On Multivalued Hybrid Contractions with Applications

Recently, a notion of b-hybrid contraction for single-valued mappings in the framework of b-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.


Introduction
The Banach contraction principle is the first most wellknown, 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][4][5][6][7][8][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 b-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 b-metric space with a view of improving the Banach fixed point theorem. Mean-while, the notion of b-metric spaces is gaining enormous generalizations (see, for example, [8,[13][14][15]). For a recent short survey on basic concepts and results in fixed point theory in the framework of b-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 bhybrid contraction in the frame of b-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][18][19][20][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][23][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 b-hybrid multivalued contraction on a b-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 b-hybrid multivalued contraction in the setting of b-metric spaces, leads to a new existence principle which extends and complements the existing literature.

Preliminaries
In this section, we collect some important notations, useful definitions, and basic results coherent with the literature. Throughout this paper, we denote by ℕ, ℝ + , and ℝ 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 b-metric space as follows.
Definition 1 (see [12]). Let X be a nonempty set and η ≥ 1 be a constant. Suppose that the mapping σ : X × X ⟶ ℝ + satisfies the following conditions for all x, y, z ∈ X: Then, the tripled ðX, σ, ηÞ is called a b-metric space. It is noteworthy that every metric is a b-metric with the parameter η = 1. Also, in general, a b-metric is not a continuous functional. Hence, the class of b-metric is larger than the class of classical metric.
Definition 2 (see [29]). Let ðX, σ, ηÞ be a b-metric space. A sequence fx n g n∈ℕ is said to be In a b-metric space, the limit of a sequence is not always unique. However, if a b-metric is continuous, then every convergent sequence has a unique limit.
Definition 3 (see [29]). Let ðX, σ, ηÞ be a b-metric space. Then, a subset A of X is called (i) compact if and only if for every sequence of elements of A, there exists a subsequence that converges to an element of A (ii) closed if and only if for every sequence fx n g n∈ℕ of elements of A that converges to an element x, we have x ∈ A Definition 4 (see [25]). A nonempty subset A of X is called proximal if, for each x ∈ X, there exists a ∈ A such that σðx, aÞ = σðx, AÞ. Throughput this paper, we shall denote by CBðXÞ, P r ðXÞ, P r b ðXÞ, and KðXÞ, the set of all nonempty closed and bounded subsets of X, the family of all nonempty proximal subsets of X, the set of all bounded proximal subsets of X, and the class of nonempty compact subsets of X, respectively.
is called Hausdorff-Pompeiu b-metric on P r ðXÞ induced by the b-metric σ, where Definition 5. Let ðX, σ, ηÞ be a metric space, and N ðXÞ denotes the family of nonempty subsets of X. A set-valued mapping T : X ⟶ N ðXÞ is called a multivalued map. A point u ∈ X is said to be a fixed point of T if u ∈ Tu.
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 φ : 1Þ and a convergent nonnegative series ∑ ∞ n=1 x n such that η k+1 φ k+1 ðtÞ ≤ λη k φ k ðtÞ + x k , for η ≥ 1, k ≥ k 0 and any t ≥ 0, where φ n denotes the n th iterate of φ Denote by Ω the family of functions φ : ℝ + ⟶ ℝ + satisfying the following conditions: [17]). A b-comparison function is a c-comparison function when η = 1. Moreover, it can be shown that a c-comparison function is a comparison function, but the converse is not always true. For further properties of comparison function, see [31].
Remark 11 (see [17]). In Lemma 10, every b-comparison function is a comparison function and thus, in Lemma 9, every b-comparison function satisfies φðtÞ < t.

Main Results
We start this section by inaugurating the following definition of b-hybrid multivalued contraction.
Definition 13. Let ðX, σ, ηÞ be a b-metric space and S, T : X ⟶ P r ðXÞ be multivalued maps. Then, the pair ðS, TÞ is said to form a b-hybrid multivalued contraction, if for all x, y ∈ X, we have where φ ∈ Ω, r ≥ 0, and Our main result runs as follows.

Theorem 14.
Let ðX, σ, ηÞ be a complete b-metric space and S, T : X ⟶ P r ðXÞ be multivalued maps. Suppose that for each x, y ∈ X, Sx and Ty are nonempty bounded proximal subsets of X. If the pair ðS, TÞ forms a b-hybrid multivalued contraction, then S and T have a common fixed point in X.
Proof. Let x 0 ∈ X, then, by hypotheses, Sx 0 ∈ P r b ðXÞ. Choose ðXÞ, by assumption. So, we can find x 2 ∈ Tx 1 such that by proximality of T, σðx 1 , x 2 Þ = σðx 1 , Tx 1 Þ. Continuing in this fashion, we generate a sequence fx n g n∈ℕ of elements of X such that By Lemma 12 and the above relations, we obtain Suppose that x 2p = x 2p+1 , for some p ∈ ℕ and r > 0. Then, from (9), we have Therefore, using Lemma 9, we have a contradiction. It follows that for all p ∈ ℕ, So, x 2p turns out to be the common fixed point of S and T.
Hence, from (7) and (17), we have Since φ is a b-comparison function, therefore, (18) implies that which is a contradiction. Consequently, it follows that σðx 2p+1 , x 2p Þ ≤ σðx 2p , x 2p−1 Þ. Thus, from (18), we obtain Setting n = 2p ∈ ℕ in (20) yields From (21), by triangle inequality on ðX, σ, ηÞ, for all k ≥ 1, we have Letting n ⟶ ∞ in (22) and applying Lemma 10, we find that lim n→∞ σðx n+k , x n Þ = 0. Therefore, fx n g n∈ℕ is a Cauchy sequence of points of ðX, σ, ηÞ. The completeness of this space implies that there exists u ∈ X such that Now, we show that u is the expected common fixed point of S and T. First, assume that u ∉ Su so that σðu, SuÞ > 0. Then, by Lemma 12 and the case r > 0 in the contractive inequality (7), we have

Journal of Function Spaces
Letting n ⟶ ∞ in (24) and using the properties of φ ∈ Ω give and as r ⟶ ∞, Notice that taking η = 1 in (26) yields a contradiction. Thus, σðu, SuÞ = 0, which further implies that u ∈ Su. On similar steps, by assuming that u is not a fixed point of T and considering we can show that u ∈ Tu. Consequently, for r > 0, there exists u ∈ X such that u ∈ Su ∩ Tu.
Proof. Put S = T in Theorem 14.
The following example is provided to support the hypotheses of Theorem 14 for S = T.

Journal of Function Spaces
Thus, H b ðS1, S3Þ = 1 ≤ φðC r ðSÞ ð1, 3ÞÞ. Consequently, for all r > 0 and x, y ∈ X with x ≠ y, we have Now, we check the case for r = 0 and x, y ∈ X \ F ix ðSÞ. Obviously, x, y ∈ f3g, and Hence, all the hypotheses of Theorem 14 are satisfied with S = T. We can see that the set of all fixed points of S is given by F ix ðSÞ = f1, 2g.

Corollary 16.
Let ðX, σ, ηÞ be a complete b-metric space and S : X ⟶ P r ðXÞ be a multivalued map. Suppose that for each x ∈ X, Sx is a nonempty bounded proximal subsets of X. If for all x, y ∈ X, where φ ∈ Ω and then, there exists u ∈ X such that u ∈ Su.
Corollary 17. Let ðX, σ, ηÞ be a complete b-metric space and S, T : X ⟶ P r ðXÞ be multivalued maps. Suppose that for each x, y ∈ X, Sx and Ty are nonempty bounded proximal subsets of X. If there exists λ ∈ ½0, 1Þ such that then S and T have a common fixed point in X.
Corollary 18. Let ðX, σ, ηÞ be a complete b-metric space and S : X ⟶ P r ðXÞ be a multivalued map. Suppose that for each x ∈ X, Sx is a nonempty bounded proximal subset of X. If there exists λ ∈ ½0, 1Þ such that then, there exists u ∈ X such that u ∈ Su.
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 singlevalued mapping.
Proof. We know that fxg ∈ KðXÞ ⊆ P r b ðXÞ for every x ∈ X. Consider a mapping Ξ : X ⟶ P r ðXÞ defined as Ξx = fFxg, x ∈ X. Then, all the conditions of Corollary 15 reduce to the conditions of Corollary 19 with S = F and H b ðSx, SyÞ = σðFx, FyÞ, for all x, y ∈ X. Thus, by application of Corollary 15, there exists u ∈ X such that fug = Ξu. The definition of Ξ implies that Ξu = fFug. Consequently, u = Fu.

Journal of Function Spaces
Proof. Alternative proof of Corollary 19. Let F : X ⟶ X be the single-valued mapping in Corollary 19; then, define a multivalued mapping S : X ⟶ P r ðXÞ by Sx = fFxg, for all x ∈ X. Clearly, Sx ∈ P r b ðXÞ. Consequently, Corollary 15 can be applied to find u ∈ X such that u ∈ Su = fFug, which further implies that Fu = u.
Remark 20. It is clear that if we take η = 1 in all the above results, we can deduce their analogs in the setting of metric spaces.

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 t ∈ ½a, b, where x ∈ Cð½a, b, ℝÞ is an unknown function, f ∈ Cð½a, b, ℝÞ is a given real-valued function, and L : ½a, b × ½a, b × ℝ ⟶ F cv ðℝÞ is a given multivalued map, where we denote the family of nonempty compact and convex subsets of ℝ by F cv ðℝÞ 1 . The set of all real-valued continuous functions on ½a, b shall be represented by Cð½a, b, ℝÞ. Now, we study the existence of solutions of (55) under the following assumptions.
Then, the integral inclusion (55) has at least one solution in Cð½a, b, ℝÞ.
Proof. Let X = Cð½a, b, ℝÞ and σ : X × X ⟶ ℝ + be defined by Then, ðX, σ, ηÞ is a complete b-metric space with the parameter η = 2. Note that X endowed with this metric σ is not a metric space. Let S : X ⟶ P r ðXÞ be a multivalued map defined as Obviously, the set of solutions of (55) coincides with the set of fixed points of S. Therefore, we have to show that under the given suppositions, S has at least one fixed point in X. For this, we shall verify that all the hypotheses of Corollary 16 are satisfied.
Take x 1 , x 2 ∈ X; then Sx 1 and Sx 2 are nonempty bounded proximal subsets of X. Let y 1 ∈ Sx 1 be arbitrary such that

Conclusion
It is well-known that in some abstract spaces, the triangle inequality does not hold. But, by multiplying the constant η ≥ 1 on the right-hand side of the triangle inequality, one can obtain a more useful abstract structure, now called a b -metric space in the literature. Following this improvement, in this work, an idea of b-hybrid multivalued contraction using the Hausdorff distance function is introduced in the setting of b-metric spaces. The established concept herein merges several results in one theorem. A few of these particular cases are stated. Thereafter, existence conditions for solutions of an integral inclusion of Fredholm type are discussed by applying one of the presented results herein. While the presented results in this paper are theoretical, it is noteworthy that Hausdorff metric has numerous and useful applications in everyday life. For example, in computer vision, the Hausdorff distance can be applied to locate a given template in an arbitrary target image. The template and image are often preprocessed through an edge detector giving a binary image. Next, each one (activated) point in the binary image of the template is treated as a point in a set, the structure of the template. Similarly, an area of the binary target image is treated as a set of points. The algorithm then tries 10 Journal of Function Spaces to minimize the Hausdorff distance between the template and some areas of the target image. The area in the target image with the minimal Hausdorff distance to the template can be taken as the best candidate for locating the template in the target. Moreover, in computer graphics, the Hausdorff metric is employed to measure the difference between two distinct representations of the same 3D objects, particularly when generating level of detail for efficient display of complex 3D models. For more applications of Hausdorff distance, see [34][35][36] and the references therein.

Data Availability
No data were used to support this study.

Conflicts of Interest
The authors declare that they have no competing interests.