# Hβ-Hausdorff Functions and Common Fixed Points of Multivalued Operators in a b-Metric Space and Their Applications

Authors: Fahad Sameer Alshammari; Naif R. Alrashedi; Reny George
Journal: Journal of Function Spaces
 (2022)

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## Abstract

Hβ-Hausdorff functions for β∈0,1 are introduced, and common fixed-point theorems for a pair of multivalued operators satisfying generalized contraction conditions are proven in a b-metric space. Our results are proper extensions and new variants of many contraction conditions existing in literature. In order to demonstrate applications of our result, we have proven an existence theorem for a unique common multivalued fractal of a pair of iterated multifunction systems and also an existence theorem for a common solution of a pair of Volterra-type integral equations.


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## Body

## 1. Introduction

In the last few decades, a wide range of extensions, generalizations, and applications of the infamous Banach contraction principle came into existence. In the sequel, Bakhtin [1] initiated the idea of a b-metric space followed by Czerwik [2], in which the author by weakening the triangular inequality formally defined a b-metric space and proved the Banach contraction principle in a b-metric space. Some examples and other details of a b-metric space can be found in Kirk and Shahzad [3] whereas a wide range of generalized fixed-point theorems in a b-metric space can be found in [4–7]. On the other hand, the study of a metric function on the set of closed and bounded subsets of a metric space was initiated by Pompeiu in [8] and then continued by Hausdorff [9]. Such a metric function is referred to as the Hausdorff-Pompeiu metric. Banach’s contraction principle was extended to a multivalued function in a metric space by Nadler [10] and in a b-metric space by Czerwik [2] using the Hausdorff-Pompeiu metric H. Further generalized results of multivalued contractions can be found in ([11–14]). Czerwik’s contraction was also generalized in many directions to name a few: q-quasi-contraction [15], Hardy-Rogers contraction [16], weak quasi-contraction [17], Ciric contraction [18], etc. More results on multivalued contraction mappings in a b-metric space can be found in [19–23]. Very recently, Debnath [24] proved the set-valued Meir–Keeler-type as well as Geraghty- and Edelstein-type fixed-point theorems in a b-metric space whereas Altun et al. [25] and Kumar and Luambano [26] proved fixed-point results for multivalued F-contraction mappings in complete metric space and partial metric space, respectively. In [27], the authors introduced the concept of Hβ-Hausdorff-Pompeiu b-metric for some 0≤β≤1 and proved fixed-point theorems for multivalued mappings belonging to various classes of multivalued Hβ-contractions in a b-metric space. Applications of fixed-point results in dealing with solutions of nonlinear problems arising in engineering and science are an important area in present-day research. Fruitful applications of fixed-point problems in solution of various types of integral equations, fractional differential equations, and optimization problems can be found in [28–32]. Barnsley [33] introduced the idea of data interpolation using the fractal methodology of iterated function systems. Nowadays, fractal functions constitute a method of approximation of nondifferentiable mappings, providing suitable tools for the description of irregular signals (see [34–39]). The aim of this work is to prove common fixed-point theorems for a pair of multivalued mappings in a b-metric space using Hβ-Hausdorff-Pompeiu b-metric and thereby extend and introduce new variants of various fixed-point results for multivalued mappings existing in literature. We have provided two applications of our main results: one to prove the existence of a unique common multivalued fractal of a pair of iterated multifunction system defined on a b-metric space and the second to prove the existence of a common solution of a pair of Volterra-type nonlinear integral equations.


## 2. Preliminaries

In this section, we provide some preliminary definitions, lemmas, and propositions required in our main results.Definition 1 (see [1]).
LetX be a nonempty set and ds:X×X⟶0,∞ satisfy the following:
(1)
dsı,ȷ=0 if and only if ı=ȷ for all ı,ȷ∈X(2)
dsı,ȷ=dsȷ,ı for all ı,ȷ∈X(3)
There exists a real numbers≥1 such that dsı,ȷ≤sdsı,ℓ+dsℓ,ȷ for all ı,ȷ,ℓ∈X
Then,ds is a b-metric on X and X,ds is a b-metric space with coefficient s.LetCBdsX be the collection of all nonempty closed and bounded subsets of a b-metric space X,ds. For A,B∈CBdsX, define dsx,A=infdsx,a:a∈A, δdsA,B=supa∈Adsa,B, and HdsA,B=maxδdsA,B,δdsB,A. Czerwik [2] has shown that Hds is a b-metric in the set CBdsX and is called the Hausdorff-Pompeiu b-metric induced by ds. In [27], the authors introduced the function HβA,B=maxβδdsA,B+1−βδdsB,A,βδdsB,A+1−βδdsA,B for some β∈0,1 and showed that Hβ is a b-metric for the set CBdsX. They called this function the Hβ-Hausdorff-Pompeiu b-metric induced by the b-metric ds. Note that for β=0 or 1, Hβ is the Hausdorff-Pompeiu metric Hds.Proposition 2 (see [27]).
For anyx,y∈X, Hβx,y=dsx,y.Definition 3 (see [18]).
Theb-metric ds is ∗-continuous if and only if for any A∈CBdsX and sequence xn in X,ds with limn⟶∞xn=x, we have limn⟶∞dsxn,A=dsx,A.Proposition 4 (see [19]).
For anyA⊆X,
(1)a∈A¯⇔dsa,A=0.Lemma 5 (see [18]).
Letxn be a sequence in (X,ds). If there exists λ∈0,1 such that dsxn,xn+1≤λdsxn−1,xn for all n∈N, then xn is a Cauchy sequence.The following lemma follows immediately from the above lemma.Lemma 6.
If for someλ,ϵ∈0,1, with λ<ϵ, dsxn,xn+1≤λdsxn−1,xn+ϵn for all n∈N, then xn is a Cauchy sequence.


## 3. Main Results

We introduce pairwiseHβ-Hausdorff functions as follows:Definition 7.
LetS,T:X⟶CBdsX. For any ı∈X, ȷ∈TıorSı and any ϵ>0, if there exist ℓ∈SȷorTȷ such that
(2)dsȷ,ℓ≤HβTı,Sȷ+ϵorrespectivelydsȷ,ℓ≤HβSı,Tȷ+ϵ,then we say that T and S are pairwise Hβ-Hausdorff functions.ForS=T, we get the following.Definition 8.
For anyı∈X,ȷ∈Tı and any ϵ>0 if there exist ℓ∈Tȷ such that
(3)dsȷ,ℓ≤HβTı,Tȷ+ϵ,then we say that T is a Hβ-Hausdorff function.Remark 9.
(i)
Forβ=1, T:X⟶CBX is always a Hβ-Hausdorff function(ii)
If for any0≤β1≤1, the function T:X⟶CBX is a Hβ1-Hausdorff function, then for any 0≤β1≤β2≤1, the function T:X⟶CBX is a Hβ2-Hausdorff functionExample 10.
LetX=0,33/48⋃1,
(4)dsı,ȷ=ı−ȷ2forallı,ȷ∈X.and S,T:X⟶CBX be as follows:
(5)Sı=ı4,forı∈0,3348,3348,1,forı∈0,1,Tı=ı2,forı∈0,3348,13,3348,1,forı∈0,1.We will show that the functionsS and T satisfy (2). We will consider the values of ı in X as follows:
(i)
ı∈0,33/48. In this case, Sı and Tȷ are singleton sets and so (2) is obviously true(ii)
ı=0. Sı=33/48,1. If ȷ=33/48,ȷ=33/96, then we have ℓ=33/96 and dsȷ,ℓ=1089/9216, δsSı,Tȷ=3969/9216, δsTȷ,Sı=1089/9216, and H3/4Sı,Tȷ=3249/9216. Thus, (2) is true for all ϵ>0. If j=1,Tj=1/3,33/48,1, then inequality (2) holds with ℓ=1(iii)
ı=1. Sı=33/48,1, and the result follows in the same way as in (ii) above.(iv)
ı=0. Tı=1/3,33/48,1. If ȷ=1/3,Sȷ=1/12, then we have ℓ=1/12 and dsȷ,ℓ=9/144, δsTı,Sȷ=121/144, δsSȷ,Tı=9/144, and H3/4Sı,Tȷ=93/144. Thus, (2) is true for all ϵ>0. If ȷ=33/48,Sȷ=33/192, then we take ℓ=33/192 and then dsȷ,ℓ=1089/4096, δsTı,Sȷ=2809/4096, δsSȷ,Tı=961/36864, and H3/4Tı,Sȷ=19201/36864. Thus, (2) is true for all ϵ>0. If ȷ=1,Sȷ=33/48,1, inequality (2) holds with ℓ=1Thus,S and T are pairwise Hβ-Hausdorff functions for β=3/4. However, S and T are not pairwise Hβ-Hausdorff functions for β=1/2, as we see that inequality (2) is not satisfied for i=0, Tı=1/3,33/48,1, and ȷ=33/48. In fact, S and T are not pairwise Hβ-Hausdorff functions for 34/95<β<61/95.We now present our main result as follows:Theorem 11.
LetX,ds be a complete b-metric space with constant s≥1, ds be ∗-continuous, and T,S:X⟶CBdsX be multivalued pairwise Hβ-Hausdorff functions for some 1/2≤β≤1 and satisfying the following condition:
(6)HβTı,Sȷ≤α1dsı,Tı+α2dsȷ,Sȷ+α3dsı,Sȷ+α4dsȷ,Tı+α5dsı,Sȷ+dsȷ,Tı2+α6dsı,Tıdsȷ,Sȷ1+dsı,ȷ+α7dsı,ȷ,for all ı,j∈X and some αk≥0, k=1,2⋯7, with α1+α2+sα5+α6+α7+max2sα3,2sα4<1, sα1+α4+α5/2<β, and sα2+α3+α5/2<β. Then, S and T have a common fixed point.Proof.
Letı0∈X, ı1∈Tı0, and 0<ϵ<1. By (2), there exist ı2∈Sı1, such that dsı1,ı2≤HβTı0,Sı1+ϵ. By (2) again, there exist ı3∈Tı2, such that dsı2,ı3≤HβSı1,Tı2+ϵ2.
Continuing these ways, we construct the sequence<ın> such that
(7)ı2n+1∈Tı2n,ı2n+2∈Sı2n+1,dsı2n+1,ı2n+2≤HβTı2n,Sı2n+1+ϵ2n+1,dsı2n+2,ı2n+3≤HβSı2n+1,Tı2n+2+ϵ2n+2.
Now,(8)dsı2n+1,ı2n+2≤HβTı2n,Sı2n+1+ϵ2n+1≤α1dsı2n,Tı2n+α2dsı2n+1,Sı2n+1+α3dsı2n,Sı2n+1+α4dsı2n+1,Tı2n+α5dsı2n,Sı2n+1+dsı2n+1,Tı2n2+α6dsı2n,Tı2n∗dsı2n+1,Sı2n+11+dsı2n,ı2n+1+α7dsı2n,ı2n+1+ϵ2n+1≤α1dsı2n,ı2n+1+α2dsı2n+1,ı2n+2+α3dsı2n,ı2n+2+α40+α5dsı2n,ı2n+2+02+α6dsı2n,ı2n+1∗dsı2n+1,ı2n+21+dsı2n,ı2n+1+α7dsı2n,ı2n+1+ϵ2n+1≤α1dsı2n,ı2n+1+α2dsı2n+1,ı2n+2+α3sdsı2n,ı2n+1+dsı2n+1,ı2n+2+α5sdsı2n,ı2n+1+dsı2n+1,ı2n+22+α6dsı2n+1,ı2n+2+α7dsı2n,ı2n+1+ϵ2n+1.
Therefore,(9)dsı2n+1,ı2n+2≤α1+sα3+sα5/2+α71−α2−sα3−sα5/2−α6dsı2n,ı2n+1+ϵ2n+1.
Again,(10)dsı2n+2,ı2n+3≤HβSı2n+1,Tı2n+2+ϵ2n+2≤αmaxdsı2n+1,ı2n+2,dsı2n+2,Tı2n+2,dsı2n+1,Sı2n+1,dsı2n+2,Sı2n+1,dsı2n+1,Tı2n+2+Lmindsı2n+2,Sı2n+1,dsı2n+1,Tı2n+2+ϵ2n+2≤αmaxdsı2n+1,ı2n+2,dsı2n+2,ı2n+3,dsı2n+1,ı2n+2,dsı2n+2,ı2n+2,dsı2n+1,ı2n+3+Lmindsı2n+2,ı2n+2,dsı2n+1,ı2n+3+ϵ2n+2,(11)ordsı2n+2,ı2n+3≤α2+sα4+sα5/2+α71−α1−sα4−sα5/2−α6dsı2n+1,ı2n+3+ϵ2n+2.
Thus, we have(12)dsın,ın+1≤λdsın−1,ın+ϵn,where λ=maxα1+sα3+sα5/2+α7/1−α2−sα3−sα5/2−α6,α2+sα4+sα5/2+α7/1−α1−sα4−sα5/2−α6<1.
By Lemma6, the sequence <ın> is a Cauchy sequence. Since X,ds is complete, there exists ℏ∈X such that the Cauchy sequence <ın> is convergent to ℏ. We will show that ℏ∈Tℏ⋂Sℏ. By the definition of Hβ, we have
(13)βδsSı2n+1,Tℏ+1−βδsTℏ,Sı2n+1≤HβSı2n+1,Tℏ≤α1dsℏ,Tℏ+α2dsı2n+1,Sı2n+1+α3dsℏ,Sı2n+1+α4dsı2n+1,Tℏ+α5dsℏ,Sı2n+1+dsı2n+1,Tℏ2+α6dsℏ,Tℏ∗dsı2n+1,Sı2n+11+dsℏ,ı2n+1+α7dsℏ,ı2n+1+ϵ2n+1≤α1dsℏ,Tℏ+α2dsı2n+1,ı2n+2+α3dsℏ,ı2n+2+α4dsı2n+1,Tℏ+α5dsℏ,ı2n+2+dsı2n+1,Tℏ2+α6dsℏ,Tℏdsı2n+1,ı2n+21+dsℏ,ı2n+1+α7dsℏ,ı2n+1+ϵ2n+1.It follows that(14)limn⟶∞βsδsSı2n+1,Tℏ+1−βδsTℏ,Sı2n+1≤limα1dsℏ,Tℏ+α2dsı2n+1,ı2n+2+α3dsℏ,ı2n+2+α4dsı2n+1,Tℏ+α5dsℏ,ı2n+1+dsı2n+1,Tℏ2+α6dsℏ,Tℏdsı2n+1,ı2n+21+dsℏ,ı2n+1+α7dsℏ,ı2n+1≤α1dsℏ,Tℏ+α4dsℏ,Tℏ+α5dsℏ,Tℏ2≤α1+α4+α52dsℏ,Tℏ.Since(15)limn⟶∞βδsSı2n+1,Tℏ+limn⟶∞1−βδsTℏ,Sı2n+1≤limn⟶∞βsδsSı2n+1,Tℏ+1−βδsTℏ,Sı2n+1,we have
(16)limn⟶∞βδsSı2n+1,Tℏ+limn⟶∞1−βδsTℏ,Sı2n+1≤α1+α4+α52dsℏ,Tℏ.This implies(17)limn⟶∞βδsSı2n+1,Tℏ≤α1+α4+α52dsℏ,Tℏ.Again, we have(18)βδsTı2n,Sℏ+1−βδsSℏ,Tı2n≤HβTı2n,Sℏ≤α1dsı2n,Tı2n+α2dsℏ,Sℏ+α3dsı2n,Sℏ+α4dsℏ,Tı2n+α5dsℏ,Tı2n+dsı2n,Sℏ2+α6dsℏ,Sℏdsı2n,Tı2n1+dsℏ,ı2n+α7dsℏ,ı2n+ϵ2n+1≤α1dsı2n,ı2n+1+α2dsℏ,Sℏ+α3dsı2n,Sℏ+α4dsℏ,ı2n+1+α5dsℏ,ı2n+1+dsı2n,Sℏ2+α6dsℏ,Sℏdsı2n,ı2n+11+dsℏ,ı2n+α7dsℏ,ı2n+ϵ2n+1.It follows that(19)limn⟶∞βsδsTı2n,Sℏ+1−βδsSℏ,Tı2n≤limα1dsı2n,ı2n+1+α2dsℏ,Sℏ+α3dsı2n,Sℏ+α4dsℏ,i2n+1+α5dsℏ,ı2n+1+dsı2n,Sℏ2+α6dsℏ,Sℏdsı2n,ı2n+11+dsℏ,ı2n+α7dsℏ,ı2n≤α2dsℏ,Sℏ+α3dsℏ,Sℏ+α5dsℏ,Sℏ2≤α2+α3+α52dsℏ,Sℏ.Since(20)limn⟶∞βδsTı2n,Sℏ+limn⟶∞1−βδsSℏ,Tı2n≤limn⟶∞βsδsTı2n,Sℏ+1−βδsSℏ,Tı2n,we have
(21)limn⟶∞βδsTı2n,Sℏ+limn⟶∞1−βδsSℏ,Tı2n≤α2+α3+α52dsℏ,Sℏ.This implies(22)limn⟶∞βδsTı2n,Sℏ≤α2+α3+α52dsℏ,Sℏ.Now(23)dsℏ,Tℏ≤sdsℏ,ı2n+2+δsSı2n+1,Tℏ,dsℏ,Sℏ≤sdsℏ,ı2n+1+δsTı2n,Sℏ.Using (17) and (22) in the above two inequalities, we get
(24)dsℏ,Tℏ≤slimn⟶∞dsℏ,ı2n+2+slimn⟶∞δsSı2n+1,Tℏ≤sα1+α4+α5/2βdsℏ,Tℏ,dsℏ,Sℏ≤slimn⟶∞dsℏ,ı2n+1+slimn⟶∞δsTı2n,Sℏ≤sα2+α3+α5/2βdsℏ,Sℏ.This givesdsℏ,Tℏ=0 and dsℏ,Sℏ=0. Since T and S are closed, we have ℏ∈T and ℏ∈S.Our next result provides an extension and new variants of Ciric’s quasi-contraction [15] and multivalued weak quasi-contraction [17], for a pair of multivalued mappings in a b-metric space.Theorem 12.
LetX,ds be a complete b-metric space with constant s≥1, ds be ∗-continuous, and T,S:X⟶CBdsX be multivalued pairwise Hβ-Hausdorff functions for some 1/2≤β≤1 and satisfying the following condition:
(25)HβTı,Sȷ≤αmaxdsı,ȷ,dı,Tı,dsȷ,Sȷ,dsı,Sȷ,dsȷ,Tı+Lmindsı,Sȷ,dsȷ,Tı,for all ı,ȷ∈X, some α≥0 with 0≤sα<1/2 and L≥0. Then, S and T have a common fixed point.Proof.
Proceeding as in the proof of Theorem11, for some ı0∈X, ı1∈Tı0, and 0<ϵ<1, we construct the sequence <ın> satisfying (7). Then, we have
(26)dsı2n+1,ı2n+2≤HβTı2n,Sı2n+1+ϵ2n+1≤αmaxdsı2n,ı2n+1,dsı2n,Tı2n,dsı2n+1,Sı2n+1,dsı2n,Sı2n+1,dsı2n+1,Tı2n+Lmindsı2n,Sı2n+1,dsı2n+1,Tı2n+ϵ2n+1≤αmaxdsı2n,ı2n+1,dsı2n,ı2n+1,dsı2n+1,ı2n+2,dsı2n,ı2n+2,dsı2n+1,ı2n+1+Lmindsı2n,ı2n+2,dsı2n+1,ı2n+1+ϵ2n+1.
Therefore,(27)dsı2n+1,ı2n+2≤sα1−sαdsı2n,ı2n+1+ϵ2n+1.
Again,(28)dsı2n+2,ı2n+3≤HβSı2n+1,Tı2n+2+ϵ2n+2≤αmaxdsı2n+1,ı2n+2,dsı2n+2,Tı2n+2,dsı2n+1,Sı2n+1,dsı2n+2,Sı2n+1,dsı2n+1,Tı2n+2+Lmindsı2n+2,Sı2n+1,dsı2n+1,Tı2n+2+ϵ2n+2≤αmaxdsı2n+1,ı2n+2,dsı2n+2,ı2n+3,dsı2n+1,ı2n+2,dsı2n+2,ı2n+2,dsı2n+1,ı2n+3+Lmindsı2n+2,ı2n+2,dsı2n+1,ı2n+3+ϵ2n+2,and we get
(29)dsı2n+2,ı2n+3≤sα1−sαdsı2n+1s,ı2n+3+ϵ2n+2.
Thus, we have(30)dsın,ın+1≤λdsın−1,ın+ϵn,where λ=sα/1−sα<1.By Lemma6, the sequence <ın> is a Cauchy sequence. Since X,ds is complete, there exists ℏ∈X such that the Cauchy sequence <ın> is convergent to ℏ. We will show that ℏ∈Tℏ⋂Sℏ. By the definition of Hβ, we have
(31)βδsSı2n+1,Tℏ+1−βδsTℏ,Sı2n+1≤HβSı2n+1,Tℏ≤αmaxdsı2n+1,ℏ,dsℏ,Tℏ,dsı2n+1,Si2n+1,dsℏ,Sı2n+1,dsı2n+1,Tℏ+Lmindsℏ,Sı2n+1,dsı2n+1,Tℏ+ϵ2n+2≤αmaxdsı2n+1,ℏ,dsℏ,Tℏ,dsı2n+1,ı2n+2,dsℏ,ı2n+2,dsı2n+1,Tℏ+Lmindsℏ,ı2n+2,dsı2n+1,Tℏ+ϵ2n+2.It follows that(32)limn⟶∞βsδsSı2n+1,Tℏ+1−βδsTℏ,Sı2n+1≤limαmaxdsı2n+1,ℏ,dsℏ,Tℏ,dsı2n+1,ı2n+2,dsℏ,ı2n+2,dsı2n+1,Tℏ+Lmindsℏ,ı2n+2,dsı2n+1,Tℏϵ2n+2≤αdsℏ,Tℏ.Since(33)limn⟶∞βδsSı2n+1,Tℏ+limn⟶∞1−βδsTℏ,Sı2n+1≤limn⟶∞βsδsSı2n+1,Tℏ+1−βδsTℏ,Sı2n+1,we have
(34)limn⟶∞βδsSı2n+1,Tℏ+limn⟶∞1−βδsTℏ,Sı2n+1≤αdsℏ,Tℏ.This implies(35)limn⟶∞βδsSı2n+1,Tℏ≤αdsℏ,Tℏ.Again, we have(36)βδsTı2n,Sℏ+1−βδsSℏ,Tı2n≤HβTı2n,Sℏ≤αmaxdsı2n,ℏ,dsı2n,Tı2n,dsℏ,Sℏ,dsı2n,Sℏ,dsℏ,Tı2n+Lmindsı2n,Sℏ,dsℏ,Tı2n+ϵ2n+1≤αmaxdsı2n,ℏ,dsı2n,ı2n+1,dsℏ,Sℏ,dsı2n,Sℏ,dsℏ,ı2n+1+Lmindsı2n,Sℏ,dsℏ,ı2n+1+ϵ2n+1.It follows that(37)limn⟶∞βsδsTı2n,Sℏ+1−βδsSℏ,Tı2n≤limαmaxdsı2n,ℏ,dsı2n,ı2n+1,dsℏ,Sℏ,dsı2n,Sℏ,dsℏ,ı2n+1+Lmindsı2n,Sℏ,dsℏ,ı2n+1+ϵ2n+1≤αdsℏ,Sℏ.Since(38)limn⟶∞βδsTı2n,Sℏ+limn⟶∞1−βδsSℏ,Tı2n≤limn⟶∞βsδsTı2n,Sℏ+1−βδsSℏ,Tı2n,we have
(39)limn⟶∞βδsTı2n,Sℏ+limn⟶∞1−βδsSℏ,Tı2n≤αdsℏ,Sℏ.This implies(40)limn⟶∞βδsTı2n,Sℏ≤αdsℏ,Sℏ.Now,(41)dsℏ,Tℏ≤sdsℏ,ı2n+2+δsSı2n+1,Tℏ,dsℏ,Sℏ≤sdsℏ,ı2n+1+δsTı2n,Sℏ.Using (35) and (40) in the above two inequalities, we get
(42)dsℏ,Tℏ≤slimn⟶∞dsℏ,ı2n+2+slimn⟶∞δsSı2n+1,Tℏ≤sαβdsℏ,Tℏ,dsℏ,Sℏ≤slimn⟶∞dsℏ,ı2n+1+slimn⟶∞δsTı2n,Sℏ≤sαβdsℏ,Tℏ.Sincesα<1/2 and 1/2≤β≤1, we get dsℏ,Tℏ=0 and dsℏ,Sℏ=0. As T and S are closed, we have ℏ∈T and ℏ∈S.Applying the same technique as in the proof of Theorem12, we can prove the following extension and new variant of Ciric’s contraction for a pair of multivalued mappings in a b-metric space.Theorem 13.
LetX,ds be a complete b-metric space with constant s≥1, ds be ∗-continuous, and T,S:X⟶CBdsXX be multivalued pairwise Hβ-Hausdorff functions for some 1/2≤β≤1 and satisfying the following condition:
(43)HβTı,Sȷ≤αmaxdsı,ȷ,dsı,Tı,dsȷ,Sȷ,dsı,Sȷ+dsȷ,Tı2,for all i,j∈X and some α≥0 with sα<β. Then, S and T have a common fixed point.ForS=T in Theorem 11, we get the following result:Corollary 14.
LetX,ds be a complete b-metric space with constant s≥1, ds be ∗-continuous, and T:X⟶CBdsX be a multivalued Hβ-Hausdorff function for some 1/2≤β≤1 and satisfying the following condition:
(44)HβTı,Tȷ≤α1dsı,Tı+α2dsȷ,Tȷ+α3dsı,Tȷ+α4dsȷ,Tı+α5dsı,Tȷ+dsȷ,Tı2+α6dsı,Tıdsȷ,Tȷ1+dsı,ȷ+α7dsı,ȷ,for all ı,ȷ∈X and some αk≥0, 1≤k≤7, with α1+α2+sα5+α6+α7+max2sα3,2sα4<1, sα1+α4+α5/2<β, and sα2+α3+α5/2<β. Then, T has a fixed point.Example 15.
LetX=0,5/12⋃2, dsı,ȷ=ı−ȷ2forallı,ȷ∈X, and S,T:X⟶CBX be as follows:
(45)Sı=ı4,forı∈0,512,0,13,2,forı=2,Tı=ı4,forı∈0,512,0,512,2,forı=2.
We will show that the functionsS and T satisfy contraction condition (6) for β=1/2.Case 1.
ı,ȷ∈0,5/12. By Proposition 2, we have
(46)H1/2Sı,Tȷ=H1/2ı4,ȷ4=dsı4,ȷ4=ı4−ȷ42≤α1ı−ȷ2,foranyα7≥116=α7dxı,ȷ.Case 2.
ı∈0,5/12,ȷ=2. We have dsı,ȷ=2−ı2. The minimum value of dsı,ȷ for ı∈0,5/12 is 361/144.
(47)δsSı,Tȷ=δsı4,0,512,2=ı216,δsTȷ,Sı=δs0,512,1,ı4=2−ı42,H1/2Sı,Tȷ=12ı216+2−ı42.
The maximum value ofH1/2Sı,Tj for ı∈0,5/12 is 2 (at ı=0). Thus, H1/2Sı,Tȷ≤α7dsı,ȷ for any α7≥288/361.Case 3.
ı=2,ȷ∈0,5/12. We have dsı,ȷ=2−ȷ2. The minimum value of dsı,ȷ for ȷ∈0,5/12 is 361/144. δsSı,Tȷ=δs0,5/12,2,ȷ/4=2−ȷ/42.
(48)δsTȷ,Sı=δsı4,0,512,1=ȷ216,H1/2Sı,Tȷ=12ȷ216+2−ȷ42.
The maximum value ofH1/2Sı,Tȷ for ȷ∈0,5/12 is 2 (at ȷ=0). Thus, H1/2Sı,Tȷ≤α7dsı,ȷ for any α7≥288/361.Thus,S and T satisfy contraction condition (6) for β=1/2, 288/361≤α7<1 and α1=α2=α3=α4=α5=α6=0. Simple calculations show that S and T are pairwise Hβ-Hausdorff functions. All conditions of Theorem 11 are satisfied, and 0 is a common fixed point of S and T. However, we see that at ı=0,ȷ=2, S and T do not satisfy contraction condition (6) for β=1 and so do not satisfy Nadler’s contraction and Czerwik’s contraction.Remark 16.
In Example15, simple calculations show that S and T do not satisfy contraction condition (6) for 62/100<β≤1. However, in view of Remark 9 (i), there may exist functions S and T which satisfy contraction condition (6) for β=1 but may not satisfy for β<1. Thus, for β=1, Theorem 11 is an extension of Nadler’s contraction [10], Czerwik’s contraction [2], and many of their generalizations. For β<1, Theorem 11 provides new variants of Nadler’s contraction [10], Czerwik’s contraction [2], and many of their generalizations.Example 17.
LetX=0,1/4,1,
(49)dsı,ȷ=ı−ȷ2forallı,ȷ∈X.and T:X⟶CBX be as follows:
(50)Tx=0,forı∈0,14,0,1,forı=1.
We will show thatT satisfies (44) with β∈7/16,9/16.
For ifı,ȷ∈0,1/4, then the result is clear. Suppose ı∈0,1/4 and ȷ=1. Then, δdsTı,T1=0 and δdsT1,Tı=1 so that HβTı,T1=maxβ,1−β. Also, we have dsı,1=1 or 9/16.Ifβ∈7/16,1/2, then HβTı,T1=1−β. Now 1−β∈8/16,9/16. So 1−β=16/91−β9/16 and 1−β<16/91−β1, that is, 1−β≤16/91−βdsı,1. Thus, we have HβTı,T1=1−β≤kdsı,1, where k=16/91−β<1.Similarly, ifβ∈1/2,9/16, we get HβTı,T1=β≤kdsı,1, where k=16/9β<1.However, forı=1/4 and ȷ=1, we have
(51)HT14,T1=maxδdsT14,T1,δdsT1,T14=1andds14,1=916.We see thatT does not satisfy condition (2.2) of [24] and condition (2.1) of [26]. Thus, Theorem 2.2 of Debnath [24] and Theorem 2.3 of Kumar and Luambano [26] are not applicable.Remark 18 (an open question).
Obtain the version of results in fixed points in the sense of Debnath [24], Kumar and Luambano [26], and Altun et al. [25] for two or more mappings using Hβ-Hausdorff-Pompeiu b-metric, which will give extension and new variants of the respective results and will also generalize Corollary 19.By taking different values ofαk in Theorem 11, we get the following extension and new variants of well-known contraction principles:Forαk=0, k=1,2,3,4,5,6, we have the following.Corollary 19 (Nadler’s and Czerwik’s contraction).
LetX,ds be a complete b-metric space with constant s≥1 and T,S:X⟶CBdsX be multivalued pairwise Hβ-Hausdorff functions for some 1/2≤β≤1 and satisfying the following condition:
(52)HβTı,Sȷ≤αdsı,ȷ,for all ı,ȷ∈X and 0≤α<1. Then, S and T have a common fixed point.Forαk=0, k=3,4,5,6,7, we have the following.Corollary 20 (Kannan’s contraction).
LetX,ds be a complete b-metric space with constant s≥1 and T,S:X⟶CBdsX be multivalued pairwise Hβ-Hausdorff functions for some 1/2≤β≤1 and satisfying the following condition:
(53)HβTı,Sȷ≤α1dsı,Tı+α2dsȷ,Sȷ,for all ı,ȷ∈X and 0≤α1+α2<1. Then, S and T have a common fixed point.Forαk=0, k=1,2,5,6,7, we have the following.Corollary 21 (Chattarjee contraction).
LetX,ds be a complete b-metric space with constant s≥1 and T,S:X⟶CBdsX be multivalued pairwise Hβ-Hausdorff functions for some 1/2≤β≤1 and satisfying the following condition:
(54)HβTı,Sȷ≤α3dsı,Sı+α4dsȷ,Tı,for all ı,ȷ∈X and maxsα3,sα4<1/2. Then, S and T have a common fixed point.Forαk=0, k=5,6, we have the following.Corollary 22 (Hardy-Rogers contraction).
LetX,ds be a complete b-metric space with constant s≥1 and T,S:X⟶CBdsX be multivalued pairwise Hβ-Hausdorff functions for some 1/2≤β≤1 and satisfying the following condition:
(55)HβTı,Sȷ≤α1dsı,Tı+α2dsȷ,Sȷ+α3dsı,Sı+α4dsȷ,Tı+α7dsı,ȷ,for all ı,ȷ∈X and α1+α2+α7+max2sα3,2sα4<1, sα1+α4<β, and sα2+α3<β. Then, S and T have a common fixed point.Remark 23.
Corollary19 is an extension and new variant of the results of Nadler [10] and Czerwik [2], Corollaries 20 and 21 are extended and new variants of the set-valued versions of the Kannan contraction and Chatterjee contraction, respectively, whereas Corollary 22 is an extended and new variant of the result of Mirmostaffae [16].
IfT,S:X⟶X are single-valued mappings and then by Proposition 2, HβTı,Sȷ=dsTı,Sȷ for all 1/2≤β≤1. So taking β=1 in Theorem 11, we get the following results for single-valued mappings.Corollary 24.
LetX,ds be a complete b-metric space with constant s≥1 and T,S:X⟶X be single-valued mappings satisfying the following condition:
(56)dsTı,Sȷ≤α1dsı,Tı+α2dsȷ,Sȷ+α3dsı,Sı+α4dsȷ,Tı+α5dsı,Sȷ+dsȷ,Tı2+α6dsı,Tıdsȷ,Sȷ1+dsı,ȷ+α7dsı,ȷ,for all ı,ȷ∈X and α1+α2+sα5+α6+α7+max2sα3,2sα4<1, sα1+α4+sα5/2<1, and sα2+α3+sα5/2<1. Then, S and T have a common fixed point.Remark 25.
Corollary24 is an extension and b-metric version of the result of Wong [40].


## 4. Applications

In this section, we provide two applications of our results.


### 4.1. Application to Multivalued Fractals

In this section inspiring from some recent works in [20, 41, 42], we will apply our result to prove the existence of a unique common multivalued fractal for a pair of iterated multifunction systems. Let Pi,Qi:X⟶CBdsX, i=1,2,⋯n, be upper semicontinuous mappings. Then, P=P1,P2,⋯Pn and Q=Q1,Q2,⋯Qn form a pair of iterated multifunction systems defined on the b-metric space X,ds. The extended multifractal operators generated by the iterated multifunction systems P=P1,P2,⋯Pn and Q=Q1,Q2,⋯Qn are the operators TP,TQ:CBdsX⟶CBdsX defined by TPY=⋃i=1nPiY and TQY=⋃i=1nQiY, respectively. A common fixed point of TP and TQ is called the common multivalued fractal of the iterated multifunction systems P=P1,P2,⋯Pn and Q=Q1,Q2,⋯Qn.Theorem 26.
LetPi,Qi:X⟶CBdsX, i=1,2,⋯n, be upper semicontinuous mappings satisfying the following condition:
Fori=1,2,⋯n, there exist β∈1/2,1 and ai,ei∈0,1, ai+2sei<1, such that for all x,y∈X,
(57)HβPix,Qiy≤ai.dsx,y+eidsx,Qiy+dsy,Pix.
Then,(i)
For allU1,U2∈CBX, HβTPU1,TQU2≤a.HβU1,U2+b.HβU1,TPU1+c.HβU2,TQU2+eHβU1,TQU2+HβU2,TPU1(v)
The pair of systemsP=P1,P2,⋯Pn and Q=Q1,Q2,⋯Qn has a unique common multivalued fractalProof.
Suppose condition (57) holds. Then, for U1,U2∈CBX, we have
(58)βδPiU1,QiU2+1−βδQiU2,PiU1=βsupx∈U1infy∈U2HβPix,Qiy+1−βsupy∈U2infx∈U1HβPix,Qiy≤βsupx∈U1infy∈U2ai.dsx,y+eidsx,Qiy+dsy,Pix+1−βsupy∈U2infx∈U1ai.dsx,y+eidsx,Qiy+dsy,Pix=ai.HβU1,U2+eiHβU1,QiU2+HβU2,PiU1.
Similarly, we get(59)βδδQiU2,PiU1+1−βPiU1,QiU2≤ai.HβU2,U1+eiHβU2,PiU1+HβU1,QiU2.
Then, we have(60)HβPiU1,QiU2≤ai.HβU1,U2+eiHβU2,PiU1+HβU1,QiU2i=1,2,⋯n≤a.HβU2,U1+eHβU2,PiU1+HβU1,QiU2,where a=maxa1,a2,⋯an and e=maxe1,e2,⋯n. Note that
(61)Hβ⋃i=1nPiU1,⋃i=1nQiU2≤maxHβP1U1,Q1U2,HβP2U1,Q2U2,⋯HβPnU1,QnU2,and so
(62)HβTPU1,TQU2≤a.HβU1,U2+eHβU1,TQU2+HβU2,TPU1.
Thus,TP,TQ:CBX⟶CBX satisfies the conditions of Corollary 24 in the metric space CBX,Hβ and hence has a common fixed point U∗ in CBX, which in turn is the unique common multivalued fractal of the pair of iterated multifunction systems P=P1,P2,⋯Pn and Q=Q1,Q2,⋯Qn.Remark 27.
SinceHβA,B≤HA,B, Theorem 26 is a proper improvement and generalization of Theorem 3.4 of [20], Theorem 3.1 of [41], and Theorem 3.8 of [42].


### 4.2. Application to the Integral Equation

In this section, motivated by the applications given in [28–30] and [31], we establish the sufficient conditions for the existence of a common solution of a pair of nonlinear Volterra-type integral equations.For some real numbersa,b with 0≤a<b and I=a,b, let X=CI,ℝ be the Banach space of real continuous functions defined on I equipped with a norm given by ı=maxt∈Iıt. For some p≥1, define a b-metric ds on X by
(63)dsı,ȷ=maxt∈Iıt−ȷtp,forallı,ȷ∈X.Then,X,ds,2p−1 is a complete b-metric space. Consider the following pair of Volterra-type integral equations:
(64)ıt=qt+∫aμtPt,sK1t,s,ısds+∫aσtQt,sK2t,s,ȷsds,ȷt=qt+∫aμtPt,sK2t,s,ȷsds+∫aσtQt,sK1t,s,ȷsds,for all t,s∈I=a,b⊆ℝ, λ>0, Ki=1,2:I×I×X⟶ℝ, and q:I⟶ℝ, and P,Q:I×I⟶ℝ are continuous functions and μ,σ:I⟶I.SupposeT,S:X⟶X is self-mappings defined by
(65)Tıt=qt+∫aμtPt,sK1t,s,ısds+∫aσtQt,sK2t,s,ȷsds,Sȷt=qt+∫aμtPt,sK2t,s,ȷsds+∫aσtQt,sK1t,s,ȷsds,for all ı,ȷ∈X, where t∈I. It is obvious that ℏt is a solution of (64) if and only if it has a common fixed point of T and S.Theorem 28.
Suppose that the following hypotheses hold:
(H1) TX and SX are closed in X
(H2) There exist nonnegative real numbers α1,α2,α3,α4,α5,α6,α7 with α1+α2+2pmaxα3,α4+2p−1α5+α6+α7<1 such that
(66)K1t,s,ıs−K2t,s,ysp≤NT,S,p,t,where
(67)NT,S,p,t=α1ıt−Tıtp+α2ȷt−Sȷtp+α3ıt−Sȷtp+α4ȷt−Tıtp+α7ıt−ȷtp+α5ıt−Sȷtp+ȷt−Tıtp2+α6ıt−Tıtpȷt−Sȷtp1+ıt−ȷtp.
(H3) ∫aμtPt,sds+∫aσtQt,sds≤1/2p−1
Then, the system (64) of integral equations has unique common solutions in X.Proof.
UsingH2 and H3, we have
(68)dsTı,Sȷ=maxt∈ITıt−Sȷtp≤maxt∈I∫aμtPt,sK1t,s,ısds+∫aσtQt,sK2t,s,ȷsds−∫aμtPt,sK2t,s,ȷsds−∫aσtQt,sK1t,s,ȷsdsp≤maxt∈I2p−1∫aμtPt,sK1t,s,ısds−∫aμtPt,sK2t,s,ȷsdsp+∫aσtQt,sK2t,s,ȷsds−∫aσtQt,sK1t,s,ȷsdsp≤maxt∈I2p−1∫aμtPt,sK1t,s,ıs−K2t,s,ȷsdsp+∫aσtQt,sK2t,s,ȷs−K1t,s,ȷsdsp≤maxt∈I2p−1∫aμtPt,spK1t,s,ıs−K2t,s,ȷspds+∫aσtQt,spK2t,s,ȷs−K1t,s,ȷspds≤maxt∈I2p−1∫aμtPt,spNT,S,p,tds+∫aσtQt,spNT,S,p,tds≤maxt∈I2p−1NT,S,p,t∫aμtPt,spds+∫aσtQt,spds≤maxt∈INT,S,p,t≤α1dsı,Tı+α2dsȷ,Sȷ+α3dsı,Sȷ+α4dsȷ,Tı+α5dsı,Sȷ+dsȷ,Tı2+α6dsı,Tıdsȷ,Sȷ1+dsı,ȷ+α7dsı,ȷ.
Thus, conditions of Theorem11 are satisfied. Theorem 11 therefore ensures a common fixed point of T and S, which in turn is a common solution of the pair of integral equations (64).Remark 29.
TakingQt,s=0,Pt,s=1,qt=0,μt=t and a=0 in (64), we get the Volterra-type integral equations considered in Rasham et al. [31] and Alshoraify et al. [30].Remark 30.
TakingQt,s=0,μt=1 and a=0 in (64), we get the Fredholm-type integral equations (III.3) considered in Shoaib et al. [29].Remark 31.
TakingQt,s=0,Pt,s=1 and μt=b in (64), we get the Fredholm-type integral equations (III.1) considered in Shoaib et al. [29].


## 4.1. Application to Multivalued Fractals

In this section inspiring from some recent works in [20, 41, 42], we will apply our result to prove the existence of a unique common multivalued fractal for a pair of iterated multifunction systems. Let Pi,Qi:X⟶CBdsX, i=1,2,⋯n, be upper semicontinuous mappings. Then, P=P1,P2,⋯Pn and Q=Q1,Q2,⋯Qn form a pair of iterated multifunction systems defined on the b-metric space X,ds. The extended multifractal operators generated by the iterated multifunction systems P=P1,P2,⋯Pn and Q=Q1,Q2,⋯Qn are the operators TP,TQ:CBdsX⟶CBdsX defined by TPY=⋃i=1nPiY and TQY=⋃i=1nQiY, respectively. A common fixed point of TP and TQ is called the common multivalued fractal of the iterated multifunction systems P=P1,P2,⋯Pn and Q=Q1,Q2,⋯Qn.Theorem 26.
LetPi,Qi:X⟶CBdsX, i=1,2,⋯n, be upper semicontinuous mappings satisfying the following condition:
Fori=1,2,⋯n, there exist β∈1/2,1 and ai,ei∈0,1, ai+2sei<1, such that for all x,y∈X,
(57)HβPix,Qiy≤ai.dsx,y+eidsx,Qiy+dsy,Pix.
Then,(i)
For allU1,U2∈CBX, HβTPU1,TQU2≤a.HβU1,U2+b.HβU1,TPU1+c.HβU2,TQU2+eHβU1,TQU2+HβU2,TPU1(v)
The pair of systemsP=P1,P2,⋯Pn and Q=Q1,Q2,⋯Qn has a unique common multivalued fractalProof.
Suppose condition (57) holds. Then, for U1,U2∈CBX, we have
(58)βδPiU1,QiU2+1−βδQiU2,PiU1=βsupx∈U1infy∈U2HβPix,Qiy+1−βsupy∈U2infx∈U1HβPix,Qiy≤βsupx∈U1infy∈U2ai.dsx,y+eidsx,Qiy+dsy,Pix+1−βsupy∈U2infx∈U1ai.dsx,y+eidsx,Qiy+dsy,Pix=ai.HβU1,U2+eiHβU1,QiU2+HβU2,PiU1.
Similarly, we get(59)βδδQiU2,PiU1+1−βPiU1,QiU2≤ai.HβU2,U1+eiHβU2,PiU1+HβU1,QiU2.
Then, we have(60)HβPiU1,QiU2≤ai.HβU1,U2+eiHβU2,PiU1+HβU1,QiU2i=1,2,⋯n≤a.HβU2,U1+eHβU2,PiU1+HβU1,QiU2,where a=maxa1,a2,⋯an and e=maxe1,e2,⋯n. Note that
(61)Hβ⋃i=1nPiU1,⋃i=1nQiU2≤maxHβP1U1,Q1U2,HβP2U1,Q2U2,⋯HβPnU1,QnU2,and so
(62)HβTPU1,TQU2≤a.HβU1,U2+eHβU1,TQU2+HβU2,TPU1.
Thus,TP,TQ:CBX⟶CBX satisfies the conditions of Corollary 24 in the metric space CBX,Hβ and hence has a common fixed point U∗ in CBX, which in turn is the unique common multivalued fractal of the pair of iterated multifunction systems P=P1,P2,⋯Pn and Q=Q1,Q2,⋯Qn.Remark 27.
SinceHβA,B≤HA,B, Theorem 26 is a proper improvement and generalization of Theorem 3.4 of [20], Theorem 3.1 of [41], and Theorem 3.8 of [42].


## 4.2. Application to the Integral Equation

In this section, motivated by the applications given in [28–30] and [31], we establish the sufficient conditions for the existence of a common solution of a pair of nonlinear Volterra-type integral equations.For some real numbersa,b with 0≤a<b and I=a,b, let X=CI,ℝ be the Banach space of real continuous functions defined on I equipped with a norm given by ı=maxt∈Iıt. For some p≥1, define a b-metric ds on X by
(63)dsı,ȷ=maxt∈Iıt−ȷtp,forallı,ȷ∈X.Then,X,ds,2p−1 is a complete b-metric space. Consider the following pair of Volterra-type integral equations:
(64)ıt=qt+∫aμtPt,sK1t,s,ısds+∫aσtQt,sK2t,s,ȷsds,ȷt=qt+∫aμtPt,sK2t,s,ȷsds+∫aσtQt,sK1t,s,ȷsds,for all t,s∈I=a,b⊆ℝ, λ>0, Ki=1,2:I×I×X⟶ℝ, and q:I⟶ℝ, and P,Q:I×I⟶ℝ are continuous functions and μ,σ:I⟶I.SupposeT,S:X⟶X is self-mappings defined by
(65)Tıt=qt+∫aμtPt,sK1t,s,ısds+∫aσtQt,sK2t,s,ȷsds,Sȷt=qt+∫aμtPt,sK2t,s,ȷsds+∫aσtQt,sK1t,s,ȷsds,for all ı,ȷ∈X, where t∈I. It is obvious that ℏt is a solution of (64) if and only if it has a common fixed point of T and S.Theorem 28.
Suppose that the following hypotheses hold:
(H1) TX and SX are closed in X
(H2) There exist nonnegative real numbers α1,α2,α3,α4,α5,α6,α7 with α1+α2+2pmaxα3,α4+2p−1α5+α6+α7<1 such that
(66)K1t,s,ıs−K2t,s,ysp≤NT,S,p,t,where
(67)NT,S,p,t=α1ıt−Tıtp+α2ȷt−Sȷtp+α3ıt−Sȷtp+α4ȷt−Tıtp+α7ıt−ȷtp+α5ıt−Sȷtp+ȷt−Tıtp2+α6ıt−Tıtpȷt−Sȷtp1+ıt−ȷtp.
(H3) ∫aμtPt,sds+∫aσtQt,sds≤1/2p−1
Then, the system (64) of integral equations has unique common solutions in X.Proof.
UsingH2 and H3, we have
(68)dsTı,Sȷ=maxt∈ITıt−Sȷtp≤maxt∈I∫aμtPt,sK1t,s,ısds+∫aσtQt,sK2t,s,ȷsds−∫aμtPt,sK2t,s,ȷsds−∫aσtQt,sK1t,s,ȷsdsp≤maxt∈I2p−1∫aμtPt,sK1t,s,ısds−∫aμtPt,sK2t,s,ȷsdsp+∫aσtQt,sK2t,s,ȷsds−∫aσtQt,sK1t,s,ȷsdsp≤maxt∈I2p−1∫aμtPt,sK1t,s,ıs−K2t,s,ȷsdsp+∫aσtQt,sK2t,s,ȷs−K1t,s,ȷsdsp≤maxt∈I2p−1∫aμtPt,spK1t,s,ıs−K2t,s,ȷspds+∫aσtQt,spK2t,s,ȷs−K1t,s,ȷspds≤maxt∈I2p−1∫aμtPt,spNT,S,p,tds+∫aσtQt,spNT,S,p,tds≤maxt∈I2p−1NT,S,p,t∫aμtPt,spds+∫aσtQt,spds≤maxt∈INT,S,p,t≤α1dsı,Tı+α2dsȷ,Sȷ+α3dsı,Sȷ+α4dsȷ,Tı+α5dsı,Sȷ+dsȷ,Tı2+α6dsı,Tıdsȷ,Sȷ1+dsı,ȷ+α7dsı,ȷ.
Thus, conditions of Theorem11 are satisfied. Theorem 11 therefore ensures a common fixed point of T and S, which in turn is a common solution of the pair of integral equations (64).Remark 29.
TakingQt,s=0,Pt,s=1,qt=0,μt=t and a=0 in (64), we get the Volterra-type integral equations considered in Rasham et al. [31] and Alshoraify et al. [30].Remark 30.
TakingQt,s=0,μt=1 and a=0 in (64), we get the Fredholm-type integral equations (III.3) considered in Shoaib et al. [29].Remark 31.
TakingQt,s=0,Pt,s=1 and μt=b in (64), we get the Fredholm-type integral equations (III.1) considered in Shoaib et al. [29].



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*Source: 1931861-2022-07-09.xml*