Abstract

In 2012, Samet et al. introduced the notion of α-ψ-contractive mapping and gave sufficient conditions for the existence of fixed points for this class of mappings. The purpose of our paper is to study the existence of fixed points for multivalued mappings, under an α-ψ-contractive condition of Ćirić type, in the setting of complete b-metric spaces. An application to integral equation is given.

1. Introduction

The theory of multivalued mappings has an important role in various branches of pure and applied mathematics because of its many applications, for instance, in real and complex analysis as well as in optimal control problems. Over the years, this theory has increased its significance and hence in the literature there are many papers focusing on the discussion of abstract and practical problems involving multivalued mappings. As a matter of fact, amongst the various approaches utilized to develop this theory, one of the most interesting is based on methods of fixed point theory, often in view of the constructive character of fixed point theorems, especially in its metric branch (see, for instance, [1]). Thus, Nadler [2] was the first author who combined the notion of contraction (see condition (1) below) with multivalued mapping by establishing the following fixed point theorem.

Theorem 1 (see [2]). Let be a complete metric space and let be a multivalued mapping satisfyingfor all , where is a constant such that and denotes the family of nonempty closed subsets of . Then has a fixed point; that is, there exists a point such that .

Later on, many authors discussed this result and gave their generalizations, extensions, and applications; see, for instance, [3–7].

On the other hand, the concept of metric space has been generalized in different directions to better cover much more general situations, arising in computer science and others (see, for instance, [3, 8]). Here, we deal with the notion of -metric space, which is a metric space satisfying a relaxed form of triangle inequality; see Bakhtin [9] and Czerwick [10]. Many researchers followed this idea and proved various results in the -metric setting [11–14].

In 2012, Samet et al. [15] introduced the notion of --contractive mapping and gave sufficient conditions for the existence of fixed points for this class of mappings. In this paper, we study the existence of fixed points for multivalued mappings, under an --contractive condition of Ćirić type, in the setting of complete -metric spaces. Also, we consider a complete -metric space endowed with a partial ordering. An inspiration for the paper is the recent work of Mohammadi et al. [16] in which some good ideas are suggested to the reader. Finally, an application to integral equation is given.

2. Preliminaries

Let denote the set of all nonnegative real numbers and let denote the set of positive integers. From [9, 10, 17, 18] we get some basic definitions, lemmas, and notations concerning the -metric space.

Definition 2. Let be a nonempty set and let be a given real number. A function is said to be a -metric if and only if for all , , the following conditions are satisfied:(1) if and only if ;(2);(3).
Then, the triplet is called a -metric space.

It is an obvious fact that a metric space is also a -metric space with , but the converse is not generally true. To support this fact, we have the following example.

Example 3. Consider the set endowed with the function defined by for all , . Clearly, is a -metric space but it is not a metric space.

Let be a -metric space. The following notions are natural deductions from their metric counterparts.(i)A sequence converges to if .(ii)A sequence is said to be a Cauchy sequence if, for every given , there exists such that for all .(iii)A -metric space is said to be complete if and only if each Cauchy sequence converges to some .

From the literature on -metric spaces, we choose the following significant example.

Example 4 (see [11]). Let . Consider the space of all real functions such that , endowed with the functional defined by Then, is a -metric space.

Next, we collect some lemmas and notions concerning the theory of multivalued mappings on -metric spaces. We recall that denotes the class of nonempty closed and bounded subsets of . For , define the function by wherewith Note that is called the Hausdorff -metric induced by the -metric .

We recall the following properties from [10, 13, 18]; see also [12] and the references therein.

Lemma 5. Let be a -metric space. For any , , and any , , one has the following: (i), for any ;(ii);(iii), for any ;(iv);(v);(vi);(vii).

Remark 6. The function is a generalized Hausdorff -metric; that is, if does not exist.

Lemma 7. Let be a -metric space. For and , one has where denotes the closure of the set .

Lemma 8. Let be a -metric space and . Then, for each and for each there exists such that if .

Finally, to prove our results we need the following class of functions.

Let be a real number; we denote by the family of strictly increasing functions such that where denotes th iterate of the function . It is well known that for all . An example of function is given by for all , where .

Definition 9. A multivalued mapping is said to be -admissible, with respect to a function , if for each and with , we have for all .

Definition 10. Let be a -metric space and let be as in (4). 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 -upper semicontinuous at every .

3. Fixed Point Theory in -Metric Spaces

We study the existence of fixed points for multivalued mappings, by adapting the ideas in [16] to the -metric setting.

Definition 11. Let be a -metric space. A multivalued mapping is said to be an --contraction of Ćirić type if there exist a function and a function such that, for all , with , the following condition holdswhere

Now, we are ready to state and prove our first main theorem.

Theorem 12. Let be a complete -metric space and let . Assume that there exist two functions and such that is an --contraction of Ćirić type. Also, suppose that the following conditions are satisfied: (i) is an -admissible multivalued mapping;(ii)there exist and such that ;(iii)T is -upper semicontinuous.
Then has a fixed point.

Proof. By condition (ii) there exist and such that . Clearly, if or , we deduce that is a fixed point of and so we can conclude the proof. Now, we assume that and and hence . First, from (9), we deduceIf , then we have which is a contradiction. Thus, and since is strictly increasing, we have where is a real number. This ensures that there exists (obviously, ) such that Since is -admissible, from condition (ii) and , we have . If , then is a fixed point. Assume that ; that is, .
Next, from (9), we deduceIf , then we have which is a contradiction. Thus, and since is strictly increasing, we have This ensures that there exists (obviously, ) such that Iterating this procedure, we construct a sequence such that Let . Then and so is a Cauchy sequence in . Hence, there exists such that .
Fromsince is -upper semicontinuous, passing to limit as , we get which implies . Finally, since is closed we obtain that ; that is, is a fixed point of .

In view of Theorem 12, we have the following corollary.

Corollary 13. Let be a complete -metric space and let . Assume that there exist two functions and such thatAlso, suppose that the following conditions are satisfied: (i) is an -admissible multivalued mapping;(ii)there exist and such that ;(iii) is -upper semicontinuous.
Then has a fixed point.

Proof. Condition (25) ensures that condition (9) holds for all with . Thus, is an --contraction of Ćirić type and by Theorem 12 the multivalued mapping has a fixed point.

Notice that one can relax the -upper semicontinuity hypothesis on , by introducing another regularity condition as shown in the next theorem.

Theorem 14. Let be a complete -metric space and let . Assume that there exist two functions and such that is an --contraction of Ćirić type. Also, suppose that the following conditions are satisfied: (i) is an -admissible multivalued mapping;(ii)there exist and such that ;(iii)for a sequence in with for all and , then for all .
If for all , then has a fixed point.

Proof. By condition (ii) there exist and such that . Proceeding as in the proof of Theorem 12, we obtain a sequence that converges to some such that , and for all . By condition (iii), we get for all . If , then the proof is concluded. Assume . From , we deduce that(i)the sequences , , and converge to ;(ii).
These facts ensure that there exists such that for all with . Since is an --contraction of Ćirić type, for all , we haveFrom , letting , we get which implies . Finally, since is closed we obtain that ; that is, is a fixed point of .

In view of Theorem 14, we have the following corollary.

Corollary 15. Let be a complete -metric space and let . Assume that there exist two functions and such that Also, suppose that the following conditions are satisfied: (i) is an -admissible multivalued mapping;(ii)there exist and such that ;(iii)for a sequence in with for all and , then for all .
If for all , then has a fixed point.

All results in the paper may be stated with respect to a self-mapping . For instance, and for our further use, we consider the following version of Theorem 14.

Corollary 16. Let be a complete -metric space and let . Assume that there exist two functions and such that, for all with , the following condition holdsAlso, suppose that the following conditions are satisfied: (i), implies ;(ii)there exists such that ;(iii)for a sequence in with for all and , then for all .
If for all , then has a fixed point.

4. Fixed Point Theory in Ordered -Metric Spaces

The study of fixed points in partially ordered sets has been developed in [15, 19–21] as a useful tool for applications on matrix equations and boundary value problems. In this section, we give some results of fixed point for multivalued mappings in the setting of ordered -metric spaces. In fact, a -metric space may be naturally endowed with a partial ordering; that is, if is a partially ordered set, then is called an ordered -metric space. We say that are comparable if or holds. Also, let , ; then whenever for each there exists such that .

Theorem 17. Let be a complete ordered -metric space and let . Assume that there exists a function such thatfor all with . Also, suppose that the following conditions are satisfied: (i)there exist and such that ;(ii)for each and with , we have for all ;(iii) is -upper semicontinuous.
Then has a fixed point.

Proof. Define the function by Clearly, the multivalued mapping is -admissible. In fact, for each and with , we have and by condition (ii) we obtain that for all . This implies that for all . Also, by condition (31), is an --contraction of Ćirić type. Thus all the hypotheses of Theorem 12 are satisfied and has a fixed point.

Also in this case, one can relax the -upper semicontinuity hypothesis on , by using condition (iii) in Theorem 14. Precisely we state the following result.

Theorem 18. Let be a complete ordered -metric space and let . Assume that there exists a function such that for all with . Also, suppose that the following conditions are satisfied: (i)there exist and such that ;(ii)for each and with , we have for all ;(iii)for a sequence in with for all and , then for all .
If for all , then has a fixed point.