# Fixed point theorems in convex metric spaces

- Mohammad Moosaei
^{1}Email author

**2012**:164

https://doi.org/10.1186/1687-1812-2012-164

© Moosaei; licensee Springer 2012

**Received: **28 February 2012

**Accepted: **30 August 2012

**Published: **25 September 2012

## Abstract

In this paper, we study some fixed point theorems for self-mappings satisfying certain contraction principles on a convex complete metric space. In addition, we investigate some common fixed point theorems for a Banach operator pair under certain generalized contractions on a convex complete metric space. Finally, we also improve and extend some recent results.

**MSC:**47H09, 47H10, 47H19, 54H25.

## Keywords

## 1 Introduction

In 1970, Takahashi [1] introduced the notion of convexity in metric spaces and studied some fixed point theorems for nonexpansive mappings in such spaces. A convex metric space is a generalized space. For example, every normed space and cone Banach space is a convex metric space and convex complete metric space, respectively. Subsequently, Beg [2], Beg and Abbas [3, 4], Chang, Kim and Jin [5], Ciric [6], Shimizu and Takahashi [7], Tian [8], Ding [9], and many others studied fixed point theorems in convex metric spaces.

*C*of a cone Banach space

*X*with the norm ${\parallel x\parallel}_{p}=d(x,0)$, if a mapping $T:C\to C$ satisfies the condition

for all $x,y\in C$, where $2\le q<4$, then *T* has at least one fixed point. Letting $x=y$ in the above inequality, it is easy to see that *T* is an identity mapping. In this paper, the above result is improved and extended to a convex complete metric space.

## 2 Preliminaries

**Definition 2.1** (see [11])

*X*if for each $(x,y,\lambda )\in X\times X\times I$ and $u\in X$,

A metric space $(X,d)$ together with a convex structure *W* is called a convex metric space, which is denoted by $(X,d,W)$.

**Example 2.2** Let $(X,\parallel \phantom{\rule{0.25em}{0ex}}\parallel )$ be a normed space. The mapping $W:X\times X\times I\to X$ defined by $W(x,y,\lambda )=\lambda x+(1-\lambda )y$ for each $x,y\in X$, $\lambda \in I$ is a convex structure on *X*.

**Definition 2.3** (see [11])

Let $(X,d,W)$ be a convex metric space. A nonempty subset *C* of *X* is said to be convex if $W(x,y,\lambda )\in C$ whenever $(x,y,\lambda )\in C\times C\times I$.

**Definition 2.4** (see [3])

Let $(X,d,W)$ be a convex metric space and *C* be a convex subset of *X*. A self-mapping *f* on *C* has a property (I) if $f(W(x,y,\lambda ))=W(f(x),f(y),\lambda )$ for each $x,y\in C$ and $\lambda \in I$.

**Example 2.5** If $(X,\parallel \phantom{\rule{0.25em}{0ex}}\parallel )$ is a normed space, then every affine mapping on a convex subset of *X* has the property (I).

**Definition 2.6**Let $f,g:X\to X$. A point $x\in X$ is called

- (i)
a fixed point of

*f*if $f(x)=x$, - (ii)
a coincidence point of a pair $(f,g)$ if $f(x)=g(x)$,

- (iii)
a common fixed point of a pair $(f,g)$ if $f(x)=g(x)=x$.

$F(f)$, $C(f,g)$, and $F(f,g)$ denote the set of all fixed points of *f*, coincidence points of the pair $(f,g)$, and common fixed points of the pair $(f,g)$, respectively.

The ordered pair $(f,g)$ of two self-maps of a metric space $(X,d)$ is called a Banach operator pair if $F(g)$ is *f*-invariant, namely $f(F(g))\subseteq F(g)$.

**Example 2.8**(i) Let $(X,d)$ be a metric space and $k\ge 0$. If the self-maps

*f*,

*g*of

*X*satisfy $d(g(f(x)),f(x))\le kd(g(x),x)$ for all $x\in X$, then $(f,g)$ is a Banach operator pair.

- (ii)
It is obvious that a commuting pair $(f,g)$ of self-maps on

*X*(namely $fg(x)=gf(x)$ for all $x\in X$) is a Banach operator pair, but the converse is generally not true. For example, let $X=\mathbb{R}$ with the usual norm, and let $f(x)={x}^{2}-2x$, $g(x)={x}^{2}-x-3$ for all $x\in X$, then $F(g)=\{-1,3\}$. Moreover, $f(F(g))\subseteq F(g)$ implies that $(f,g)$ is a Banach operator pair, but the pair $(f,g)$ does not commute.

In [10], Karapinar obtained the following theorems.

**Theorem 2.9** (see Theorem 2.4 of [10])

*Let*

*C*

*be a closed and convex subset of a cone Banach space*

*X*

*with the norm*${\parallel x\parallel}_{p}=d(x,0)$,

*and*$T:C\to C$

*be a mapping which satisfies the condition*

*for all* $x,y\in C$, *where* $2\le q<4$. *Then*, *T* *has at least one fixed point*.

**Theorem 2.10** (see Theorem 2.6 of [10])

*Let*

*C*

*be a closed and convex subset of a cone Banach space*

*X*

*with the norm*${\parallel x\parallel}_{p}=d(x,0)$,

*and*$T:C\to C$

*be a mapping which satisfies the condition*

*for all* $x,y\in C$, *where* $2\le r<5$. *Then*, *T* *has at least one fixed point*.

## 3 Main results

To prove the next theorem, we need the following lemma.

**Lemma 3.1**

*Let*$(X,d,W)$

*be a convex metric space*,

*then the following statements hold*:

- (i)
$d(x,y)=d(x,W(x,y,\lambda ))+d(y,W(x,y,\lambda ))$

*for all*$(x,y,\lambda )\in X\times X\times I$. - (ii)
$d(x,W(x,y,\frac{1}{2}))=d(y,W(x,y,\frac{1}{2}))=\frac{1}{2}d(x,y)$

*for all*$x,y\in X$.

*Proof*(i) For any $(x,y,\lambda )\in X\times X\times I$, we have

- (ii)Let $x,y\in X$. By the definition of
*W*and using (i), we have$d(x,W(x,y,\frac{1}{2}))\le \frac{1}{2}d(x,y)=\frac{1}{2}d(x,W(x,y,\frac{1}{2}))+\frac{1}{2}d(y,W(x,y,\frac{1}{2})).$

for all $x,y\in C$, and the proof of the lemma is complete. □

The following theorem improves and extends Theorem 2.6 in [10].

**Theorem 3.2**

*Let*

*C*

*be a nonempty closed convex subset of a convex complete metric space*$(X,d,W)$

*and*

*f*

*be a self*-

*mapping of*

*C*.

*If there exist*

*a*,

*b*,

*c*,

*k*

*such that*

*for all* $x,y\in C$, *then* *f* *has at least one fixed point*.

*Proof*Suppose ${x}_{0}\in C$ is arbitrary. We define a sequence ${\{{x}_{n}\}}_{n=1}^{\mathrm{\infty}}$ in the following way:

*x*with ${x}_{n}$ and

*y*with ${x}_{n-1}$ in (3.2), we get

*c*be a nonnegative number. Using the triangle inequality, (3.4) and (3.5), we obtain

*C*. Therefore, it is a Cauchy sequence. Since

*C*is a closed subset of a complete space, there exists $v\in C$ such that ${lim}_{n\to \mathrm{\infty}}{x}_{n}=v$. Therefore, the triangle inequality and (3.4) imply ${lim}_{n\to \mathrm{\infty}}f({x}_{n})=v$. Now, by substituting

*x*with

*v*and

*y*with ${x}_{n}$ in (3.2), we obtain

Since $a+c$ is positive from (3.1), it follows that $d(v,f(v))=0$. Therefore, $f(v)=v$ and the proof of the theorem is complete. □

The following corollary improves and extends Theorem 2.4 in [10].

**Corollary 3.3**

*Let*$(X,d,W)$

*be a convex complete metric space and*

*C*

*be a nonempty closed convex subset of*

*X*.

*Suppose that*

*f*

*is a self*-

*map of*

*C*.

*If there exist*

*a*,

*b*,

*k*

*such that*

*for all* $x,y\in C$, *then* $F(f)$ *is a nonempty set*.

*Proof* Set $c=0$ in Theorem 3.2. □

**Theorem 3.4**

*Let*$(X,d,W)$

*be a convex complete metric space and*

*C*

*be a nonempty subset of*

*X*.

*Suppose that*

*f*,

*g*

*are self*-

*mappings of*

*C*,

*and there exist*

*a*,

*b*,

*c*,

*k*

*such that*

*for all* $x,y\in C$. *If* $(f,g)$ *is a Banach operator pair*, *g* *has the property* (*I*) *and* $F(g)$ *is a nonempty closed subset of* *C*, *then* $F(f,g)$ *is nonempty*.

*Proof*From (3.9), we obtain

for all $x,y\in F(g)$. $F(g)$ is convex because *g* has the property (I). It follows from Theorem 3.2 that $F(f,g)$ is nonempty. □

**Theorem 3.5**

*Let*$(X,d,W)$

*be a convex complete metric space and*

*C*

*be a nonempty subset of*

*X*.

*Suppose that*

*f*,

*g*

*are self*-

*mappings of*

*C*, $F(g)$

*is a nonempty closed subset of*

*C*,

*and there exist*

*a*,

*b*,

*c*,

*k*

*such that*

*for all* $x,y\in C$. *If* $(f,g)$ *is a Banach operator pair and* *g* *has the property* (*I*), *then* $F(f,g)$ *is nonempty*.

*Proof*Since $(f,g)$ is a Banach operator pair from (3.12), we have

for all $x,y\in F(g)$. Because *g* has the property (I) and $F(g)$ is closed, Theorem 3.2 guaranties that $F(f,g)$ is nonempty. □

## Declarations

### Acknowledgements

The author is grateful to Bu-Ali Sina University for supporting this research.

## Authors’ Affiliations

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