- Research Article
- Open Access
Generalized Caristi's Fixed Point Theorems
© Abdul Latif. 2009
- Received: 27 December 2008
- Accepted: 9 February 2009
- Published: 5 March 2009
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.
- Real Number
- Point Theorem
- Nonlinear Analysis
- Generalize Version
- Close Subset
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 , 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.  and Suzuki  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. , Suzuki [8, 10], Khamsi , and many of others.
In 1976, Caristi  obtained the following fixed point theorem on complete metric spaces, known as Caristi's fixed point theorem.
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 , Kada et al. introduced a concept of -distance on a metric space as follows.
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.
Using the concept of -distance, Kada et al.  generalized Caristi's fixed point theorem as follows.
Applying Theorem 1.2, first we prove the following generalization of Theorem 1.1.
Now, applying Theorem 2.1, we obtain generalized Caristi's fixed point results.
Thus, by Theorem 2.1, the result follows.
Applying Theorem 2.3, we prove the following fixed point result.
The following result can be seen as a generalization of [5, Theorem 4].
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.
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.
The author is thankful to the referees for their valuable comments and suggestions.
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