JMATHJournal of Mathematics2314-47852314-4629Hindawi10.1155/2021/99458429945842Research ArticleCommon α-Fuzzy Fixed Point Results with Applications to Volterra Integral InclusionsAl-MazrooeiAbdullah Eqalhttps://orcid.org/0000-0003-4755-7044AhmadJamshaidKocinacLjubisaDepartment of MathematicsUniversity of JeddahP.O.Box 80327Jeddah 21589Saudi Arabiauj.edu.sa2021125202120211632021284202112520212021Copyright © 2021 Abdullah Eqal Al-Mazrooei and Jamshaid Ahmad.This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

The purpose of this paper is to establish some common α-fuzzy fixed point theorems for a pair fuzzy mappings and obtain some results of literature for multivalued mappings. For it, we define the notion of generalized Θ-contractions in the context of b-metric spaces. As applications, we investigate the solutions of Volterra integral inclusions by our established results.

1. Introduction and Preliminaries

Among all the impressive and inspiring generalizations of metric spaces, b-metric space has an integral place. Czerwik  in 1993 extended the notion of metric space by introducing the conception of b-metric space in this way.

Definition 1.

Let . A mapping db: ×0+ is called b-metric if it satisfies these assertions:

b1dbω,ϖ=0ω=ϖ

b2dbω,ϖ=dbϖ,ω

b3dbω,υsdbω,ϖ+dbϖ,υ

for all ω,ϖ,υ, where s1.

Then, ,db,s is called a b-metric space. A standard example of b-metric space which is not metric space is the following:

= and db:× defined by(1)dbω,ϖ=ωϖ2,for all ω,ϖ with s=2.

Let Pcb represent the class of all nonempty, bounded, and closed subsets of . For Ξ1,Ξ2,Ξ3Pcb, we define Hb:Pcb×Pcb+ by(2)HbΞ1,Ξ2=maxδbΞ1,Ξ2,δbΞ2,Ξ1,where(3)δbΞ1,Ξ2=supdbω,ϖ:ωΞ1,ϖΞ2,Dbω,Ξ3=Dbω,Ξ3=infdbω,υ:ωΞ1,υΞ3.

Note that Hb is called the Hausdorff b-metric induced by the b-metric db. We recall the following properties from .

Lemma 1.

(see ). Let ,db,s be a b-metric space. For any Ξ1,Ξ2,Ξ3Pcb and any ω,ϖ, we have the following:

Dbω,Ξ2dbω,b for any bΞ2

δbΞ1,Ξ2HbΞ1,Ξ2

Dbω,Ξ2HbΞ1,Ξ2 for any ωΞ1

HbΞ1,Ξ2=0

HbΞ1,Ξ2=HbΞ2,Ξ1

HbΞ1,Ξ3sHbΞ1,Ξ2+HbΞ2,Ξ3

Dbω,Ξ1sdbω,ϖ+dbϖ,Ξ1.

Later on, many authors (see ) worked in this way. Recently Jleli and Samet  gave the notion of Θ-contractions and proved a contemporary result for such contractions in generalized metric spaces. Afterwards, Hancer et al.  revised the foregoing definitions by including a broad condition (Θ4). Inspired by Jleli and Samet  and Hancer et al. , Alamri et al.  initiated the above notions in the context of b-metric spaces and introduced a more general condition (Θ5) along with above axioms.

Definition 2.

(see ). We represent by Ωss1 the family of all mappings Θ:+1, satisfying these properties:

(Θ1) 0<ϱ1<ϱ2Θϱ1Θϱ2

(Θ2) for ϱn+, limnΘϱn=1 if and only if limnϱn=0

(Θ3) there exists h0,1 and q0, such that limϱ0+Θϱ1/ϱh=q

(Θ4) ΘinfΞ=infΘΞ for all Ξ0, with infΞ>0

(Θ5) for all ϱn+ such that ΘsϱnΘϱn1k, n and some k0,1, then ΘsnnϱΘsn1ϱn1k, for all n

They supported this condition by the following nontrivial example.

Example 1.

(see ). Let Θ:0,1, be given by θη=eηeη. Clearly, Θ satisfies (Θ1)-(Θ5). Here we show only (Θ5). Assume that, for all n and some k0,1, we have θsϱnθϱn1k, which implies that(4)esϱnesϱneϱn1eϱn1k,sϱnesϱnkϱn1eϱn1.

This implies that(5)sϱnesϱnϱn1kϱn1.

As θsϱnθϱn1kθϱn1. Also θ is nondecreasing, so sϱnϱn1 and sϱnϱn10 implies esn1sϱnϱn1esϱnϱn1. Therefore, (5) implies(6)sϱnesn1sϱnϱn1kϱn1sϱnesnϱnesn1ϱn1kϱn1sϱnesnϱnkϱn1esn1ϱn1snϱnesnϱnksn1ϱn1esn1ϱn1esnϱnesnϱneksn1ϱn1esn1ϱn1θsnϱnθsn1ϱn1k,and hence (Θ5) holds.

On the other side, Kumam et al.  utilized the concept of b-metric space and obtained common α- fuzzy fixed points for fuzzy mappings under generalized rational contractions. For more details in the direction of fixed point results for fuzzy mappings, we refer  to the readers.

We need the following lemma of Czerwik .

Lemma 2.

(see ). Let ,db,s be a b-metric space and Ξ1,Ξ2CB, then ωΞ1,(7)dbω,Ξ2HbΞ1,Ξ2.

In this paper, we obtain common α-fuzzy fixed point results for a pair of fuzzy mappings and establish some theorems to generalize some results from the literature. We solve the Volterra integral inclusions as application of our established results.

2. Main Results

In this way, we state our main result.

Theorem 1.

Let ,db,s be a complete b-metric space with coefficient s1 such that db is continuous. Assume that O1,O2: and for each ω,ϖ, there exist αO1ω,αO2ϖ0,1 such that O1ωαO1ω,O2ϖαO2ϖPcb. If there exist ΘΩs and k0,1 such that(8)ΘsHbO1ωαO1ω,O2ϖαO2ϖΘdbω,ϖk,for all ω,ϖ with HbO1ωαO1ω,O2ϖαO2ϖ>0, then there exists ω such that ωO1ωαO1ωO2ωαO2ω.

Proof.

Let ω0, then by assumption there exists αO1ω00,1 such that O1ω0αO1ω0CB. Let ω1O1ω0αO1ω0. For this ω1, there exists αO2ω10,1 such that O2ω1αO2ω1Pcb. By Lemma 2, (Θ1) and (8), we have(9)Θsdω1,O2ω1αO2ω1ΘsHbO1ω0αO1ω0,O2ω1αO2ω1Θdbω0,ω1k.

Thus,(10)Θsdω1,O2ω1αO2ω1Θdbω0,ω1k.

From (Θ4), we know that(11)Θsdω1,O2ω1αO2ω1=infyO2ω1αO2ω1Θsdbω1,y.

Thus from (10), we get(12)infyO2ω1αO2ω1Θsdbω1,yΘdbω0,ω1k.

Then, from (12), there exists ω2O2ω1αO2ω1 (obviously, ω2ω1) such that(13)Θsdbω1,ω2Θdbω0,ω1k.

For this ω2, there exists αO1ω20,1 such that O1ω2αO1ω2Pcb. By Lemma 2, (Θ1), and (8), we have(14)Θsdbω2,O1ω2αO1ω2ΘsHbO2ω1αO2ω1,O1ω2αO1ω2Θdbω1,ω2k.

Thus,(15)Θsdbω2,O1ω2αO1ω2Θdbω1,ω2k.

From (Θ4), we know that(16)Θsdbω2,O1ω2αO1ω2=infyO1ω2αO1ω2Θsdbω2,y.

Thus from (15), we get(17)infyO1ω2αO1ω2Θsdbω2,yΘdbω1,ω2k.

Then, from (17), there exists ω3O1ω2αO1ω2 (obviously, ω3ω2) such that(18)Θsdbω2,ω3Θdbω1,ω2k.

So, continuing in the same way, we construct ωn in such that(19)ω2n+1O1ω2nαO1ω2n,ω2n+2O2ω2n+1αO2ω2n+1,(20)Θsdbω2n+1,ω2n+2Θdbω2n,ω2n+1k,(21)Θsdbω2n+2,ω2n+3Θdbω2n+1,ω2n+2k,for all n. From (20) and (21), we get(22)Θsdbωn,ωn+1Θdbωn1,ωnk,for all n. It follows by (22) and property (Θ5) that(23)Θsndbωn,ωn+1Θsn1dbωn1,ωnk,which further implies that(24)Θsndbωn,ωn+1Θsn1dbωn1,ωnkΘsn2dbωn2,ωn1k2Θdbω0,ω1kn,for all n. Thus,(25)Θsndbωn,ωn+1Θdbω0,ω1kn,for all n. Since ΘΩ, so letting n in (25), we get(26)limnΘsndbωn,ωn+1=1.

This implies(27)limnsndbωn,ωn+1=0,by (Θ2). By (Θ3), there exists 0<r<1 and q0, so that(28)limnΘsndbωn,ωn+11sndbωn,ωn+1r=q.

Suppose that q<. For this case, let ς2=q/2>0. By definition of the limit, there exists n0 so that(29)Θsndbωn,ωn+11sndbωn,ωn+1rqς2,for all n>n0. This implies that(30)Θsndbωn,ωn+11sndbωn,ωn+1rqς2=q2=ς2,for all n>n0. Then(31)nsndbωn,ωn+1rς1nΘsndbωn,ωn+11,for all n>n0, where ς1=1/ς2. Now we assume that q=. Let ς2>0. From the definition of the limit, there exists n0 such that(32)ς2Θsndbωn,ωn+11sndbωn,ωn+1r,for all n>n0, which implies(33)nsndbωn,ωn+1rς1nΘsndbωn,ωn+11,for all n>n0, where ς1=1/ς2. Hence, in all cases, there exists ς1>0 and n0 such that(34)nsndbωn,ωn+1rς1nΘsndbωn,ωn+11,for all n>n0. Hence by (25) and (34), we obtain(35)nsndbωn,ωn+1rς1nΘdbω0,ω1rn1.

Taking the limit n, we get(36)limnnsndbωn,ωn+1r=0.

Thus limnn1/rsndbωn,ωn+1=0 which implies that n=1sndbωn,ωn+1 is convergent. Thus ωn is a Cauchy sequence in . Since ,db,s is a complete b-metric space, so there exists a ω such that(37)limnωn=ω.

Now, we prove that ωO2ωαO2ω. We suppose on the contrary that ωO2ωαO2ω, then there exist n0 and ωnk of ωn such that dbω2nk+1,O2ωαO2ω>0, for all nkn0. Now, using (8) with ω=ω2nk+1 and ϖ=ω, we obtain(38)Θdbω2nk+1,O2ωαO2ωΘsdbω2nk+1,O2ωαO2ωΘsHbO1ω2nkαO1ω2nk,O2ωαO2ωΘdω2nk,ωk.

As k0,1, so by (Θ1) so we obtain(39)dbω2nk+1,O2ωαO2ω<dbω2nk,ω.

Letting n, we have(40)dbω,O2ωαO2ω0.

Hence, ωO2ωαO2ω. Likewise, one can straightforwardly prove that ωO1ωαO1ω. Thus, ωO1ωαO1ωO2ωαO2ω.

Note: From now to onwards, we consider db as continuous functional and ,db,s as complete b-metric space.

The following corollary follows from Theorem 1 by considering Θη=eη for η>0.

Theorem 2.

Let O1,O2:, and for each ω,ϖ, αO1ω,αO2ϖ0,1 such that O1ωαO1ω,O2ϖαO2ϖPcb. If k0,1 such that(41)sHbO1ωαO1ω,O2ϖαO2ϖkdbω,ϖfor all ω,ϖ, then there exists ω such that ωO1ωαO1ωO2ωαO2ω.

Theorem 3.

Let O:, and for each ω,ϖ, there exist αOω,αOϖ0,1 such that OωαOω,OϖαOϖPcb. If there exists k0,1 such that(42)sHbOωαOω,OϖαOϖkdbω,ϖ,for all ω,ϖ, then there exists ω such that ωOωαOω.

Example 2.

Let =0,1,2 and db:×0+ by(43)dbω,ϖ=0,if ω=ϖ,16,if ωϖ and ω,ϖ0,1,12,if ωϖ and ω,ϖ0,2,1,if ωϖ and ω,ϖ1,2.

It is easy to see that ,d is a complete b-metric space with coefficient s=3/2. Define(44)O0η=O1η=12,if η=0,0,if η=1,2,O2η=0,if η=0,2,12,if η=1.

Define α:0,1 by αω=1/2 for all ω. Now we obtain that(45)Oω1/2=0,if ω=0,11,if ω=2.

For ω,ϖ, we get(46)HbO11/2,O21/2=HbO11/2,O21/2=Hb0,1=16.

Taking Θη=eη for η>0 and k=1/2. Then(47)ΘsHbO01/2,O21/2=e141/2<e121/4=Θdb0,2k,also(48)ΘsHbO11/2,O21/2=e141/2<e11/4=Θdb1,2k,for all ω,ϖ. As a result, all assertions of Theorem 6 hold and there exists 0 such that 0O01/2is an α-fuzzy fixed point of O.

3. Set-Valued ResultsTheorem 4.

Let G1,G2:XCBX. Suppose that k0,1 such that(49)ΘsHbG1ω,G2ϖΘdbω,ϖk,for all ω,ϖ. Then there exists ω such that ωG1ωG2ω.

Proof.

Define α:0,1 and O1,O2: by(50)O1ωη=αω,if ηG1ω,0,if ηG1ω,O2ωη=αω,if ηG2ω,0,if ηG2ω.

Then(51)O1ωαω=η:O1ωηαω=G1ω,O2ωαω=η:O2ωηαω=G2ω.

Thus, Theorem 4 can be applied to get ω such that(52)ωO1ωαωO2ωαω=G1ωG2ω.

Corollary 1.

Let G:XCBX be multivalued mapping. Assume that there exists k0,1 such that(53)ΘsHbGω,GϖΘdbω,ϖk,for all ω,ϖ. Then there exists ω such that ωGω.

Remark 1.

If s=1, then b-metric spaces turns into complete metric space and we obtain some new results for fuzzy mappings as well as multivalued mappings in metric spaces.

4. Applications

Consider the Volterra integral inclusion(54)ωκhκ+0κJκ,τ,ωτdτ,κ0,1,where J:0,1×0,1×χcv a given set-valued mapping and h,ωC0,1 be such that h is given and ω is unknown function.

Now, for p1, consider the b-metric db on C0,1 defined by(55)dbω,ϖ=maxκ0,1|ωκϖκ|p=maxκ0,1ωκϖκp,for all ω,ϖC0,1. Then, C0,1,db,2p1 is a complete b-metric space.

We will assume the following:

For each ωC0,1, the mapping J:0,1×0,1×χcv is such that Jκ,τ,ωτ is lower semicontinuous in 0,1×0,1

There exists l:0,10,+ which is continuous such that(56)Jκ,τ,ωJκ,τ,ϖplτωτϖτ

for all κ,τ0,1, ω,ϖC0,1.

There exists k0,1 so that

(57)0κlτdτpk2p1.

Theorem 5.

Under the assumptions (a)–(c), the integral inclusion (54) has a solution in C0,1.

Proof.

Let =C0,1. Define O: by(58)OωαOω=ϖ:ϖκhκ+0κJκ,τ,ωτdτ,for all κ0,1. Let ω be arbitrary, then there exists αOω0,1. For Jωκ,τ:0,1×0,1χcv, it follows from Michael’s selection theorem that there exists jωκ,τ:0,1×0,1 such that jωκ,τJωκ,τ for each κ,τ0,1. It follows that hκ+0κjωκ,τdτOωαOω. Hence, OωαOω. It is a simple matter to show that OωαOω is closed, and so details are excluded (see also ). Moreover, since h is continuous on 0,1, and Jωκ,τ is continuous, their ranges are bounded. This means that OωαOω is bounded. Thus, OωαOωCB. ■

Let ω1,ω2, then there exists αOω1,αOω10,1 such that Oω1αOω1,Oω2αOω2CB. Let ϖ1Oω1αOω1 be arbitrary such that(59)ϖ1κhκ+0κJκ,τ,ω1τdτ,for κ0,1 holds. This means that for all κ,τ0,1, there exists jω1κ,τJω1κ,τ=Jκ,τ,ω1τ such that(60)ϖ1κ=hκ+0κjω1κ,τdτ,for κ0,1. For all ω1,ω2, it follows from (b) that(61)Jκ,τ,ω1Jκ,τ,ω2plτω1τω2τ.

It means that there exists zκ,τJω2κ,τ such that(62)jω1κ,τzκ,τplτω1τω2τ,for all κ,τ0,1.

Now, we can take the set-valued operator U defined by(63)Uκ,τ=Jω2κ,τu:jω1κ,τulτω1τω2τ.

Hence, by (a), U is lower semicontinuous. It follows that there exists a continuous operator jω2κ,τ:0,1×0,1 such that jω2κ,τUκ,τ for κ,τ0,1. Then, ϖ2κ=hκ+0κjω1κ,τdτ satisfies that(64)ϖ2κhκ+0κJκ,τ,ω2τdτ,κ0,1.

That is ϖ2Oω2αOω2 and(65)ϖ1κϖ2κp0κjω1κ,τjω2κ,τdτp0κlτω1τω2τdτpmaxτ0,1pωτϖτp0κlτdτk22p1dbω1,ω2,for all κ,τ0,1. Thus, we obtain that(66)2p1dbϖ1,ϖ2k2dbω1,ω2.

Interchanging the roles of ω1 and ω2, we obtain that(67)sHbOω1αOω1,Oω2αOω2k2dbω1,ω2.

This implies that(68)sHbOω1αOω1,Oω2αOω2kdbω1,ω2.

Taking exponential, we have(69)esHbOω1αOω1,Oω2αOω2ekdbω1,ω2.

Taking the function ΘΩs defined by Θη=eη for η>0, we get that the condition (8) is satisfied. Using the result 6, we conclude that (54) has a solution.

Data Availability

The data used to support the findings of this study are available from the corresponding author upon request.

Conflicts of Interest

The authors declare that they have no conflicts of interest.

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