#### Abstract

Recently, a notion of -hybrid contraction for single-valued mappings in the framework of -metric spaces which unify and improve several significant existing results in the corresponding literature was introduced. This paper presents a multivalued generalization for such contraction. Moreover, one of our obtained results is applied to analyze some solvability conditions of Fredholm-type integral inclusions. Nontrivial examples are also provided to support the assertions of our theorems.

#### 1. Introduction

The Banach contraction principle is the first most well-known, simple, and versatile classical result in fixed point theory with metric space structure. More than a handful of literature embraces applications and generalizations of this principle from different perspectives, for example, by weakening the hypotheses, employing different mappings and various forms of metric spaces. In this context, the work of Rhoades [1] is useful for visiting important modifications of Banach-type contractive definitions. In 1969, Nadler [2] gave a generalization of the Banach contraction principle for multivalued contraction mappings by using the Hausdorff metric and established the first fixed point theorem for multivalued mappings defined on metric space. Since then, a number of generalizations in diverse frames of Nadler’s fixed point result have been investigated by several authors (see, for example, [3–9] and references therein).

The analysis of new spaces and their properties has been an interesting topic among the mathematical research community. In this direction, the notion of -metric spaces is presently thriving. The idea commenced with the work of Bakhtin [10] and Bourbaki [11]. Thereafter, Czerwik [12] gave a postulate which is weaker than the classical triangle inequality and formally established a -metric space with a view of improving the Banach fixed point theorem. Meanwhile, the notion of -metric spaces is gaining enormous generalizations (see, for example, [8, 13–15]). For a recent short survey on basic concepts and results in fixed point theory in the framework of -metric spaces, we refer the interested reader to Karapinar [16]. On similar development, one of the active branches of fixed point theory that is also currently drawing the attentions of researchers is the study of hybrid contractions. The concept has been viewed in two directions; viz., first, hybrid contraction deals with contractions involving both single-valued and multivalued mappings, and the second merges linear and nonlinear contractions. Recently, Karapinar and Fulga [17] introduced a new notion of -hybrid contraction in the frame of -metric space and studied the existence and uniqueness of fixed points for such contraction. Their ideas merged several existing results in the corresponding literature. Interestingly, hybrid fixed point theory has potential applications in functional inclusions, optimization theory, fractal graphics, discrete dynamics for set-valued operators, and other areas of nonlinear functional analysis. For some work on this line, the reader may consult [17–21].

Integral inclusions arise in several problems in mathematical physics, control theory, critical point theory for nonsmooth energy functionals, differential variational inequalities, fuzzy set arithmetic, traffic theory, etc. (see, for instance, [22–24]). Usually, the first most concerned problem in the study of integral inclusions is the conditions for existence of its solutions. In this direction, several authors have applied different fixed point approaches and topological methods to obtain existence results of integral inclusions in abstract spaces (see, for example, Appell et al. [22], Cardinali and Papageorgiou [24], Kannan and O’Regan [25], Pathak et al. [9], Sintamarian [26], and the references therein). Most of the results established in the above papers are based on the multivalued analogs of the Banach, Leray-Schauder, Matelli, Schauder, and Sadovskii-type fixed point theorems. In addition, the ambient space of the existence theorems is either a Banach space or classical metric space.

Following the above development, we define in this paper the idea of -hybrid multivalued contraction on a -metric space and analyze conditions for existence of fixed points for such contraction. A nontrivial example which supports the hypotheses of our results is provided. Thereafter, a few significant particular cases are deduced which include the recent results of Karapinar and Fulga [17] and many others. Furthermore, one of our results is applied to investigate sufficient conditions for existence of solutions to an integral inclusion of Fredholm type. The latter concept is adapted from Sintamarian [26]. However, our technique, being obtained through a -hybrid multivalued contraction in the setting of -metric spaces, leads to a new existence principle which extends and complements the existing literature.

#### 2. Preliminaries

In this section, we collect some important notations, useful definitions, and basic results coherent with the literature. Throughout this paper, we denote by , , and the sets of natural numbers, nonnegative real numbers, and real numbers, respectively. These preliminary concepts are recorded from [2, 12, 17].

In 1993, Czerwik [12] introduced the notion of a -metric space as follows.

*Definition 1 (see [12]). *Let be a nonempty set and be a constant. Suppose that the mapping satisfies the following conditions for all :
(i) if and only if (self-distancy)(ii) (symmetry)(iii) (weighted triangle inequality)Then, the tripled is called a -metric space.

It is noteworthy that every metric is a -metric with the parameter . Also, in general, a -metric is not a continuous functional. Hence, the class of -metric is larger than the class of classical metric.

*Example 1 (see [27]). *Let with , where
Define as
where and . Then, is a -metric with parameter and hence, is a -metric space.

*Example 2 (see [28]). *Let and be defined by
Then, is a -metric space with parameter , but is not a continuous functional.

*Definition 2 (see [29]). *Let be a -metric space. A sequence is said to be
(i)convergent if and only if there exists such that as , and we write this as (ii)Cauchy if and only if as (iii)complete if every Cauchy sequence in is convergentIn a -metric space, the limit of a sequence is not always unique. However, if a -metric is continuous, then every convergent sequence has a unique limit.

*Definition 3 (see [29]). *Let be a -metric space. Then, a subset of is called
(i)compact if and only if for every sequence of elements of , there exists a subsequence that converges to an element of (ii)closed if and only if for every sequence of elements of that converges to an element , we have

*Definition 4 (see [25]). *A nonempty subset of is called proximal if, for each , there exists such that .

Throughput this paper, we shall denote by , , , and , the set of all nonempty closed and bounded subsets of , the family of all nonempty proximal subsets of , the set of all bounded proximal subsets of , and the class of nonempty compact subsets of , respectively.

Let be a -metric space. For , the function , defined by
is called Hausdorff-Pompeiu -metric on induced by the -metric , where

*Definition 5. *Let be a metric space, and denotes the family of nonempty subsets of . A set-valued mapping is called a multivalued map. A point is said to be a fixed point of if .

*Remark 6. *Since every compact set is proximal and every proximal set is closed (see [25]), we have the inclusions

*Definition 7 (see [17, 30]). *A nondecreasing function is called
(i)a -comparison function if as for every (ii)a -comparison function if there exist , and a convergent nonnegative series such that , for and any , where denotes the iterate of Denote by the family of functions satisfying the following conditions:
(i) is a -comparison function(ii) if and only if (iii) is continuous

*Remark 8 (see [17]). *A -comparison function is a -comparison function when . Moreover, it can be shown that a -comparison function is a comparison function, but the converse is not always true. For further properties of comparison function, see [31].

Lemma 9 (see [30]). *For a comparison function , the following properties hold:
*(i)*Each iterate , is also a comparison function*(ii)* for all *

Lemma 10 (see [30]). *Let be a -comparison function. Then, the series converges for every .*

*Remark 11 (see [17]). *In Lemma 10, every -comparison function is a comparison function and thus, in Lemma 9, every -comparison function satisfies .

Lemma 12 (see [32]). *Let be a -metric space. For and , the following conditions hold:
*(i)*, for any *(ii)*(iii)**(iv)**(v)**(vi)*

#### 3. Main Results

We start this section by inaugurating the following definition of -hybrid multivalued contraction.

*Definition 13. *Let be a -metric space and be multivalued maps. Then, the pair is said to form a -hybrid multivalued contraction, if for all , we have
where , , and , with and
where

Our main result runs as follows.

Theorem 14. *Let be a complete -metric space and be multivalued maps. Suppose that for each , and are nonempty bounded proximal subsets of . If the pair forms a -hybrid multivalued contraction, then and have a common fixed point in .*

*Proof. *Let , then, by hypotheses, . Choose such that . Similarly, , by assumption. So, we can find such that by proximality of , . Continuing in this fashion, we generate a sequence of elements of such that
By Lemma 12 and the above relations, we obtain
Suppose that , for some and . Then, from (9), we have

Therefore, using Lemma 9, we have a contradiction. It follows that for all ,

So, turns out to be the common fixed point of and .

Again, for and , for some , we get , for all . Therefore, by property (ii) of , one obtains , for all , from which, on similar arguments as above, the same conclusion follows that . Hereafter, we presume that for all , if and only if .

Now, in view of (9), setting and , we have

That is,

Now, we consider the following two cases.

*Case 1. *. Suppose that , then from (16), we have
Hence, from (7) and (17), we have
Since is a -comparison function, therefore, (18) implies that
which is a contradiction. Consequently, it follows that . Thus, from (18), we obtain
Setting in (20) yields
From (21), by triangle inequality on , for all , we have
Letting in (22) and applying Lemma 10, we find that . Therefore, is a Cauchy sequence of points of . The completeness of this space implies that there exists such that
Now, we show that is the expected common fixed point of and . First, assume that so that . Then, by Lemma 12 and the case in the contractive inequality (7), we have
Letting in (24) and using the properties of give
and as ,
Notice that taking in (26) yields a contradiction. Thus, , which further implies that . On similar steps, by assuming that is not a fixed point of and considering
we can show that . Consequently, for , there exists such that .

*Case 2. *. Using the inequality (16) on account of -comparison of , we have
Assume that ; then, (28) gives
a contradiction. Therefore,
Using (28) and (30), we obtain
Notice that (31) is equivalent to (21). So, on similar steps, we deduce that the sequence is Cauchy in . Thus, the completeness of this space guarantees that as , for some .

To see that is a common fixed point of and , we apply Lemma 12 and inequality (7) as follows:
where

(33)(70)

We see that . Hence, under this limiting case, (33) becomes
By condition (ii) of , (33) implies that . Therefore, . On similar steps, we can show that . Consequently, and have a common fixed point .

Corollary 15. *Let be a complete -metric space and be a multivalued map. Suppose that for each , is a nonempty bounded proximal subset of . If for all , the following conditions are satisfied:
where , , and , with and
where
then, there exists such that .*

*Proof. *Put in Theorem 14.

The following example is provided to support the hypotheses of Theorem 14 for .

*Example 3. *Let and , for all . Then, is a complete -metric space. Note that is not a metric space, since for , and , we have
Define a multivalued map by

Let , for all . Clearly, . Now, we verify inequality (7) as follows: For and with , consider the following cases.

*Case 1. * and . For this, we have ; ; and . Define as
Take and ; then, (39) becomes
Letting in (40), we have
Therefore,

*Case 2. *For and , we have
and ; ; and . Define as
Take and ; then,
Thus, . Consequently, for all and with , we have
Now, we check the case for and . Obviously, , and

Hence, all the hypotheses of Theorem 14 are satisfied with . We can see that the set of all fixed points of is given by .

Corollary 16. *Let be a complete -metric space and be a multivalued map. Suppose that for each , is a nonempty bounded proximal subsets of . If
for all , where and
then, there exists such that .*

*Proof. *Take , , and in Theorem 14.

Corollary 17. *Let be a complete -metric space and be multivalued maps. Suppose that for each , and are nonempty bounded proximal subsets of . If there exists such that
then and have a common fixed point in .*

*Proof. *Take and for all and in Theorem 14.

Corollary 18. *Let be a complete -metric space and be a multivalued map. Suppose that for each , is a nonempty bounded proximal subset of . If there exists such that
then, there exists such that .*

*Proof. *Put , , , and , , in Theorem 14.

In the following corollary, as an application of Corollary 15, we deduce the main result of Karapinar and Fulga [17] without using the continuity of the considered single-valued mapping.

Corollary 19. *(see [17], Theorem 1). Let be a complete -metric space and be a single-valued mapping. If
for all , where and , , , with and
where
**Then, there exists such that .*

*Proof. *We know that for every . Consider a mapping defined as . Then, all the conditions of Corollary 15 reduce to the conditions of Corollary 19 with and , for all . Thus, by application of Corollary 15, there exists such that . The definition of implies that . Consequently, .

*Proof. *Alternative proof of Corollary 19.

Let be the single-valued mapping in Corollary 19; then, define a multivalued mapping by , for all . Clearly, . Consequently, Corollary 15 can be applied to find such that , which further implies that .

*Remark 20. *It is clear that if we take in all the above results, we can deduce their analogs in the setting of metric spaces.

#### 4. Application to Fredholm Integral Inclusions

In this section, we apply one of the results in the previous section to study some sufficient conditions for existence of solutions of a Fredholm Integral inclusion. For basic concepts of integral inclusions, we refer the interested reader to [22, 23, 25] and references therein.

Consider the following integral inclusion of Fredholm type: for , where is an unknown function, is a given real-valued function, and is a given multivalued map, where we denote the family of nonempty compact and convex subsets of by . The set of all real-valued continuous functions on shall be represented by .

Now, we study the existence of solutions of (55) under the following assumptions.

Theorem 21. *Suppose that**(C _{1}): the multivalued map is such that for every , the map is lower semicontinuous.*

*(C*

_{2}): there exists a -comparison function such that for all