Coupled Fixed Point Theorems for Nonlinear Contractions Satisfied Mizoguchi-Takahashi's Condition in Quasiordered Metric Spaces
© Wei-Shih Du. 2010
Received: 10 November 2009
Accepted: 14 March 2010
Published: 30 March 2010
The main aim of this paper is to study and establish some new coupled fixed point theorems for nonlinear contractive maps that satisfied Mizoguchi-Takahashi's condition in the setting of quasiordered metric spaces or usual metric spaces.
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 single-valued 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 Rodríguez-López 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  proved the following interesting fixed point theorem for a weak contraction which is a partial answer of Problem 9 in Reich  (see also [14–16] and references therein).
Theorem MT. (Mizoguchi and Takahashi ).
Then there exists such that .
In fact, Mizoguchi-Takahashi's fixed point theorem is a generalization of Nadler's fixed point theorem [17, 18] which extended the Banach contraction principle (see, e.g., ) to multivalued maps, but its primitive proof is different. Recently, Suzuki  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 Mizoguchi-Takahashi'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 Bhaskar-Lakshmikantham's Coupled Fixed Point Theorems and Others
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 .
Definition 2.1 . (see ).
It is quite obvious that if has the mixed monotone property on , then for any , with (i.e., and ), .
Definition 2.2 . (see ).
Let be a metric space with a quasi-order 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 ).
A function is said to be a - if it satisfies Mizoguchi-Takahashi's condition (i.e., for all ).
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 ).
Let be a -function. Then defined by is also a -function.
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.
for each . If and , then is -nondecreasing and is -nonincreasing.
Hence, by induction, we prove that is -nondecreasing and is -nonincreasing.
If there exist such that and , then there exist , such that and
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 .
Since is arbitrary, or . Similarly, we can also prove that . The proof is completed.
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.
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 ;
If there exist , such that and , then there exist , , such that and
[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  for -functions.
Finally, we discuss the following coupled fixed point theorem in (usual) complete metric spaces.
Then has a unique coupled fixed point in ; that is, there exists unique such that and .
a contradiction. The proof is completed.
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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