New Existence Results for Fixed Point Problem and Minimization Problem in Compact Metric Spaces

Abstract

We first present some new existence theorems for fixed point problem and minimization problem in compact metric spaces without assuming that mappings possess convexity property. Some applications of our results to new fixed point theorems for nonself mappings in the setting of strictly convex normed linear spaces and usual metric spaces are also given.

1. Introduction and Preliminaries

Let be a metric space. Denote by the family of all nonempty subsets of . The symbols , , and are used to denote the sets of positive integers, real numbers, and complex numbers, respectively. Let be a nonempty subset of , let be a single-valued mapping, and let be a multivalued mapping. A point in is said to be a fixed point of (resp. ) if (resp. ). The set of fixed points of (resp. ) is denoted by (resp. ). An extended real valued function is said to be lower semicontinuous at if, for any sequence in with , we have . The function is called to be lower semicontinuous on if is lower semicontinuous at every point of . The function is said to be proper if .

Let be a normed linear space over the field or . is said to be strictly convex if whenever ; in other words, the unit sphere of does not contain nontrivial segments. It is worth mentioning that the strict convexity of a normed linear space can be characterized by the properties: for any nonzero vectors , if , then for some real . The following four types of line segments between two distinct points and of are defined as the sets: Clearly, is a closed subset of .

The celebrated Banach contraction principle [1] plays an important role in various fields of nonlinear analysis and applied mathematical analysis.

Theorem 1 (Banach [1]). Let be a complete metric space and let be a selfmap. Assume that there exists a nonnegative number such that Then has a unique fixed point in . Moreover, for each , the iterative sequence converges to the unique fixed point of .

Let be a nonempty subset of a metric space and let be a mapping. Recall that is said to be contractive [2] if

The following interesting fixed point theorem in the setting of compact metric spaces is due to Edelstein in [2].

Theorem 2 (Edelstein [2]). Let be a nonempty compact metric space and let be contractive. Then has a unique fixed point in .

In 1976, Caristi proved the following famous fixed point theorem to extend Banach contraction principle.

Theorem 3 (Caristi [30]). Let be a complete metric space and let be 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 Caristi’s fixed point theorem is equivalent to Ekeland’s variational principle, to Takahashi’s nonconvex minimization theorem, to Daneš’ drop theorem, to petal theorem, and to Oettli-Théra’s theorem; see, for example, [3, 4] and references therein for more details. In view of the important contribution of Caristi’s fixed point theorem on nonlinear analysis, a great deal of generalizations in various different directions of the Caristi’s fixed point theorem has been investigated by several authors. For more details on these generalizations, one can refer to [3–19] and references therein.

During the last few decades, an interesting and important direction of research in metric fixed point theory is to study the existence and uniqueness of fixed points for single-valued nonself mappings or multivalued nonself mappings satisfying certain nonlinear conditions. A mass of such research has been investigated by many authors; see, for example, [20–29] and the references therein.

In this work, we first present some new existence theorems for fixed point problem and minimization problem in compact metric spaces without assuming that mappings possess convexity property. Some applications of our results to new fixed point theorems for nonself mappings in the setting of strictly convex normed linear spaces and usual metric spaces are also given.

2. Existence Results for Fixed Point Problem and Minimization Problem without Convexity

We start with the following crucial and useful existence result for fixed point problem and minimization problem which is one of the main results of this paper.

Theorem 4. Let be a nonempty compact metric space, let be a proper lower semicontinuous function bounded from below, and let be a multivalued mapping. Suppose that()for any with , there exists such that Then, there exists such that(a),(b).

Proof. Since is bounded from below, Since is proper, there exists such that . It follows that Hence, by (6) and (7), we know . One can find a sequences in such that By the compactness of , there exists subsequences and such that By the lower semicontinuity of and (8), we have which implies Next, we claim that . On the contrary, assume that . Then, by our hypothesis , there exists such that which is a contradiction. Therefore and the conclusion (a) is proved. Due to we show the conclusion (b). The proof is completed.

The following existence theorem is obviously an immediate result from Theorem 4.

Theorem 5. Let be a nonempty compact metric space, let be a proper lower semicontinuous function bounded from below, and let be a single-valued selfmapping. Suppose that() for any with .Then there exists such that(a),(b).

In fact, we have the following important fact.

Theorem 6. Theorems 4 and 5 are equivalent.

Proof. It suffices to show that Theorem 5 implies Theorem 4. Under the assumption of Theorem 4, for any with , there exists such that So, we can define a single-valued selfmap by It is easy to see that satisfies for any with . So, all the hypotheses of Theorem 5 are fulfilled. It is therefore possible to apply Theorem 5 to get such that(a),(b).By (a) and the definition of , we have . From (b) and , we get Therefore Theorem 5 implies Theorem 4 and hence the proof is completed.

Applying Theorem 4, we establish the following compactness version of Caristi’s type fixed point theorem for multivalued mappings.

Theorem 7. Let be a nonempty compact metric space, let be a proper lower semicontinuous function bounded from below, and let be a multivalued mapping. Suppose that, for any , there exists such that Then there exists such that(a),(b).

Proof. For any with , by our hypothesis, there exists such that which implies So as in Theorem 4 is satisfied. Therefore the conclusion follows from Theorem 4.

As a direct consequence of Theorem 7 we obtain the following result which is a compactness version of Caristi’s fixed point theorem.

Theorem 8. Let be a nonempty compact metric space, let be a proper lower semicontinuous function bounded from below, and let be a single-valued selfmapping. Suppose that is a Caristi-type map on dominated by ; that is, satisfies Then there exists such that(a),(b).

Theorem 9. Theorems 7 and 8 are equivalent.

By applying Theorem 5 (or Theorem 4), we obtain the following new fixed point theorem for nonself mappings in metric spaces.

Theorem 10. Let be a nonempty compact subset of a metric space and let be a continuous mapping. Suppose that ()for any with there exists such that Then admits a fixed point in .

Proof. Define by By the continuity of , is continuous and bounded below by . By the assumption , for any with , there exists such that so we can define a single-valued selfmap by For any with , by (23) and the definition of , we obtain Hence we prove that implies in Theorem 5. Applying Theorem 5, there exists such that , which deduce . The proof is completed.

Remark 11. Edelstein’s fixed point theorem [2] (i.e., Theorem 2) is a special case of Theorem 10. Indeed, since is contractive, it is easy to see that is continuous on . For any with , let . Then and Hence as in Theorem 10 is satisfied. Therefore the conclusion follows from Theorem 10.

3. Some Applications of Theorem 10

In this section, we study some applications of Theorem 10 to fixed point theory. We first establish a new fixed point theorem without assuming that nonself mappings possess convexity property in the setting of strictly convex normed linear spaces by exploiting Theorem 10.

Theorem 12. Let be a strictly convex normed linear space, let be a nonempty compact subset of , and let be a continuous mapping. Suppose that()for any with there exists and such that Then admits a fixed point in .

Proof. We first claim that the condition holds, where()for any with there exists such that .Indeed, let with be given. By , there exists and such that It follows that If , then our claim is finished.
Suppose . Since is strictly convex, , , and , there exists such that Let . Then . By (30), we have Hence . Put Since , . Let . Then We can choose a sequence , such that Since and is a nonempty compact subset of , there exist a subsequence of and a vector such that By taking into account (34) and (35), we get which implies . So and hence there exists such that On the other hand, by the continuity of , we obtain For any , since , we have Thus, by (37), we obtain By taking the limit from both sides of the last inequality, we get If , then our claim is proved when we take . Suppose . Let Then . Let . Then . We can find a sequence , such that By the compactness of , there exist a subsequence of and a vector such that From (43) and (44), we get By (44) and the continuity of , we have For any , Since taking into account (44), (46), and (47), we get We will verify . Assume . Then . So for some . Thus, by (37), we have which deduces Since , we have and . Because , , and , we know and hence Since and , by (36), (45), and (52), we get which leads a contradiction. Hence it must be . So our claim is proved when we take . Now, all the hypotheses of Theorem 10 are fulfilled, so it is therefore possible to apply Theorem 10 to get the thesis.

As another interesting application of Theorem 10, we give the following new fixed point result for nonself mappings in usual metric spaces. It is worth mentioning that condition as in Theorem 13 is different from condition as in Theorem 12.

Theorem 13. Let be a nonempty compact subset of a metric space and let be a continuous mapping. Suppose that()for any with , there exists such that Then admits a fixed point in .

Proof. Let with be given. Then, by , there exists such that It follows from the last inequalities that So as in Theorem 10 is satisfied. Hence the conclusion follows from Theorem 10.

Let be a nonempty subset of a metric space . A mapping is said to be metrically inward [30] if, for each , there exists such that where if and only if .

Theorem 14. Let be a nonempty compact subset of a metric space and let be a metrically inward contractive mapping. Then admits a unique fixed point in .

Proof. Applying Theorem 13, has a fixed point in . To see the uniqueness of fixed points of , let . If , since is contractive, we have a contradiction. Hence and is a singleton set. The proof is completed.

Finally, the following example is given to illustrate Theorem 14.

Example 15 (see [26, Example 3.1]). Let . Define a norm on by Then is a Banach space and the norm is equivalent to the Euclidean norm on . Let So is a nonempty compact subset of . Define a mapping by Hence is a metrically inward contractive mapping (see [26, Example 3.1]). By applying Theorem 14, we know that has a unique fixed point in . In fact, precisely speaking, is the unique fixed point of .

Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

Acknowledgment

The second author was supported by Grant no. NSC 102-2115-M-017-001 of the National Science Council of the Republic of China.