Coupled fixed point theorems for generalized symmetric Meir-Keeler contractions in ordered metric spaces
© Berinde and Păcurar; licensee Springer 2012
Received: 14 January 2012
Accepted: 2 July 2012
Published: 20 July 2012
In this paper we introduce generalized symmetric Meir-Keeler contractions and prove some coupled fixed point theorems for mixed monotone operators 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.
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 , the Picard iteration converges to p.
In compensation, the authors in  assumed that T satisfies a certain monotonicity condition.
This new approach has been then followed by several authors: Agarwal et al. , Nieto and Lopez [16, 17], O’Regan and Petruşel , who obtained fixed point theorems, and also by Bhaskar and Lakshmikantham , Lakshmikantham and Ćirić , Luong and Thuan , Samet  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.
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.
The next theorem is the main existence result in .
Theorem 1 (Samet )
for allsatisfying, .
In the same paper  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 , our main aim in this paper is to obtain more general coupled fixed point theorems for mixed monotone operators 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  and . We thus extend, unify, generalize and complement several related results in literature, amongst which we mention the ones in [1, 9, 13, 20] and .
2 Main results
Theorem 2 Letbe a partially ordered set and suppose there is a metric d on X such thatis a complete metric space. Assumeis continuous and has the mixed monotone property and is also a generalized symmetric Meir-Keeler operator, that is, for each, there existssuch that for allsatisfying, ,
where , .
which shows that T is monotone and the sequence is nondecreasing.
is non-increasing, hence convergent to some .
that is, is a coupled fixed point of F. □
Remark 1 Theorem 2 is more general than Theorem 1 (i.e., Theorem 2.1 in ), 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 .
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 . Thus, condition (2.1) holds. Note also that , satisfy (2.2).
So Theorem 2 can be applied to F in this example to conclude that F has a (unique) coupled fixed point , while Theorem 1 cannot be applied since (1.6) is not satisfied.
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 . In search for a contradiction, assume that is a coupled fixed point of F, different from . This means that . We discuss two cases:
Case 1. is comparable to .
Case 2. and are not comparable.
is non-increasing, hence convergent to some .
as , which leads to the contradiction . □
Similarly to  and , 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.
Proof Let be a coupled fixed point of F (ensured by Theorem 2). Suppose, to the contrary, that . Without any loss of generality, we can assume . We consider again two cases.
where T was defined in the proof of Theorem 3.
which shows that , that is . □
Similarly, one can obtain the same conclusion under the following alternative assumption.
Remark 3 Note that our contractive condition (2.1) is symmetric, while the contractive condition (1.6) used in  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  and .
then, as pointed out by Proposition 2.1 in , F also satisfies the contractive condition (1.6) and hence (2.1).
This follows by simply taking .
In view of the results in  and , the coupled fixed point theorems established in the present paper are also generalizations of all results in [1, 13, 20] and . See also [3–8] and [11, 23] for other recent results.
The research was supported by the Grant PN-II-RU-TE-2011-3-0239 of the Romanian Ministry of Education and Research.
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