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

- SJ Hosseini Ghoncheh
^{1}and - A Razani
^{1}Email author

**2014**:471

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

© Hosseini Ghoncheh and Razani; licensee Springer. 2014

**Received:**16 February 2014**Accepted:**13 November 2014**Published:**26 November 2014

## 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

- $CAT(0)$ 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]

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 $CAT(0)$ 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

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*.

- (i)
*T*is said to satisfy the $SCC$ condition if$\frac{1}{2}d(x,Tx)\le d(x,y)\phantom{\rule{1em}{0ex}}\text{implies}\phantom{\rule{1em}{0ex}}d(Tx,Ty)\le M(x,y),$where$M(x,y)=max\{d(x,y),d(x,Tx),d(Ty,y),d(Tx,y),d(x,Ty)\}\phantom{\rule{1em}{0ex}}\text{for all}x,y\in K.$ - (ii)
*T*is said to satisfy the $SKC$ condition if$\frac{1}{2}d(x,Tx)\le d(x,y)\phantom{\rule{1em}{0ex}}\text{implies}\phantom{\rule{1em}{0ex}}d(Tx,Ty)\le N(x,y),$where$N(x,y)=max\{d(x,y),\frac{1}{2}\{d(x,Tx)+d(Ty,y)\},\frac{1}{2}\{d(Tx,y)+d(x,Ty)\}\}\phantom{\rule{1em}{0ex}}\text{for all}x,y\in K.$ - (iii)
*T*is said to satisfy the $KSC$ condition if$\frac{1}{2}d(x,Tx)\le d(x,y)\phantom{\rule{1em}{0ex}}\text{implies}\phantom{\rule{1em}{0ex}}d(Tx,Ty)\le \frac{1}{2}\{d(x,Tx)+d(Ty,y)\}.$ - (iv)
*T*is said to satisfy the $CSC$ condition if$\frac{1}{2}d(x,Tx)\le d(x,y)\phantom{\rule{1em}{0ex}}\text{implies}\phantom{\rule{1em}{0ex}}d(Tx,Ty)\le \frac{1}{2}\{d(Tx,y)+d(x,Ty)\}.$

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]

*T*on $[0,3]$ with $d(x,y)=|x-y|$ 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* $F(T)$ *is closed*. *Moreover*, *if* *X* *is strictly convex and* *K* *is convex*, *then* $F(T)$ *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* $d(x,Ty)\le 5d(Tx,x)+d(x,y)$ *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.

*X*be a metric space and $\{{x}_{n}\}$ be a bounded sequence in

*X*. For $x\in X$, let

*K*is given by

*K*is the set

**Definition 1.8** [9]

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

**Lemma 1.9**

- (i)
*Every bounded sequence in**X**has a*Δ-*convergent subsequence*[[10],*p*.3690]. - (ii)
*If**C**is a closed convex subset of**X**and if*$\{{x}_{n}\}$*is a bounded sequence in**C*,*then the asymptotic center of*$\{{x}_{n}\}$*is in**C*[[11],*Proposition*2.1]. - (iii)
*If**C**is a closed convex subset of**X**and if*$f:C\to X$*is a nonexpansive mapping*,*then the conditions*, $\{{x}_{n}\}\mathrm{\Delta}$-*converges to**x**and*$d({x}_{n},f({x}_{n}))\to 0$,*imply*$x\in C$*and*$f(x)=x$ [[10],*Proposition*3.7].

**Lemma 1.10** [12]

*If* $\{{x}_{n}\}$ *is a bounded sequence in* *X* *with* $A(\{{x}_{n}\})=\{x\}$ *and* $\{{u}_{n}\}$ *is a subsequence of* $\{{x}_{n}\}$ *with* $A(\{{u}_{n}\})=\{u\}$ *and the sequence* $\{d({x}_{n},u)\}$ *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* $\{{z}_{n}\}$ *and* $\{{w}_{n}\}$ *be bounded sequences in metric space* *K* *and* $\lambda \in (0,1)$. *Suppose* ${z}_{n+1}=\lambda {w}_{n}+(1-\lambda ){z}_{n}$ *and* $d({w}_{n+1},{w}_{n})\le d({z}_{n+1},{z}_{n})$ *for all* $n\in \mathbb{N}$. *Then* ${lim\hspace{0.17em}sup}_{n\to \mathrm{\infty}}d({w}_{n},{z}_{n})=0$.

**Theorem 1.12** [5]

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

## 2 Main results

*X*be complete geodesic Ptolemy space and $P(X)$ 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 $T:K\to {P}_{f}(X)$ 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]

*K*be a subset of a $CAT(0)$ space

*X*. A map $T:X\to P(X)$ is said to satisfy the

*C*condition if for each $x\in K$, ${u}_{x}\in Tx$, and $y\in K$

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* $T:K\to {P}_{cp}(K)$ *is a multi*-*valued mapping satisfying the* *C* *condition*, *then* $F(T)\ne \mathrm{\varnothing}$.

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 $T:X\to P(X)$ is said to satisfy conditions (i) $SCC$, (ii) $SKC$, (iii) $KSC$, (iv) $CSC$ if for each $x\in K$, ${u}_{x}\in Tx$, and $y\in K$

- (i)$d({u}_{x},{u}_{y})\le {M}^{\prime}(x,y)$, where${M}^{\prime}(x,y)=max\{d(x,y),d(x,{u}_{x}),d({u}_{y},y),d({u}_{x},y),d(x,{u}_{y})\},$
- (ii)$d({u}_{x},{u}_{y})\le {N}^{\prime}(x,y)$, where${N}^{\prime}(x,y)=max\{d(x,y),\frac{1}{2}\{d(x,{u}_{x})+d({u}_{y},y)\},\frac{1}{2}\{d({u}_{x},y)+d(x,{u}_{y})\}\},$
- (iii)
$d({u}_{x},{u}_{y})\le \frac{1}{2}\{d(x,{u}_{x})+d({u}_{y},y)\}$,

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

**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*$T:K\to {P}_{cp}(K)$

*is a multi*-

*valued mapping satisfying the*$SKC$

*condition*,

*then for every*$x,y\in K$, ${u}_{x}\in T(x)$

*and*${u}_{xx}\in T({u}_{x})$

*the following hold*:

- (i)
$d({u}_{x},{u}_{xx})\le d(x,{u}_{x})$,

- (ii)
*either*$\frac{1}{2}d(x,{u}_{x})\le d(x,y)$*or*$\frac{1}{2}d({u}_{x},{u}_{xx})\le d({u}_{x},y)$, - (iii)
*either*$d({u}_{x},{u}_{y})\le {N}^{\prime}(x,y)$*or*$d({u}_{y},{u}_{xx})\le {N}^{\prime}({u}_{x},y)$,

*where*

*Proof*The first statement follows from the $SKC$ condition. Indeed, we always have

□

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

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* $T:K\to {P}_{cp}(K)$ *is a multi*-*valued mapping satisfying* $SCC$ *condition*, *then* $d(x,{u}_{y})\le 7d({u}_{x},x)+d(x,y)$ *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* $T:K\to {P}_{cp}(K)$ *is a multi*-*valued mapping satisfying the* $SKC$ *condition and* ${x}_{n}$ *is a sequence in* *K* *with* ${lim}_{n\to \mathrm{\infty}}d({x}_{n},{u}_{{x}_{n}})=0$, *where* ${u}_{{x}_{n}}\in T{x}_{n}$, *then* $F(T)\ne \mathrm{\varnothing}$.

*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 $d({x}_{n},{u}_{{z}_{n}})\le 7d({x}_{n},{u}_{{x}_{n}})+d({x}_{n},x)$.

taking the superior limit as $k\to \mathrm{\infty}$ and knowing that the asymptotic center of $\{{x}_{{n}_{k}}\}$ 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 $T:X\to P(X)$, which is $SKC$ and has a fixed point.

**Example 2.9**Consider the space

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

*T*on

*X*by

*T*satisfies the $SKC$ condition. Suppose $x=(0,0)$ and $y=(1,1)$, thus $Tx=\{(0,1)\}$, then ${u}_{x}=(0,1)$, so

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

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

**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* $T:K\to {P}_{cp}(K)$ *is a multi*-*valued mapping satisfying the condition* $SCC$ *and* ${x}_{n}$ *is a sequence in* *K* *with* ${lim}_{n\to \mathrm{\infty}}d({x}_{n},T{x}_{n})=0$, *then* $F(T)\ne \mathrm{\varnothing}$.

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 $(X,d)$. A map $T:K\to {P}_{cl,bd}(X)$ is said to satisfy the ${E}_{\mu}^{\prime}$ condition provided that

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 $(X,d)$ and $\lambda \in (0,1)$. A map $T:K\to P(X)$ is said to satisfy the ${C}_{\lambda}^{\prime}$ condition if for each $x,y\in K$,

where $H(\cdot ,\cdot )$ 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* $T:K\to {P}_{cl,bd}(K)$ *is a multi*-*valued mapping satisfying* ${E}^{\prime}$ *and* ${C}_{\lambda}^{\prime}$ *conditions*, *then* $F(T)\ne \mathrm{\varnothing}$.

*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

*K*is convex, ${x}_{1}\in K$. Let ${y}_{1}\in T{x}_{1}$ be chosen such that

*T*satisfies the ${C}_{\lambda}^{\prime}$ condition,

*Tv*is compact, the sequence $\{{z}_{n}\}$ has a convergent subsequence $\{{z}_{{n}_{k}}\}$ 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

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

**Example 2.14** [15]

*T*on

*D*with $d(x,y)=|x-y|$ by

*T*satisfies the ${C}_{\lambda}^{\prime}$ condition. Let $x,y\in [0,\frac{7}{2})$, then

*T*satisfies the ${C}_{\lambda}^{\prime}$ condition with $\lambda =\frac{1}{2}$. Let $x,y\in D$, then

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

## Declarations

## Authors’ Affiliations

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