# Coupled fixed point theorems for generalized symmetric Meir-Keeler contractions in ordered metric spaces

- Vasile Berinde
^{1}Email author and - Mădălina Păcurar
^{2}

**2012**:115

**DOI: **10.1186/1687-1812-2012-115

© Berinde and Păcurar; licensee Springer 2012

**Received: **14 January 2012

**Accepted: **2 July 2012

**Published: **20 July 2012

## Abstract

In this paper we introduce generalized symmetric Meir-Keeler contractions and prove some coupled fixed point theorems for mixed monotone operators $F:X\times X\to X$ in partially ordered metric spaces. The obtained results extend, complement and unify some recent coupled fixed point theorems due to Samet (Nonlinear Anal. 72:4508-4517, 2010), Bhaskar and Lakshmikantham (Nonlinear Anal. 65:1379-1393, 2006) and some other very recent papers. An example to show that our generalizations are effective is also presented.

## 1 Introduction

*T*is a contraction,

*i.e.*, there exists a constant $a\in [0,1)$ such that

then, by Banach contraction mapping principle, which is a classical and powerful tool in nonlinear analysis, we know that *T* has a unique fixed point *p* and, for any ${x}_{0}\in X$, the Picard iteration $\{{T}^{n}{x}_{0}\}$ converges to *p*.

In compensation, the authors in [20] assumed that *T* satisfies a certain monotonicity condition.

This new approach has been then followed by several authors: Agarwal *et al.* [1], Nieto and Lopez [16, 17], O’Regan and Petruşel [18], who obtained fixed point theorems, and also by Bhaskar and Lakshmikantham [9], Lakshmikantham and Ćirić [12], Luong and Thuan [15], Samet [22] and many others, who obtained coupled fixed point theorems or coincidence point theorems. These results also found important applications to the existence of solutions for matrix equations or ordinary differential equations and integral equations, see [9, 10, 15–20] and some of the references therein.

*mixed monotone property*if $F(x,y)$ is monotone nondecreasing in

*x*and is monotone non-increasing in

*y*, that is, for any $x,y\in X$,

We say *F* has the *strict mixed monotone property* if the strict inequality in the left-hand side of (1.4) and (1.5) implies the strict inequality in the right-hand side, respectively.

*coupled fixed point*of

*F*if

The next theorem is the main existence result in [22].

**Theorem 1** (Samet [22])

*Let*$(X,\le )$

*be a partially ordered set and suppose there is a metric*

*d*

*on*

*X*

*such that*$(X,d)$

*is a complete metric space*.

*Let*$F:X\times X\to X$

*be a continuous mapping having the strict mixed monotone property on*

*X*.

*Assume also that*

*F*

*is a generalized Meir*-

*Keeler operator*,

*that is*,

*for each*$\u03f5>0$,

*there exists*$\delta (\u03f5)>0$

*such that*

*for all*$x,y\in X$*satisfying*$x\ge u$, $y\le v$.

*If there exist*${x}_{0},{y}_{0}\in X$

*such that*

*then there exist*$x,y\in X$

*such that*

In the same paper [22] the author also established other existence as well as existence and uniqueness results for coupled fixed points of mixed strict monotone generalized Meir-Keeler operators.

Starting from the results in [22], our main aim in this paper is to obtain more general coupled fixed point theorems for mixed monotone operators $F:X\times X\to X$ satisfying a generalized Meir-Keeler contractive condition which is significantly weaker than (1.6). Our technique of proof is different and slightly simpler than the ones used in [22] and [9]. We thus extend, unify, generalize and complement several related results in literature, amongst which we mention the ones in [1, 9, 13, 20] and [22].

## 2 Main results

The first main result in this paper is the following coupled fixed point result which generalizes Theorem 1 (Theorem 2.1 in [22]) and Theorem 2.1 in [9] and some other related results.

**Theorem 2** *Let*$(X,\le )$*be a partially ordered set and suppose there is a metric* *d* *on* *X* *such that*$(X,d)$*is a complete metric space*. *Assume*$F:X\times X\to X$*is continuous and has the mixed monotone property and is also a generalized symmetric Meir*-*Keeler operator*, *that is*, *for each*$\u03f5>0$, *there exists*$\delta (\u03f5)>0$*such that for all*$x,y\in X$*satisfying*$x\ge u$, $y\le v$,

*implies*

*If there exist*${x}_{0},{y}_{0}\in X$

*such that*

*or*

*then there exist*$\overline{x},\overline{y}\in X$

*such that*

*Proof*Consider the functional ${d}_{2}:{X}^{2}\times {X}^{2}\to {\mathbb{R}}_{+}$ defined by

*T*and to the initial approximation ${Z}_{0}$, that is, the sequence $\{{Z}_{n}\}\subset {X}^{2}$ defined by

where ${Z}_{n}=({x}_{n},{y}_{n})\in {X}^{2}$, $n\ge 0$.

*F*is mixed monotone, we have

which shows that *T* is monotone and the sequence ${\{{Z}_{n}\}}_{n=0}^{\mathrm{\infty}}$ is nondecreasing.

is non-increasing, hence convergent to some $\u03f5\ge 0$.

*p*such that

*ϵ*. Therefore $\u03f5=0$, that is,

*k*such that

*k*, consider now the set

*T*is continuous in $({X}^{2},{d}_{2})$, and hence by (2.5) it follows that $\overline{Z}$ is a fixed point of

*T*, that is,

*T*, this means

that is, $(\overline{x},\overline{y})$ is a coupled fixed point of *F*. □

**Remark 1** Theorem 2 is more general than Theorem 1 (*i.e.*, Theorem 2.1 in [22]), since the contractive condition (2.1) is weaker than (1.6), a fact which is clearly illustrated by Example 1.

Apart from these improvements, we note that our proof is significantly simpler and shorter than the one in [22].

**Example 1**Let $X=\mathbb{R}$, $d(x,y)=|x-y|$ and $F:X\times X\to X$ be defined by

Then *F* is mixed monotone and satisfies condition (2.1) but does not satisfy condition (1.6).

a contradiction. Hence *F* does not satisfy condition (1.6).

which holds if we simply take $\delta (\u03f5)<\frac{5}{4}\u03f5$. Thus, condition (2.1) holds. Note also that ${x}_{0}=-3$, ${y}_{0}=3$ satisfy (2.2).

So Theorem 2 can be applied to *F* in this example to conclude that *F* has a (unique) coupled fixed point $(0,0)$, while Theorem 1 cannot be applied since (1.6) is not satisfied.

**Remark 2**One can prove that the coupled fixed point ensured by Theorem 2 is in fact unique, like in Example 1, provided that: every pair of elements in ${X}^{2}$ has either a lower bound or an upper bound, which is known, see [9], to be equivalent to the following condition: for all $Y=(x,y),\overline{Y}=(\overline{x},\overline{y})\in {X}^{2}$

**Theorem 3** *Adding condition* (2.12) *to the hypotheses of Theorem* 2, *we obtain the uniqueness of the coupled fixed point of* *F*.

*Proof* By Theorem 2 there exists a coupled fixed point $(\overline{x},\overline{y})$. In search for a contradiction, assume that ${Z}^{\ast}=({x}^{\ast},{y}^{\ast})\in {X}^{2}$ is a coupled fixed point of *F*, different from $\overline{Z}=(\overline{x},\overline{y})$. This means that ${d}_{2}({Z}^{\ast},\overline{Z})>0$. We discuss two cases:

Case 1. ${Z}^{\ast}$ is comparable to $\overline{Z}$.

a contradiction.

Case 2. ${Z}^{\ast}$ and $\overline{Z}$ are not comparable.

*T*, ${T}^{n}(Z)$ is comparable to ${T}^{n}({Z}^{\ast})={Z}^{\ast}$ and to ${T}^{n}(\overline{Z})=\overline{Z}$. Assume, without any loss of generality, that $\overline{x}<{z}_{1}$, $\overline{y}\ge {z}_{2}$ and ${x}^{\ast}<{z}_{1}$, ${y}^{\ast}\ge {z}_{2}$, which means $\overline{Z}<Z$ and ${Z}^{\ast}<Z$. By the monotonicity of

*T*, we have

is non-increasing, hence convergent to some $\u03f5\ge 0$.

*p*such that

*ϵ*. Therefore $\u03f5=0$, that is,

as $n\to \mathrm{\infty}$, which leads to the contradiction $0<{d}_{2}({Z}^{\ast},\overline{Z})\le 0$. □

Similarly to [9] and [22], by assuming a similar condition to (2.12), but this time with respect to the ordered set *X*, that is, by assuming that every pair of elements of *X* have either an upper bound or a lower bound in *X*, one can show that even the components of the coupled fixed points are equal.

**Theorem 4**

*In addition to the hypotheses of Theorem*3,

*suppose that every pair of elements of*

*X*

*has an upper bound or a lower bound in*

*X*.

*Then for the coupled fixed point*$(\overline{x},\overline{y})$

*we have*$\overline{x}=\overline{y}$,

*that is*,

*F*

*has a fixed point*

*Proof* Let $(\overline{x},\overline{y})$ be a coupled fixed point of *F* (ensured by Theorem 2). Suppose, to the contrary, that $\overline{x}\ne \overline{y}$. Without any loss of generality, we can assume $\overline{x}>\overline{y}$. We consider again two cases.

a contradiction.

*Y*,

*V*in ${X}^{2}$, one has

where *T* was defined in the proof of Theorem 3.

which shows that $d(\overline{x},\overline{y})=0$, that is $\overline{x}=\overline{y}$. □

Similarly, one can obtain the same conclusion under the following alternative assumption.

**Theorem 5**

*In addition to the hypotheses of Theorem*3,

*suppose that*${x}_{0},{y}_{0}\in X$

*are comparable*.

*Then for the coupled fixed point*$(\overline{x},\overline{y})$

*we have*$\overline{x}=\overline{y}$,

*that is*,

*F*

*has a fixed point*:

**Remark 3** Note that our contractive condition (2.1) is symmetric, while the contractive condition (1.6) used in [22] is not. Our generalization is based in fact on the idea of making the last one symmetric, which is very natural, as the great majority of contractive conditions in metrical fixed point theory are symmetric, see [2] and [21].

**Remark 4**Note also that if

*F*satisfies the contractive condition in [9], that is, there exists a constant $k\in [0,1)$ with

then, as pointed out by Proposition 2.1 in [22], *F* also satisfies the contractive condition (1.6) and hence (2.1).

This follows by simply taking $\delta (\u03f5)=(\frac{1}{k}-1)\u03f5$.

In view of the results in [14] and [24], the coupled fixed point theorems established in the present paper are also generalizations of all results in [1, 13, 20] and [22]. See also [3–8] and [11, 23] for other recent results.

## Declarations

### Acknowledgement

The research was supported by the Grant PN-II-RU-TE-2011-3-0239 of the Romanian Ministry of Education and Research.

## Authors’ Affiliations

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