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# Multi-valued version of $SCC$, $SKC$, $KSC$, and $CSC$ conditions in Ptolemy metric spaces

Journal of Inequalities and Applications20142014:471

https://doi.org/10.1186/1029-242X-2014-471

• Accepted: 13 November 2014
• Published:

## Abstract

In this paper, multi-valued version of $SCC$, $SKC$, $KSC$, and $CSC$ conditions in Ptolemy metric space are presented. Then the existence of a fixed point for these mappings in a Ptolemy metric space are proved. Finally, some examples are presented.

MSC:47H10.

## Keywords

• $CAT\left(0\right)$ spaces
• fixed point
• C condition
• Ptolemy metric space

## 1 Introduction

The definition of a Ptolemy metric space is introduced by Schoenberg [1, 2]. In order to define it, we need to recall the definition of a Ptolemy inequality as follows.

Definition 1.1 [1]

Let $\left(X,d\right)$ be a metric space, the inequality
$d\left(x,y\right)d\left(z,p\right)\le d\left(x,z\right)d\left(y,p\right)+d\left(x,p\right)d\left(y,z\right)$

is called a Ptolemy inequality, where $x,y,z,p\in X$.

Now, the definition of Ptolemy metric space is as follows.

Definition 1.2 [1]

A Ptolemy metric space is a metric space where the Ptolemy inequality holds.

Schoenberg proved that every pre-Hilbert space is Ptolemaic and each linear quasinormed Ptolemaic space is a pre-Hilbert space (see [1] and [3]). Moreover, Burckley et al. [4] proved that $CAT\left(0\right)$ spaces are Ptolemy metric spaces. They presented an example to show the converse is not true. Espinola and Nicolae in [5] proved a geodesic Ptolemy space with a uniformly continuous midpoint map is reflexive. With respect to this, they proved some fixed point theorems.

In 2008, Suzuki [6] introduced the C condition.

Definition 1.3 Let T be a mapping on a subset K of a metric space X, then T is said to satisfy C condition if
$\frac{1}{2}d\left(x,Tx\right)\le d\left(x,y\right)\phantom{\rule{1em}{0ex}}\text{implies}\phantom{\rule{1em}{0ex}}d\left(Tx,Ty\right)\le d\left(x,y\right),$

for all $x,y\in K$.

Karapınar and Taş [7] presented some new definitions which are modifications of Suzuki’s C condition as follows.

Definition 1.4 Let T be a mapping on a subset K of a metric space X.
1. (i)
T is said to satisfy the $SCC$ condition if
$\frac{1}{2}d\left(x,Tx\right)\le d\left(x,y\right)\phantom{\rule{1em}{0ex}}\text{implies}\phantom{\rule{1em}{0ex}}d\left(Tx,Ty\right)\le M\left(x,y\right),$
where

2. (ii)
T is said to satisfy the $SKC$ condition if
$\frac{1}{2}d\left(x,Tx\right)\le d\left(x,y\right)\phantom{\rule{1em}{0ex}}\text{implies}\phantom{\rule{1em}{0ex}}d\left(Tx,Ty\right)\le N\left(x,y\right),$
where

3. (iii)
T is said to satisfy the $KSC$ condition if
$\frac{1}{2}d\left(x,Tx\right)\le d\left(x,y\right)\phantom{\rule{1em}{0ex}}\text{implies}\phantom{\rule{1em}{0ex}}d\left(Tx,Ty\right)\le \frac{1}{2}\left\{d\left(x,Tx\right)+d\left(Ty,y\right)\right\}.$

4. (iv)
T is said to satisfy the $CSC$ condition if
$\frac{1}{2}d\left(x,Tx\right)\le d\left(x,y\right)\phantom{\rule{1em}{0ex}}\text{implies}\phantom{\rule{1em}{0ex}}d\left(Tx,Ty\right)\le \frac{1}{2}\left\{d\left(Tx,y\right)+d\left(x,Ty\right)\right\}.$

It is clear that every nonexpansive mapping satisfies the $SKC$ condition [[7], Proposition 9]. There exist mappings which do not satisfy the C condition, but they satisfy the $SCC$ condition as the following example shows.

Example 1.5 [8]

Define a mapping T on $\left[0,3\right]$ with $d\left(x,y\right)=|x-y|$ by

Karapınar and Taş [7] proved some fixed point theorems as follows.

Theorem 1.6 Let T be a mapping on a closed subset K of a metric space X. Assume T satisfies the $SKC$, $KSC$, $SCC$ or $CSC$ condition, then $F\left(T\right)$ is closed. Moreover, if X is strictly convex and K is convex, then $F\left(T\right)$ is also convex.

Theorem 1.7 Let T be a mapping on a closed subset K of a metric space X which satisfying the $SKC$, $KSC$, $SCC$ or $CSC$ condition, then $d\left(x,Ty\right)\le 5d\left(Tx,x\right)+d\left(x,y\right)$ holds for $x,y\in K$.

Hosseini Ghoncheh and Razani [8] proved some fixed point theorems for the $SCC$, $SKC$, $KSC$, and $CSC$ conditions in a single-valued version in Ptolemy metric space. In this paper, the notation of $SCC$, $SKC$, $KSC$, and $CSC$ conditions are generalized for multi-valued mappings and some new fixed point theorems are obtained in Ptolemy metric spaces.

Let X be a metric space and $\left\{{x}_{n}\right\}$ be a bounded sequence in X. For $x\in X$, let
$r\left(x,\left\{{x}_{n}\right\}\right)=\underset{n\to \mathrm{\infty }}{lim sup}d\left(x,{x}_{n}\right).$
The asymptotic radius $r\left(\left\{{x}_{n}\right\}\right)$ of $\left\{{x}_{n}\right\}$ in K is given by
$r\left(K,\left\{{x}_{n}\right\}\right)=\underset{x\in K}{inf}r\left(x,\left\{{x}_{n}\right\}\right),$
and the asymptotic center $A\left(\left\{{x}_{n}\right\}\right)$ of $\left\{{x}_{n}\right\}$ in K is the set
$A\left(K,\left\{{x}_{n}\right\}\right)=\left\{x\in K:r\left(x,\left\{{x}_{n}\right\}\right)\right\}=r\left(K,\left\{{x}_{n}\right\}\right).$

Definition 1.8 [9]

A sequence $\left\{{x}_{n}\right\}$ in a $CAT\left(0\right)$ space X is said to be Δ-convergent to $x\in X$, if x is the unique asymptotic center of every subsequence of $\left\{{x}_{n}\right\}$.

Lemma 1.9
1. (i)

Every bounded sequence in X has a Δ-convergent subsequence [[10], p.3690].

2. (ii)

If C is a closed convex subset of X and if $\left\{{x}_{n}\right\}$ is a bounded sequence in C, then the asymptotic center of $\left\{{x}_{n}\right\}$ is in C [[11], Proposition  2.1].

3. (iii)

If C is a closed convex subset of X and if $f:C\to X$ is a nonexpansive mapping, then the conditions, $\left\{{x}_{n}\right\}\mathrm{\Delta }$-converges to x and $d\left({x}_{n},f\left({x}_{n}\right)\right)\to 0$, imply $x\in C$ and $f\left(x\right)=x$ [[10], Proposition  3.7].

Lemma 1.10 [12]

If $\left\{{x}_{n}\right\}$ is a bounded sequence in X with $A\left(\left\{{x}_{n}\right\}\right)=\left\{x\right\}$ and $\left\{{u}_{n}\right\}$ is a subsequence of $\left\{{x}_{n}\right\}$ with $A\left(\left\{{u}_{n}\right\}\right)=\left\{u\right\}$ and the sequence $\left\{d\left({x}_{n},u\right)\right\}$ converges, then $x=u$.

The next lemma and theorem play main roles for obtaining a fixed point in the Ptolemy metric spaces.

Lemma 1.11 [13]

Let $\left\{{z}_{n}\right\}$ and $\left\{{w}_{n}\right\}$ be bounded sequences in metric space K and $\lambda \in \left(0,1\right)$. Suppose ${z}_{n+1}=\lambda {w}_{n}+\left(1-\lambda \right){z}_{n}$ and $d\left({w}_{n+1},{w}_{n}\right)\le d\left({z}_{n+1},{z}_{n}\right)$ for all $n\in \mathbb{N}$. Then ${lim sup}_{n\to \mathrm{\infty }}d\left({w}_{n},{z}_{n}\right)=0$.

Theorem 1.12 [5]

Let X be a complete geodesic Ptolemy space with a uniformly continuous midpoint map, $\left\{{x}_{n}\right\}\subseteq X$ a bounded sequence and $K\subseteq X$ nonempty closed and convex. Then $\left\{{x}_{n}\right\}$ has a unique asymptotic center in K.

## 2 Main results

Let X be complete geodesic Ptolemy space and $P\left(X\right)$ denote the class of all subsets of X. Denote

Thus ${P}_{bd}$, ${P}_{cl}$, ${P}_{cv}$, ${P}_{cp}$, ${P}_{cl,bd}$, ${P}_{cp,cv}$ denote the classes of bounded, closed, convex, compact, closed bounded, and compact convex subsets of X, respectively. Also $T:K\to {P}_{f}\left(X\right)$ is called a multi-valued mapping on X. A point $u\in X$ is called a fixed point of T if $u\in Tu$.

Definition 2.1 [14]

Let K be a subset of a $CAT\left(0\right)$ space X. A map $T:X\to P\left(X\right)$ is said to satisfy the C condition if for each $x\in K$, ${u}_{x}\in Tx$, and $y\in K$
$\frac{1}{2}d\left(x,{u}_{x}\right)\le d\left(x,y\right)$
there exists a ${u}_{y}\in Ty$ such that
$d\left({u}_{x},{u}_{y}\right)\le d\left(x,y\right).$

Espinola and Nicolae [5] used the C condition as follows.

Theorem 2.2 Let X be a complete geodesic Ptolemy space with a uniformly continuous midpoint map, and K a nonempty bounded, closed, and convex subset of X. Suppose $T:K\to {P}_{cp}\left(K\right)$ is a multi-valued mapping satisfying the C condition, then $F\left(T\right)\ne \mathrm{\varnothing }$.

Now, we extend the $SCC$, $SKC$, $KSC$, and $CSC$ conditions to multi-valued versions.

Definition 2.3 Let K be a subset of a geodesic Ptolemy space X. A map $T:X\to P\left(X\right)$ is said to satisfy conditions (i) $SCC$, (ii) $SKC$, (iii) $KSC$, (iv) $CSC$ if for each $x\in K$, ${u}_{x}\in Tx$, and $y\in K$
$\frac{1}{2}d\left(x,{u}_{x}\right)\le d\left(x,y\right)$
there exists a ${u}_{y}\in Ty$ such that
1. (i)
$d\left({u}_{x},{u}_{y}\right)\le {M}^{\prime }\left(x,y\right)$, where
${M}^{\prime }\left(x,y\right)=max\left\{d\left(x,y\right),d\left(x,{u}_{x}\right),d\left({u}_{y},y\right),d\left({u}_{x},y\right),d\left(x,{u}_{y}\right)\right\},$

2. (ii)
$d\left({u}_{x},{u}_{y}\right)\le {N}^{\prime }\left(x,y\right)$, where
${N}^{\prime }\left(x,y\right)=max\left\{d\left(x,y\right),\frac{1}{2}\left\{d\left(x,{u}_{x}\right)+d\left({u}_{y},y\right)\right\},\frac{1}{2}\left\{d\left({u}_{x},y\right)+d\left(x,{u}_{y}\right)\right\}\right\},$

3. (iii)

$d\left({u}_{x},{u}_{y}\right)\le \frac{1}{2}\left\{d\left(x,{u}_{x}\right)+d\left({u}_{y},y\right)\right\}$,

4. (iv)

$d\left({u}_{x},{u}_{y}\right)\le \frac{1}{2}\left\{d\left({u}_{x},y\right)+d\left(x,{u}_{y}\right)\right\}$.

Remark 2.4 Notice that any $KSC$ or $CSC$ map is a $SKC$ map.

Lemma 2.5 Let X be a complete geodesic Ptolemy space, and K a nonempty closed subset of X. Suppose $T:K\to {P}_{cp}\left(K\right)$ is a multi-valued mapping satisfying the $SKC$ condition, then for every $x,y\in K$, ${u}_{x}\in T\left(x\right)$ and ${u}_{xx}\in T\left({u}_{x}\right)$ the following hold:
1. (i)

$d\left({u}_{x},{u}_{xx}\right)\le d\left(x,{u}_{x}\right)$,

2. (ii)

either $\frac{1}{2}d\left(x,{u}_{x}\right)\le d\left(x,y\right)$ or $\frac{1}{2}d\left({u}_{x},{u}_{xx}\right)\le d\left({u}_{x},y\right)$,

3. (iii)

either $d\left({u}_{x},{u}_{y}\right)\le {N}^{\prime }\left(x,y\right)$ or $d\left({u}_{y},{u}_{xx}\right)\le {N}^{\prime }\left({u}_{x},y\right)$,

where
${N}^{\prime }\left({u}_{x},y\right)=max\left\{d\left({u}_{x},y\right),\frac{1}{2}\left\{d\left({u}_{xx},{u}_{x}\right)+d\left({u}_{y},y\right)\right\},\frac{1}{2}\left\{d\left({u}_{xx},y\right)+d\left({u}_{y},{u}_{x}\right)\right\}\right\}.$
Proof The first statement follows from the $SKC$ condition. Indeed, we always have
$\frac{1}{2}d\left(x,{u}_{x}\right)\le d\left(x,{u}_{x}\right),$
which yields
$d\left({u}_{x},{u}_{xx}\right)\le {N}^{\prime }\left(x,{u}_{x}\right),$
(2.1)
where
$\begin{array}{rl}{N}^{\prime }\left(x,{u}_{x}\right)& =max\left\{d\left(x,{u}_{x}\right),\frac{1}{2}\left\{d\left({u}_{x},x\right)+d\left({u}_{xx},{u}_{x}\right)\right\},\frac{1}{2}\left\{d\left({u}_{x},{u}_{x}\right)+d\left({u}_{xx},x\right)\right\}\right\}\\ =max\left\{d\left(x,{u}_{x}\right),\frac{1}{2}\left\{d\left({u}_{x},x\right)+d\left({u}_{xx},{u}_{x}\right)\right\},\frac{1}{2}d\left({u}_{xx},x\right)\right\}.\end{array}$
If ${N}^{\prime }\left(x,{u}_{x}\right)=d\left(x,{u}_{x}\right)$ we are done. If ${N}^{\prime }\left(x,{u}_{x}\right)=\frac{1}{2}\left\{d\left({u}_{x},x\right)+d\left({u}_{xx},{u}_{x}\right)\right\}$ then (2.1) turns into
$d\left({u}_{x},{u}_{xx}\right)\le {N}^{\prime }\left(x,{u}_{x}\right)=\frac{1}{2}\left\{d\left({u}_{x},x\right)+d\left({u}_{xx},{u}_{x}\right)\right\}.$
(2.2)
By simplifying (2.2), one can get (i). For the case ${N}^{\prime }\left(x,{u}_{x}\right)=\frac{1}{2}d\left({u}_{xx},x\right)$ (2.1) turns into
$d\left({u}_{x},{u}_{xx}\right)\le {N}^{\prime }\left(x,{u}_{x}\right)=\frac{1}{2}d\left({u}_{xx},x\right)\le \frac{1}{2}\left\{d\left({u}_{x},x\right)+d\left({u}_{xx},{u}_{x}\right)\right\},$
which implies (i). It is clear that (iii) is a consequence of (ii). To prove (ii), assume the contrary, that is,
$\frac{1}{2}d\left({u}_{x},x\right)>d\left(x,y\right)\phantom{\rule{1em}{0ex}}\text{and}\phantom{\rule{1em}{0ex}}\frac{1}{2}d\left({u}_{xx},{u}_{x}\right)>d\left({u}_{x},y\right)$
hold for all $x,y\in K$. Thus by triangle inequality and (i), we have
$\begin{array}{rl}d\left(x,{u}_{x}\right)& \le d\left(x,y\right)+d\left(y,{u}_{x}\right)\\ <\frac{1}{2}\left\{d\left({u}_{x},x\right)+d\left({u}_{xx},{u}_{x}\right)\right\}\\ \le \frac{1}{2}d\left({u}_{x},x\right)+\frac{1}{2}d\left({u}_{x},x\right)=d\left(x,{u}_{x}\right).\end{array}$

□

Theorem 2.6 Let X be a complete geodesic Ptolemy space, K a nonempty closed subset of X. Suppose $T:K\to {P}_{cp}\left(K\right)$ is a multi-valued mapping satisfying $SKC$ condition, then $d\left(x,{u}_{y}\right)\le 7d\left({u}_{x},x\right)+d\left(x,y\right)$ for all $x,y\in K$, ${u}_{x}\in Tx$, and ${u}_{y}\in Ty$.

Proof The proof is based on Lemma 2.5; it is proved that
$d\left({u}_{x},{u}_{y}\right)\le {N}^{\prime }\left(x,y\right)\phantom{\rule{1em}{0ex}}\text{or}\phantom{\rule{1em}{0ex}}d\left({u}_{y},{u}_{xx}\right)\le {N}^{\prime }\left({u}_{x},y\right)$
holds, where
${N}^{\prime }\left({u}_{x},y\right)=max\left\{d\left({u}_{x},y\right),\frac{1}{2}\left\{d\left({u}_{xx},{u}_{x}\right)+d\left({u}_{y},y\right)\right\},\frac{1}{2}\left\{d\left({u}_{xx},y\right)+d\left({u}_{y},{u}_{x}\right)\right\}\right\}.$
Consider the first case. If ${N}^{\prime }\left(x,y\right)=d\left(x,y\right)$, then we have
$d\left(x,{u}_{y}\right)\le d\left(x,{u}_{x}\right)+d\left({u}_{x},{u}_{y}\right)\le d\left(x,{u}_{x}\right)+d\left(x,y\right).$
For ${N}^{\prime }\left(x,y\right)=\frac{1}{2}\left\{d\left({u}_{x},x\right)+d\left({u}_{y},y\right)\right\}$ one can observe
$\begin{array}{rl}d\left(x,{u}_{y}\right)& \le d\left(x,{u}_{x}\right)+d\left({u}_{x},{u}_{y}\right)\\ \le d\left(x,{u}_{x}\right)+\frac{1}{2}\left\{d\left({u}_{x},x\right)+d\left({u}_{y},y\right)\right\}\\ \le \frac{3}{2}d\left({u}_{x},x\right)+\frac{1}{2}d\left({u}_{y},y\right)\\ \le \frac{3}{2}d\left({u}_{x},x\right)+\frac{1}{2}\left\{d\left({u}_{y},x\right)+d\left(x,y\right)\right\}.\end{array}$
Thus,
$\frac{1}{2}d\left(x,{u}_{y}\right)\le \frac{3}{2}d\left(x,{u}_{x}\right)+\frac{1}{2}d\left(x,y\right)\phantom{\rule{1em}{0ex}}\text{if and only if}\phantom{\rule{1em}{0ex}}d\left(x,{u}_{y}\right)\le 3d\left(x,{u}_{x}\right)+d\left(x,y\right).$
For ${N}^{\prime }\left(x,y\right)=\frac{1}{2}\left\{d\left({u}_{x},y\right)+d\left({u}_{y},x\right)\right\}$ one can obtain
$\begin{array}{rl}d\left(x,{u}_{y}\right)& \le d\left(x,{u}_{x}\right)+d\left({u}_{x},{u}_{y}\right)\\ \le d\left(x,{u}_{x}\right)+\frac{1}{2}\left\{d\left({u}_{x},y\right)+d\left({u}_{y},x\right)\right\}\\ \le d\left({u}_{x},x\right)+\frac{1}{2}\left\{d\left({u}_{x},x\right)+d\left(x,y\right)\right\}+\frac{1}{2}d\left({u}_{y},x\right).\end{array}$
Thus
$\frac{1}{2}d\left(x,{u}_{y}\right)\le \frac{3}{2}d\left(x,{u}_{x}\right)+\frac{1}{2}d\left(x,y\right)\phantom{\rule{1em}{0ex}}\text{if and only if}\phantom{\rule{1em}{0ex}}d\left(x,{u}_{y}\right)\le 3d\left(x,{u}_{x}\right)+d\left(x,y\right).$
Take the second case into account. For ${N}^{\prime }\left({u}_{x},y\right)=d\left({u}_{x},y\right)$
$\begin{array}{rl}d\left(x,{u}_{y}\right)& \le d\left(x,{u}_{x}\right)+d\left({u}_{x},{u}_{xx}\right)+d\left({u}_{xx},{u}_{y}\right)\\ \le d\left(x,{u}_{x}\right)+\left({u}_{x},x\right)+d\left({u}_{x},y\right)\\ =2d\left(x,{u}_{x}\right)+d\left({u}_{x},y\right)\\ \le 2d\left(x,{u}_{x}\right)+d\left({u}_{x},x\right)+d\left(x,y\right)\\ =3d\left(x,{u}_{x}\right)+d\left(x,y\right).\end{array}$
If ${N}^{\prime }\left({u}_{x},y\right)=\frac{1}{2}\left\{d\left({u}_{xx},{u}_{x}\right)+d\left({u}_{y},y\right)\right\}$ then
$\begin{array}{rl}d\left(x,{u}_{y}\right)& \le d\left(x,{u}_{x}\right)+d\left({u}_{x},{u}_{xx}\right)+d\left({u}_{xx},{u}_{y}\right)\\ \le 2d\left(x,{u}_{x}\right)+\frac{1}{2}\left\{d\left({u}_{xx},{u}_{x}\right)+d\left({u}_{y},y\right)\right\}\\ \le \frac{5}{2}d\left(x,{u}_{x}\right)+\frac{1}{2}d\left({u}_{y},y\right)\\ \le \frac{5}{2}d\left(x,{u}_{x}\right)+\frac{1}{2}\left\{d\left({u}_{y},x\right)+d\left(x,y\right)\right\};\end{array}$
then
$\frac{1}{2}d\left(x,{u}_{y}\right)\le \frac{5}{2}d\left(x,{u}_{x}\right)+\frac{1}{2}d\left(x,y\right)\phantom{\rule{1em}{0ex}}\text{if and only if}\phantom{\rule{1em}{0ex}}d\left(x,{u}_{y}\right)\le 5d\left(x,{u}_{x}\right)+d\left(x,y\right).$
For the last case, ${N}^{\prime }\left({u}_{x},y\right)=\frac{1}{2}\left\{d\left({u}_{xx},y\right)+d\left({u}_{y},{u}_{x}\right)\right\}$ and we have
$\begin{array}{rl}d\left(x,{u}_{y}\right)& \le d\left(x,{u}_{x}\right)+d\left({u}_{x},{u}_{xx}\right)+d\left({u}_{xx},{u}_{y}\right)\\ \le 2d\left(x,{u}_{x}\right)+\frac{1}{2}\left\{d\left({u}_{xx},y\right)+d\left({u}_{y},{u}_{x}\right)\right\}\\ \le 2d\left(x,{u}_{x}\right)+\frac{1}{2}\left\{d\left({u}_{xx},{u}_{x}\right)+d\left({u}_{x},x\right)+d\left(x,y\right)\right\}+\frac{1}{2}\left\{d\left({u}_{y},x\right)+d\left(x,{u}_{x}\right)\right\}\\ \le \frac{7}{2}d\left(x,{u}_{x}\right)+\frac{1}{2}d\left({u}_{y},x\right)+\frac{1}{2}d\left(x,y\right).\end{array}$
Thus
$\frac{1}{2}d\left(x,{u}_{y}\right)\le \frac{7}{2}d\left(x,{u}_{x}\right)+\frac{1}{2}d\left(x,y\right)\phantom{\rule{1em}{0ex}}\text{if and only if}\phantom{\rule{1em}{0ex}}d\left(x,{u}_{y}\right)\le 7d\left(x,{u}_{x}\right)+d\left(x,y\right).$

Hence, the result follows from all the above cases. □

Corollary 2.7 Let X be a complete geodesic Ptolemy space, K a nonempty closed subset of X. Suppose $T:K\to {P}_{cp}\left(K\right)$ is a multi-valued mapping satisfying $SCC$ condition, then $d\left(x,{u}_{y}\right)\le 7d\left({u}_{x},x\right)+d\left(x,y\right)$ for all $x,y\in K$, ${u}_{x}\in Tx$, and ${u}_{y}\in Ty$.

Theorem 2.8 Let X be a complete geodesic Ptolemy space with a uniformly continuous midpoint map, and K a nonempty, bounded, closed, and convex subset of X. Suppose $T:K\to {P}_{cp}\left(K\right)$ is a multi-valued mapping satisfying the $SKC$ condition and ${x}_{n}$ is a sequence in K with ${lim}_{n\to \mathrm{\infty }}d\left({x}_{n},{u}_{{x}_{n}}\right)=0$, where ${u}_{{x}_{n}}\in T{x}_{n}$, then $F\left(T\right)\ne \mathrm{\varnothing }$.

Proof By Theorem 1.12, ${x}_{n}$ has unique asymptotic center denoted by x. Let $n\in \mathbb{N}$. Applying Theorem 2.6 for ${x}_{n}$, x, and ${u}_{{x}_{n}}$, respectively, it follows that there exists ${u}_{{z}_{n}}\in Tx$ such that $d\left({x}_{n},{u}_{{z}_{n}}\right)\le 7d\left({x}_{n},{u}_{{x}_{n}}\right)+d\left({x}_{n},x\right)$.

Let ${u}_{{z}_{{n}_{k}}}$ be a subsequence of ${u}_{{z}_{n}}$ that converges to some ${u}_{z}\in Tx$, then
$\begin{array}{rl}d\left({x}_{{n}_{k}},{u}_{z}\right)& \le d\left({x}_{{n}_{k}},{u}_{{z}_{{n}_{k}}}\right)+d\left({u}_{{z}_{{n}_{k}}},{u}_{z}\right)\\ \le 7d\left({x}_{{n}_{k}},{u}_{{x}_{{n}_{k}}}\right)+d\left({x}_{{n}_{k}},x\right)+d\left({u}_{{z}_{{n}_{k}}},{u}_{z}\right),\end{array}$

taking the superior limit as $k\to \mathrm{\infty }$ and knowing that the asymptotic center of $\left\{{x}_{{n}_{k}}\right\}$ is precisely x. Thus we obtain $x={u}_{z}\in Tx$. Hence the proof is complete. □

By the same idea of [[4], p.6] we construct a function $T:X\to P\left(X\right)$, which is $SKC$ and has a fixed point.

Example 2.9 Consider the space
$X=\left\{\left(0,0\right),\left(0,1\right),\left(1,1\right),\left(1,2\right)\right\}$
with ${l}^{\mathrm{\infty }}$ metric,
$d\left(\left({x}_{1},{y}_{1}\right),\left({x}_{2},{y}_{2}\right)\right)=max\left\{|{x}_{1}-{x}_{2}|,|{y}_{1}-{y}_{2}|\right\}.$

X is a geodesic Ptolemy space, but it is not a $CAT\left(0\right)$ space (see [4]).

Define a mapping T on X by
T satisfies the $SKC$ condition. Suppose $x=\left(0,0\right)$ and $y=\left(1,1\right)$, thus $Tx=\left\{\left(0,1\right)\right\}$, then ${u}_{x}=\left(0,1\right)$, so
$\frac{1}{2}d\left(x,{u}_{x}\right)=\frac{1}{2}d\left(\left(0,0\right),\left(0,1\right)\right)=\frac{1}{2}\le d\left(x,y\right)=d\left(\left(0,0\right),\left(1,1\right)\right)=1,$
and we can choose ${u}_{y}=\left(0,0\right)$,
$\begin{array}{rl}{N}^{\prime }\left(\left(0,0\right),\left(1,1\right)\right)=& max\left\{d\left(\left(0,0\right),\left(1,1\right)\right),\frac{1}{2}\left[d\left(\left(0,0\right),\left(0,1\right)\right)+d\left(\left(0,0\right),\left(1,1\right)\right)\right],\\ \frac{1}{2}\left[d\left(\left(0,1\right),\left(1,1\right)\right)+d\left(\left(0,0\right),\left(0,0\right)\right)\right]\right\}=1;\end{array}$
thus
$d\left({u}_{x},{u}_{y}\right)=d\left(\left(0,1\right),\left(0,0\right)\right)=1\le {N}^{\prime }\left(x,y\right)={N}^{\prime }\left(\left(0,0\right),\left(1,1\right)\right)=1.$

One can check the $SKC$ condition holds for the other points of the space X.

Note that $\left(1,1\right)\in T\left(1,1\right)$; thus $F\left(T\right)=\left\{\left(1,1\right)\right\}\ne \mathrm{\varnothing }$.

Corollary 2.10 Let X be a complete geodesic Ptolemy space with a uniformly continuous midpoint map, and K a nonempty bounded, closed, and convex subset of X. Suppose $T:K\to {P}_{cp}\left(K\right)$ is a multi-valued mapping satisfying the condition $SCC$ and ${x}_{n}$ is a sequence in K with ${lim}_{n\to \mathrm{\infty }}d\left({x}_{n},T{x}_{n}\right)=0$, then $F\left(T\right)\ne \mathrm{\varnothing }$.

One can find in [15] the multi-valued version of the ${E}_{\mu }$ and ${C}_{\lambda }$ conditions.

Definition 2.11 Let K be a subset of a metric space $\left(X,d\right)$. A map $T:K\to {P}_{cl,bd}\left(X\right)$ is said to satisfy the ${E}_{\mu }^{\prime }$ condition provided that
$dist\left(x,Ty\right)\le \mu dist\left(x,Tx\right)+d\left(x,y\right),\phantom{\rule{1em}{0ex}}x,y\in K;$

we say that T satisfies the ${E}^{\prime }$ condition whenever T satisfies ${E}_{\mu }^{\prime }$ for some $\mu \ge 1$.

One can replace the metric space with a Ptolemy space in the following definition.

Definition 2.12 Let K be a subset of a metric space $\left(X,d\right)$ and $\lambda \in \left(0,1\right)$. A map $T:K\to P\left(X\right)$ is said to satisfy the ${C}_{\lambda }^{\prime }$ condition if for each $x,y\in K$,
$\lambda dist\left(x,Tx\right)\le d\left(x,y\right)$
implies
$H\left(Tx,Ty\right)\le d\left(x,y\right),$

where $H\left(\cdot ,\cdot \right)$ stands for the Hausdorff distance.

Theorem 2.13 Let X be a complete geodesic Ptolemy space with a uniformly continuous midpoint map, and K be a nonempty bounded, closed, and convex subset of X. Suppose $T:K\to {P}_{cl,bd}\left(K\right)$ is a multi-valued mapping satisfying ${E}^{\prime }$ and ${C}_{\lambda }^{\prime }$ conditions, then $F\left(T\right)\ne \mathrm{\varnothing }$.

Proof We find an approximate fixed point for T. Take ${x}_{0}\in K$, since $T{x}_{0}\ne \mathrm{\varnothing }$ we can choose ${y}_{0}\in T{x}_{0}$. Define
${x}_{1}=\left(1-\lambda \right){x}_{0}\oplus \lambda {y}_{0}.$
Since K is convex, ${x}_{1}\in K$. Let ${y}_{1}\in T{x}_{1}$ be chosen such that
$d\left({y}_{0},{y}_{1}\right)=dist\left({y}_{0},T{x}_{1}\right).$
Similarly, set
${x}_{2}=\left(1-\lambda \right){x}_{1}\oplus \lambda {y}_{1}.$
Again we choose ${y}_{2}\in T{x}_{2}$ such that
$d\left({y}_{1},{y}_{2}\right)=dist\left({y}_{1},T{x}_{2}\right).$
By the same argument, we get ${y}_{2}\in K$. In this way we find a sequence $\left\{{x}_{n}\right\}\subset K$ such that
${x}_{n+1}=\left(1-\lambda \right){x}_{n}\oplus \lambda {y}_{n},$
where ${y}_{n}\in T{x}_{n}$ and
$d\left({y}_{n-1},{y}_{n}\right)=dist\left({y}_{n-1},T{x}_{n}\right).$
For every $n\in \mathbb{N}$
$\lambda d\left({x}_{n},{y}_{n}\right)=d\left({x}_{n},{x}_{n+1}\right),$
for which it follows that
$\lambda dist\left({x}_{n},T{x}_{n}\right)\le \lambda d\left({x}_{n},{y}_{n}\right)=d\left({x}_{n},{x}_{n+1}\right);$
since T satisfies the ${C}_{\lambda }^{\prime }$ condition,
$H\left(T{x}_{n},T{x}_{n+1}\right)\le d\left({x}_{n},{x}_{n+1}\right),$
this implies
$\begin{array}{rl}d\left({y}_{n+1},{y}_{n}\right)& =dist\left({y}_{n},T{x}_{n+1}\right)\\ \le H\left(T{x}_{n},T{x}_{n+1}\right)\\ \le d\left({x}_{n},{x}_{n+1}\right).\end{array}$
Now, we apply Lemma 1.11 to conclude ${lim}_{n\to \mathrm{\infty }}d\left({x}_{n},{y}_{n}\right)=0$, where ${y}_{n}\in T{x}_{n}$. The bounded sequence $\left\{{x}_{n}\right\}$ is Δ-convergent, hence by passing to a subsequence $\mathrm{\Delta }\text{-}{lim}_{n}{x}_{n}=v\in K$. We choose ${z}_{n}\in Tv$ such that
$d\left({x}_{n},{z}_{n}\right)=dist\left({x}_{n},Tv\right).$
Since Tv is compact, the sequence $\left\{{z}_{n}\right\}$ has a convergent subsequence $\left\{{z}_{{n}_{k}}\right\}$ with ${lim}_{k\to \mathrm{\infty }}{z}_{{n}_{k}}=w\in Tv$. Moreover, ${z}_{n}\in K$, and K is closed; then $w\in K$. By the ${E}^{\prime }$ condition
Note that
$\begin{array}{rl}d\left({x}_{{n}_{k}},w\right)& \le d\left({x}_{{n}_{k}},{z}_{{n}_{k}}\right)+d\left({z}_{{n}_{k}},w\right)\\ \le \mu dist\left({x}_{{n}_{k}},T{x}_{{n}_{k}}\right)+d\left({x}_{{n}_{k}},v\right)+d\left({z}_{{n}_{k}},w\right);\end{array}$
this implies
$\underset{n\to \mathrm{\infty }}{lim sup}d\left({x}_{{n}_{k}},w\right)\le \underset{n\to \mathrm{\infty }}{lim sup}d\left({x}_{{n}_{k}},v\right).$

Thus by the Opial property, $w=v\in Tv$. □

Example 2.14 [15]

Let $X=\mathbb{R}$ and $D=\left[0,\frac{7}{2}\right]$. Define a mapping T on D with $d\left(x,y\right)=|x-y|$ by
First we show T satisfies the ${C}_{\lambda }^{\prime }$ condition. Let $x,y\in \left[0,\frac{7}{2}\right)$, then
$H\left(Tx,Ty\right)=|\frac{x-y}{7}|\le |x-y|.$
Let $x\in \left[0,\frac{5}{2}\right]$ and $y=\frac{7}{2}$, then
$H\left(Tx,Ty\right)=1\le \frac{7}{2}-x.$
Let $x\in \left(\frac{5}{2},\frac{7}{2}\right)$ and $y=\frac{7}{2}$, then $dist\left(x,Tx\right)=\frac{6x}{7}$, thus
$\frac{1}{2}dist\left(x,Tx\right)=\frac{6x}{14}>\frac{30}{28}>1>|x-y|$
and
$\frac{1}{2}dist\left(y,Ty\right)=\frac{5}{4}>1>|x-y|.$
Thus T satisfies the ${C}_{\lambda }^{\prime }$ condition with $\lambda =\frac{1}{2}$. Let $x,y\in D$, then
$dist\left(x,Ty\right)\le 3d\left(x,Tx\right)+|x-y|,$

this shows T satisfies the ${E}^{\prime }$ condition. Since $T\left(0\right)=\left\{0\right\}$, $0\in F\left(T\right)\ne \mathrm{\varnothing }$.

## Authors’ Affiliations

(1)
Department of Mathematics, Science and Research Branch, Islamic Azad University, Tehran, Iran

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