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Generalized Caristi's Fixed Point Theorems
Fixed Point Theory and Applications volume 2009, Article number: 170140 (2009)
Abstract
We present generalized versions of Caristi's fixed point theorem for multivalued maps. Our results either improve or generalize the corresponding generalized Caristi's fixed point theorems due to Bae (2003), Suzuki (2005), Khamsi (2008), and others.
1. Introduction
A number of extensions of the Banach contraction principle have appeared in literature. One of its most important extensions is known as Caristi's fixed point theorem. It is well known that Caristi's fixed point theorem is equivalent to Ekland variational principle [1], which is nowadays an important tool in nonlinear analysis. Many authors have studied and generalized Caristi's fixed point theorem to various directions. For example, see [2–8]. Kada et al. [9] and Suzuki [10] introduced the concepts of -distance and -distance on metric spaces, respectively. Using these generalized distances, they improved Caristi's fixed point theorem and Ekland variational principle for single-valued maps. In this paper, using the concepts of -distance and -distance, we present some generalizations of the Caristi's fixed point theorem for multivalued maps. Our results either improve or generalize the corresponding results due to Bae [4, 11], Kada et al. [9], Suzuki [8, 10], Khamsi [5], and many of others.
Let be a metric space with metric . We use to denote the collection of all nonempty subsets of . A point is called a fixed point of a map () if ().
In 1976, Caristi [12] obtained the following fixed point theorem on complete metric spaces, known as Caristi's fixed point theorem.
Theorem 1.1.
Let be a complete metric space with metric . Let be a lower semicontinuous function, and let be a single-valued map such that for any ,
Then has a fixed point.
To generalize Theorem 1.1, one may consider the weakening of one or more of the following hypotheses: (i) the metric ; (ii) the lower semicontinuity of the real-valued function ; (iii) the inequality (1.1); (iv) the function .
In [9], Kada et al. introduced a concept of -distance on a metric space as follows.
A function is a -distance on if it satisfies the following conditions for any :
(w 1 )
(w 2 ) the map is lower semicontinuous;
(w 3 ) for any there exists such that and imply
Clearly, the metric is a -distance on . Let be a normed space. Then the functions defined by and for all are -distances. Many other examples of -distance are given in [9, 13]. Note that, in general, for , , and neither of the implications necessarily holds.
In the sequel, otherwise specified, we shall assume that is a complete metric space with metric , is a lower semicontinuous function and is a -distance on .
Using the concept of -distance, Kada et al. [9] generalized Caristi's fixed point theorem as follows.
Theorem 1.2.
Let be a single-valued self map on such that for every
Then, there exists such that and
2. The Results
Applying Theorem 1.2, first we prove the following generalization of Theorem 1.1.
Theorem 2.1.
Let be any function such that for some ,
Let be a multivalued map such that for each there exists satisfying
Then has a fixed point such that
Proof.
Define a function by Note that for each we have
Now, since it follows that Put
Note that is nonempty, and by the lower semicontinuity of and , is closed subset of a complete metric space , and hence it is complete. Now, we show that Let , and , then we have
and thus , and hence is a self map on . Note that is lower semicontinuous and for each , we have
By Theorem 1.2, there exists such that and
Now, applying Theorem 2.1, we obtain generalized Caristi's fixed point results.
Theorem 2.2.
Let be a multivalued map such that for each there exists satisfying
where is an upper semicontinuous function from the right. Then has a fixed point such that
Proof.
Put . By the definition of the function , there exist some positive real numbers , such that for all Now, for all , we define
Clearly, maps into . Note that for all , we get , and thus for any with , we have
Now, clearly, and hence we obtain
By Theorem 2.1, has a fixed point such that
Theorem 2.3.
Let be a multivalued map such that for each there exists satisfying
where is nondecreasing function. Then has a fixed point such that
Proof.
For each , define . Clearly, does carry into . Now, since the function is nondecreasing, for any real number we have
Thus, by Theorem 2.1, the result follows.
Corollary 2.4.
Let be a multivalued map such that for each there exists satisfying
where is a nondecreasing function. Then has a fixed point such that
Proof.
Since for each there is such that and the function is nondecreasing, we have . Thus the result follows from Theorem 2.3.
Applying Theorem 2.3, we prove the following fixed point result.
Theorem 2.5.
Let be a multivalued map such that for each there exists satisfying and
where is an upper semicontinuous function. Then has a fixed point such that
Proof.
Define a function from into by
Clearly, is nondecreasing function. Now, since , we have . Thus by Theorem 2.3, the result follows.
The following result can be seen as a generalization of [5, Theorem 4].
Corollary 2.6.
Let be a lower semicontinuous function such that
Let be a multivalued map such that for each there exists satisfying and
Then has a fixed point such that
Proof.
Define a function by
Then is upper semicontinuous. Also note that
Thus by Theorem 2.5, has a fixed point such that
Now, let be a distance on [8], using the same technique as in the proof of Theorem 2.1, and applying [8, Theorem 3], we can obtain the following result.
Theorem 2.7.
Let be any function such that for some ,
Let be a multivalued map such that for each there exists satisfying
Then has a fixed point such that
Now, following similar methods as in the proofs of Theorems 2.2, 2.3, 2.5, and Corollaries 2.4 and 2.6, we can obtain the following generalizations of Caristi's fixed point theorem with respect to -distance.
Theorem 2.8.
Let be a multivalued map such that for each there exists satisfying
where is an upper semicontinuous from the right. Then has a fixed point such that
Theorem 2.9.
Let be a multivalued map such that for each there exists satisfying
where is a nondecreasing function. Then has a fixed point such that
Corollary 2.10.
Let be a multivalued map such that for each there exists satisfying
where is a nondecreasing function. Then has a fixed point such that
Theorem 2.11.
Let be a multivalued map such that for each there exists satisfying and
where is an upper semicontinuous function. Then has a fixed point such that
Corollary 2.12.
Let be a lower semicontinuous function such that
Let be a multivalued map such that for each there exists satisfying and
Then has a fixed point such that
Similar generalizations of Caristi's fixed point theorem in the setting of quasi-metric spaces with respect to -distance and with respect to -function are studied in [3, Theorem 5.1(iii), Theorem 5.2] and in [2, Theorem 4.1], respectively.
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The author is thankful to the referees for their valuable comments and suggestions.
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Latif, A. Generalized Caristi's Fixed Point Theorems. Fixed Point Theory Appl 2009, 170140 (2009). https://doi.org/10.1155/2009/170140
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DOI: https://doi.org/10.1155/2009/170140