- Research Article
- Open Access

# Generalized Caristi's Fixed Point Theorems

- Abdul Latif
^{1}Email author

**2009**:170140

https://doi.org/10.1155/2009/170140

© Abdul Latif. 2009

**Received:**27 December 2008**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.

## Keywords

- Real Number
- Point Theorem
- Nonlinear Analysis
- Generalize Version
- Close Subset

## 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.

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* _{
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.

## 2. The Results

Applying Theorem 1.2, first we prove the following generalization of Theorem 1.1.

Theorem 2.1.

Then has a fixed point such that

Proof.

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.

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

Proof.

By Theorem 2.1, has a fixed point such that

Theorem 2.3.

where is nondecreasing function. Then has a fixed point such that

Proof.

Thus, by Theorem 2.1, the result follows.

Corollary 2.4.

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.

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.

Then has a fixed point such that

Proof.

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.

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.

where is an upper semicontinuous from the right. Then has a fixed point such that

Theorem 2.9.

where is a nondecreasing function. Then has a fixed point such that

Corollary 2.10.

where is a nondecreasing function. Then has a fixed point such that

Theorem 2.11.

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

Corollary 2.12.

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

## References

- 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

## Copyright

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