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 [1] in 1993 extended the notion of metric space by introducing the conception of b-metric space in this way.

Definition 1.

Let ℳ≠∅. A mapping db: ℳ×ℳ⟶ℝ0+ is called b-metric if it satisfies these assertions:

b1dbω,ϖ=0⇔ω=ϖ

b2dbω,ϖ=dbϖ,ω

b3dbω,υ≤sdbω,ϖ+dbϖ,υ

for all ω,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.

Let Pcbℳ 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 that Hb 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:

Dbω,Ξ2≤dbω,b for any b∈Ξ2

δbΞ1,Ξ2≤HbΞ1,Ξ2

Dbω,Ξ2≤HbΞ1,Ξ2 for any ω∈Ξ1

HbΞ1,Ξ2=0

HbΞ1,Ξ2=HbΞ2,Ξ1

HbΞ1,Ξ3≤sHbΞ1,Ξ2+HbΞ2,Ξ3

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.

As θsϱn≤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 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 ℳ,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.

Let ω0∈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 this ω2, 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 construct ωn 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 that q<∞. 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 limit n⟶∞, we get(36)limn⟶∞nsndbωn,ωn+1r=0.

Thus limn⟶∞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 that ω∗∈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.

As k∈0,1, so by (Θ1) so we obtain(39)dbω2nk+1,O2ω∗αO2ω∗<dbω2nk,ω∗.

Letting n⟶∞, 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 consider db as continuous functional and ℳ,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: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.

Let O: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.

Let ℳ=0,1,2 and db:ℳ×ℳ⟶ℝ0+ by(43)dbω,ϖ=0,if ω=ϖ,16,if ω≠ϖ and ω,ϖ∈0,1,12,if ω≠ϖ and ω,ϖ∈0,2,1,if ω≠ϖ and ω,ϖ∈1,2.

It is easy to see that ℳ,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:ℳ⟶0,1 by αω=1/2 for all ω∈ℳ. Now we obtain that(45)Oω1/2=0,if ω=0,11,if ω=2.

For ω,ϖ∈ℳ, 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∈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 ResultsTheorem 4.

Let G1,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.

Define α:ℳ⟶0,1 and O1,O2:ℳ⟶ℱℳ by(50)O1ωη=αω,if η∈G1ω,0,if η∉G1ω,O2ωη=αω,if η∈G2ω,0,if η∉G2ω.

Then(51)O1ωαω=η:O1ωη≥αω=G1ω,O2ωαω=η:O2ωη≥αω=G2ω.

Thus, Theorem 4 can be applied to get ω∗∈ℳ such that(52)ω∗∈O1ω∗αω∗∩O2ω∗αω∗=G1ω∗∩G2ω∗.

Corollary 1.

Let G: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.

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)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, for p≥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:

For each ω∈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

There exists l:0,1⟶0,+∞ which is continuous such that(56)Jκ,τ,ω−Jκ,τ,ϖp≤lτωτ−ϖτ

for all κ,τ∈0,1, ω,ϖ∈C0,1.

There exists k∈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.

Let ℳ=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. ■

Let ω1,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 exists zκ,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 operator U 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 is ϖ2∈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 of ω1 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 function Θ∈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.

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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