# Commonα-Fuzzy Fixed Point Results with Applications to Volterra Integral Inclusions

Authors: Abdullah Eqal Al-Mazrooei; Jamshaid Ahmad
Journal: Journal of Mathematics
 (2021)

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## Abstract

The purpose of this paper is to establish some commonα-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.


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## Body

## 1. Introduction and Preliminaries

Among all the impressive and inspiring generalizations of metric spaces,b-metric space has an integral place. Czerwik [1] in 1993 extended the notion of metric space by introducing the conception of b-metric space in this way.Definition 1.
Letℳ≠∅. A mapping db: ℳ×ℳ⟶ℝ0+ is called b-metric if it satisfies these assertions:b1dbω,ϖ=0⇔ω=ϖb2dbω,ϖ=dbϖ,ωb3dbω,υ≤sdbω,ϖ+dbϖ,υ
for allω,ϖ,υ∈ℳ, where s≥1.
Then,ℳ,db,s is called a b-metric space. A standard example of b-metric space which is not metric space is the following:
ℳ=ℝ and db:ℳ×ℳ⟶ℝ defined by(1)dbω,ϖ=ω−ϖ2,for all ω,ϖ∈ℳ with s=2.
LetPcbℳ represent the class of all nonempty, bounded, and closed subsets of ℳ. For Ξ1,Ξ2,Ξ3∈Pcbℳ, we define Hb:Pcbℳ×Pcbℳ⟶ℝ+ by(2)HbΞ1,Ξ2=maxδbΞ1,Ξ2,δbΞ2,Ξ1,where(3)δbΞ1,Ξ2=supdbω,ϖ:ω∈Ξ1,ϖ∈Ξ2,Dbω,Ξ3=Dbω,Ξ3=infdbω,υ:ω∈Ξ1,υ∈Ξ3.
Note thatHb is called the Hausdorff b-metric induced by the b-metric db. We recall the following properties from [1–3].Lemma 1.
(see [2]). Let ℳ,db,s be a b-metric space. For any Ξ1,Ξ2,Ξ3∈Pcbℳ and any ω,ϖ∈ℳ, we have the following:(i)
Dbω,Ξ2≤dbω,b for any b∈Ξ2(ii)
δbΞ1,Ξ2≤HbΞ1,Ξ2(iii)
Dbω,Ξ2≤HbΞ1,Ξ2 for any ω∈Ξ1(iv)
HbΞ1,Ξ2=0(v)
HbΞ1,Ξ2=HbΞ2,Ξ1(vi)
HbΞ1,Ξ3≤sHbΞ1,Ξ2+HbΞ2,Ξ3(vii)
Dbω,Ξ1≤sdbω,ϖ+dbϖ,Ξ1.
Later on, many authors (see [4–7]) worked in this way. Recently Jleli and Samet [8] gave the notion of Θ-contractions and proved a contemporary result for such contractions in generalized metric spaces. Afterwards, Hancer et al. [9] revised the foregoing definitions by including a broad condition (Θ4). Inspired by Jleli and Samet [8] and Hancer et al. [9], Alamri et al. [10] 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 [10]). We represent by Ωss≥1 the family of all mappings Θ:ℝ+⟶1,∞ satisfying these properties:(Θ1) 0<ϱ1<ϱ2⟶Θϱ1≤Θϱ2(Θ2) for ϱn⊆ℝ+, limn⟶∞Θϱn=1 if and only if limn⟶∞ϱn=0(Θ3) there exists h∈0,1 and q∈0,∞ such that limϱ⟶0+Θϱ−1/ϱh=q(Θ4) ΘinfΞ=infΘΞ for all Ξ⊂0,∞ with infΞ>0(Θ5) for all ϱn⊆ℝ+ such that Θsϱn≤Θϱn−1k, ∀n∈ℕ and some k∈0,1, then Θsnnϱ≤Θsn−1ϱn−1k, for all n∈ℕ
They supported this condition by the following nontrivial example.Example 1.
(see [10]). Let Θ:0,∞⟶1,∞ be given by θη=eηeη. Clearly, Θ satisfies (Θ1)-(Θ5). Here we show only (Θ5). Assume that, for all n∈ℕ and some k∈0,1, we have θsϱn≤θϱn−1k, which implies that(4)esϱnesϱn≤eϱn−1eϱn−1k,sϱnesϱn≤kϱn−1eϱn−1.
This implies that(5)sϱnesϱn−ϱn−1≤kϱn−1.
Asθsϱn≤θϱn−1k≤θϱn−1. Also θ is nondecreasing, so sϱn≤ϱn−1 and sϱn−ϱn−1≤0 implies esn−1sϱn−ϱn−1≤esϱn−ϱn−1. Therefore, (5) implies(6)sϱnesn−1sϱn−ϱn−1≤kϱn−1⇒sϱnesnϱnesn−1ϱn−1≤kϱn−1⇒sϱnesnϱn≤kϱn−1esn−1ϱn−1⇒snϱnesnϱn≤ksn−1ϱn−1esn−1ϱn−1⇒esnϱnesnϱn≤eksn−1ϱn−1esn−1ϱn−1⇒θsnϱn≤θsn−1ϱn−1k,and hence (Θ5) holds.
On the other side, Kumam et al. [11] 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 [9–20] to the readers.
We need the following lemma of Czerwik [2].Lemma 2.
(see [11]). Let ℳ,db,s be a b-metric space and Ξ1,Ξ2∈CBℳ, then ∀ω∈Ξ1,(7)dbω,Ξ2≤HbΞ1,Ξ2.
In this paper, we obtain commonα-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.
Letℳ,db,s be a complete b-metric space with coefficient s≥1 such that db is continuous. Assume that O1,O2: ℳ⟶ℱℳ and for each ω,ϖ∈ℳ, there exist αO1ω,αO2ϖ∈0,1 such that O1ωαO1ω,O2ϖαO2ϖ∈Pcbℳ. If there exist Θ∈Ωs and k∈0,1 such that(8)ΘsHbO1ωαO1ω,O2ϖαO2ϖ≤Θdbω,ϖk,for all ω,ϖ∈ℳ with HbO1ωαO1ω,O2ϖαO2ϖ>0, then there exists ω∗∈ℳ such that ω∗∈O1ω∗αO1ω∗∩O2ω∗αO2ω∗.Proof.Letω0∈ℳ, then by assumption there exists αO1ω0∈0,1 such that O1ω0αO1ω0∈CBℳ. Let ω1∈O1ω0αO1ω0. For this ω1, there exists αO2ω1∈0,1 such that O2ω1αO2ω1∈Pcbℳ. By Lemma 2, (Θ1) and (8), we have(9)Θsdω1,O2ω1αO2ω1≤ΘsHbO1ω0αO1ω0,O2ω1αO2ω1≤Θdbω0,ω1k.Thus,(10)Θsdω1,O2ω1αO2ω1≤Θdbω0,ω1k.From (Θ4), we know that(11)Θsdω1,O2ω1αO2ω1=infy∈O2ω1αO2ω1Θsdbω1,y.Thus from (10), we get(12)infy∈O2ω1αO2ω1Θsdbω1,y≤Θdbω0,ω1k.Then, from (12), there exists ω2∈O2ω1αO2ω1 (obviously, ω2≠ω1) such that(13)Θsdbω1,ω2≤Θdbω0,ω1k.For thisω2, there exists αO1ω2∈0,1 such that O1ω2αO1ω2∈Pcbℳ. By Lemma 2, (Θ1), and (8), we have(14)Θsdbω2,O1ω2αO1ω2≤ΘsHbO2ω1αO2ω1,O1ω2αO1ω2≤Θdbω1,ω2k.Thus,(15)Θsdbω2,O1ω2αO1ω2≤Θdbω1,ω2k.From (Θ4), we know that(16)Θsdbω2,O1ω2αO1ω2=infy∈O1ω2αO1ω2Θsdbω2,y.Thus from (15), we get(17)infy∈O1ω2αO1ω2Θsdbω2,y≤Θdbω1,ω2k.Then, from (17), there exists ω3∈O1ω2αO1ω2 (obviously, ω3≠ω2) such that(18)Θsdbω2,ω3≤Θdbω1,ω2k.So, continuing in the same way, we constructωn in ℳ such that(19)ω2n+1∈O1ω2nαO1ω2n,ω2n+2∈O2ω2n+1αO2ω2n+1,(20)Θsdbω2n+1,ω2n+2≤Θdbω2n,ω2n+1k,(21)Θsdbω2n+2,ω2n+3≤Θdbω2n+1,ω2n+2k,for all n∈ℕ. From (20) and (21), we get(22)Θsdbωn,ωn+1≤Θdbωn−1,ωnk,for all n∈ℕ. It follows by (22) and property (Θ5) that(23)Θsndbωn,ωn+1≤Θsn−1dbωn−1,ωnk,which further implies that(24)Θsndbωn,ωn+1≤Θsn−1dbωn−1,ωnk≤Θsn−2dbωn−2,ωn−1k2≤⋯≤Θdbω0,ω1kn,for all n∈ℕ. Thus,(25)Θsndbωn,ωn+1≤Θdbω0,ω1kn,for all n∈ℕ. Since Θ∈Ω, so letting n⟶∞ in (25), we get(26)limn⟶∞Θsndbωn,ωn+1=1.This implies(27)limn⟶∞sndbωn,ωn+1=0,by (Θ2). By (Θ3), there exists 0<r<1 and q∈0,∞ so that(28)limn⟶∞Θsndbωn,ωn+1−1sndbωn,ωn+1r=q.Suppose thatq<∞. For this case, let ς2=q/2>0. By definition of the limit, there exists n0∈ℕ so that(29)Θsndbωn,ωn+1−1sndbωn,ωn+1r−q≤ς2,for all n>n0. This implies that(30)Θsndbωn,ωn+1−1sndbωn,ωn+1r≥q−ς2=q2=ς2,for all n>n0. Then(31)nsndbωn,ωn+1r≤ς1nΘsndbωn,ωn+1−1,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∈ℕ such that(32)ς2≤Θsndbωn,ωn+1−1sndbωn,ωn+1r,for all n>n0, which implies(33)nsndbωn,ωn+1r≤ς1nΘsndbωn,ωn+1−1,for all n>n0, where ς1=1/ς2. Hence, in all cases, there exists ς1>0 and n0∈ℕ such that(34)nsndbωn,ωn+1r≤ς1nΘsndbωn,ωn+1−1,for all n>n0. Hence by (25) and (34), we obtain(35)nsndbωn,ωn+1r≤ς1nΘdbω0,ω1rn−1.Taking the limitn⟶∞, we get(36)limn⟶∞nsndbωn,ωn+1r=0.Thuslimn⟶∞n1/rsndbωn,ωn+1=0 which implies that ∑n=1∞sndbωn,ωn+1 is convergent. Thus ωn is a Cauchy sequence in ℳ. Since ℳ,db,s is a complete b-metric space, so there exists a ω∗∈ℳ such that(37)limn⟶∞ωn=ω∗.Now, we prove thatω∗∈O2ω∗αO2ω∗. We suppose on the contrary that ω∗∉O2ω∗αO2ω∗, then there exist n0∈ℕ and ωnk of ωn such that dbω2nk+1,O2ω∗αO2ω∗>0, for all nk≥n0. Now, using (8) with ω=ω2nk+1 and ϖ=ω∗, we obtain(38)Θdbω2nk+1,O2ω∗αO2ω∗≤Θsdbω2nk+1,O2ω∗αO2ω∗≤ΘsHbO1ω2nkαO1ω2nk,O2ω∗αO2ω∗≤Θdω2nk,ω∗k.Ask∈0,1, so by (Θ1) so we obtain(39)dbω2nk+1,O2ω∗αO2ω∗<dbω2nk,ω∗.Lettingn⟶∞, we have(40)dbω∗,O2ω∗αO2ω∗≤0.Hence,ω∗∈O2ω∗αO2ω∗. Likewise, one can straightforwardly prove that ω∗∈O1ω∗αO1ω∗. Thus, ω∗∈O1ω∗αO1ω∗∩O2ω∗αO2ω∗.Note: From now to onwards, we considerdb as continuous functional and ℳ,db,s as complete b-metric space.The following corollary follows from Theorem1 by considering Θη=eη for η>0.Theorem 2.
LetO1,O2:ℳ⟶ℱℳ, and for each ω,ϖ∈ℳ, ∃αO1ω,αO2ϖ∈0,1 such that O1ωαO1ω,O2ϖαO2ϖ∈Pcbℳ. If ∃k∈0,1 such that(41)sHbO1ωαO1ω,O2ϖαO2ϖ≤kdbω,ϖfor all ω,ϖ∈ℳ, then there exists ω∗∈ℳ such that ω∗∈O1ω∗αO1ω∗∩O2ω∗αO2ω∗.Theorem 3.
LetO:ℳ⟶ℱℳ, and for each ω,ϖ∈ℳ, there exist αOω,αOϖ∈0,1 such that OωαOω,OϖαOϖ∈Pcbℳ. If there exists k∈0,1 such that(42)sHbOωαOω,OϖαOϖ≤kdbω,ϖ,for all ω,ϖ∈ℳ, then there exists ω∗∈ℳ such that ω∗∈Oω∗αOω∗.Example 2.
Letℳ=0,1,2 and db:ℳ×ℳ⟶ℝ0+ by(43)dbω,ϖ=0,ifω=ϖ,16,ifω≠ϖandω,ϖ∈0,1,12,ifω≠ϖandω,ϖ∈0,2,1,ifω≠ϖandω,ϖ∈1,2.
It is easy to see thatℳ,d is a complete b-metric space with coefficient s=3/2. Define(44)O0η=O1η=12,ifη=0,0,ifη=1,2,O2η=0,ifη=0,2,12,ifη=1.
Defineα:ℳ⟶0,1 by αω=1/2 for all ω∈ℳ. Now we obtain that(45)Oω1/2=0,ifω=0,11,ifω=2.
Forω,ϖ∈ℳ, we get(46)HbO11/2,O21/2=HbO11/2,O21/2=Hb0,1=16.
TakingΘη=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∈ℳ. As a result, all assertions of Theorem 6 hold and there exists 0∈ℳ such that 0∈O01/2is an α-fuzzy fixed point of O.


## 3. Set-Valued Results

Theorem 4.
LetG1,G2:X⟶CBX. Suppose that ∃k∈0,1 such that(49)ΘsHbG1ω,G2ϖ≤Θdbω,ϖk,for all ω,ϖ∈ℳ. Then there exists ω∗∈ℳ such that ω∗∈G1ω∗∩G2ω∗.Proof.
Defineα:ℳ⟶0,1 and O1,O2:ℳ⟶ℱℳ by(50)O1ωη=αω,ifη∈G1ω,0,ifη∉G1ω,O2ωη=αω,ifη∈G2ω,0,ifη∉G2ω.
Then(51)O1ωαω=η:O1ωη≥αω=G1ω,O2ωαω=η:O2ωη≥αω=G2ω.
Thus, Theorem4 can be applied to get ω∗∈ℳ such that(52)ω∗∈O1ω∗αω∗∩O2ω∗αω∗=G1ω∗∩G2ω∗.Corollary 1.
LetG:X⟶CBX be multivalued mapping. Assume that there exists k∈0,1 such that(53)ΘsHbGω,Gϖ≤Θdbω,ϖk,for all ω,ϖ∈ℳ. Then there exists ω∗∈ℳ such that ω∗∈Gω∗.Remark 1.
Ifs=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)ωκ∈hκ+∫0κJκ,τ,ωτdτ,κ∈0,1,where J:0,1×0,1×ℝ⟶χcvℝ a given set-valued mapping and h,ω∈C0,1 be such that h is given and ω is unknown function.Now, forp≥1, consider the b-metric db on C0,1 defined by(55)dbω,ϖ=maxκ∈0,1|ωκ−ϖκ|p=maxκ∈0,1ωκ−ϖκp,for all ω,ϖ∈C0,1. Then, C0,1,db,2p−1 is a complete b-metric space.We will assume the following:(a)
For eachω∈C0,1, the mapping J:0,1×0,1×ℝ⟶χcvℝ is such that Jκ,τ,ωτ is lower semicontinuous in 0,1×0,1(b)
There existsl:0,1⟶0,+∞ which is continuous such that(56)Jκ,τ,ω−Jκ,τ,ϖp≤lτωτ−ϖτfor allκ,τ∈0,1, ω,ϖ∈C0,1.(c)
There existsk∈0,1 so that(57)∫0κlτdτp≤k2p−1.Theorem 5.
Under the assumptions (a)–(c), the integral inclusion (54) has a solution in C0,1.Proof.
Letℳ=C0,1. Define O:ℳ⟶ℱℳ by(58)OωαOω=ϖ∈ℳ:ϖκ∈hκ+∫0κJκ,τ,ωτdτ,for all κ∈0,1. Let ω∈ℳ be arbitrary, then there exists αOω∈0,1. For Jωκ,τ:0,1×0,1⟶χcvℝ, it follows from Michael’s selection theorem that there exists jωκ,τ:0,1×0,1⟶ℝ such that jωκ,τ∈Jωκ,τ for each κ,τ∈0,1. It follows that hκ+∫0κjωκ,τdτ∈OωαOω. Hence, OωαOω≠∅. It is a simple matter to show that OωαOω is closed, and so details are excluded (see also [17]). Moreover, since h is continuous on 0,1, and Jωκ,τ is continuous, their ranges are bounded. This means that OωαOω is bounded. Thus, OωαOω∈CBℳ. ■
Letω1,ω2∈ℳ, then there exists αOω1,αOω1∈0,1 such that Oω1αOω1,Oω2αOω2∈CBℳ. Let ϖ1∈Oω1αOω1 be arbitrary such that(59)ϖ1κ∈hκ+∫0κJκ,τ,ω1τdτ,for κ∈0,1 holds. This means that for all κ,τ∈0,1, there exists jω1κ,τ∈Jω1κ,τ=Jκ,τ,ω1τ such that(60)ϖ1κ=hκ+∫0κjω1κ,τdτ,for κ∈0,1. For all ω1,ω2∈ℳ, it follows from (b) that(61)Jκ,τ,ω1−Jκ,τ,ω2p≤lτω1τ−ω2τ.
It means that there existszκ,τ∈Jω2κ,τ such that(62)jω1κ,τ−zκ,τp≤lτω1τ−ω2τ,for all κ,τ∈0,1.
Now, we can take the set-valued operatorU defined by(63)Uκ,τ=Jω2κ,τ∩u∈ℝ:jω1κ,τ−u≤lτω1τ−ω2τ.
Hence, by (a),U is lower semicontinuous. It follows that there exists a continuous operator jω2κ,τ:0,1×0,1⟶ℝ such that jω2κ,τ∈Uκ,τ for κ,τ∈0,1. Then, ϖ2κ=hκ+∫0κjω1κ,τdτ satisfies that(64)ϖ2κ∈hκ+∫0κJκ,τ,ω2τdτ,κ∈0,1.
That isϖ2∈Oω2αOω2 and(65)ϖ1κ−ϖ2κp≤∫0κjω1κ,τ−jω2κ,τdτp≤∫0κlτω1τ−ω2τdτp≤maxτ∈0,1pωτ−ϖτp∫0κlτdτ≤k22p−1dbω1,ω2,for all κ,τ∈0,1. Thus, we obtain that(66)2p−1dbϖ1,ϖ2≤k2dbω1,ω2.
Interchanging the roles ofω1 and ω2, we obtain that(67)sHbOω1αOω1,Oω2αOω2≤k2dbω1,ω2.
This implies that(68)sHbOω1αOω1,Oω2αOω2≤kdbω1,ω2.
Taking exponential, we have(69)esHbOω1αOω1,Oω2αOω2≤ekdbω1,ω2.
Taking the functionΘ∈Ωs 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.



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