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# Common fixed points for $(\psi ,\alpha ,\beta )$-weakly contractive mappings in generalized metric spaces

- Hüseyin Işık
^{1, 2}Email author and - Duran Türkoğlu
^{1, 3}

**2013**:131

https://doi.org/10.1186/1687-1812-2013-131

© Işık and Türkoğlu; licensee Springer 2013

**Received:**1 August 2012**Accepted:**29 April 2013**Published:**16 May 2013

## Abstract

We establish some common fixed point theorems for mappings satisfying a $(\psi ,\alpha ,\beta )$-weakly contractive condition in generalized metric spaces. Presented theorems extend and generalize many existing results in the literature.

**MSC:** Primary 54H25; secondary 47H10.

## Keywords

- fixed point
- generalized metric
- weakly contractive condition
- contraction of integral type
- common fixed points

## 1 Introduction and preliminaries

In 2000, Branciari [1] introduced the concept of a generalized metric space where the triangle inequality of a metric space was replaced by an inequality involving three terms instead of two. As such, any metric space is a generalized metric space, but the converse is not true [1]. He proved the Banach fixed point theorem in such a space. After that, many fixed point results have been established for this interesting space. For more, the reader can refer to [2–12].

It is also known that common fixed point theorems are generalizations of fixed point theorems. Recently, many researchers have interested in generalizing fixed point theorems to coincidence point theorems and common fixed point theorems. In a recent paper, Choudhury and Kundu [13] established the $(\psi ,\alpha ,\beta )$-weak contraction principle to coincidence point and common fixed point results in partially ordered metric spaces.

The purpose of this paper is to extend the results in [13] to the set of generalized metric spaces.

**Definition 1** ([1])

*X*be a non-empty set and let $d:X\times X\to [0,+\mathrm{\infty})$ be a mapping such that for all $x,y\in X$ and for all distinct points $u,v\in X$, each of them different from

*x*and

*y*, one has

- (i)
$d(x,y)=0$ if and only if $x=y$,

- (ii)
$d(x,y)=d(y,x)$,

- (iii)
$d(x,y)\le d(x,u)+d(u,v)+d(v,y)$ (the rectangular inequality).

Then $(X,d)$ is called a generalized metric space (or for short g.m.s.).

**Definition 2** ([1])

*X*and $x\in X$.

- (i)
We say that $\{{x}_{n}\}$ is a g.m.s. convergent to

*x*if and only if $d({x}_{n},x)\to 0$ as $n\to +\mathrm{\infty}$. We denote this by ${x}_{n}\to x$. - (ii)
We say that $\{{x}_{n}\}$ is a g.m.s. Cauchy sequence if and only if for each $\epsilon >0$ there exists a natural number $n(\epsilon )$ such that $d({x}_{n},{x}_{m})<\epsilon $ for all $n>m>n(\epsilon )$.

- (iii)
$(X,d)$ is called a complete g.m.s. if every g.m.s. Cauchy sequence is g.m.s. convergent in

*X*.

We denote by Ψ the set of functions $\psi :[0,+\mathrm{\infty})\to [0,+\mathrm{\infty})$ satisfying the following hypotheses:

(*ψ* 1) *ψ* is continuous and monotone nondecreasing,

(*ψ* 2) $\psi (t)=0$ if and only if $t=0$.

We denote by Φ the set of functions $\alpha :[0,+\mathrm{\infty})\to [0,+\mathrm{\infty})$ satisfying the following hypotheses:

(*α* 1) *α* is continuous,

(*α* 2) $\alpha (t)=0$ if and only if $t=0$.

We denote by Γ the set of functions $\beta :[0,+\mathrm{\infty})\to [0,+\mathrm{\infty})$ satisfying the following hypotheses:

(*β* 1) *β* is lower semi-continuous,

(*β* 2) $\beta (t)=0$ if and only if $t=0$.

## 2 Main results

**Definition 3** ([14])

Let *X* be a non-empty set and let $T,F:X\to X$. The mappings *T*, *F* are said to be weakly compatible if they commute at their coincidence points, that is, if $Tx=Fx$ for some $x\in X$ implies that $TFx=FTx$.

**Lemma 1**

*Let*$\{{a}_{n}\}$

*be a sequence of non*-

*negative real numbers*.

*If*

*for all*$n\in \mathbb{N}$,

*where*$\psi \in \mathrm{\Psi}$, $\alpha \in \mathrm{\Phi}$, $\beta \in \mathrm{\Gamma}$

*and*

*then the following hold*:

- (i)
${a}_{n+1}\le {a}_{n}$

*if*${a}_{n}>0$, - (ii)
${a}_{n}\to 0$

*as*$n\to +\mathrm{\infty}$.

*Proof*(i) Let, if possible, ${a}_{n}<{a}_{n+1}$ for some $n\in \mathbb{N}$. Then, using the monotone property of

*ψ*and (2.1), we have

(ii) By (i) the sequence $\{{a}_{n}\}$ is non-increasing, hence there is $a\ge 0$ such that ${a}_{n}\to a$ as $n\to +\mathrm{\infty}$. Letting $n\to +\mathrm{\infty}$ in (2.1), using the lower semi-continuity of *β* and the continuities of *ψ* and *α*, we obtain $\psi (a)\le \alpha (a)-\beta (a)$, which by (2.2) implies that $a=0$. □

**Theorem 1**

*Let*$(X,d)$

*be a Hausdorff and complete g*.

*m*.

*s*.

*and let*$T,F:X\to X$

*be self*-

*mappings such that*$TX\subseteq FX$,

*and*

*FX*

*is a closed subspace of*

*X*,

*and that the following condition holds*:

*for all* $x,y\in X$, *where* $\psi \in \mathrm{\Psi}$, $\alpha \in \mathrm{\Phi}$, $\beta \in \mathrm{\Gamma}$ *and satisfy condition* (2.2). *Then* *T* *and* *F* *have a unique coincidence point in* *X*. *Moreover*, *if* *T* *and* *F* *are weakly compatible*, *then* *T* *and* *F* *have a unique common fixed point*.

*Proof*Let ${x}_{0}$ be an arbitrary point in

*X*. Since $TX\subseteq FX$, we can define the sequence $\{{x}_{n}\}$ in

*X*by

*β*and the continuities of

*ψ*and

*α*, we obtain

*FX*is closed and by (2.4), $T{x}_{n}=F{x}_{n+1}$ for all $n\ge 0$, we have that there exists $w\in FX$ for which

*y*in

*X*such that $Fy=w$. From (2.3), we get

Therefore, *w* is a point of coincidence of *T* and *F*. The uniqueness of the point of coincidence is a consequence of condition (2.3).

*T*and

*F*. Since

*T*and

*F*are weakly compatible, by (2.12), we have that $TFy=FTy$, and

*y*is a common fixed point. If $y\ne w$, then by (2.3) we have

From (2.2), $Fy=Fw$. Then, by (2.12) and (2.13), we have $w=Fw=Tw$. Consequently, *w* is the unique common fixed point of *T* and *F*. □

Denote by Λ the set of functions $\gamma :[0,+\mathrm{\infty})\to [0,+\mathrm{\infty})$ satisfying the following hypotheses:

(${h}_{1}$) *γ* is a Lebesgue-integrable mapping on each compact of $[0,+\mathrm{\infty})$.

We have the following result.

**Theorem 2**

*Let*$(X,d)$

*be a Hausdorff and complete g*.

*m*.

*s*.

*and let*$T,F:X\to X$

*be self*-

*mappings such that*$TX\subseteq FX$,

*and*

*FX*

*is a closed subspace of*

*X*,

*and that the following condition holds*:

*for all* $x,y\in X$, *where* ${\gamma}_{1},{\gamma}_{2},{\gamma}_{3}\in \mathrm{\Lambda}$ *and satisfy condition* (2.2). *If* *T* *and* *F* *are weakly compatible*, *then* *T* *and* *F* *have a unique fixed point*.

*Proof* Follows from Theorem 1 by taking $\psi (t)={\int}_{0}^{t}{\gamma}_{1}(s)\phantom{\rule{0.2em}{0ex}}ds$, $\alpha (t)={\int}_{0}^{t}{\gamma}_{2}(s)\phantom{\rule{0.2em}{0ex}}ds$ and $\beta (t)={\int}_{0}^{t}{\gamma}_{3}(s)\phantom{\rule{0.2em}{0ex}}ds$. □

Taking ${\gamma}_{3}(s)=(1-k){\gamma}_{2}(s)$ for $k\in [0,1)$ in Theorem 2, we obtain the following result.

**Corollary 1**

*Let*$(X,d)$

*be a Hausdorff and complete g*.

*m*.

*s*.

*and let*$T,F:X\to X$

*be self*-

*mappings such that*$TX\subseteq FX$,

*and*

*FX*

*is a closed subspace of*

*X*,

*and that the following condition holds*:

*for all* $x,y\in X$, *where* ${\gamma}_{1},{\gamma}_{2}\in \mathrm{\Lambda}$ *and* $k\in [0,1)$ *and satisfy condition* (2.2). *If* *T* *and* *F* *are weakly compatible*, *then* *T* *and* *F* *have a unique fixed point*.

## Declarations

## Authors’ Affiliations

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