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.
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.
2. The Results
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.
- Ekeland I: Nonconvex minimization problems. Bulletin of the American Mathematical Society 1979,1(3):443–474. 10.1090/S0273-0979-1979-14595-6MathSciNetView ArticleMATHGoogle Scholar
- Al-Homidan S, Ansari QH, Yao J-C: Some generalizations of Ekeland-type variational principle with applications to equilibrium problems and fixed point theory. Nonlinear Analysis: Theory, Methods & Applications 2008,69(1):126–139. 10.1016/j.na.2007.05.004MathSciNetView ArticleMATHGoogle Scholar
- Ansari QH: Vectorial form of Ekeland-type variational principle with applications to vector equilibrium problems and fixed point theory. Journal of Mathematical Analysis and Applications 2007,334(1):561–575. 10.1016/j.jmaa.2006.12.076MathSciNetView ArticleMATHGoogle Scholar
- Bae JS: Fixed point theorems for weakly contractive multivalued maps. Journal of Mathematical Analysis and Applications 2003,284(2):690–697. 10.1016/S0022-247X(03)00387-1MathSciNetView ArticleMATHGoogle Scholar
- Khamsi MA: Remarks on Caristi's fixed point theorem. Nonlinear Analysis: Theory, Methods & Applications. In pressGoogle Scholar
- Khamsi MA, Kirk WA: An Introduction to Metric Spaces and Fixed Point Theory, Pure and Applied Mathematics. Wiley-Interscience, New York, NY, USA; 2001:x+302.View ArticleMATHGoogle Scholar
- Park S: On generalizations of the Ekeland-type variational principles. Nonlinear Analysis: Theory, Methods & Applications 2000,39(7):881–889. 10.1016/S0362-546X(98)00253-3MathSciNetView ArticleMATHGoogle Scholar
- Suzuki T: Generalized Caristi's fixed point theorems by Bae and others. Journal of Mathematical Analysis and Applications 2005,302(2):502–508. 10.1016/j.jmaa.2004.08.019MathSciNetView ArticleMATHGoogle Scholar
- Kada O, Suzuki T, Takahashi W: Nonconvex minimization theorems and fixed point theorems in complete metric spaces. Mathematica Japonica 1996,44(2):381–391.MathSciNetMATHGoogle Scholar
- Suzuki T: Generalized distance and existence theorems in complete metric spaces. Journal of Mathematical Analysis and Applications 2001,253(2):440–458. 10.1006/jmaa.2000.7151MathSciNetView ArticleMATHGoogle Scholar
- Bae JS, Cho EW, Yeom SH: A generalization of the Caristi-Kirk fixed point theorem and its applications to mapping theorems. Journal of the Korean Mathematical Society 1994,31(1):29–48.MathSciNetMATHGoogle Scholar
- Caristi J: Fixed point theorems for mappings satisfying inwardness conditions. Transactions of the American Mathematical Society 1976, 215: 241–251.MathSciNetView ArticleMATHGoogle Scholar
- Takahashi W: Nonlinear Functional Analysis: Fixed Point Theory and Its Application. Yokohama Publishers, Yokohama, Japan; 2000:iv+276.MATHGoogle Scholar
This article is published under license to BioMed Central Ltd. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.