# On Multivalued Hybrid Contractions with Applications

Authors: Monairah Alansari; Shehu Shagari Mohammed; Akbar Azam; Nawab Hussain
Journal: Journal of Function Spaces
 (2020)

---

## Abstract

Recently, a notion ofb-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.


---

## Body

## 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 ofb-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. Meanwhile, the notion of b-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 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 b-hybrid 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 [17–21].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 ofb-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.


## 2. Preliminaries

In this section, we collect some important notations, useful definitions, and basic results coherent with the literature. Throughout this paper, we denote byℕ, 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]).
LetX 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:
(i)
σx,y=0 if and only if x=y (self-distancy)(ii)
σx,y=σy,x (symmetry)(iii)
σx,y≤ησx,z+σz,y (weighted triangle inequality)
Then, the tripledX,σ,η is called a b-metric space.
It is noteworthy that every metric is ab-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.Example 1 (see [27]).
LetX=lpℝ with 0<p<1, where
(1)lpℝ=xnn∈ℕ⊆ℝ:∑n=1∞xnp<∞.
Defineσ:X×X⟶ℝ+ as
(2)σx,y=∑n=1∞xn−ynp1/p,where x=xnn∈ℕ and y=ynn∈ℕ. Then, σ is a b-metric with parameter η=21/p and hence, X,σ,21/p is a b-metric space.Example 2 (see [28]).
LetX=ℕ∪∞ and σ:X×X⟶ℝ+ be defined by
(3)σx,y=0,ifx=y,1x−1y,ifx,yareevenorxy=∞,5,ifx,yareoddandx≠y,2,otherwise.
Then,X,σ is a b-metric space with parameter η=3, but σ is not a continuous functional.Definition 2 (see [29]).
LetX,σ,η be a b-metric space. A sequence xnn∈ℕ is said to be
(i)
convergent if and only if there existsx∈X such that σxn,x⟶0 as n⟶∞, and we write this as limn→∞σxn,x=0(ii)
Cauchy if and only ifσxn,xm⟶0 as n,m⟶∞(iii)
complete if every Cauchy sequence inX is convergent
In ab-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]).
LetX,σ,η be a b-metric space. Then, a subset A of X is called
(i)
compact if and only if for every sequence of elements ofA, there exists a subsequence that converges to an element of A(ii)
closed if and only if for every sequencexnn∈ℕ of elements of A that converges to an element x, we have x∈ADefinition 4 (see [25]).
A nonempty subsetA of X is called proximal if, for each x∈X, there exists a∈A such that σx,a=σx,A.
Throughput this paper, we shall denote byCBX, PrX, PbrX, and KX, 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.
LetX,σ,η be a b-metric space. For A,B∈PrX, the function Hb:PrX×PrX⟶ℝ+, defined by
(4)HbA,B=maxsupx∈Aσx,B,supx∈Bσx,A,ifitexists,∞,otherwise,is called Hausdorff-Pompeiu b-metric on PrX induced by the b-metric σ, where
(5)σx,A=infy∈Aσx,y.Definition 5.
LetX,σ,η be a metric space, and NX denotes the family of nonempty subsets of X. A set-valued mapping T:X⟶NX 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
(6)KX⊆PrX⊆CBX⊆NX.Definition 7 (see [17, 30]).
A nondecreasing functionφ:ℝ+⟶ℝ+ is called
(i)
ac-comparison function if φnt⟶0 as n⟶∞ for every t∈ℝ+(ii)
ab-comparison function if there exist k0∈ℕ, λ∈0,1 and a convergent nonnegative series ∑n=1∞xn such that ηk+1φk+1t≤ληkφkt+xk, for η≥1,k≥k0 and any t≥0, where φn denotes the nth iterate of φ
Denote byΩ the family of functions φ:ℝ+⟶ℝ+ satisfying the following conditions:
(i)
φ is a b-comparison function(ii)
φt=0 if and only if t=0(iii)
φ is continuousRemark 8 (see [17]).
Ab-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].Lemma 9 (see [30]).
For a comparison functionφ:ℝ+⟶ℝ+, the following properties hold:
(i)
Each iterateφn,n∈ℕ, is also a comparison function(ii)
φt<t for all t>0Lemma 10 (see [30]).
Letφ:ℝ+⟶ℝ+ be a b-comparison function. Then, the series ∑k=0∞ηkφkt converges for every t∈ℝ+.Remark 11 (see [17]).
In Lemma10, every b-comparison function is a comparison function and thus, in Lemma 9, every b-comparison function satisfies φt<t.Lemma 12 (see [32]).
LetX,σ,η be a b-metric space. For A,B∈KX and x,y∈X, the following conditions hold:
(i)
σx,B≤HbA,B, for any x∈A(ii)
σx,A≤ησx,y+σy,A(iii)
σx,A=0⇔x∈A(iv)
HbA,B=0⇔A=B(v)
HbA,B=HB,A(vi)
HbA,B≤ηHbA,C+HbC,B


## 3. Main Results

We start this section by inaugurating the following definition ofb-hybrid multivalued contraction.Definition 13.
LetX,σ,η be a b-metric space and S,T:X⟶PrX be multivalued maps. Then, the pair S,T is said to form a b-hybrid multivalued contraction, if for all x,y∈X, we have
(7)HbSx,Ty≤φCS,Trx,y,where φ∈Ω, r≥0, and ai≥0, i=1,2,3,4 with ∑i=14ai=1 and
(8)CS,Trx,y=a1σx,yr+a2σx,Sxr+a3σy,Tyr+a4σy,Sx+σx,Ty2ηr1/r,forr>0,x,y∈X,σx,ya1σx,Sxa2σy,Tya3σx,Ty+σy,Sx2ηa4,forr=0,x,y∈X\FixS,T,where
(9)FixS,T=x,y∈X:x∈Sx,y∈Ty.Our main result runs as follows.Theorem 14.
LetX,σ,η be a complete b-metric space and S,T:X⟶PrX be multivalued maps. Suppose that for each x,y∈X, Sx and Ty are nonempty bounded proximal subsets of X. If the pair S,T forms a b-hybrid multivalued contraction, then S and T have a common fixed point in X.Proof.
Letx0∈X, then, by hypotheses, Sx0∈PbrX. Choose x1∈Sx0 such that σx0,x1=σx0,Sx0. Similarly, Tx1∈PbrX, by assumption. So, we can find x2∈Tx1 such that by proximality of T, σx1,x2=σx1,Tx1. Continuing in this fashion, we generate a sequence xnn∈ℕ of elements of X such that
(10)x2p+1∈Sx2p,x2p+2∈Tx2p+1σx2p,x2p+1=σx2p,Sx2p,σx2p+1,x2p+2=σx2p+1,Tx2p+1,p∈ℕ.
By Lemma12 and the above relations, we obtain
(11)σx2p,x2p+1≤HbSx2p,Tx2p−1,σx2p+1,x2p+2≤HbSx2p,Tx2p+1.
Suppose thatx2p=x2p+1, for some p∈ℕ and r>0. Then, from (9), we have
(12)CS,Trx2p,x2p+1=a1σx2p,x2p+1r+a2σx2p,Sx2pr+a3σx2p+1,Tx2p+1r+a4σx2p+1,Sx2p+σx2p,Tx2p+12ηr1/r=a1σx2p,x2p+1r+a2σx2p,x2p+1r+a3σx2p+1,x2p+2r+a4σx2p+1,x2p+1+σx2p,x2p+22ηr1/r≤a3σx2p+1,x2p+2r+a4ησx2p,x2p+1+σx2p+1,x2p+22ηr1/r≤a3σx2p+1,x2p+2r+a4σx2p+1,x2p+2r1/r=a3+a41/rσx2p+1,x2p+2=σx2p+1,x2p+2asr⟶∞.Therefore, using Lemma9, we have
(13)σx2p+1,x2p+2≤HbSx2p,Tx2p+1≤φσx2p+1,x2p+2<σx2p+1,x2p+2,a contradiction. It follows that for all p∈ℕ,
(14)x2p=x2p+1∈Sx2p,x2p=x2p+1=x2p+2∈Tx2p+1=Tx2p.So,x2p turns out to be the common fixed point of S and T.Again, forr=0 and x2p=x2p+1, for some p∈ℕ, we get CS,Trx2p,x2p+1=0, for all p∈ℕ. Therefore, by property (ii) of Ω, one obtains σx2p+1,x2p+2=0, for all p∈ℕ, from which, on similar arguments as above, the same conclusion follows that x2p∈Sx2p∩Tx2p. Hereafter, we presume that for all p∈ℕ, xp+1≠xp if and only if σxp+1,xp>0.Now, in view of (9), setting x=x2p and y=x2p−1, we have
(15)CS,Trx2p,x2p−1=a1σx2p,x2p−1r+a2σx2p,Sx2pr+a3σx2p−1,Tx2p−1r+a4σx2p−1,Sx2p+σx2p,Tx2p−12ηr1/r,forr>0,σx2p,x2p−1a1σx2p,Sx2pa2σx2p−1,Tx2p−1a3×σx2p,Tx2p−1+σx2p−1,Sx2p2ηa4,forr=0.That is,(16)CS,Trx2p,x2p−1=a1σx2p,x2p−1r+a2σx2p,x2p+1r+a3σx2p−1,x2pr+a4σx2p−1,x2p+1+σx2p,x2p2ηr1/r,forr>0,σx2p,x2p−1a1σx2p,x2p+1a2σx2p−1,x2pa3×σx2p,x2p+σx2p−1,x2p+12ηa4,forr=0Now, we consider the following two cases.Case 1.
r>0. Suppose that σx2p,x2p+1≥σx2p−1,x2p, then from (16), we have
(17)CS,Trx2p,x2p−1≤a1σx2p+1,x2pr+a2σx2p+1,x2pr+a3σx2p+1,x2pr+a4ησx2p+1,x2p+σx2p,x2p−12ηr1/r≤a1σx2p+1,x2pr+a2σx2p+1,x2pr+a3σx2p+1,x2pr+a4ησx2p+1,x2p+σx2p+1,x2p2ηr1/r≤a1σx2p+1,x2pr+a2σx2p+1,x2pr+a3σx2p+1,x2pr+a4σx2p+1,x2pr1/r=a1+a2+a3+a4σx2p+1,x2pr1/r=σx2p+1,x2p∑i=14ai1/r=σx2p+1,x2p.
Hence, from (7) and (17), we have
(18)σx2p+1,x2p≤φσx2p+1,x2p.
Sinceφ is a b-comparison function, therefore, (18) implies that
(19)σx2p+1,x2p<σx2p+1,x2p,which is a contradiction. Consequently, it follows that σx2p+1,x2p≤σx2p,x2p−1. Thus, from (18), we obtain
(20)σx2p+1,x2p≤φσx2p,x2p−1.
Settingn=2p∈ℕ in (20) yields
(21)σxn+1,xn≤φσxn,xn−1≤φ2σxn−1,xn−2≤φ3σxn−2,xn−3⋮⋮≤φnσx1,x0.
From (21), by triangle inequality on X,σ,η, for all k≥1, we have
(22)σxn+k,xn≤ησxn+k,xn+1+σxn+1,xn≤1ηn−1∑i=nn+k−1ηkσxi,xi+1≤1ηn−1∑i=nn+k−1ηkφkσx1,x0≤1ηn−1∑i=n∞ηiφiσx1,x0.
Lettingn⟶∞ in (22) and applying Lemma 10, we find that limn→∞σxn+k,xn=0. Therefore, xnn∈ℕ is a Cauchy sequence of points of X,σ,η. The completeness of this space implies that there exists u∈X such that
(23)limn→∞σxn,u=0.
Now, we show thatu is the expected common fixed point of S and T. First, assume that u∉Su so that σu,Su>0. Then, by Lemma 12 and the case r>0 in the contractive inequality (7), we have
(24)σu,Su≤ησu,xn+ησxn,Su≤ησu,xn+ηHbSu,Txn−1≤ησu,xn+ηφCS,Tru,xn−1=ησu,xn+ηφa1σu,xn−1r+a2σu,Sur+a3σxn−1,Txn−1r+a4σxn−1,Su+σu,Txn−12ηr1/r=ησu,xn+ηφa1σu,xn−1r+a2σu,Sur+a3σxn−1,xnr+a4σxn−1,Su+σu,xn2ηr1/r.
Lettingn⟶∞ in (24) and using the properties of φ∈Ω give
(25)σu,Su<ησu,Sua2+a41/r,and as r⟶∞,
(26)σu,Su<ησu,Su.
Notice that takingη=1 in (26) yields a contradiction. Thus, σu,Su=0, which further implies that u∈Su. On similar steps, by assuming that u is not a fixed point of T and considering
(27)σu,Tu≤ησu,xn+ησxn,Tu≤ησu,xn+ηHbSxn−1,Tu≤ησu,xn+ηφCS,Trxn−1,u,we can show that u∈Tu. Consequently, for r>0, there exists u∈X such that u∈Su∩Tu.Case 2.
r=0. Using the inequality (16) on account of b-comparison of φ, we have
(28)σx2p,x2p−1≤HbSx2p−1,Tx2p−2≤φCS,Trx2p−1,x2p−2<σx2p−1,x2p−2a1x2p−1,Sx2p−1a2σx2p−2,Tx2p−2a3×σx2p−1,Tx2p−2+σx2p−2,Sx2p−12ηa4=σx2p−1,x2p−2a1σx2p−1,x2pa2σx2p−2,x2p−1a3×σx2p−1,x2p−1+σx2p−2,x2p2ηa4≤σx2p−1,x2p−2a1σx2p−1,x2pa2σx2p−2,x2p−1a3×σx2p,x2p−1+σx2p−1,x2p−22a4.=σx2p−1,x2p−2a1+a3σx2p−1,x2pa2×σx2p,x2p−1+σx2p−1,x2p−221−a1−a2−a3.
Assume thatσx2p−1,x2p−2≤σx2p,x2p−1; then, (28) gives
(29)σx2p,x2p−1≤φCS,Trx2p−1,x2p−2<σx2p,x2p−1a1+a2+a3σx2p,x2p−11−a1−a2−a3=σx2p,x2p−1,
a contradiction. Therefore,(30)σx2p,x2p−1≤σx2p−1,x2p−2.
Using (28) and (30), we obtain
(31)σx2p,x2p−1≤φσx2p−1,x2p−2.
Notice that (31) is equivalent to (21). So, on similar steps, we deduce that the sequence xnn∈ℕ is Cauchy in X,σ,η. Thus, the completeness of this space guarantees that σxn,u⟶0 as n⟶∞, for some u∈X.
To see thatu is a common fixed point of T and S, we apply Lemma 12 and inequality (7) as follows:
(32)σu,Tu≤ησu,xn+ησxn,Tu≤ησu,xn+ηHbSxn−1,Tu≤ησu,xn+ηφCS,Trxn−1,u,where
CS,Trxn−1,u=σxn−1,ua1σxn−1,Sxn−1a2σu,Tua3×σxn−1,Tu+σu,Sxn−1/2ηa4=σxn−1,ua1σxn−1,xna2σu,Tua3×σxn−1,Tu+σu,xn/2ηa4. (33)(70)
We see thatlimn→∞CS,Trxn−1,u=0. Hence, under this limiting case, (33) becomes
(33)σu,Tu≤ηφ0.
By condition (ii) ofφ, (33) implies that σu,Tu=0. Therefore, u∈Tu. On similar steps, we can show that u∈Su. Consequently, S and T have a common fixed point u∈X.Corollary 15.
LetX,σ,η be a complete b-metric space and S:X⟶PrX be a multivalued map. Suppose that for each x∈X, Sx is a nonempty bounded proximal subset of X. If for all x,y∈X, the following conditions are satisfied:
(34)HbSx,Sy≤φCSrx,y,where φ∈Ω, r≥0, and ai≥0, i=1,2,3,4 with ∑i=14ai=1 and
(35)CSrx,y=a1σx,yr+a2σx,Sxr+a3σy,Syr+a4σy,Sx+σx,Sy2ηr1/r,forr>0,x,y∈X,σx,ya1σx,Sxa2σy,Sya3σx,Sy+σy,Sx2ηa4,forr=0,x,y∈X\FixS,where
(36)FixS=x∈X:x∈Sx,then, there exists u∈X such that u∈Su.Proof.
PutS=T in Theorem 14.The following example is provided to support the hypotheses of Theorem14 for S=T.Example 3.
LetX=1,2,3 and σx,y=x−y2, for all x,y∈X. Then, X,σ,η=2 is a complete b-metric space. Note that X,σ,η=2 is not a metric space, since for x=1,y=3, and z=2, we have
(37)σx,y=4>2=σx,z+σz,y.
Define a multivalued mapS:X⟶PrX by
(38)Sx=1,2,ifx=1,2,3,ifx=2,2,ifx=3.Letφt=t/2, for all t≥0. Clearly, φ∈Ω. Now, we verify inequality (7) as follows: For r>0 and x,y∈X with x≠y, consider the following cases.Case 1.
x=1 and y=2. For this, we have HbS1,S2=Hb1,2,2,3=1; σ1,2=σ1,S2=1; and σ1,S1=σ2,S2=σ2,S1=0. Define Δσ11,2 as
(39)Δσ11,2=a1σ1,2r+a2σ1,S1r+a3σ2,S2r+a4σ2,S1+σ1,S22×2r1/r.
Takea1=a4=1/2 and a2=a3=0; then, (39) becomes
(40)Δσ11,2=12+1212×2r1/r=121/r+1r121/r−1×1212×2r+⋯+121/r12×2=12121/r−1+12r121/r−112×2r+⋯+12121/r−112×2.
Lettingr⟶∞+ in (40), we have
(41)Δσ11,2=122+12=φCSr1,2.
Therefore,(42)HbT1,T2=1≤φCSr1,2.Case 2.
Forx=1 and y=3, we have
(43)HbT1,T3=Hb1,2,2=1,and σ1,3=4; σ1,S1=0; and σ3,S3=σ3,S1=σ1,S3=1. Define Δσ21,3 as
(44)Δσ21,3=a1σ1,3r+a2σ1,S1r+a3σ2,S3r+a4σ3,S1+σ1,S32×2r1/r.
Takea1=a4=1/2 and a2=a3=0; then,
(45)Δσ21,3=124r+121+12×2r1/r=12121/r−14r+12r1/r=12121/r−14+⋯+12=φCSr1,3.
Thus,HbS1,S3=1≤φCSr1,3. Consequently, for all r>0 and x,y∈X with x≠y, we have
(46)HbSx,Sy=1≤φCSrx,y.
Now, we check the case forr=0 and x,y∈X\FixS. Obviously, x,y∈3, and
(47)HbSx,Sy=HbS3,S3=Hb2,2=0≤φCSrx,y.Hence, all the hypotheses of Theorem14 are satisfied with S=T. We can see that the set of all fixed points of S is given by FixS=1,2.Corollary 16.
LetX,σ,η be a complete b-metric space and S:X⟶PrX be a multivalued map. Suppose that for each x∈X, Sx is a nonempty bounded proximal subsets of X. If
(48)HbSx,Sy≤φ14CSrx,y,for all x,y∈X, where φ∈Ω and
(49)CSr=σx,y+σx,Sx+σy,Sy+σy,Sx+σx,Sy2η,then, there exists u∈X such that u∈Su.Proof.
TakeS=T, r=1, and a1=a2=a3=a4=1/4 in Theorem 14.Corollary 17.
LetX,σ,η be a complete b-metric space and S,T:X⟶PrX be multivalued maps. Suppose that for each x,y∈X, Sx and Ty are nonempty bounded proximal subsets of X. If there exists λ∈0,1 such that
(50)HbSx,Ty≤λσx,yσx,Sxσy,Tyσx,Ty+σy,Sx2η4,then S and T have a common fixed point in X.Proof.
Takea1=a2=a3=a4=1/4 and φt=λt for all t≥0 and r=0 in Theorem 14.Corollary 18.
LetX,σ,η be a complete b-metric space and S:X⟶PrX be a multivalued map. Suppose that for each x∈X, Sx is a nonempty bounded proximal subset of X. If there exists λ∈0,1 such that
(51)HbSx,Sy≤λσx,y,then, there exists u∈X such that u∈Su.Proof.
PutS=T, a1=r=1, a2=a3=a4=0, and φt=λt, t≥0, in Theorem 14.In the following corollary, as an application of Corollary15, 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 X,σ,η be a complete b-metric space and F:X⟶X be a single-valued mapping. If
(52)σFx,Fy≤φCFrx,y,for all x,y∈X, where φ∈Ω and r≥0, ai≥0, i=1,2,3,4, with ∑i=14ai=1 and
(53)CFrx,y=a1σx,yr+a2σx,Fxr+a3σy,Fyr+a4σy,Fx+σx,Fy2ηr1/r,forr>0,x,y∈X,σx,ya1σx,Fxa2σy,Fya3σx,Fy+σy,Fx2ηa4,forr=0,x,y∈X\FixF,where
(54)FixF=x∈X:x=Fx.
Then, there existsu∈X such that u=Fu.Proof.
We know thatx∈KX⊆PbrX for every x∈X. Consider a mapping Ξ:X⟶PrX defined as Ξx=Fx,x∈X. Then, all the conditions of Corollary 15 reduce to the conditions of Corollary 19 with S=F and HbSx,Sy=σFx,Fy, for all x,y∈X. Thus, by application of Corollary 15, there exists u∈X such that u=Ξu. The definition of Ξ implies that Ξu=Fu. Consequently, u=Fu.Proof.
Alternativeproof of Corollary 19.
LetF:X⟶X be the single-valued mapping in Corollary 19; then, define a multivalued mapping S:X⟶PrX by Sx=Fx, for all x∈X. Clearly, Sx∈PbrX. Consequently, Corollary 15 can be applied to find u∈X such that u∈Su=Fu, which further implies that Fu=u.Remark 20.
It is clear that if we takeη=1 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:(55)xt∈ft+∫abLt,s,xsσs,for t∈a,b, where x∈Ca,b,ℝ is an unknown function, f∈Ca,b,ℝ is a given real-valued function, and L:a,b×a,b×ℝ⟶Fcvℝ is a given multivalued map, where we denote the family of nonempty compact and convex subsets of ℝ by Fcvℝ1. The set of all real-valued continuous functions on a,b shall be represented by Ca,b,ℝ.Now, we study the existence of solutions of (55) under the following assumptions.Theorem 21.
Suppose that
(C1): the multivalued map L:a,b×a,b×ℝ⟶Fcvℝ is such that for every x∈Ca,b,ℝ, the map Lxt,s≔Lt,s,xs is lower semicontinuous.
(C2): there exists a b-comparison function φ:ℝ+⟶ℝ+ such that for all x,y∈Ca,b,ℝ,
(56)HbLxt,s,Lyt,s≤πt,sφ14xs−ys2+x−Lt,s,xs2+ys−Lt,s,ys2+ys−Lt,s,xs2+xs−Lt,s,ys22η,for each t,s∈a,b and sup∫abπt,s2σs≤1, where πt,.∈L1a,b and for some constants η≥1.Then, the integral inclusion (55) has at least one solution in Ca,b,ℝ.Proof.
LetX=Ca,b,ℝ and σ:X×X⟶ℝ+ be defined by
(57)σx,y=x−y2,forallx,y∈X.
Then,X,σ,η 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⟶PrX be a multivalued map defined as
(58)Sx=y∈X:yt∈ft+∫abLt,s,xsσs,t∈a,b.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.Letx∈X be arbitrary. Since the multivalued map Lx:a,b×a,b⟶⊑cvℝ is lower semicontinuous, it follows from Michael’s selection theorem ([33], Theorem 1]) that there exists a continuous map ρx:a,b×a,b⟶ℝ such that ρxt,s∈Lxt,s, for each t,s∈a,b×a,b. Therefore, ft+∫abρxt,sσs∈Sx. So, Sx is nonempty. One can easily see that Sx is a closed subset of X. Further, since f∈Ca,b,ℝ and Lxt,s is continuous on a,b×a,b, their range sets are compact. Hence, Sx∈KX⊆PbrX⊆PrX.Takex1,x2∈X; then Sx1 and Sx2 are nonempty bounded proximal subsets of X. Let y1∈Sx1 be arbitrary such that
(59)y1t∈ft+∫abLt,s,x1sσs,t∈a,b.This means for eacht,s∈a,b×a,b, there exists ρx1∈Lx1t,s such that
(60)y1t=ft+∫abρx1t,sσs,t∈a,b.Since from (C2),
(61)HbLt,s,x1s,Lt,s,x2s≤πt,sφ14x1s−x2s2+x1s−Lt,s,x1s2+x2s−Lt,s,x2s2+x2s−Lt,s,x1s2+x1−Lt,s,x2s22η,for each t,s∈a,b and all constants η≥1, so, there exists ρx2∈Lx2t,s such that
(62)ρx1t,s−ρx2t,s2≤πt,sφ14x1s−x2s2+x1s−Lt,s,x1s2+x2s−Lt,s,x2s2+x2s−Lt,s,x1s2+x1−Lt,s,x2s22η,for allt,s∈a,b×a,b.Now, consider the multivalued mapM defined by
(63)Mt,s=Lx2t,s∩ω∈ℝ:ρx1t,s−ω2≤πt,sφ14x1s−x2s2.Taking into account the fact that from (C1), M is lower semicontinuous, therefore, there exists a continuous map ρx2:a,b×a,b⟶ℝ such that ρx2t,s∈Mt,s, for all t,s∈a,b×a,b. Then,
(64)y2t=ft+∫abρx1t,sσs∈ft+∫abLt,s,x2sσs,t∈a,b.Thus,y2∈Sx2, and
(65)y1t−y2t2≤∫abρx1t,s−ρx2t,sσs2≤sup∫abπt,sσs2φ14x1s−x2s2+x1s−Lt,s,x1s2+x2s−Lt,s,x2s2+x2s−Lt,s,x1s2+x1s−Lt,s,x2s22η≤φ14x1s−x2s2+x1s−Lt,s,x1s2+x2s−Lt,s,x2s2+x2s−Lt,s,x1s2+x1s−Lt,s,x2s22η≤φ14σx1,x2+σx1,Sx1+σx2,Sx2+σx2,Sx1+σx1,Sx22η.Therefore,σy1,y2≤φ1/4CSrx1,x2; that is,
(66)HbSx1,Sx2≤φ14CSrx1,x2.Settingx=x1 and y=x2 in the above inequation gives
(67)CSrx1,x2=CSrx,y=σx,y+σx,Sx+σy,Sy+σy,Sx+σx,Sy2η,and HbSx,Sy≤φ1/4CSrx,y. Thus, all the hypotheses of Corollary 16 are satisfied. It follows that the integral inclusion (55) has at least one solution in X.Example 4.
LetL:0,20×0,20×ℝ⟶Fcvℝ be defined by
(68)Lt,s,xs=−16,4,ifx≠0,s,t∈0,3×0,3,0,ifx=0,s,t∈3,20×3,20.
By takingft=cost and φt=t4/1+t4, for all t∈0,20, then all the conditions of Theorem 21 are satisfied. Therefore, there exists at least one solution to the Fredholm integral inclusion:
(69)xt∈cost+∫020Lt,s,xsσs,t∈0,20.


## 5. Conclusion

It is well-known that in some abstract spaces, the triangle inequality does not hold. But, by multiplying the constantη≥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 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 [34–36] and the references therein.



---

*Source: 8401403-2020-07-11.xml*