The main aim of this paper is to study and establish some new coupled fixed point theorems for nonlinear contractive maps that satisfied MizoguchiTakahashi's condition in the setting of quasiordered metric spaces or usual metric spaces.
1. Introduction
Let be a metric space. For each and , let . Denote by the class of all nonempty subsets of and the family of all nonempty closed and bounded subsets of . A function defined by
is said to be the Hausdorff metric on induced by the metric on . A point in is a fixed point of a map if (when is a singlevalued map) or (when is a multivalued map). Throughout this paper we denote by and the set of positive integers and real numbers, respectively.
The existence of fixed point in partially ordered sets has been investigated recently in [1–11] and references therein. In [6, 8], Nieto and RodríguezLópez used Tarski's theorem to show the existence of solutions for fuzzy equations and fuzzy differential equations, respectively. The existence of solutions for matrix equations or ordinary differential equations by applying fixed point theorems is presented in [2, 4, 7, 9, 10]. The authors in [3, 11] proved some fixed point theorems for a mixed monotone mapping in a metric space endowed with partial order and applied their results to problems of existence and uniqueness of solutions for some boundary value problems.
The various contractive conditions are important to find the existence of fixed point. There is a trend to weaken the requirement on the contraction. In 1989, Mizoguchi and Takahashi [12] proved the following interesting fixed point theorem for a weak contraction which is a partial answer of Problem 9 in Reich [13] (see also [14–16] and references therein).
Theorem MT. (Mizoguchi and Takahashi [12]).
Let be a complete metric space and a map from into . Assume that
for all , , where is a function from into satisfying
Then there exists such that .
In fact, MizoguchiTakahashi's fixed point theorem is a generalization of Nadler's fixed point theorem [17, 18] which extended the Banach contraction principle (see, e.g., [18]) to multivalued maps, but its primitive proof is different. Recently, Suzuki [19] gave a very simple proof of Theorem MT.
The purpose of this paper is to present some new coupled fixed point theorems for weakly contractive maps that satisfied MizoguchiTakahashi's condition (i.e., for all ) in the setting of quasiordered metric spaces or usual metric spaces. Our results generalize and improve some results in [2, 7, 9] and references therein.
2. Generalized BhaskarLakshmikantham's Coupled Fixed Point Theorems and Others
Let be a nonempty set and "" a quasiorder (preorder or pseudoorder, i.e., a reflexive and transitive relation) on . Then is called a quasiordered set. A sequence is called  (resp., ) if (resp., ) for each . Let be a metric space with a quasiorder ( for short). We endow the product space with the metric defined by
A map is said to be continuous at if any sequence with implies . is said to be continuous on if is continuous at every point of .
In this paper, we also endow the product space with the following quasiorder :
Definition 2.1 . (see [2]).
Let be a quasiordered set and a map. We say that has the mixed monotone property on if is monotone nondecreasing in and is monotone nonincreasing in that is, for any ,
It is quite obvious that if has the mixed monotone property on , then for any , with (i.e., and ), .
Definition 2.2 . (see [2]).
Let be a nonempty set and a map. We call an element a coupled fixed point of if
Definition 2.3.
Let be a metric space with a quasiorder A nonempty subset of is said to be
 if every nondecreasing Cauchy sequence in converges;
 if every nonincreasing Cauchy sequence in converges;
 if it is both complete and .
Definition 2.4 . (see [20]).
A function is said to be a  if it satisfies MizoguchiTakahashi's condition (i.e., for all ).
Remark 2.5.
Obviously, if is defined by where , then is a function.
If is a nondecreasing function, then is a function.
Notice that is a function if and only if for each there exist and such that for all Indeed, if is a function, then for all So for each there exists such that . Therefore we can find such that , and hence for all . The converse part is obvious.
The following lemmas are crucial to our proofs.
Lemma 2.6 . (see [20]).
Let be a function. Then defined by is also a function.
Proof.
Clearly, and for all . Let be fixed. Since is a function, there exist and such that for all Let . Then for all and hence is a function.
Lemma 2.7.
Let be a quasiordered set and a map having the mixed monotone property on . Let , . Define two sequences and by
for each . If and , then is nondecreasing and is nonincreasing.
Proof.
Since and , by (2.5), and the mixed monotone property of , we have
Let and assume that and is already known. Then
Hence, by induction, we prove that is nondecreasing and is nonincreasing.
Theorem 2.8.
Let be a sequentially complete metric space and a continuous map having the mixed monotone property on . Assume that there exists a function such that for any with
If there exist such that and , then there exist , such that and
Proof.
By Lemma 2.6, we can define a function by . Then and for all . For any , let and be defined as in Lemma 2.7. Then, by Lemma 2.7, is nondecreasing and is nonincreasing. So and for each . By (2.8), we obtain
It follows that
For each , let Then By induction, we can obtain the following: for each ,
Since for all , the sequence is strictly decreasing in from (2.13). Let . Since is a function, there exists and such that for all . Also, there exists such that
for all with . So for each . Let and , . We claim that is a nondecreasing Cauchy sequence in and is a nonincreasing Cauchy sequence in . Indeed, from our hypothesis, for each , we have
Similarly,
Hence we get
So it follows from (2.17) that
Let For , with , we have
Since and hence
So is a nondecreasing Cauchy sequence in and is a nonincreasing Cauchy sequence in . By the sequentially completeness of , there exist , such that and as . Hence and as .
Let be given. Since is continuous at , there exists such that
whenever with . Since and as , for , there exist such that
So, for each with , by (2.22),
and hence we have from (2.21) that
Therefore
Since is arbitrary, or . Similarly, we can also prove that . The proof is completed.
Remark 2.9.
Theorem 2.8 generalizes and improves BhaskarLakshmikantham's coupled fixed points theorem [2, Theorem 2.1] and some results in [7, 9].
Following a similar argument as in the proof of [2, Theorem 2.2] and applying Theorem 2.8, one can verify the following result where is not necessarily continuous.
Theorem 2.10.
Let be a sequentially complete metric space and a map having the mixed monotone property on . Assume that
any nondecreasing sequence with implies for each ;
any nonincreasing sequence with implies for each ;
there exists a function such that for any , with ,
If there exist , such that and , then there exist , , such that and
Remark 2.11.
[2, Theorem 2.2] is a special case of Theorem 2.10.
Similarly, we can obtain the generalizations of Theorems 2.4–2.6 in [2] for functions.
Finally, we discuss the following coupled fixed point theorem in (usual) complete metric spaces.
Theorem 2.12.
Let be a complete metric space and a map. Assume that there exists a function such that for any
Then has a unique coupled fixed point in ; that is, there exists unique such that and .
Proof.
Let be given. For any define and . By our hypothesis, we know that is continuous. Following the same argument as in the proof of Theorem 2.8, there exists , such that and . We prove the uniqueness of the coupled fixed point of . On the contrary, suppose that there exists , such that and . Then we obtain
It follows from (2.28) that
a contradiction. The proof is completed.
Acknowledgment
This paper is dedicated to Professor Wataru Takahashi in celebration of his retirement. This research was supported by the National Science Council of the Republic of China.
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