- Research Article
- Open Access

# Existence Theorems for Generalized Distance on Complete Metric Spaces

- JeongSheok Ume
^{1}Email author

**2010**:397150

https://doi.org/10.1155/2010/397150

© Jeong Sheok Ume. 2010

**Received:**20 September 2009**Accepted:**20 May 2010**Published:**20 June 2010

## Abstract

We first introduce the new concept of a distance called -distance, which generalizes -distance, Tataru's distance, and -distance. Then we prove a new minimization theorem and a new fixed point theorem by using a -distance on a complete metric space. Our results extend and unify many known results due to Caristi, Ćirić, Ekeland, Kada-Suzuki-Takahashi, Kannan, Ume, and others.

## Keywords

- Variational Principle
- Point Theorem
- Fixed Point Theorem
- Existence Theorem
- Cauchy Sequence

## 1. Introduction

The Banach contraction principle [1], Ekeland's -variational principle [2], and Caristi's fixed point theorem [3] are very useful tools in nonlinear analysis, control theory, economic theory, and global analysis. These theorems are extended by several authors in different directions.

Takahashi [4] proved the following minimization theorem. Let be a complete metric space and let be a proper lower semicontinuous function, bounded from below. Suppose that, for each with there exists such that and Then there exists such that Some authors [5–7] have generalized and extended this minimization theorem in complete metric spaces.

In 1996, Kada et al. [5] introduced the concept of -distance on a metric space as follows. Let be a metric space with metric . Then a function is called a -distance on if the followings are satisfied.

(1) for any .

(2)For any is lower semicontinuous.

(3)For any , there exists such that and imply

They gave some examples of -distance and improved Caristi's fixed point theorem [3], Ekeland's variational principle [2], and Takahashi's nonconvex minimization theorem [4]. The fixed point theorems with respect to a -distance were proved in [8–12].

Throughout this paper we denote by the set of all positive integers, by the set of all real numbers, and by the set of all nonnegative real numbers.

Recently, Suzuki [6] introduced the concept of -distance on a metric space, which generalizes Tataru's distance [13] as follows. Let be a metric space with metric .

Then a function from into is called -distance on if there exists a function from into and the followings are satisfied:

(*τ*1)
for all
;

(*τ*2)
and
for all
and
, and
is concave and continuous in its second variable;

(*τ*3)
and
imply
for all
;

(*τ*4)
and
imply
;

(*τ*5)
and
imply
.

In this paper, we first introduce the new concept of a distance called -distance, which generalizes -distance, Tataru's distance, and -distance. Then we prove a new minimization theorem and a new fixed point theorem by using -distance on a complete metric space. Our results extend and unify many known results due to Caristi [3], Ćirić [14], Ekeland [2], Takahashi [4], Kada et al. [5], Kannan [15], Suzuki [6], and Ume [7, 12] and others.

## 2. Preliminaries

Definition 2.1.

Let be metric space with metric . Then a function from into is called -distance on if there exists a function from into such that

(u1) for all ;

for all ;

Remark 2.2.

Suppose that is a mapping satisfying (u2) (u5). Then there exists a mapping from into such that is nondecreasing in its third and fourth variable, respectively, satisfying (u2) (u5) , where (u2) (u5) stand for substituting for in (u2) (u5), respectively.

Proof.

By (2.12), we have and for all and . Also it follows from (2.12) that is nondecreasing in its third and fourth variable, respectively.

By virtue of (2.15), (2.19), (2.20), (2.22), and (2.25), we have which is a contradiction. Hence (u2) holds. From (2.12) and (u2)~(u5), it follows that (u3) ~(u5) are satisfied.

Remark 2.3.

From Remark 2.2, we may assume that is nondecreasing in its third and fourth variables, respectively, for a function satisfying (u2) (u5).

We give some examples of -distance.

Example 2.4.

Let be the set of real numbers with the usual metric and let be defined by . Then is a -distance on but not a -distance on .

Proof.

since the limit of the sequence and the limit of the sequence do not depend on and , the limit of the sequence may not be . This does not satisfy ( 5). Hence is not a -distance on . Therefore is a -distance on but not a -distance on .

Example 2.5.

Let be a -distance on a metric space . Then is also a -distance on .

Proof.

Then it is easy to see that and satisfy (u2)~(u5). Thus is a -distance on .

Example 2.6.

Let be a normed space with norm . Then a function defined by for every is a -distance on but not a -distance.

Proof.

Let be as in the proof of Example 2.4. Then it is clear that satisfies and satisfies (u2)~(u5) on but does not satisfy . Thus is a -distance on but not a -distance.

Example 2.7.

Let be a normed space with norm . Then a function defined by for every is a -distance on .

Proof.

Define by for all and . Then satisfies and satisfies (u2)~(u5). Thus is a -distance on .

Example 2.8.

Let be a -distance on a metric space and let be a positive real number. Then a function from into defined by for every is also a -distance on .

Proof.

Since is a -distance on , there exists a function satisfying (u2) ~(u5) and satisfies (u1). Define by for all and . Then it is clear that satisfies and satisfies (u2)~(u5). Thus is a -distance on .

The following examples can be easily obtained from Remark 2.3.

Example 2.9.

Let be a metric space with metric and let be a -distance on such that is a lower semicontinuous in its first variable. Then a function defined by for all is a -distance on .

Example 2.10.

is a -distance on .

Remark 2.11.

It follows from Example 2.4 to Example 2.10 that -distance is a proper extension of -distance.

Definition 2.12.

The following lemmas play an important role in proving our theorems.

Lemma 2.13.

Let be a metric space with a metric and let be a -distance on . If is a -Cauchy sequence, then is a Cauchy sequence.

Proof.

Then from (u5), we have . This means that is a Cauchy sequence.

Lemma 2.14.

Let be a metric space with a metric and let be a -distance on .

(1)If sequences and of satisfy and for some , then .

(2)If and , then .

(3)Suppose that sequences and of satisfy and for some , then .

(4)If and , then .

- (2)
In (1), putting and for all , (2) holds.

By method similar to (1) and (2), results of (3) and (4) follow.

Lemma 2.15.

Then is a -Cauchy sequence and is a Cauchy sequence.

Proof.

This implies that is a -Cauchy sequence. By Lemma 2.13, is a Cauchy sequence. Similarly, if we can prove that is also a Cauchy sequence.

## 3. Minimization Theorems and Fixed Point Theorems

The following theorem is a generalization of Takahashi's minimization theorem [4].

Theorem 3.1.

Let be a metric space with metric , let be a proper function which is bounded from below, and let be a function such that, one has the following.

(i) for all .

(iii) imply .

Proof.

for all .

This is a contradiction from (3.26).

Corollary 3.2.

Let be a complete metric space with metric , and let be a proper lower semicontinuous function which is bounded from below. Assume that there exists a -distance on such that for each with , there exists with and . Then there exists such that .

Proof.

for all . It follows easily from Definition 2.12, Lemmas 2.13, 2.14, and 2.15, and (u3) that conditions of Corollary 3.2 satisfy all conditions of Theorem 3.1. Thus, we obtain result of Corollary 3.2.

Remark 3.3.

Corollary 3.2 is a generalization of Kadaet al. [5, Theorem ] and Suzuki [6, Theorem ].

From Lemmas 2.13, 2.14, and 2.15, we have the following fixed point theorem.

Theorem 3.4.

for every with . Then there exists such that and . Moreover, if , then , .

Proof.

By Lemma 2.14, we have .

From Theorem 3.4, we have the following corollary which generalizes the results of Ćirić [14], Kannan [15], and Ume [12].

Corollary 3.5.

for every with . Then there exists such that and . Moreover, if , then and .

Proof.

Since a -distance is a -distance, Corollary 3.5 follows from Theorem 3.4.

The following corollary is a generalization of Suzuki's fixed point theorem [6].

Corollary 3.6.

then . Then there exists such that and . Moreover, if , then and .

Proof.

Let and be as in Theorem 3.4. Then from Theorem 3.4 and hypotheses of Corollary 3.6, we have the following properties.

(1) is a Cauchy sequence.

(2)There exists such that .

By (1)~(5) and hypotheses, we have . The remainders are same as Theorem 3.4.

The following theorem is a generalization of Caristi's fixed point theorem [3].

Theorem 3.7.

Proof.

This is a contradiction.

Corollary 3.8.

Proof.

for all . Then, by Definition 2.12 and Lemmas 2.13, 2.14, and 2.15, we can easily show that conditions of Corollary 3.8 satisfy all conditions of Theorem 3.7. Thus, Corollary 3.8 follows from Theorem 3.7.

Remark 3.9.

Since a -distance and a -distance are a -distance, Corollary 3.8 is a generalization of Kada-Suzuki-Takahashi [5, Theorem ] and Suzuki [6, Theorem ].

The following theorem is a generalization of Ekeland's -variational principle [2].

Theorem 3.10.

Let be a complete metric space with metric , let be a proper lower semicontinuous function which is bounded from below, and let be a function satisfying (i), (ii), and (iii) of Theorem 3.1. Then the following (1) and (2) hold.

for all .

for all with

Proof.

This completes the proof of (2).

Corollary 3.11.

Let and be as in Corollary 3.8. Then the following (1) and (2) hold.

for all with

for all with

Proof.

By method similar to Corollary 3.8, Corollary 3.11 follows from Theorem 3.10.

Remark 3.12.

Corollary 3.11 is a generalization of Suzuki [6, Theorem ].

## Declarations

### Acknowledgments

The author would like to thank the referees for useful comments and suggestions. This work was supported by the Korea Research Foundation (KRF) Grant funded by the Korea government (MEST) (2009-0073655).

## Authors’ Affiliations

## References

- Banach S:
**Sur les opérations dans les ensembles abstraits et leur application aux équations intégrales.***Fundamenta Mathematicae*1992,**3:**133–181.MATHGoogle Scholar - 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 - 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:
**Minimization theorems and fixed point theorems.**In*Nonlinear Analysis and Mathematical Economics*.*Volume 829*. Edited by: Maruyama T. RIMS Kokyuroku; 1993:175–191.Google 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 - Ume J-S:
**Some existence theorems generalizing fixed point theorems on complete metric spaces.***Mathematica Japonica*1994,**40**(1):109–114.MathSciNetMATHGoogle Scholar - Shioji N, Suzuki T, Takahashi W:
**Contractive mappings, Kannan mappings and metric completeness.***Proceedings of the American Mathematical Society*1998,**126**(10):3117–3124. 10.1090/S0002-9939-98-04605-XMathSciNetView ArticleMATHGoogle Scholar - Suzuki T:
**Fixed point theorems in complete metric spaces.**In*Nonlinear Analysis and Convex Analysis*.*Volume 939*. Edited by: Takahashi W. RIMS Kokyuroku; 1996:173–182.Google Scholar - Suzuki T:
**Several fixed point theorems in complete metric spaces.***Yokohama Mathematical Journal*1997,**44**(1):61–72.MathSciNetMATHGoogle Scholar - Suzuki T, Takahashi W:
**Fixed point theorems and characterizations of metric completeness.***Topological Methods in Nonlinear Analysis*1996,**8**(2):371–382.MathSciNetMATHGoogle Scholar - Ume J-S:
**Fixed point theorems related to Ćirić's contraction principle.***Journal of Mathematical Analysis and Applications*1998,**225**(2):630–640. 10.1006/jmaa.1998.6030MathSciNetView ArticleMATHGoogle Scholar - Tataru D:
**Viscosity solutions of Hamilton-Jacobi equations with unbounded nonlinear terms.***Journal of Mathematical Analysis and Applications*1992,**163**(2):345–392. 10.1016/0022-247X(92)90256-DMathSciNetView ArticleMATHGoogle Scholar - Ćirić LjB:
**A generalization of Banach's contraction principle.***Proceedings of the American Mathematical Society*1974,**45:**267–273.MathSciNetMATHGoogle Scholar - Kannan R:
**Some results on fixed points. II.***The American Mathematical Monthly*1969,**76:**405–408. 10.2307/2316437MathSciNetView ArticleMATHGoogle Scholar

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