Abstract and Applied Analysis

Volume 2013 (2013), Article ID 407219, 8 pages

http://dx.doi.org/10.1155/2013/407219

Research Article

On Caristi Type Maps and Generalized Distances with Applications

Wei-Shih Du

Department of Mathematics, National Kaohsiung Normal University, Kaohsiung 824, Taiwan

Received 3 July 2013; Accepted 5 August 2013

Academic Editor: E. Karapinar

Copyright © 2013 Wei-Shih Du. 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.

Abstract.

We prove some new existence theorems of fixed points for Caristi type maps and some suitable generalized distances without lower semicontinuity assumptions on dominated functions. As applications of our results, some new fixed point theorems and new generalizations of the Banach contraction principle are given.

1. Introduction

In 1972, Caristi proved the following famous fixed point theorem.

Theorem 1 (Caristi [1]). Let be a complete metric space and a lower semicontinuous and bounded below function. Suppose that is a Caristi type map on dominated by ; that is, satisfies

Then has a fixed point in .

It is well-known that the Caristi’s fixed point theorem is one of the most valuable generalization of the Banach contraction principle [2], and it is equivalent to the Ekeland’s variational principle, to the Takahashi’s nonconvex minimization theorem, to the Daneš’ drop theorem, to the petal theorem, and to the Oettli-Théra’s theorem; see [3–26] and references therein for more details. A number of generalizations in various different directions of the Caristi’s fixed point theorem have been investigated by several authors; see, for example, [4–30] and references therein. An interesting direction of research is the extension of Caristi’s fixed point theorem, Ekeland’s variational principle, and Takahashi’s nonconvex minimization theorem to generalized distances, for example, -distances [5, 10, 14, 19], -distances [11, 12, 22], -functions [13, 15, 18, 22–25, 31–36], weak -functions [24, 25], -distances [26], -functions [21], generalized pseudodistances [22, 23], and others. For more details on these generalizations, one can refer to [5, 10–26] and references therein.

Let us recall how we can exploit Caristi’s fixed point theorem to prove the Banach contraction principle. A selfmap on a metric space is called contractive or Banach type if there exists a real number such that

It is obvious that if is a contractive map on , then is continuous on and (2) will deduce the following inequality:

The inequality (3) admits that is a Caristi type map on dominated by defined by . From the continuity of , the function is continuous on , and therefore the Caristi’s fixed point theorem is applicable to prove the Banach contraction principle. It is quite obvious that for any map and any generalized distances , the function is not necessarily to be continuous even lower semicontinuous, so such well-known generalized versions of Caristi’s fixed point theorem with lower semicontinuity are not easily applicable to any generalized version of Banach contraction principle for generalized distances. Motivated by the reason, in the recently paper [20], the author established some new versions of Caristi type fixed point theorem such that they can be applicable to prove generalized versions of Banach contraction principle for suitable generalized distances.

This work can be considered as a continuation of the paper [20]. In this paper, we first establish some new fixed point theorems for Caristi type maps and some suitable generalized distances without assuming that the dominated functions possess lower semicontinuity property. As applications of our results, some new fixed point theorems and new generalizations of the Banach contraction principle are given. We have already succeeded in utilizing our new versions of Caristi type fixed point theorem to deal with the existence results for any map satisfying

where is a function satisfying for all .

2. Preliminaries

We recall in this section the notations, definitions, and results needed. Let be a metric space. An extended real valued function is said to be lower semicontinuous (l.s.c., for short) at if for any sequence in with as , we have . The function is called to be l.s.c. on if is l.s.c. at every point of . The function is said to be proper if . Let be a selfmap. is said to be closed if , the graph of , is closed in . A point in is a fixed point of if . The set of fixed points of is denoted by . Throughout this paper we denote by and , the set of positive integers and nonnegative real numbers, respectively.

Recall that a function is called a -distance [5, 10–20, 31], first introduced and defined by Kada, Suzuki, and Takahashi, if the following are satisfied:

for any ; for any , is l.s.c.; for any , there exists such that and imply .

A function is said to be a -function [13, 15, 18, 22–25, 31–36], first introduced and studied by Lin and Du, if the following conditions hold:

for all ; if and in with such that for some , then ; for any sequence in with , if there exists a sequence in such that , then ; for , and imply .

Note that not either of the implications necessarily holds and is nonsymmetric in general. It is well known that the metric is a -distance and any -distance is a -function, but the converse is not true; see [13, 31] for more detail.

Example 2 (see [31, Example  A]). Let with the metric for , and . Define the function by

Then is a -function.

The following result is crucial in this paper.

Theorem 3 (see [18, Lemma 2.1]). Let be a metric space and a function. Assume that satisfies the condition . If a sequence in with , then is a Cauchy sequence in .

Recently, the concepts of weak -function and generalized pseudodistance were introduced and studied by Khanh and Quy [24, 25] and Wodarczyk and Plebaniak [22] as follows.

Definition 4. Let be a metric space. A function is called

(i)a weak -function [24, 25] on if conditions , , and hold;(ii)a generalized pseudodistance [22] on if conditions and hold.

It is obvious that any -function is a weak -function and every weak -function is a generalized pseudodistance, but the converse parts are not always true. The first observation is that there exists a weak -function which is not a -function.

Example 5 (see [24, Example  2.5]). Let , , and be defined by

Then is a weak -function which is neither a -function nor a -distance.

The following example shows that there exists a generalized pseudodistance which is not a weak -function.

Example 6 (see [22, Example  1.3]). Define a function by

Then is a generalized pseudodistance but not a weak -function.

Very recently, the author first introduced the following concepts.

Definition 7 (see [20]). Let be a metric space, and let , , and be functions. A single-valued selfmap is called

(i)Caristi type on dominated by , , and (abbreviated as -Caristi type on ) if

(ii)Caristi type on dominated by and (abbreviated as -Caristi type on ) if

(iii)Caristi type on dominated by and (abbreviated as - on ) if

(iv)Caristi type on dominated by (abbreviated as -Caristi type on ) if

Clearly, if is -Caristi type (resp. -Caristi type) on , then is -Caristi type (resp. -Caristi type) on with for all . The following example illustrates that their converse are not always true.

Example 8. Let with the usual metric . Then is a complete metric space. Let be defined by

for all , . By Example 2, we know that is a -function. Let be defined by , . Define by

Then is not lower semicontinuous at . For , let be defined by

respectively. For , we have

For , we have

Hence, for any , we show

So, is -Caristi type on as well as -Caristi type on for all . Moreover, we know that is -Caristi type on , but it is neither -Caristi type nor -Caristi type on based on the following fact

Definition 9 (see [31–36]). A function is said to be an -function (or -function) if for all .

It is obvious that if is a nondecreasing function or a nonincreasing function, then is an -function. So the set of -functions is a rich class. But it is worth to mention that there exist functions which are not -functions.

Example 10 (see [32]). Let be defined by

Since is not an -function.

Recently, Du [32] first proved the following characterizations of -functions.

Theorem 11 (see [32]). Let be a function. Then the following statements are equivalent.

(a) is an -function.(b)For each , there exist and such that for all .(c)For each , there exist and such that for all .(d)For each , there exist and such that for all .(e)For each , there exist and such that for all .(f)For any nonincreasing sequence in , one has .(g) is a function of contractive factor; that is, for any strictly decreasing sequence in , one has .

3. New Results for Caristi Type Maps and Their Applications

We start with the following useful auxiliary result.

Theorem 12. Let be a metric space, a proper and bounded below function, a nondecreasing function, a function, and a selfmap on . Let with . Define and for each . If satisfies and is -Caristi type on , then

Moreover, if we futher assume that satisfies , then is a Cauchy sequence in .

Proof. For , . Since is -Caristi type on , we get

which implies

Similarly, we have

Hence, by induction, we can obtain the following inequalities

Since is bounded below,

By (25), since is nondecreasing, we have

For with , taking into account , (24), (26), and (27), we get

Let , . Then for each . Since , we obtain , and hence . Moreover, if satisfies , then the desired conclusion follows from Theorem 3 immediately. The proof is completed.

Applying Theorem 12, we prove a new fixed point theorem for Caristi type maps and generalized pseudodistances. It is worth to mention that in Theorem 13 we pose some suitable assumptions on the map without assuming that the dominated functions possess lower semicontinuity property.

Theorem 13. Let be a complete metric space, a proper and bounded below function, a nondecreasing function, and a generalized pseudodistance on . Suppose that is a -Caristi type selfmap on and one of the following conditions is satisfied:

(H1) is continuous; (H2) is closed;(H3) implies for all and the map defined by is l.s.c.;(H4) the map defined by is l.s.c.;(H5) for any sequence in with , and , we have .

Then admits a fixed point in . Moreover, for any with , the sequence converges to a fixed point of .

Proof. Let . Since is proper, . Let . Define and for each . Since is a generalized pseudodistance on , by applying Theorem 12, we know that is a Cauchy sequence in and

The last equality implies

By the completeness of , there exists such that as .

Now, we verify . If (H1) holds, since is continuous on , for each and as , we get

which means . If (H2) holds, since is closed, for each and as , we have . Suppose that (H3) holds. By the lower semicontinuity of , as and (30), we obtain

which implies . By the hypothesis in (H3), we get . Suppose that (H4) holds. Since is convergent in ,

Since

we obtain , and hence . Finally, assume (H5) holds. Since and , there exists with and for all , such that . By , . Since as and , we get as . So or . Therefore, in any case, we prove . Since is arbitrary, the sequence converges to a fixed point of . This completes the proof.

Here, we give an example illustrating Theorem 13. This example also gives a negative answer to the uniqueness of fixed point.

Example 14. Let with the usual metric . Then is a complete metric space. Define by

for all , . Then is a generalized pseudodistance on . Let and be defined by

respectively. So for all . Note that is not lower semicontinuous at , so is not lower semicontinuous on . Since for all , is a bounded below function on . Let be defined by

Then is continuous on and . It is also easy to see that is closed and the map is l.s.c. Hence (H1), (H2), and (H4) as in Theorem 13 hold. We deduce that for any ,

so is -Caristi type on . Let . Since

we know that converges and the limit of belongs to . On the other hand, since all the assumptions of Theorem 13 are satisfied, by applying Theorem 13, we also prove that has a fixed point in and for any , the sequence converges to a fixed point of . It is worth noticing that any well-known generalized version of Caristi’s fixed point theorem is not applicable here.

The following conclusions are immediate from Theorem 13.

Corollary 15. Let be a complete metric space, a proper and bounded below function, and a generalized pseudodistance on . Suppose that is a -Caristi type selfmap on and one of the conditions (H1), (H2), (H3), (H4), and (H5) as in Theorem 13 holds. Then admits a fixed point in . Moreover, for any with , the sequence converges to a fixed point of  T.

Corollary 16. Let be a complete metric space, a proper and bounded below function, and a nondecreasing function. Suppose that is a -Caristi type selfmap on and one of the following conditions is satisfied:

(D1) is continuous; (D2) is closed;(D3)the map defined by is l.s.c.

Then admits a fixed point in . Moreover, for any with , the sequence converges to a fixed point of .

Corollary 17. Let be a complete metric space and a proper and bounded below function. Suppose that is a -Caristi type selfmap on and one of the conditions (D1), (D2), and (D3) as in Corollary 16 holds. Then admits a fixed point in . Moreover, for any with , the sequence converges to a fixed point of .

Now, we give another quite useful auxiliary theorem for our applications.

Theorem 18. Let be a metric space, a function, and a selfmap. Suppose that there exists an -function such that

Then there exists a function such that for each ,

Here, we denote (the identity map).

Proof. Let be given. From our hypothesis, we have

for each . So the sequence is strictly decreasing in . Since is an -function, by (g) of Theorem 11, we obtain

Since is arbitrary, we can define a new function by

It is obvious that for each , we have

As an interesting application of Theorem 13, we prove the following new fixed point theorems for Banach type maps.

Theorem 19. Let be a complete metric space, a generalized pseudodistance on with implies for all , and a selfmap. Suppose that

(a)there exists an -function such that

(b)one of the conditions (H1), (H2), (H3), (H4), and (H5) as in Theorem 13 holds.

Then admits a unique fixed point in . Moreover, for each , the sequence converges to the unique fixed point of .

Proof. Denote (the identity map). Applying Theorem 18, there exists a function such that for each ,

For each , by (46), we get

By exploiting the inequalities (47) and (48), we obtain

Let and be defined by

respectively. Then is a nondecreasing function, and is a bounded below function. Clearly, for all . By (49), we obtain

Hence we prove that is a -Caristi type selfmap on . Applying Theorem 13, we know . We claim that is a singleton set. Let . Then and . From (46), we have

or

which implies . By our hypothesis, we get and our claim is proved. By the uniqueness of fixed point of and applying Theorem 13 again, the sequence converges to the unique fixed point of for any .

As a direct consequence of Theorem 19, we obtain the following result.

Corollary 20. Let be a complete metric space, be a generalized pseudodistance on with implies for all , and a selfmap. Suppose that

(a)there exists such that

(b)one of the conditions (H1), (H2), (H3), (H4), and (H5) as in Theorem 13 holds.

Then admits a unique fixed point in . Moreover, for each , the sequence converges to the unique fixed point of .

Applying Theorem 19, we obtain a generalization of the celebrated Banach contraction principle.

Corollary 21. Let be a complete metric space and a selfmap. Suppose that there exists an -function such that

Then admits a unique fixed point in . Moreover, for each , the sequence converges to the unique fixed point of .

Proof. By (55), we know that is continuous on . Hence the conclusion follows from Theorem 19 immediately.

Remark 22. Theorems 13 and 19, Corollaries 15–21 all generalize and improve the celebrated Banach contraction principle.

Acknowledgment

In this research, the author was supported by Grant no. NSC 102–2115-M-017-001 of the National Science Council of the Republic of China.

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