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.
The Banach contraction mapping principle has been generalized in several directions, see for example  and  for recent surveys. One of these generalizations, known as the Meir-Keeler fixed point theorem , has been obtained by replacing the contraction condition (1.1) by the following more general assumption: for all there exists such that
Recently, Ran and Reurings  have initiated another important direction in generalizing the Banach contraction mapping principle by considering a partial ordering on the metric space and by requiring that the contraction condition (1.1) is satisfied only for comparable elements, that is, we have
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.
In order to state the main result in , we recall the following notions. Let be a partially ordered set and endow the product space with the following partial order:
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 )
Letbe a partially ordered set and suppose there is a metricdonXsuch thatis a complete metric space. Letbe a continuous mapping having the strict mixed monotone property onX. Assume also thatFis a generalized Meir-Keeler operator, that is, for each, there existssuch that
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 2Letbe a partially ordered set and suppose there is a metricdonXsuch 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, ,
Since F is mixed monotone, we have
and, by induction,
Note that (2.4) implies the strict contractive condition
which, by the Meir-Keeler condition (2.4), yields
For this fixed number k, consider now the set
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).
which by (1.6) would imply
and this in turn, by (2.11), would imply
a contradiction. Hence F does not satisfy condition (1.6).
Now we prove that (2.1) holds. Indeed, we have
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 has either a lower bound or an upper bound, which is known, see , to be equivalent to the following condition: for all
Theorem 3Adding condition (2.12) to the hypotheses of Theorem 2, we obtain the uniqueness of the coupled fixed point ofF.
In this case, there exists an upper bound or a lower bound of and . Then, in view of the monotonicity of T, is comparable to and to . Assume, without any loss of generality, that , and , , which means and . By the monotonicity of T, we have
Note that, like in the proof of Theorem 2, condition (2.4) implies the strict contractive condition
which, by the Meir-Keeler condition (2.4), yields
Similarly, one obtains
Now, by (2.16) and (2.17), we have
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.
Theorem 4In addition to the hypotheses of Theorem 3, suppose that every pair of elements ofXhas an upper bound or a lower bound inX. Then for the coupled fixed pointwe have, that is, Fhas a fixed point
where T was defined in the proof of Theorem 3.
Now, by using the triangle inequality and (2.20), (2.21), (2.22), one has
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 .
Remark 4 Note also that if F satisfies the contractive condition in , that is, there exists a constant with
then, as pointed out by Proposition 2.1 in , F also satisfies the contractive condition (1.6) and hence (2.1).
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 authors declare that they have no competing interests.
Both authors contributed equally to the writing of this paper. They read and approved the final manuscript.
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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