# A Solution of Fredholm Integral Inclusions via Suzuki-Type Fuzzy Contractions

Authors: Izhar Uddin; Atiya Perveen; Hüseyin Işık; Ramakant Bhardwaj
Journal: Mathematical Problems in Engineering
 (2021)

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## Abstract

In this study, we introduce fuzzy weakϕ-contraction and Suzuki-type fuzzy weak ϕ-contraction and employ these to prove some fuzzy fixed point results for fuzzy mappings in the setting of metric spaces, which is followed by an example to support our claim. Next, we deduce some corollaries and fixed point results for multivalued mappings from our main result. Finally, as an application of our result, we provide the existence of a solution for a Fredholm integral inclusion.


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## Body

## 1. Introduction and Preliminaries

The idea of fuzzy mapping was inspired by the fuzzy set theory given by Zadeh [1]. It was initiated by Heilpern [2] in 1981, defined to be a mapping from an arbitrary set to a subfamily of fuzzy sets in metric linear spaces. He established a fuzzy expansion of Banach contraction principle. It broadens and develops the concept of fuzzy fixed point theory, and several authors worked in this field afterward ([3–7] and references therein).We describe some related concepts in short in the successive lines.Here,M,d depicts a metric space. A fuzzy set in M is a function with domain M and codomain 0,1. If F is a fuzzy set and μ∈M, the function value Fμ is called grade of membership of μ in F. The collection of all fuzzy set in M is denoted by FM.LetF∈FM and α∈0,1. The α-level set of F, which we denote here by Fα, is defined by(1)Fα=μ:Fμ≥α,α∈0,1,F0=μ:Fμ>0¯,where B¯ denotes closure of set B.Definition 1 (see [2]).
A fuzzy subsetF on M is said to be an approximate quantity if and only if its α-level set is a compact convex subset of M, for each α∈0,1 and supμ∈MFμ=1.We denote byWM⊆FM, the subcollection of approximate quantities. We also denote WαM=F∈FM:Fα is nonempty compact convex subset of a metric space M,d. If F∈WM and Fμ0=1, for some μ0∈M, then F is an approximation of μ0.Definition 2 (see [2]).
[2] For F1,F2∈WM and α∈0,1, define(2)pαF1,F2=infμ∈F1α,ν∈F2αdμ,ν;pF1,F2=supαpαF1,F2;DαF1,F2=HF1α,F2α=maxsupa∈F1αda,F2α,supb∈F2αdb,F1α;DF1,F2=supαDαF1,F2.Remark 1 (see [2]).
D is a metric on WM. Let F1,F2∈WM. Then, F1 is more accurate than F2, denoted by F1⊂F2 iff F1μ≤F2μ, for each μ∈M.Definition 3 (see [2]).
LetM be a nonempty set and N any metric linear space. A mapping S is called fuzzy mapping if and only if S is a mapping from M into WN (or WαN), i.e., Sμ∈WN (or WαN), for each μ∈M.Lemma 1 (see [2]).
The following conditions hold for a metric spaceM,d:(a)
Ifpαμ,F=0, then μα⊂F(b)
pαμ,F≤dμ,ν+pαν,F(c)
Ifμα⊂F1, then pαμ,F2≤DαF1,F2
For allz,ν∈M and F,F1,F2∈WM.A fuzzy mappingS is a fuzzy subset on M×N with membership function Sμ,ν. The function value Sμ,ν is the grade of membership of ν in Sμ.Definition 4 (see [8]).
Letα∈0,1 and μ∈M. The fuzzy point μα of M is the fuzzy set μα:M⟶0,1 given by(3)μαν=α,ifμ=ν,0,otherwise.
Forα=1, we have(4)μ1ν=μ=1,ifμ=ν,0,otherwise.Definition 5 (see [9]).
A fuzzy pointμα in M is a fixed fuzzy point of the mapping S over M if μα⊂Sμ, i.e., Sμμ≥α or μ∈Sμα.Remark 2 (see [2]).
Ifμ⊂Sμ, then μ is a fixed point of fuzzy mapping S.Also, the generalization of Banach contraction principle has been done in many ways along with providing their applications in different fields. One of them is by generalizing the contraction condition, specially using nonlinear contractions, e.g., Suzuki-type contraction,F-contraction, and θ-contraction [10–16]. One of such generalizations, namely, ϕ-weak contraction was done by Alber and Guerre-Delabriere [17] in 1997 to prove fixed point result in the setting of Hilbert space, which was further utilized by Rhoades [18] in metric fixed point theory. Recently, a generalization of the same was furnished by Xue [19]. He used the class of mappings Γ=class of all continuous nondecreasing functions ϕ:0,∞⟶0,∞ with ϕ0=0 and defined generalized ϕ-weak contraction as follows:(5)dSμ,Sν≤dμ,ν−ϕdSμ,Sν,∀μ,ν∈M,where S is a self-mapping on a metric space M,d and ϕ∈Γ. Also, he showed that this contraction condition is more weaker than ϕ-weak contraction condition (viz. dSμ,Sν≤dμ,ν−ϕdμ,ν). After that, Perveen et al. [20, 21] used his idea and proved results using weaker conditions.In this study, we utilize the above ideas and define fuzzy weakϕ-contraction and Suzuki-type fuzzy weak ϕ-contraction, which we use to prove the existence of fuzzy fixed point supported by an example. Last, we furnish an application of our result to prove the existence of a solution of integral inclusion of Fredholm type.


## 2. Main Result

First, we define the same class of mappings used in [20, 21].LetΦ denotes the set of all mappings ϕ:0,∞⟶0,∞ satisfying the following:(a)
Φ1ϕ is nondecreasing(b)
Φ2ϕτ=0 iff τ=0 and liminfn⟶∞ϕτn>0, whenever limn⟶∞τn>0We have noticed that [19] used the continuity of ϕ. Inspired by [22], we dropped the continuity condition and use a weaker condition, which is given in Φ2. In fact, Φ2 is also weaker than the condition that ϕ is lower semicontinuous. Indeed, if ϕ is lower semicontinuous, then for a sequence τn with limn⟶∞τn=τ>0, we have liminfn⟶∞ϕτn≥ϕτ>0.Using the class defined above, we define the following contraction for fuzzy mapping.Definition 6.
LetM,d be a metric space. A fuzzy mapping S:M⟶WαM is(a)
a fuzzy weakϕ-contraction mapping if(6)DαSμ,Sν≤dμ,ν−ϕDαSμ,Sν(b)
a Suzuki-type fuzzy weakϕ-contraction mapping if the following condition is satisfied:(7)12pαμ,Sμ≤dμ,ν⇒DαSμ,Sν≤dμ,ν−ϕDαSμ,Sν,for allμ,ν∈M, where ϕ∈ΦRemark 3.
IfS is fuzzy weak ϕ-contraction, then S is Suzuki-type fuzzy weak ϕ-contraction.Now, we are ready to commence our main theorem.Theorem 1.
Let a complete metric spaceM,d and S:M⟶WαM be a Suzuki-type fuzzy weak ϕ-contraction, such that for every μ∈M, Sμα is closed. Then, there exists μ∗∈M, such that μα∗ is a fuzzy fixed point of S, i.e., μα∗⊂Sμ∗.Proof.
Letμ1∈M be any arbitrary point. Since Sμ1∈WαM, we can choose μ2∈Sμ1α, such that dμ1,μ2=pαμ1,Sμ1. If μ1=μ2∈Sμ1α, then we are done. Suppose that μ1≠μ2. Since Sμ2∈WαM, there exists μ3∈Sμ2α, such that(8)dμ2,μ3=pαμ2,Sμ2≤DαSμ1,Sμ2.
Again, ifμ2=μ3, we are done. Otherwise, we continue this process and obtain a sequence μn satisfying the following conditions:(9)μn+1∈Sμnα,μn+2∈Sμn+1α,dμn+1,μn+2=pαμn+1,Sμn+1≤DαSμn,Sμn+1,∀n∈ℕ.
Thus, we easily obtain(10)12pαμn,Sμn<dμn,μn+1,which implies that(11)dμn+1,μn+2≤DαSμn,Sμn+1≤dμn,μn+1−ϕDαSμn,Sμn+1≤dμn,μn+1−ϕdμn+1,μn+2,which further implies(12)dμn+1,μn+2≤dμn,μn+1.
Thus, we see thatμn is a nonincreasing sequence of positive real number bounded below by 0. Hence, μn converges to a point r≥0. We assert that r=0. Suppose it is not so, then taking limit in (11), we obtain(13)r≤r−liminfn⟶∞ϕdμn+1,μn+2,which is a contradiction. Therefore, we have(14)limn⟶∞dμn,μn+1=0.
Next, we prove thatμn is a Cauchy sequence. Suppose on contrary that it is not so, then there exist two subsequences μmk and μnk of μn, such that nk is the smallest positive integer for which(15)nk>mk>k,dμmk,μnk≥ɛ,dμmk,μnk−1<ɛ.
Now, utilizing triangular inequality, we obtain(16)ɛ≤dμmk,μnk≤dμmk,μnk−1+dμnk−1,μnk<ɛ+dμnk−1,μnk.
Lettingn⟶∞, we obtain(17)limn⟶∞dμmk,μnk=ɛ.
Again, by triangular inequality,(18)dμmk,μnk≤dμmk,μmk+1+dμmk+1,μnk+1+dμnk+1,μnk,dμmk+1,μnk+1≤dμmk+1,μmk+dμmk,μnk+dμnk,μnk+1,which on letting n⟶∞, we obtain(19)limn⟶∞dμmk+1,μnk+1=ɛ.
Next, by (14) and (17), there exists n0≥1, such that(20)12pαμmk,Sμmk<12ɛ<dμmk,μnk,∀k≥n0.
Thus, forμ=μmk and ν=μnk, by (7), we obtain(21)DαSμmk,Sμnk≤dμmk,μnk−ϕDαSμmk,Sμnk,which on letting n⟶∞ yields ɛ≤ɛ−liminfn⟶∞ϕDαSμmk,Sμnk, a contradiction. Thus, μn is Cauchy in M. The completeness of M,d implies that μn⟶μ∗, for some μ∗∈M.
Next, we show thatμα∗⊂Sμ∗. As μn⟶μ∗, there exists n1∈ℕ, such that for all n≥n1,(22)dμn,μ∗≤13dμ,μ∗,∀μ∈M.
Using the above inequality, we obtain (for alln≥n1)(23)12pαμn,Sμn≤pαμn,Sμn≤dμn,μn+1≤dμn,μ∗+dμ∗,μn+1≤dμn,μ∗+pαμ∗,μn+1≤13dμ,μ∗+13dμ,μ∗=dμ,μ∗−13dμ,μ∗≤dμ,μ∗−dμ∗,μn≤dμn,μ,i.e.,(24)12pαμn,Sμn≤dμn,μ,∀n≥n1.
Thus, by (7), we obtain(25)pαμn+1,Sμ≤DαSμn,Sμ≤dμn,μ−ϕDαSμn,Sμ,which implies(26)pαμn+1,Sμ≤dμn,Sμ.
Taking limitn⟶∞, we obtain(27)pαμ∗,Sμ≤dμ∗,μ.
Furthermore, we prove that(28)DαSμ,Sμ∗≤dμ,μ∗−ϕDαSμ,Sμ∗,∀μ∈M.
The above equation holds trivially forμ=μ∗. Suppose μ≠μ∗. Then, for every n∈ℕ, there exists νn∈Sμα, such that(29)dμ∗,νn≤pαμ∗,Sμ+1ndμ,μ∗.
Thus, with the help of the above inequality and (27),(30)pαμ,Sμ≤dμ,νn≤dμ,μ∗+dμ∗,νn≤dμ,μ∗+pαμ∗,Sμ+1ndμ,μ∗≤dμ,μ∗+dμ,μ∗+1ndμ,μ∗=2+1ndμ,μ∗.
On taking limitn⟶∞, we obtain(31)12pαμ,Sμ≤dμ,μ∗.
So, (28) holds true for all μ∈M. Now, if limn⟶∞pαμn+1,Sμ∗=0, then we are done. Assume that it is not so, then there exists ɛ0>0, such that for every k∈ℕ, we can choose nk∈ℕ, such that pαμnk+1,Sμ∗>ɛ0>0 for all nk≥k. For μ=μnk, (28) reduces to(32)pαμnk+1,Sμ∗≤DαSμnk,Sμ≤dμnk,μ∗−ϕDαSμnk,Sμ∗.
Takingk⟶∞, we obtain(33)limk⟶∞pαμnk+1,Sμ∗≤limk⟶∞DαSμnk,Sμ∗≤limk⟶∞dμnk,μ∗−ϕDαSμnk,Sμ∗≤limk⟶∞dμnk,μ∗,i.e., pαμ∗,Sμ∗≤0, a contradiction. So, μα∗⊂Sμ∗, and the proof is completed.We present the following example to illustrate the utility of our proven result.Example 1.
LetM=1,2,3, and d:M×M⟶0,∞ is defined by(34)d1,3=38,d1,2=12,d2,3=32,dμ,μ=0,∀μ∈M,dμ,ν=dν,μ,∀μ,ν∈M.
We defineϕ:0,∞⟶0,∞ by(35)ϕτ=τ2,∀τ∈0,∞,and a fuzzy mapping by(36)S1μ=0,ifμ=1,α,ifμ=2,α2,ifμ=3,S2μ=α2,ifμ=1,2α,ifμ=2,α4,ifμ=3,S3μ=2α,ifμ=1,α3,ifμ=2,0,ifμ=3.
Then,S1α=S2α=2andS3α=1, and(37)12pα1,S1≤d1,ν,ν=2,3;12pα2,S2≤d2,ν,∀ν∈M,12pα3,S3≤d3,ν,ν=1,2.
We consider three cases.Case 1.
Ifμ,ν∈1,2, then we have(38)DαSμ,Sν=0,∀μ,ν∈M.
Hence, (7) is satisfied for μ,ν∈1,2 trivially.Case 2.
Ifμ=3 and ν=1, then we have(39)DαS3,S1=d1,2=12,d3,1=38.
Then,(40)DαS3,S1=12≤38−1119=38−ϕDαS3,S1.
So condition (7) is satisfied.Case 3.
Ifμ=3 and ν=2, then we have(41)DαS3,S2=d1,2=12,d3,2=32.
Thus, we get(42)DαS3,S2=12≤32−1119=32−ϕDαS3,S1.
We see that the assumptions of Theorem1 are fulfilled in all cases, and hence, S has a fuzzy fixed point which is 2.In view of Remark3, we deduce the underlying result.Theorem 2.
LetM,d be a complete metric space and S:M⟶WαM a fuzzy weak ϕ-contraction, such that for every μ∈M, Sμα is closed. Then, there exists μ∗∈M, such that μα∗ is a fuzzy fixed point of S, i.e., μα∗⊂Sμ∗.If the fuzzy mappingS is a Suzuki-type fuzzy weak ϕ-contraction, then it immediately satisfies the following contraction condition:(43)12pαμ,Sμ≤dμ,ν⇒DαSμ,Sν≤dμ,ν−ϕdμ,ν,where μ,ν∈M and ϕ∈Φ. But the converse need not be true. We justify this claim by showing that the condition DαSμ,Sν≤dμ,ν−ϕDαμ,ν is weaker than DαSμ,Sν≤dμ,ν−ϕdμ,ν. For this, we consider the following example.LetM=a,b,c with the metric d:M×M⟶0,∞ defined by da,b=da,c=5, db,c=3; dμ,μ=0, and dμ,ν=dν,μ, ∀μ,ν∈M. Define(44)ϕτ=τ2,∀τ∈0,∞,and a fuzzy mapping S by(45)Saμ=α,ifμ=a,2α,ifμ∈b,c,Sbμ=α,ifμ∈a,b,α2,ifμ=c,Scμ=2α,ifμ=a,α,ifμ=b,0,ifμ=c.So, we getSaα=a,b,c and Sbα=Scα=a,b, and(46)DαSa,Sb=DαSa,Sc=3,DαSb,Sc=0.We observe that the conditionDαSμ,Sν≤dμ,ν−ϕDαμ,ν is satisfied for all μ,ν∈M, but for μ=a and ν=b, the condition DαSμ,Sν≤dμ,ν−ϕdμ,ν is not fulfilled.Hence, we obtain the following result.Theorem 3.
LetM,d be a complete metric space and S:M⟶WαM a fuzzy mapping satisfying (43). Then, there exists μ∗∈M, such that μα∗⊂Sμ∗.Takingϕτ=1−kτ, k∈0,1 in the above result (viz. (43)), we obtain the next result.Theorem 4.
LetM,d be a complete metric space and S:M⟶WαM a fuzzy mapping satisfying the condition(47)12pαμ,Sμ≤dμ,ν⇒DαSμ,Sν≤kdμ,ν,where μ,ν∈M. Then, there exists μ∗∈M, such that μα∗⊂Sμ∗.Remark 4.
LetS be a fuzzy mapping from M to WαM and T:M⟶KM a closed mapping (where KM denotes the set of all compact subsets of M). Define(48)Sμν=α,ifν∈Tμ,0,otherwise,for each μ∈M. Note that,(49)Sμα=ν:Sμν≥α=Tμ.In view of above remark, we obtain the fixed point results for multivalued mappingT (defined above) from Theorems 1-4.Theorem 5.
LetM,d be a complete metric space and T:M⟶KM a multivalued closed mapping satisfying(50)12dμ,Tμ≤dμ,ν⇒HTμ,Tν≤dμ,ν−ϕHTμ,Tν,∀μ,ν∈M and ϕ∈Φ. Then, T has a fixed point.Theorem 6.
LetM,d be a complete metric space and T:M⟶KM a multivalued closed mapping satisfying(51)12dμ,Tμ≤dμ,ν⇒HTμ,Tν≤dμ,ν−ϕdμ,ν,∀μ,ν∈M and ϕ∈Φ. Then, T has a fixed point.Theorem 7.
LetM,d be a complete metric space and T:M⟶KM a multivalued closed mapping satisfying(52)12dμ,Tμ≤dμ,ν⇒HTμ,Tν≤kdμ,ν,∀μ,ν∈M. Then, T has a fixed point.Similarly, we can obtain the results corresponding to Theorem2.


## 3. An Application to the Fredholm Integral Inclusion

Consider the following Fredholm integral inclusion:(53)μτ∈fτ+∫abKτ,s,μsds,τ∈a,b,where f∈Ca,b and K:a,b×a,b×ℝ⟶PCVℝ (PCVℝ denotes the class of all nonempty compact and convex subsets of ℝ), and μ∈C0,1 is an unknown function. Consider M=Ca,b and take the complete metric space M,d, where(54)dμ,ν=maxτ∈a,bμτ−ντ,∀μ,ν∈Ca,b.Before proving our claim, we note down the following lemma.Lemma 2 (see [23, 24]).
LetM,d be a metric space and P,Q∈PM. If there exists η∈ℝη>0, such that(a)
For eachp∈P, there exists q∈Q, such that dp,q≤η(b)
For eachq∈Q, there exists p∈P, such that dq,p≤η
Then,HP,Q≤η.Theorem 8.
Under the conditions given as follows:A1 for all μ∈Ca,b, the operator K:a,b×a,b×ℝ⟶PCVℝ is such that Kμτ,s=Kτ,s,μs is lower semicontinuous on a,b×a,bA2 there exists a continuous function λ:a,b×a,b⟶0,∞, such that(55)HKμτ,s,Kντ,s≤λτ,sμτ−ντ,for allτ,s∈a,b and μ,ν∈Ca,b with ∫abλτ,sds≤2/3The Fredholm integral inclusion (53) has a solution in Ca,b.Proof.
Define the fuzzy mappingS:M⟶FM in such a way that (56)Sμα=ν∈M:ντ∈fτ+∫abKμτ,sds,τ∈a,b.
It is very obvious that the set of solutions ofSμα coincides with the set of fixed points of (53). So, we need to prove that Sμα has at least one fixed point.
For this, we consider an arbitrary fixed pointμ∈M and the set-valued operator Kμ:a,b×a,b⟶PCVℝ. Using Michael’s theorem, we obtain a continuous function, such that kμτ,s∈Kμτ,s, for each τ,s∈a,b. Thus, fτ+kμτ,s∈Sμα, and so, Sμα≠∅. Clearly, Sμα is closed (hence compact) and convex. So, S∈WαM.
Now, we will check that(57)DαSμ1,Sμ2≤dμ1,μ2−ϕDαSμ1,Sμ2,∀μ1,μ2∈M.
Letν1∈Sμ1α (arbitrary), such that ν1τ∈fτ+∫abKτ,s,μ1sds, for τ∈a,b. This means for all τ,s∈a,b, there exists kμ1τ,s∈Kμ1τ,s, such that ν1τ=fτ+∫abkμ1τ,sds. Now, from A2, we have(58)HKμ1τ,s,Kμ2τ,s≤λτ,sμ1τ−μ2τ.
Then, there existsμτ,s∈Kμ2τ,s, such that(59)kμ1τ,s−μτ,s≤λτ,sμ1τ−μ2τ.
Now, we consider the multivalued operatorU defined by(60)Uτ,s=Kμ2τ,s∩u∈ℝ:kμ1τ,s−u≤λτ,sμ1τ−μ2τ.
Hence, byA1, U is lower semicontinuous which ensures the existence of a continuous operator kμ2τ,s∈Uτ,s, implying that(61)ν2τ=fτ+∫abkμ2τ,sds,and hence,(62)ν2τ∈fτ+∫abKμ2τ,sds.
So, we getν2∈Sμ2α, and(63)ν2τ−ν1τ≤∫abkμ2τ,s−kμ1τ,sds≤∫abλτ,sμ2s−μ1sds≤maxt∈a,bμ2τ−μ1τ∫abλτ,sds<23dμ2,μ1.
After interchanging the roles ofμ1 and μ2 and using Lemma 2, we obtain (for each μ1,μ2∈M)(64)HSμ1α,Sμ2α≤23dμ1,μ2,and by considering ϕτ=τ/2 (for all τ∈0,∞), all the assumptions of Theorem 1 as well as Theorem 2 are satisfied. Hence, the inclusion problem (53) has a solution.


## 4. Conclusion

In this study, inspired by the work of Suzuki [10] and Xue [19], we define two new contractions, i.e., fuzzy weak ϕ-contraction and Suzuki-type fuzzy weak ϕ-contraction and use them to prove the existence of fuzzy fixed point and well exemplify them. Also, we provide an application of our proven result to show the existence of solution of Fredholm integral inclusion problem.



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