta \left({x}_{{k}_{n}},T\left({x}_{{k}_{n}}\right)\right)+c\delta \left({x}^{\ast },T\left({x}^{\ast }\right)\right).\end{array}$
Thus

Hence, ${x}_{{k}_{n}}\to {x}^{\ast }$. □

Remark 3.2 If we replace the property (P) with the condition that T has a closed graph, we reach the same conclusion.

Next we consider the case where the operator T satisfies a φ-contraction condition.

Theorem 3.3 Let $\left(X,d\right)$ be a complete metric space and let G be a directed graph such that the triple $\left(X,d,G\right)$ satisfies the property (P).

Let $T:X\to {P}_{\mathrm{b}}\left(X\right)$ be a multivalued operator. Suppose that
1. (i)

conditions (i) and (ii) in Theorem  2.2 hold;

2. (ii)

the following implication holds: ${x}^{\ast },{y}^{\ast }\in Fix\left(T\right)$ implies $\left({x}^{\ast },{y}^{\ast }\right)\in E\left(G\right)$;

3. (iii)

the function $\psi :{\mathbb{R}}_{+}\to {\mathbb{R}}_{+}$, given by $\psi \left(t\right)=t-\phi \left(t\right)$, has the following property: if $\psi \left({t}_{n}\right)\to 0$ as $n\to \mathrm{\infty }$, then ${t}_{n}\to 0$ as $n\to \mathrm{\infty }$;

4. (iv)

for any sequence ${\left({x}_{n}\right)}_{n\in \mathbb{N}}\subset X$ with $H\left({x}_{n},T\left({x}_{n}\right)\right)\to 0$ as $n\to \mathrm{\infty }$, we have $\left({x}_{n},{x}^{\ast }\right)\in E\left(G\right)$ for all $n\in \mathbb{N}$.

In these conditions the fixed point problem is well posed for T with respect to H.

Proof From (i) and (ii) we obtain that $SFix\left(T\right)=\left\{{x}^{\ast }\right\}$. Let ${\left({x}_{n}\right)}_{n\in \mathbb{N}}$, ${x}_{n}\in X$ be a sequence which satisfies (iv). It is obvious that $H\left({x}_{n},T\left({x}_{n}\right)\right)=\delta \left({x}_{n},T\left({x}_{n}\right)\right)$,
$\begin{array}{rcl}d\left({x}_{n},{x}^{\ast }\right)& \le & \delta \left({x}_{n},T\left({x}^{\ast }\right)\right)\le \delta \left({x}_{n},T\left({x}_{n}\right)\right)+\delta \left(T\left({x}_{n}\right),T\left({x}^{\ast }\right)\right)\\ \le & \delta \left({x}_{n},T\left({x}_{n}\right)\right)+\phi \left(d\left({x}_{n},{x}^{\ast }\right)\right).\end{array}$
Thus

Using condition (iii), we get that $d\left({x}_{n},{x}^{\ast }\right)\to 0$ as $n\to \mathrm{\infty }$. Hence, ${x}_{n}\to {x}^{\ast }$. □

Remark 3.3 If we replace the property (P) with the condition that T has a closed graph, we reach the same conclusion.

The next result gives a well-posedness (in the generalized sense) criterion for the fixed point problem.

Theorem 3.4 Let $\left(X,d\right)$ be a complete metric space and let G be a directed graph such that the triple $\left(X,d,G\right)$ satisfies the property (P).

Let $T:X\to {P}_{\mathrm{b}}\left(X\right)$ be a multivalued operator. Suppose that
1. (i)

the conditions (i) and (ii) in Theorem  2.2 hold;

2. (ii)

the function $\psi :{\mathbb{R}}_{+}\to {\mathbb{R}}_{+}$, given $\psi \left(t\right)=t-\phi \left(t\right)$, has the following property: for any sequence ${\left({t}_{n}\right)}_{n\in \mathbb{N}}$, there exists a subsequence ${\left({x}_{{k}_{n}}\right)}_{n\in \mathbb{N}}$ such that if $\psi \left({t}_{{k}_{n}}\right)\to 0$ as $n\to \mathrm{\infty }$, then ${t}_{{k}_{n}}\to 0$ as $n\to \mathrm{\infty }$;

3. (iii)

for any sequence ${\left({x}_{n}\right)}_{n\in \mathbb{N}}$, ${x}_{n}\in X$ with $H\left({x}_{n},T\left({x}_{n}\right)\right)\to 0$ as $n\to \mathrm{\infty }$, there exists a subsequence ${\left({x}_{{k}_{n}}\right)}_{n\in \mathbb{N}}$ such that $\left({x}_{{k}_{n}},{x}^{\ast }\right)\in E\left(G\right)$ and $H\left({x}_{{k}_{n}},T\left({x}_{{k}_{n}}\right)\right)\to 0$.

In these conditions the fixed point problem is well posed in the generalized sense for T with respect to H.

Proof From (i) we have that $SFix\left(T\right)\ne \mathrm{\varnothing }$. Let ${\left({x}_{n}\right)}_{n\in \mathbb{N}}$, ${x}_{n}\in X$ be a sequence which satisfies (iii). Then there exists a subsequence ${\left({x}_{{k}_{n}}\right)}_{n\in \mathbb{N}}$ such that $\left({x}_{{k}_{n}},{x}^{\ast }\right)\in E\left(G\right)$.

We have $H\left({x}_{{k}_{n}},T\left({x}_{{k}_{n}}\right)\right)=\delta \left({x}_{{k}_{n}},T\left({x}_{{k}_{n}}\right)\right)$,
$\begin{array}{rcl}d\left({x}_{{k}_{n}},{x}^{\ast }\right)& \le & \delta \left({x}_{{k}_{n}},T\left({x}^{\ast }\right)\right)\le \delta \left({x}_{{k}_{n}},T\left({x}_{{k}_{n}}\right)\right)+\delta \left(T\left({x}_{{k}_{n}}\right),T\left({x}^{\ast }\right)\right)\\ \le & \delta \left({x}_{{k}_{n}},T\left({x}_{{k}_{n}}\right)\right)+\phi \left(d\left({x}_{{k}_{n}},{x}^{\ast }\right)\right).\end{array}$
Thus

Using condition (ii), we get that $d\left({x}_{{k}_{n}},{x}^{\ast }\right)\to 0$ as $n\to \mathrm{\infty }$. Hence, ${x}_{n}\to {x}^{\ast }$. □

Remark 3.4 If we replace the property (P) with the condition that T has a closed graph, we reach the same conclusion.

## Declarations

### Acknowledgements

The third author is supported by a grant of the Romanian National Authority for Scientific Research, CNCS UEFISCDI, project number PN-II-ID-PCE-2011-3-0094.

## Authors’ Affiliations

(1)
Department of Business, Babeş-Bolyai University Cluj-Napoca, Horea street, No. 7, Cluj-Napoca, Romania
(2)
Department of Mathematics, Babeş-Bolyai University Cluj-Napoca, Kogalniceanu street, No. 1, Cluj-Napoca, Romania

## References 