Open Access

# Generalized Caristi's Fixed Point Theorems

Fixed Point Theory and Applications20092009:170140

DOI: 10.1155/2009/170140

Accepted: 9 February 2009

Published: 5 March 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 [28]. 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 ,
(1.1)

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
(1.2)

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 ,
(2.1)
Let be a multivalued map such that for each there exists satisfying
(2.2)

Then has a fixed point such that

Proof.

Define a function by Note that for each we have
(2.3)
Now, since it follows that Put
(2.4)
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
(2.5)
and thus , and hence is a self map on . Note that is lower semicontinuous and for each , we have
(2.6)

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
(2.7)

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
(2.8)
Clearly, maps into . Note that for all , we get , and thus for any with , we have
(2.9)
Now, clearly, and hence we obtain
(2.10)

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
(2.11)

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
(2.12)

Thus, by Theorem 2.1, the result follows.

Corollary 2.4.

Let be a multivalued map such that for each there exists satisfying
(2.13)

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
(2.14)

where is an upper semicontinuous function. Then has a fixed point such that

Proof.

Define a function from into by

(2.15)

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
(2.16)
Let be a multivalued map such that for each there exists satisfying and
(2.17)

Then has a fixed point such that

Proof.

Define a function by
(2.18)
Then is upper semicontinuous. Also note that
(2.19)

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 ,
(2.20)
Let be a multivalued map such that for each there exists satisfying
(2.21)

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
(2.22)

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
(2.23)

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
(2.24)

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
(2.25)

where is an upper semicontinuous function. Then has a fixed point such that

Corollary 2.12.

Let be a lower semicontinuous function such that
(2.26)
Let be a multivalued map such that for each there exists satisfying and
(2.27)

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.

## Declarations

### Acknowledgment

The author is thankful to the referees for their valuable comments and suggestions.

## Authors’ Affiliations

(1)
Department of Mathematics, King Abdulaziz University

## References

1. Ekeland I: Nonconvex minimization problems. Bulletin of the American Mathematical Society 1979,1(3):443–474. 10.1090/S0273-0979-1979-14595-6
2. 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.2007.05.004
3. 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.2006.12.076
4. 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)00387-1
5. Khamsi MA: Remarks on Caristi's fixed point theorem. Nonlinear Analysis: Theory, Methods & Applications. In pressGoogle Scholar
6. 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.
7. 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)00253-3
8. 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.2004.08.019
9. Kada O, Suzuki T, Takahashi W: Nonconvex minimization theorems and fixed point theorems in complete metric spaces. Mathematica Japonica 1996,44(2):381–391.
10. 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.2000.7151
11. 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.
12. Caristi J: Fixed point theorems for mappings satisfying inwardness conditions. Transactions of the American Mathematical Society 1976, 215: 241–251.
13. Takahashi W: Nonlinear Functional Analysis: Fixed Point Theory and Its Application. Yokohama Publishers, Yokohama, Japan; 2000:iv+276.