#### Abstract

Multivalued mappings and related selection theorems are fundamental tools in many branches of mathematics and applied sciences. In this paper we continue this theory and prove the existence of Caristi type selections for generalized multivalued contractions on complete metric spaces, by using some classes of functions. Also we prove fixed point and quasi-fixed point theorems.

#### 1. Introduction and Preliminaries

In 1998, Repovs and Semenov [1] furnished a comprehensive study of the theory of continuous selections for multivalued mappings. They point out that “this interesting branch of modern topology was started by Michael [2] and, since then, has received a great amount of interest with various applications outside topology, for instance, approximation theory, control theory, convex sets, differential inclusions, economics, fixed point theory, and vector measures.” Thus an interesting matter is to obtain existence conditions for selections, under different regularity hypotheses, for instance, Lipschitz-continuity and measurability; see also [3–5]. In particular, we are interested in developing this theory for fixed point theorems, by using Caristi’s mappings. Some precise results concerning existence of fixed points for Caristi’s single-valued and multivalued mappings and data dependence of fixed points set are proved in [6]; see also the references therein. We recall that Browder [7] was the first author to use continuous selections to prove a fixed point result; but the first result of Caristi type selection was proved by Jachymski [8] for Nadler’s multivalued contraction with closed values.

*Definition 1. *Let be a metric space. A function is lower semicontinuous at if and only if, for every sequence in with as , . Also, is lower semicontinuous if and only if it is lower semicontinuous at every .

Also, is called the lower counter set defined by a point . Then the following results hold true.

Proposition 2. *Let be a metric space. Let be a function. Then, is lower semicontinuous if and only if is closed for every .*

Theorem 3 (Caristi [9]). *Let be a complete metric space and let be a mapping not necessarily continuous. Assume that there exists a function , which is lower semicontinuous, such that
**
Then, has a fixed point ; that is, .*

Also, is called Caristi’s mapping on . On the other hand, Nadler [10, 11] established the following result.

Theorem 4 (Nadler [10, 11]). *Let be a complete metric space and let be a multivalued mapping such that, for some , one has
**
where denotes the class of all nonempty closed subsets of and denotes a generalized Hausdorff metric on . Then has a fixed point ; that is, .*

For , we recall that , where . Also, a multivalued mapping satisfying the assumption of Theorem 4 is called Nadler’s multivalued contraction.

*Definition 5. *Let be a multivalued mapping and let be a (single-valued) mapping. Then, is said to be a selection for if

Also is called Caristi type selection if it is Caristi’s mapping. As mentioned above, Jachymski established existence theorems stating that certain multivalued mappings admit selections that are Caristi’s mappings, which do not need to be continuous (see, for instance, Example 1 in [8]).

Theorem 6 (Jachymski [8]). *If is Nadler’s multivalued contraction on a complete metric space , then admits a selection , which is Caristi’s mapping on generated by a Lipschitz function .*

Clearly, Theorem 3 yields Theorem 4; that is, every Nadler’s multivalued mapping admits a fixed point, but the converse does not hold in general. Obviously, if the multivalued mapping does not admit a fixed point, then a Caristi type selection cannot exist. The following example illustrates the case of a multivalued mapping which does not admit a Caristi type selection, even if it has a fixed point.

*Example 7 (Xu [12]). *Consider the complete metric space , where denotes the standard metric. Define as for all .

Trivially, 0 is a unique fixed point of . Now, assume that there exists a Caristi type selection for , say . Then, referring to notions and notations of Theorem 3, we write for all . By definition of , we have and so for all . By iteration, we can get easily that This implies that the sequence is nonincreasing and so, being bounded below, convergent to some . Also, from (4), as , we get . On the other hand, the reader can immediately prove that is a strictly increasing sequence and hence we get a contradiction with the above limit. Then we conclude that is not a Caristi type selection.

*Definition 8. *Given a function with for , a multivalued mapping is said to be a multivalued -contraction if

*Definition 9. *A function is said to be subadditive if
Also, is said to be superadditive if the reverse inequality holds true.

Theorem 10 (Jachymski [8]). *Let be a multivalued -contraction on a complete metric space such that is superadditive, and the function is nondecreasing. Then there exist a selection of and a function , which is nondecreasing and subadditive and continuous at such that . Moreover, there is an equivalent metric such that is complete and f is Caristi’s mapping on .*

In this paper we continue this study and prove the existence of Caristi type selections for generalized multivalued contractions on complete metric spaces. Our results fit into the theory of selections for multivalued mappings showing certain ways to establish selection theorems, by using some classes of functions. Also we prove fixed point and quasi-fixed point theorems. We remark that the existence of a Caristi type selection for a multivalued mapping ensures the existence of a fixed point.

#### 2. Caristi Type Selection Theorems

We start to develop our theory by using the concept of lower semicontinuity, which is one of the most important concepts in multivalued analysis.

Theorem 11. *Let be a complete metric space. Let be a multivalued mapping and let be a real number. Consider and suppose that satisfies the following conditions: *(i)*there exist two nonnegative real numbers with such that, for each , there is having the property
*(ii)*the function defined by , for all , is lower semicontinuous.**
Then has a selection that is Caristi’s mapping.*

*Proof. *By the axiom of choice and condition (i), there is a mapping with for all such that
and so
Then

By condition (ii), the function defined by , for all , is lower semicontinuous and hence is Caristi’s mapping that is a selection of .

*Example 12. *Let be endowed with the usual metric for all so that is a complete metric space. Also, let be defined by
Consider , , and such that . Then, for and , we have ; that is . Moreover,
Also, for and , we have ; that is, . Moreover,
Finally, the function defined by
is lower semicontinuous in . Thus, all the hypotheses of Theorem 11 are satisfied and so has a selection that is Caristi’s mapping. In fact, , defined by for all , is such that and also , where is given by for all .

Notice that
and hence is not Nadler’s multivalued contraction.

Analogous results to Theorem 11 can be established under different hypotheses. For instance, in the next theorem, the multivalued mapping satisfies another contractive condition.

Theorem 13. *Let be a complete metric space. Let be a multivalued mapping and let be a real number. Consider and suppose that satisfies the following conditions: *(i)*there exist nonnegative real numbers with such that, for each , there is having the property
*(ii)*the function defined by , for all , is lower semicontinuous.**
Then has a selection that is Caristi’s mapping.*

*Proof. *By the axiom of choice and condition (i), there is a mapping with for all such that
and so, by using the triangular inequality for ,
Then

By condition (ii), the function defined by , for all , is lower semicontinuous and hence is Caristi’s mapping that is a selection of .

We would like to remark that other results can be stated by involving upper semicontinuous multivalued mappings, in view of the following situation.

*Definition 14. *Let be a metric space. Then, a multivalued mapping is said to be -upper semicontinuous at , if the function
is continuous at . Clearly, is said to be -upper semicontinuous, whenever is continuous at every .

Now we present a class of multivalued mappings such that the function , for all , is lower semicontinuous.

Proposition 15. *Let be a metric space. If is -upper semicontinuous, then the function is lower semicontinuous.*

*Proof. *Given , for all , we get
From above inequalities, we deduce that and so is a lower semicontinuous function.

For instance, from Theorem 13 and Proposition 15 we get the following corollary.

Corollary 16. *Let be a complete metric space. Let be an -upper semicontinuous multivalued mapping and let be a real number. Consider and suppose that satisfies the condition of Theorem 13. Then has a selection that is Caristi’s mapping.*

#### 3. Extension to Quasi-Fixed Point Theorems

Let be a metric space. We recall that a multivalued mapping has a quasi-fixed point if there exists a point such that . Then we extend our theory by considering functions instead of constant values. Therefore, let be a metric space, and let and be functions such that

*Remark 17. *Notice that in (22) we do not need that . We will return on this fact to derive a particular situation from the following theorem.

Theorem 18. *Let be a complete metric space. Let be a multivalued mapping. Suppose that satisfies the following conditions: *(i)*there exist three functions and such that (22) holds;*(ii)*for each , with , there exists such that
*(iii)*the function defined by , for all , is lower semicontinuous;*(iv)*there exists such that
**Then has a quasi-fixed point; that is, there exists such that .*

*Proof. *Assume that for all . By the axiom of choice and condition (ii), there is a mapping with such that

Then for each
This implies
Consequently, we have
Now, let . Since, by (iii), is a closed subset of , we deduce that is complete. Denote by . For all , we get
where the function is defined by , for all . Clearly, by condition (iii), the function is lower semicontinuous. From (29), we get that whenever and hence is Caristi’s mapping. This implies that has a fixed point in , a contradiction since for all . Hence there is such that .

As a consequence of Theorem 18, in the case that is also closed, we obtain the following corollary.

Corollary 19. *Let be a complete metric space. Let be a multivalued mapping such that is closed. Suppose that satisfies the following conditions: *(i)*there exist three functions and such that (22) holds;*(ii)*for each , with , there exists such that
*(iii)*the function defined by , for all , is lower semicontinuous;*(iv)*there exists such that
**Then has a fixed point.*

In view of Remark 17, by assuming , for all , on the same lines of the proof of Theorem 18, one can prove the following result.

Theorem 20. *Let be a complete metric space. Let be a multivalued mapping. Suppose that satisfies the following conditions: *(i)*there exist three functions and such that (22) holds;*(ii)*for each , with , there exists such that
*(iii)*the function defined by , for all , is lower semicontinuous;*(iv)*there exists such that
**Then has a selection that is Caristi’s mapping on a closed subset of .*

*Remark 21. *If, in Theorems 18 and 20 and Corollary 19, we assume that the multivalued mapping is -upper semicontinuous, then (iii) holds true. In this case, we can reformulate the statements of these results, requiring that satisfies only conditions (i), (ii), and (iv).

#### 4. Generalization of Caristi’s Theorem

We denote by the set of all functions such that there exist and satisfying , for all and for all .

*Remark 22. *Given a nondecreasing function continuous at with , consider the right lower Dini derivative of at ; that is,
Then provided that ; see [8]. Also, each function that is nondecreasing, subadditive, and continuous at with belongs to .

Inspired by Khamsi [13] and Jachymski [8], we give two fixed point theorems. In particular our first theorem furnishes an alternative proof to Theorem 3 of [13] and the related Kirk’s problem, without using order relations (see Section 3 in [13] for more details).

Theorem 23. *Let be a complete metric space. Let be a mapping. Suppose that there exist a lower semicontinuous function and a function such that
**
Then has a fixed point in .*

*Proof. *Let , , and let be as stated above. Let
The set is closed since is lower semicontinuous and hence complete. Now, from (35), we get that whenever . Also
for all ; we obtain that whenever . Hence
that is,
Since the function is lower semicontinuous, by Theorem 3, the mapping has a fixed point in and so in .

*Example 24. *Let be endowed with the usual metric for all so that is a complete metric space. Also, let be defined by
It follows that
Notice that , defined by for all , is a lower semicontinuous function such that , where is given by for all , where . Thus, we can apply Theorem 23 to conclude that has a fixed point; clearly and are fixed points of .

The inspiration of our next theorem is Theorem 10. In particular, our result does not use a monotonic condition. For a comprehensive discussion, we refer the reader to the fundamental paper of Jachymski [8].

Theorem 25. *Let be a complete metric space. Let be a multivalued mapping. Suppose that is a -contraction with right upper semicontinuous such that the function for all belongs to . Then has a fixed point.*

*Proof. *Let be the function defined by , for all . Clearly, is right upper semicontinuous and for all . Therefore, the set
for all . In fact, if is a sequence such that , the right upper semicontinuity of the function ensures that
and hence there exists such that
The axiom of choice ensures that there is a mapping such that
Since , we get
where the function is defined by , for all . Since the function is lower semicontinuous, then we get that has a fixed point, which is a fixed point for .

#### 5. Conclusion

Under suitable hypotheses for multivalued mappings, we established the existence of Caristi type selections. Also we proved fixed point and quasi-fixed point theorems, by using weaker and modified hypotheses on some classes of functions present in the literature. Our results extend and complement many theorems in the literature.

#### Conflict of Interests

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

#### Authors’ Contribution

All authors contributed equally and significantly in writing this paper. All authors read and approved the final paper.

#### Acknowledgments

The first author is a Member of the Gruppo Nazionale per l’Analisi Matematica, la Probabilità e le loro Applicazioni (GNAMPA) of the Istituto Nazionale di Alta Matematica (INdAM). The second author is a Member of the Gruppo Nazionale per le Strutture Algebriche, Geometriche e le loro Applicazioni (GNSAGA) of the Istituto Nazionale di Alta Matematica (INdAM).