Abstract
In this paper we introduce generalized symmetric MeirKeeler 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. 72:45084517, 2010), Bhaskar and Lakshmikantham (Nonlinear Anal. 65:13791393, 2006) and some other very recent papers. An example to show that our generalizations are effective is also presented.
1 Introduction
Let be a metric space and a self mapping. If is complete and T is a contraction, i.e., there exists a constant 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 , the Picard iteration converges to p.
The Banach contraction mapping principle has been generalized in several directions, see for example [2] and [21] for recent surveys. One of these generalizations, known as the MeirKeeler fixed point theorem [13], 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 [20] 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 [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 [9,10,1520] and some of the references therein.
In order to state the main result in [22], we recall the following notions. Let be a partially ordered set and endow the product space with the following partial order:
We say that a mapping has the mixed monotone property if is monotone nondecreasing in x and is monotone nonincreasing in y, that is, for any ,
and, respectively,
We say F has the strict mixed monotone property if the strict inequality in the lefthand side of (1.4) and (1.5) implies the strict inequality in the righthand side, respectively.
A pair is called a coupled fixed point of F if
The next theorem is the main existence result in [22].
Theorem 1 (Samet [22])
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 MeirKeeler operator, that is, for each, there existssuch that
then there existsuch 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 MeirKeeler 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 satisfying a generalized MeirKeeler 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 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 MeirKeeler operator, that is, for each, there existssuch that for allsatisfying, ,
implies
orthen there existsuch that
Proof Consider the functional defined by
It is a simple task to check that is a metric on and, moreover, that, if is complete, then is a complete metric space, too. Now consider the operator defined by
Clearly, for , , in view of the definition of , we have
and
Hence, by the contractive condition (2.1) we obtain a usual MeirKeeler type condition: for each there exists such that
Assume (2.2) holds (the case (2.3) is similar). Then, there exists such that
Denote and consider the Picard iteration associated to T and to the initial approximation , that is, the sequence defined by
Since F is mixed monotone, we have
and, by induction,
which shows that T is monotone and the sequence is nondecreasing.
Note that (2.4) implies the strict contractive condition
which shows that the sequence of nonnegative numbers given by
is nonincreasing, hence convergent to some .
We now prove that necessarily . Suppose, to the contrary, that . Then, there exist a positive integer p such that
which, by the MeirKeeler condition (2.4), yields
a contradiction, since converges nonincreasingly to ϵ. Therefore , that is,
Let now be arbitrary and the corresponding value from the hypothesis of our theorem. By (2.8), there exists a positive integer k such that
For this fixed number k, consider now the set
which, by (2.10) and MeirKeeler type condition (2.1), is . Thus, by (2.10) we have and, by induction,
This implies that for all , we have
Therefore, is a Cauchy sequence in the complete metric space , and hence there exists such that
By hypothesis, T is continuous in , and hence by (2.5) it follows that is a fixed point of T, that is,
Let . Then, by the definition of T, this means
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 [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 , and be defined by
Then F is mixed monotone and satisfies condition (2.1) but does not satisfy condition (1.6).
Assume, to the contrary, that (1.6) holds. Let , , , such that
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
and
and, by summing up the two inequalities above, we get for all , :
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.
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 [9], 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.
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:
As is comparable to with respect to the ordering in , by taking in (2.4) and (or and ), we obtain
a contradiction.
Case 2. and are not comparable.
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 shows that the sequence of nonnegative numbers given by
is nonincreasing, hence convergent to some .
We now prove that necessarily . Suppose, to the contrary, that . Then, there exists a positive integer p such that
which, by the MeirKeeler condition (2.4), yields
a contradiction, since converges nonincreasingly to ϵ. Therefore , that is,
Similarly, one obtains
Now, by (2.16) and (2.17), we have
as , which leads to the contradiction . □
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 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
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.
Case 1. If , are comparable, then is comparable to and hence, by taking , , , , in (2.13) one obtains
a contradiction.
Case 2. If , are not comparable, then there exists a comparable to and . Suppose and (the other case is similar). Then in view of the order on , it follows that
that is , ; , ; , are comparable in . Now, similarly to the proof of Theorem 3, we obtain that, for any two comparable elements Y, V in , one has
where T was defined in the proof of Theorem 3.
Now use (2.19) for the comparable pairs , ; , ; , , respectively, to get
Now, by using the triangle inequality and (2.20), (2.21), (2.22), one has
which shows that , that is . □
Similarly, one can obtain the same conclusion under the following alternative assumption.
Theorem 5In addition to the hypotheses of Theorem 3, suppose thatare comparable. Then for the coupled fixed pointwe have, that is, Fhas 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 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 .
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 [38] and [11,23] for other recent results.
Competing interests
The authors declare that they have no competing interests.
Authors’ contributions
Both authors contributed equally to the writing of this paper. They read and approved the final manuscript.
Acknowledgement
The research was supported by the Grant PNIIRUTE201130239 of the Romanian Ministry of Education and Research.
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