# Solutions of Fractional Differential Type Equations by Fixed Point Techniques for Multivalued Contractions

Authors: Hasanen A. Hammad; Hassen Aydi; Manuel De la Sen
Journal: Complexity
 (2021)

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## Abstract

This paper involves extendedb−metric versions of a fractional differential equation, a system of fractional differential equations and two-dimensional (2D) linear Fredholm integral equations. By various given hypotheses, exciting results are established in the setting of an extended b−metric space. Thereafter, by making consequent use of the fixed point technique, short and simple proofs are obtained for solutions of a fractional differential equation, a system of fractional differential equations and a two-dimensional linear Fredholm integral equation.


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## Body

## 1. Introduction and Preliminaries

In the last years, the fractional calculus branch [1, 2] has attracted great interest. There exist many kinds of proposed fractional operators, for instance, we have the well-known Caputo, Riemann–Liouville, Grunwald–Letnikov derivative etc. Among all the papers dealing with fractional derivatives, fractional differential equations as an important research field have attained great deal of attention from many researchers (see [3–8]).There are many applications of the fractional topic in complex analysis, such as, in the sense of conformable derivatives and integrals, interesting results for fractional formulations of complex-valued functions of a real variable have been successfully introduced, which in turn open the door to the researchers to construct the theory of conformable integration by studying functions of a complex variable [9]. On the other hand, the standard definition for the Atangana–Baleanu fractional derivative involves an integral transform with a Mittag–Leffler function, where the kernel can be rewritten as a complex contour integral, which can be used to provide an analytic continuation of the definition to complex orders of differentiation [10]. These lines are very important due to their applications in the field of natural science or engineering.In the last few decades and in the branch of fractional differential equations, Riemann–Liouville and Caputo derivative ones are the mostly used. Note that several fractional differential equations have been resolved by using fixed point techniques. This paper is concerned with this fact when considering the class of extendedb-metric spaces.Letℜ,ϖ be a metric space. Denote by CBℜ a set of nonempty closed bounded subsets of ℜ. Define the function ¥:CBℜ×CBℜ⟶ℝ+ by(1)¥℧1,℧2=maxsupξ∈℧1ϖξ,℧2,supζ∈℧2ϖζ,℧1,where ϖξ,℧2=infϖξ,ζ:ζ∈℧2, for ℧1,℧2∈CBℜ.Then, ¥ is called the Hausdorff–Pompeiu metric. Consider(2)δ℧1,℧2=supϖξ,ζ:ξ∈℧1,ζ∈℧2,D℧1,℧2=infϖξ,ζ:ξ∈℧1,ζ∈℧2.The following can be deduced from the definition ofδ. For all ℧1,℧2,℧3∈CBℜ, we have the following:(a)
δ℧1,℧2=δ℧2,℧1(b)
δ℧1,℧2=0 iff ℧1=℧2=ξ(c)
δ℧1,℧3≤δ℧1,℧2+δ℧2,℧3(d)
δ℧1,℧1=diam℧1In 2017, the concept of extendedb-metric spaces has been initiated by Kamran et al. [11], by considering a control function at the right-hand side of the triangular inequality.Definition 1 (see [11]).
Letℜbe a nonempty set andθ:ℜ×ℜ⟶1,∞be a given function. An extendedb-metric is a functionϖθ:ℜ×ℜ⟶0,∞such that, for allη,ξ,σ∈ℜ, we have the following:(1)
ϖθη,ξ=0⟺η=ξ(2)
ϖθη,ξ=ϖθξ,η(3)
ϖθη,ξ≤θη,ξϖθη,σ+ϖθσ,ξ
This (generalized) metric space has attracted many researchers where many real applications have been resolved. For more details, see [12–15]. Some of the related topological concepts are as follows.Definition 2.
Letℜ,ϖθbe an extendedb−metric space. Letξmm≥0be a sequence inℜ.(1)
ξm converges to some ξ in ℜ, if for each ε>0, there is M=Mε∈ℕ so that ϖθξm,ξ<ε for each m≥M(2)
ξm is Cauchy, if for each ε>0, there is M=Mε∈ℕ so that ϖθξm,ξn<ε for all m,n≥M(3)
ℜ,ϖθ is called complete if each Cauchy sequence is convergentLemma 1 (see [12]).
Letℜ,ϖθbe an extendedb−metric space. If the sequenceξninℜis such thatlimn,m⟶∞ϖθξn,ξm≤1/∇, where∇∈0,1and0≤ϖθξn,ξn+1≤∇ϖθξn−1,ξn, thenξnis Cauchy.Example 1 (see [16]).
Takeℜ=0,1. Givenθ:ℜ×ℜ⟶1,∞asθν,μ=νμ+1/ν+μ. Consider the extendedb-metricϖθ:ℜ×ℜ⟶0,∞so that(3)ϖθν,μ=1νμ,forν,μ∈0,1,ν≠μ,0,forν=μ,and ϖθ0,μ=ϖθμ,0=1/μ for μ∈0,1.Definition 3 (see [17]).
Denote byΞthe set of functionsΥ:ℝ+⟶0,1such that(i)
ℝ+=η∈ℝ:η>0(ii)
For any sequenceηnn=0∞,Υηn⟶1implies thatηn⟶0asn⟶∞Example 2.
GivenΥ:ℝ+⟶0,1as(4)Υℏ=1−ℏ32,ifℏ≤1,τ<1,ifℏ>1.
Clearly,Υ∈Ξ.
The manuscript is organized as follows. In Section2, some fixed point results in the class of extended b-metric spaces have been provided. We also present some useful examples. By using fixed point techniques, we solve in Section 3 a fractional nonlinear differential equation, we ensure in Section 4 the existence of a unique solution of a system of nonlinear fractional differential equations, and in Section 5, we establish that a two-dimensional linear Fredholm integral equation has a unique solution. At the end, in Section 6, we give a conclusion.


## 2. Main Theorems

In this section,ℜ refers to an extended b−metric space equipped with the distance ϖθ. We begin with the following lemmas.Lemma 2.
Letℜ,ϖθbe an extendedb−metric space with the functionθ:ℜ×ℜ⟶1,∞. For any℧1,℧2,℧3∈CBℜandξ,ζ∈ℜ,the following assertions are valid:(i)
ϖθξ,℧2≤ϖθξ,ζ for ζ∈℧2(ii)
δ℧1,℧2≤¥℧1,℧2(iii)
ϖθξ,℧2≤¥℧1,℧2(iv)
¥℧1,℧1=0(v)
¥℧1,℧2=¥℧2,℧1(vi)
¥℧1,℧3≤θ℧1,℧3¥℧1,℧2+¥℧2,℧3(vii)
ϖθξ,℧1≤θξ,℧1ϖθξ,ζ+ϖθζ,℧1Proof.
The assertions (i)–(v) follow immediately by Czerwik [18] in b−metric spaces and (vi)-(vii) follow immediately by the definition of an extended b− metric space and (1) with (2).Lemma 3.
Letℜ,ϖθbe an extendedb−metric space. Then, for all℧1,℧2∈CBℜ,μ∈℧1andℏ≥1, there existsημ∈℧2such thatϖθμ,η≤ℏ¥℧1,℧2.Proof.
By a similar way as in the proof of Lemma 4 in [19], we get the result.
Now, we state and prove our main theorems.Theorem 1.
Letℜ,ϖθbe a complete extendedb−metric space and℘,ℑ:ℜ⟶CBℜbe multivalued mappings satisfying, for allξ,ℓ∈ℜ,(5)Ωκ¥℘ξ,ℑℓ≤ΩΔξ,ℓ−ΛΠξ,ℓ,where(6)Δξ,ℓ=maxϖθξ,ℓ,ϖθξ,℘ξ1+ϖθℓ,ℑℓ1+ϖθ℘ξ,ℑℓ,ϖθℓ,℘ξ1+ϖθξ,℘ξ1+ϖθξ,ℓ,Πξ,ℓ=minϖθξ,ℓϖθξ,℘ξ,ϖθξ,℘ξϖθℓ,ℑℓ,ϖθℓ,℘ξϖθξ,ℑℓ.(1)
Ω:ℝ+⟶ℝ+is continuous and nondecreasing function so thatΩτ=0iffτ=0(2)
Λ:ℝ+⟶ℝ+is a continuous function so thatΛτ=0iffτ=0
Iflimn,m⟶∞ϖθξn,ξm≤1/kfork>1, then℘andℑhave a unique common fixed point (cfp).Proof.
Letξ∈ℜ be a fixed element. Define ξ°=ξ, and let ξ1∈℘ξ°, by Lemma 3, there exists ξ2∈ℑξ1 such that ϖθξ1,ξ2≤κ¥℘ξ°,ℑξ1. For ξ2∈ℑξ1, there is ξ3∈℘ξ1 such that ϖθξ2,ξ3≤κ¥℘ξ1,ℑξ2.
Continuing with the same manner, we haveξ2n+1∈℘ξ2n, ξ2n+2∈ℑξ2n+1. If ξ2n+1=ξ2n+2, then the sequence ξn is Cauchy. Suppose that ξ2n+1≠ξ2n+2. Then, by (3), we have(7)Ωκϖθξ2n+1,ξ2n+2≤Ωκ¥℘ξ2n,ℑξ2n+1≤ΩΔξ2n,ξ2n+1−ΛΠξ2n,ξ2n+1,where(8)Δξ2n,ξ2n+1=maxϖθξ2n,ξ2n+1,ϖθξ2n,℘ξ2n1+ϖθξ2n+1,ℑξ2n+11+ϖθ℘ξ2n,ℑξ2n+1,ϖθξ2n+1,℘ξ2n1+ϖθξ2n,℘ξ2n1+ϖθξ2n,ξ2n+1=maxϖθξ2n,ξ2n+1,ϖθξ2n,ξ2n+11+ϖθξ2n+1,ξ2n+21+ϖθξ2n+1,ξ2n+2,ϖθξ2n+1,ξ2n+11+ϖθξ2n,ξ2n+11+ϖθξ2n,ξ2n+1=maxϖθξ2n,ξ2n+1,ϖθξ2n,ξ2n+1,0=ϖθξ2n,ξ2n+1,and(9)Πξ2n,ξ2n+1=minϖθξ2n,ξ2n+1ϖθξ2n,℘ξ2n,ϖθξ2n,℘ξ2nϖθξ2n+1,ℑξ2n+1,ϖθξ2n+1,℘ξ2nϖθξ2n,ℑξ2n+1≤minϖθξ2n,ξ2n+1ϖθξ2n,ξ2n+1,ϖθξ2n,ξ2n+1ϖθξ2n+1,ξ2n+2,ϖθξ2n+1,ξ2n+1ϖθξ2n,ξ2n+2=0.
Applying (8) and (9) in (7), one can write Ωκϖθξ2n+1,ξ2n+2≤Ωϖθξ2n,ξ2n+1, By definition of Ω, we conclude that(10)κϖθξ2n+1,ξ2n+2≤ϖθξ2n,ξ2n+1,for alln∈ℕ.
Similarly, if we replaceξ with ξ2n+2 and ℓ with ξ2n+3, we have(11)κϖθξ2n+2,ξ2n+3≤ϖθξ2n+1,ξ2n+2,for alln∈ℕ.
From (10) and (11), we get(12)ϖθξn,ξn+1≤∇ϖθξn−1,ξn,∇=1κ<1for alln∈ℕ.
Now, by Lemma1, we observe that ξn is Cauchy sequence. Since ℜ is complete, then there is φ∈ℜ such that limsupn⟶∞ξn=φ. Assume that φ∉ℑφ, then we have(13)Ωκϖθξ2n+1,ℑφ≤Ωκ¥℘ξ2n,ℑφ≤ΩΔξ2n,φ−ΛΠξ2n,φ,where(14)Δξ2n,φ=maxϖθξ2n,φ,ϖθξ2n,℘ξ2n1+ϖθφ,ℑφ1+ϖθ℘ξ2n,ℑφ,ϖθφ,℘ξ2n1+ϖθξ2n,℘ξ2n1+ϖθξ2n,φ,≤maxϖθξ2n,φ,ϖθξ2n,ξ2n+11+ϖθφ,ℑφ1+ϖθξ2n+1,ℑφ,ϖθφ,ξ2n+11+ϖθξ2n,ξ2n+11+ϖθξ2n,φ≤maxϖθξ2n,φ,ϖθξ2n,ξ2n+11+ϖθφ,ℑφ,ϖθφ,ξ2n+11+ϖθξ2n,ξ2n+1.
Takinglimsup as n⟶∞ in the above inequalities, we conclude that(15)limsupn⟶∞Δξ2n,φ=0andlimsupn⟶∞Πξ2n,φ=0.
It follows from definition ofΔ,Π and (11) that limsupn⟶∞Ωκϖθξ2n+1,ℑφ=0 or limsupn⟶∞ϖθξ2n+1,ℑφ=0.
Using Lemma2, we get(16)ϖθφ,ℑφ≤θφ,ℑφϖθφ,ξ2n+1+ϖθξ2n+1,ℑφ.
At the limit, we haveϖθφ,ℑφ⟶0. Thus, φ∈ℑφ. Similarly, we can show that φ∈℘φ. Hence, φ is a cfp of the two mappings ℘ and ℑ. For the uniqueness, let ν≠φ be another cfp of ℘ and ℑ, then, by our contractive condition, one can write(17)Ωκϖθφ,ν≤Ωκ¥℘φ,ℑν≤Ωmaxϖθφ,ν,ϖθφ,℘φ1+ϖθν,ℑν1+ϖθ℘φ,ℑν,ϖθν,℘φ1+ϖθφ,℘φ1+ϖθφ,ν−Λminϖθφ,νϖθφ,℘φ,ϖθφ,℘φϖθν,ℑν,ϖθν,℘φϖθφ,ℑν≤Ωmaxϖθφ,ν,ϖθφ,φ1+ϖθν,ν1+ϖθφ,ν,ϖθν,φ1+ϖθφ,φ1+ϖθφ,ν−Λminϖθφ,νϖθφ,φ,ϖθφ,φϖθν,ν,ϖθν,φϖθφ,ν=Ωϖθφ,ν−Λ0=Ωϖθφ,ν.
This leads toΩκϖθφ,ν≤Ωϖθφ,ν, or 1−κϖθφ,ν≤0; since 1−κ≰0, then ϖθφ,ν=0. Thus, φ=ν, i.e., the uniqueness holds. Then, the proof is completed.
If we consider℘=ℑ in the above theorem, we get the important below result.Corollary 1.
Letℜ,ϖθbe a complete extendedb−metric space and℘:ℜ⟶CBℜbe a multivalued mapping such that, for allξ,ℓ∈ℜ, the following hypothesis is fulfilled:(18)Ωκ¥℘ξ,℘ℓ≤ΩΔξ,ℓ−ΛΠξ,ℓ,where(19)Δξ,ℓ=maxϖθξ,ℓ,ϖθξ,℘ξ1+ϖθℓ,℘ℓ1+ϖθ℘ξ,℘ℓ,ϖθℓ,℘ξ1+ϖθξ,℘ξ1+ϖθξ,ℓ,Πξ,ℓ=minϖθξ,ℓϖθξ,℘ξ,ϖθξ,℘ξϖθℓ,℘ℓ,ϖθℓ,℘ξϖθξ,℘ℓ,(i)
Ω:ℝ+⟶ℝ+is a nondecreasing and continuous function such thatΩτ=0ifτ=0(ii)
Λ:ℝ+⟶ℝ+is a continuous function such thatΛτ=0ifτ=0
Iflimn,m⟶∞ϖθξn,ξm≤1/κwithκ>1, then℘has a unique fixed point.Example 3.
Assume thatℜ=0,1and1<e<∞. Defineϖθ:ℜ×ℜ⟶ℝ+byϖθξ,ℓ=ξ−ℓefor allξ,ℓ∈ℜ, then the pairℜ,ϖθis an extendedb−metric space with the functionθ:ℜ×ℜ⟶1,∞defined byθξ,ℓ=2e−1+ξ+ℓ/2, see [20]. Define℘,ℑ:ℜ⟶CBℜandΩ,Λ:ℝ+⟶ℝ+by(20)℘ξ=0,ξeξeκμ+5,℘ℓ=0,ℓeκμ+5e,for allμ≥κ,and Ωω=ω and Λω=μ+4/μ+5ω, for any ω∈ℜ.
Now, we have(21)κ¥℘ξ,ℑℓ=κ¥0,ξκμ+5e,0,ℓκμ+5e,≤κξκμ+5e−ξκμ+5ee=κ1κμ+5ξ−ℓe=1μ+5ϖθξ,ℓ≤1μ+5Δξ,ℓ=1−μ+4μ+5Δξ,ℓ=Δξ,ℓ−μ+4μ+5Δξ,ℓ≤ΩΔξ,ℓ−ΛΠξ,ℓ.
Thus, all required conditions of Theorem1 are fulfilled. Hence, ℘ and ℑ have a unique cfp, which is 0.Theorem 2.
Suppose that℘,ℑ:ℜ⟶CBℜare multivalued mappings defined on a complete extendedb−metric spaceℜ,ϖθ. Let for allξ,ℓ∈ℜ,(22)κ¥℘ξ,ℑℓ≤σϖθξ,ℓΔξ,ℓ+ρϖθξ,ℓΠξ,ℓ,whereΔξ,ℓandΠξ,ℓare defined in the above theorem andσξ,ρξ∈Ξsuch thatσξ+ρξ<1. Iflimn,m⟶∞ϖθξn,ξm≤1/k,k>1, then℘and have a unique cfp.Proof.
For a fixed elementξ∈ℜ, define ξ°=ξ and let ξ1∈℘ξ°, by Lemma 3 (for ℏ=1, there exists ξ2∈ℑξ1 such that ϖθξ1,ξ2≤¥℘ξ°,ℑξ1. For ξ2∈ℑξ1, there exists ξ3∈℘ξ1 such that ϖθξ2,ξ3≤¥℘ξ1,ℑξ2.
With the same scenario, we obtain thatξ2n+1∈℘ξ2n, ξ2n+2∈ℑξ2n+1. If for some n, ξ2n+1=ξ2n+2, then ξn is a Cauchy sequence. Assume that, for each n, ξ2n+1≠ξ2n+2. Then, by (22), we get(23)κϖθξ2n+1,ξ2n+2≤κ¥℘ξ2n,ℑξ2n+1,≤σϖθξ2n,ξ2n+1Δξ2n,ξ2n+1+ρϖθξ2n,ξ2n+1Πξ2n,ξ2n+1,where Δξ2n,ξ2n+1=ϖθξ2n,ξ2n+1 and Πξ2n,ξ2n+1=0 according to the above theorem.
It follows from (23) that(24)κϖθξ2n+1,ξ2n+2≤σϖθξ2n,ξ2n+1ϖθξ2n,ξ2n+1.
Similarly, replacingξ with ξ2n+2 and ℓ with ξ2n+3, we can write(25)κϖθξ2n+2,ξ2n+3≤σϖθξ2n+1,ξ2n+2ϖθξ2n+1,ξ2n+2.
From (24) and (25), we have(26)ϖθξn,ξn+1≤1κϖθξn−1,ξn.
From Lemma1, we obtain that ξn is a Cauchy sequence. The completeness of ℜ leads to the conclusion that there is φ∈ℜ such that limsupn⟶∞ξn=φ. Let φ∉ℑφ, then, by Theorem 1, one can write(27)ϖθξ2n+1,ℑφ≤ϖθ℘ξ2n,ℑφ,≤σϖθξ2n,φΔξ2n,φ+ρϖθξ2n,φΠξ2n,φ.
Passing to the upper limit, we can get(28)limsupn⟶∞ϖθξ2n+1,ℑφ≤0.
By the factϖθξ2n+1,ℑφ≥0, one can see limsupn⟶∞ϖθξ2n+1,ℑφ≥0. Thus, we conclude that(29)limsupn⟶∞ϖθξ2n+1,ℑφ=0.
From Lemma2,(30)ϖθφ,ℑφ≤θφ,ℑφϖθφ,ξ2n+1+ϖθξ2n+1,ℑφ.
By taking the upper limit, we haveϖθφ,ℑφ⟶0. Thus, φ∈ℑφ. Similarly, we can show that φ∈℘φ. Hence, φ is a cfp of ℘ and ℑ. The uniqueness comes immediately in a similar way as in Theorem 1.
If we put℘=ℑ in Theorem 2, we have the following result.Corollary 2.
let℘:ℜ⟶CBℜbe a multivalued mapping defined on a complete extendedb−metric spaceℜ,ϖθ. Let for allξ,ℓ∈ℜ,(31)κ¥℘ξ,℘ℓ≤σϖθξ,ℓΔξ,ℓ+ρϖθξ,ℓΠξ,ℓ,whereΔξ,ℓandΠξ,ℓare defined in Corollary 1andσξ,ρξ∈Ξsuch thatσξ+ρξ<1. Iflimn,m⟶∞ϖθξn,ξm≤1/κ,κ>1, then℘has a unique fixed point.Example 4.
Suppose thatℜ=0,1. Defineϖθ:ℜ×ℜ⟶ℝ+byϖθξ,ℓ=ξ−ℓ2for allξ,ℓ∈ℜ, then the pairℜ,ϖθis an extendedb−metric space with the functionθ:ℜ×ℜ⟶1,∞, which takes the formθξ,ℓ=2+ξ+ℓ/2. Define℘,ℑ:ℜ⟶CBℜandσ,ρ:ℝ+⟶0,1as follows:(32)℘ξ=ξ5κ,ξ3κ,ℑℓ=ℓ5κ,ℓ3κ,for allξ,ℓ∈ℜ,κ>1.andσω=ρω=1/9<1, for allω∈ℜ. Consider(33)κ¥℘ξ,ℑℓ=κmaxsupξ∈℘ξϖθξ,ℑℓ,supℓ∈ℑℓϖθ℘ξ,ℓ,=κmaxsupξ∈℘ξϖθξ,ℓ5κ,ℓ3κ,supℓ∈ℑℓϖθξ5κ,ξ3κ,ℓ=κmaxξ5κ−ℓ5κ2,ξ3κ−ℓ3κ2≤κ9κ2maxξ−ℓ2,ξ−ℓ2≤19maxξ−ℓ2,ξ−ℓ2,since1κ<1≤19ϖθξ,ℓ≤σϖθξ,ℓΔξ,ℓ≤σϖθξ,ℓΔξ,ℓ+ρϖθξ,ℓΠξ,ℓ.
Hence, the conditions managed by Theorem2 are fulfilled, thereby concluding 0∈ℜ is the unique cfp of ℘ and ℑ.Example 5.
Suppose that all data of Example 4are fulfilled. Define the multivalued mappings℘,ℑ:ℜ⟶CBℜandσ,ρ:ℝ+⟶0,1by(34)℘ξ=0,ξ8κ,ℑℓ=ℓ8κ,∀ξ,ℓ∈ℜ,κ>1,and σω=ρω=1/8<1, for all ω∈ℜ. Consider(35)κ¥℘ξ,ℑℓ=κmaxsupξ∈℘ξϖθξ,ℑℓ,supℓ∈ℑℓϖθ℘ξ,ℓ,=κmaxsupξ∈℘ξϖθξ,ℓ8κ,supℓ∈ℑℓϖθ0,ξ8κ,ℓ=κmaxξ8κ−ℓ8κ2,ℓ8κ2=κ8κmaxξ−ℓ2,ℓ2≤18maxξ−ℓ2,ℓ−ℓ8κ2,≤18maxϖθξ,ℓ,ϖθℓ,ℑℓ≤σϖθξ,ℓΔξ,ℓ≤σϖθξ,ℓΔξ,ℓ+ρϖθξ,ℓΠξ,ℓ.
Hence, the conditions managed by Theorem2 are fulfilled, thereby concluding 0∈ℜ is the unique cfp of ℘ and ℑ.


## 3. Solving a Fractional Nonlinear Differential Equation

Recently, by the technique of nonlinear analysis such as fixed-point results, the Leray–Schauder theorem and stability, there are some papers dealing with the existence of solutions of nonlinear initial-value problems of fractional differential equations (see [21–23]). The main advantage of using fractional nonlinear differential equations is to describe the dynamics of complex nonlocal systems with memory. This part is devoted to obtain an existence solution of the subsequent nonlinear differential equation of fractional order:(36)D∁ϑξτ=Θτ,ξτ,τ∈0,1,ϑ∈1,2,with boundary conditions(37)ξ0=0,ξ1=∫0αξμdμ,α∈0,1.The Caputo fractional derivativeD∁ϑ with ordered ϑ is defined as follows:(38)D∁ϑΘτ=1Γυ−ϑ∫0ττ−μυ−ϑ−1Θυμdμ,where υ−1≤ϑ<υ, υ=ϑ+1, and Θ:0,1×ℝ⟶ℝ+ is a continuous function. Let ℜ=C0,1 be the set of all real-valued continuous functions on 0,1. Define ϖθ:ℜ×ℜ⟶ℝ and θ:ℜ×ℜ⟶1,∞ by(39)ϖθξ,ℓ=supτ∈0,1|ξτ−ℓτ|eandθξ,ℓ=2e−1+ξτ+ℓτ1+ξτ+ℓτ,for all ξ,ℓ∈ℜ, e>1. Then, the pair ℜ,ϖθ is a complete extended b−metric space [20]. Here, we need to be reminded that the Riemann–Liouville fractional integral of order ϑ is as follows:(40)IϑΘτ=1Γϑ∫0ττ−μϑ−1Θμdμ.Now, our main theorem of this section is.Theorem 3.
The problem (36) with boundary conditions (37) has a unique solution if the following assumptions are fulfilled:(i)
Θ:0,1×ℝ⟶ℝ+ is a continuous function satisfying(41)Θτ,ξ−Θτ,ℓ≤eϕξ−ℓ,for allτ∈0,1,ξ,ℓ∈ℜ,e>1.(ii)
There is a constantϕsuch thatϕG<1, where(42)G=2−α21+ϑ+2τϑ+1+αϑ+12−α2Γϑ+2e,α∈0,1.(iii)
limn,m⟶∞ϖθξn,ξm≤1/κ, whereκ=2e−1.Proof.
Define the mapping℘:ℜ⟶ℜ by(43)℘ξτ=2τ2−α2Γϑ∫0α∫0μμ−σϑ−1Θσ,ξσdσdμ−2τ2−α2Γϑ∫011−μϑ−1Θμ,ξμdμ+1Γϑ∫0ττ−μϑ−1Θμ,ξμdμ,
Forτ∈0,1. The function ξ∈ℜ is a unique solution of problem (36) if ξ∈℘ξ, i.e., ξ is a unique fixed point of the multivalued mapping ℘. To get that, we shall prove that ℘ satisfies the contractive condition of Corollary 1. Consider(44)2e−1ϖθ℘ξτ,℘ℓτ=2e−1supτ∈0,1℘ξτ−℘ℓτe=2e−1supτ∈0,1|2τ2−α2Γϑ∫0α∫0μμ−σϑ−1Θσ,ξσdσdμ−2τ2−α2Γϑ∫011−μϑ−1Θμ,ξμdμ+1Γϑ∫0ττ−μϑ−1Θμ,ξμdμ−2τ2−α2Γϑ∫0α∫0μμ−σϑ−1Θσ,ℓσdσdμ+2τ2−α2Γϑ∫011−μϑ−1Θμ,ℓμdμ−1Γϑ∫0ττ−μϑ−1Θμ,ℓμdμ|e≤2e−1supτ∈0,12τ2−α2Γϑ∫0α∫0μμ−σϑ−1Θσ,ξσ−Θσ,ℓσdσdμ+2τ2−α2Γϑ∫011−μϑ−1Θμ,ℓμ−Θμ,ξμdμ+1Γϑ∫0ττ−μϑ−1Θμ,ξμ−Θμ,ℓμdμe≤2e−1supτ∈0,12τ2−α2Γϑ∫0α∫0μμ−σϑ−1dσdμ+2τ2−α2Γϑ∫011−μϑ−1dμ+1Γϑ∫0ττ−μϑ−1dμeϕξ−ℓe≤2e−1ϕ2−α21+ϑ+2τϑ+1+αϑ+12−α2Γϑ+2e×supτ∈0,1ξτ−ℓτe=2e−1ϕGϖθξτ,ℓτ≤2e−1ϖθξτ,ℓτ.
This implies that(45)2e−1ϖθ℘ξτ,℘ℓτ≤2e−1ϖθξτ,ℓτ,≤2e−1Δξ,ℓ≤ΩΔξ,ℓ−ΛΠξ,ℓ,where κ=2e−1, Ωϰ=4e−1ϰ, and Λϰ=ϰ/16e−1. Then, by Corollary 1, there exists a unique fixed point of the mapping (43), which is the unique solution of problem (36) in ℜ.


## 4. Solving a System of Nonlinear Fractional Differential Equations

Differential equations of fractional order have been the focus of many studies due to their frequent appearance in various applications in fluid mechanics, viscoelasticity, physics, engineering, and biology. Recently, a large amount of literature studies developed concerning the application of fractional differential equations in nonlinear dynamics [24–29]. In this part, we shall find the existence of a solution to the following system of nonlinear fractional ordered differential equations:(46)D∁ϑℷτ+℘ℶτ=0,τ∈0,1,ϑ∈1,2,D∁ϑℶτ+ℑℷτ=0,τ∈0,1,ϑ∈1,2,ℷ0=ℶ0=Ω,ℷ1=ℶ1=Ω∗,where Ω and Ω∗ are constants, ℘,ℑ:0,1×ℝ+⟶ℝ+, and D∁ϑ refers to the Caputo fractional derivative. If we apply Green’s (continuous) function Υτ,μ on 0,1×0,1, (18) is equivalent to the following system:(47)ℷτ=ℵτ+∫01Υτ,μ℘ℷμdμ,τ∈0,1,ℶτ=ℵτ+∫01Υτ,μℑℷμdμ,τ∈0,1,where Υτ,μ is defined by(48)Υτ,μ=τ,μϑ−1−τ1−μϑ−1Γϑ,0≤μ≤τ≤1,−τ1−μϑ−1Γϑ,0≤τ≤μ≤1.Moreover,supτ∈0,1∫01Υτ,μdμ≤1. Now, put ϕ1τ,μ,ξμ=Υτ,μ℘ℶμ and ϕ2τ,μ,ℓμ=Υτ,μℑℶμ, then system (19) turns into(49)ξτ=ℵτ+∫01ϕ1τ,μ,ξμdμ,τ∈0,1,ℓτ=ℵτ+∫01ϕ2τ,μ,ℓμdμ,τ∈0,1.Letℜ=C0,1 be the set of all real-valued continuous functions on 0,1. Define ϖθ:ℜ×ℜ⟶ℝ and θ:ℜ×ℜ⟶1,∞ by(50)ϖθξ,ℓ=supτ∈0,1|ξτ−ℓτ|eandθξ,ℓ=2e−1+ξτ+ℓτ1+ξτ+ℓτ,for all ξ,ℓ∈ℜ, e>1. Then, the pair ℜ,ϖθ is a complete extended b−metric space.Now, we provide the following theorem to derive an existence result for the solution of problem (49).Theorem 4.
Consider system (49) under the following hypotheses:(i)
ϕ1,ϕ2:0,1×0,1×ℝ+⟶ℝ+andℵ:0,1⟶ℝ+is continuous.(ii)
There is a continuous functionΨ:0,1×0,1⟶ℝ+such that(51)ϕ1τ,μ,p−ϕ2τ,μ,p≤Ψτ,μp−q,for allτ,μ∈0,1,p,q∈ℜ.(iii)
supτ∈0,1∫01Ψτ,μedμ≤1.(iv)
limn,m⟶∞ϖθξn,ξm≤1/κ, whereκ=2e−1.
Then, problem (49) has a unique solution onℜ, which is considered as the unique solution to system (46).Proof.
Consider two multivalued mappings℘,ℑ:ℜ⟶ℜ having the form(52)℘ξτ=ℵτ+∫01ϕ1τ,μ,ξμdμ,τ∈0,1,ℑℓτ=ℵτ+∫01ϕ2τ,μ,ℓμdμ,τ∈0,1.
Hence, the unique cfp of the mappings℘ and ℑ defined in (52) is considered as the unique solution of problem (49), which leads to the solution of problem (47) and from it to the solution of system (46).
Consider(53)2e−1ϖθ℘ξτ,ℑℓτ=2e−1supτ∈0,1℘ξτ−ℑℓτe,=2e−1supτ∈0,1ℵτ+∫01ϕ1τ,μ,ξμdμ−ℵτ−∫01ϕ2τ,μ,ℓμdμe≤2e−1supτ∈0,1∫01ϕ1τ,μ,ξμ−ϕ2τ,μ,ℓμdμe≤2e−1supτ∈0,1∫01Ψτ,μξμ−ℓμdμe≤2e−1supτ∈0,1∫01Ψτ,μdμesupτ∈0,1ξτ−ℓτe≤2e−1ϖθξτ,ℓτ≤2e−1Δξ,ℓ≤ΩΔξ,ℓ−ΛΠξ,ℓ,where κ=2e−1, Ωϰ=4e−1ϰ, and Λϰ=ϰ/16e−1. Thus, by Theorem 1, there exists a unique cfp of the mappings (52), which is the unique solution of system (46) in ℜ.


## 5. Solving a Two-Dimensional Linear Fredholm Integral Equation

Two-dimensional Fredholm integral equations of the second kind arise from many problems in engineering and mechanics by a suitable transformation. For example, in the calculation of plasma physics, it is usually required to solve Fredholm integral equations (see [30]).In this section, we consider the two-dimensional Fredholm integral equation of the form:(54)ξτ,μ=¥τ,μ+∫01∫01Gτ,μ,h,gΦτ,μ,ξh,gdhdg;τ,μ∈0,12,where ¥ and G and Φ are given continuous functions defined on L2C0,1×0,1 and ξh,g is unknown on L2C0,1×0,1.Assume thatℜ=C0,1 is the set of all real-valued continuous functions on 0,1. Define ϖθ:ℜ×ℜ⟶ℝ and θ:ℜ×ℜ⟶1,∞ by(55)ϖθξ,ℓ=maxτ∈0,1|ξτ−ℓτ|2andθξ,ℓ=2+ξτ+ℓτ1+ξτ+ℓτ,for all ξ,ℓ∈ℜ. Then, the pair ℜ,ϖθ is a complete extended b−metric space.Now, we consider problem (46) under the following assumptions:(i)
G:0,14⟶ℝ+, and Φ:0,12×ℝ+⟶ℝ+ and ¥:0,12⟶ℝ+ are continuous functions.(ii)
For allξ,ℓ∈ℜ,(56)Φτ,μ,ξh,g−Φτ,μ,ℓh,g≤σ1/2ϖθξh,g,ℓh,g2ξh,g−ℓh,g.(iii)
There is a constantz∈0,1 such that ∫01∫01Gτ,μ,h,gdhdg≤z.(iv)
limn,m⟶∞ϖθξn,ξm≤1/2.Our main theorem in this section is presented as follows.Theorem 5.
Problem (54) has a unique solution onL2C0,1×0,1if the hypothesesi1−iv4hold.Proof.
As usual, define the multivalued operator℘:ℜ⟶ℜ by(57)℘ξτ,μ=¥τ,μ+∫01∫01Gτ,μ,h,gΦτ,μ,ξh,gdhdg;τ,μ∈0,1×0,1.
Then, we have(58)2℘ξτ,μ−℘ℓτ,μ2=2∫01∫01Gτ,μ,h,gΦτ,μ,ξh,gdhdg−∫01∫01Gτ,μ,h,gΦτ,μ,ℓh,gdhdg2,≤2∫01∫01Gτ,μ,h,gΦτ,μ,ξh,g−Φτ,μ,ℓh,gdhdg2≤2∫01∫01Gτ,μ,h,gdhdg2Φτ,μ,ξh,g−Φτ,μ,ℓh,g2≤2z2σ1/2ϖθξh,g,ℓh,g2ξh,g−ℓh,g2≤σϖθξh,g,ℓh,gξh,g−ℓh,g2.
Taking the maximum, we have(59)2max℘ξτ,μ−℘ℓτ,μ2≤σϖθξh,g,ℓh,gmaxξh,g−ℓh,g2,=σϖθξh,g,ℓh,gϖθξh,g,ℓh,g≤σϖθξ,ℓΔξ,ℓ+ρϖθξ,ℓΠξ,ℓ.
By takingκ=2, from Corollary 2, operator (57) has a unique fixed point in L2C0,1×0,1, which is considered the unique solution of problem (54).


## 6. Conclusion

In this manuscript, we gave some common fixed point theorems involving generalized multivalued contraction mappings in the class of extendedb-metric spaces. Applying our obtained results, we ensure the existence of (unique) solutions of a fractional differential equation, a system of fractional differential equations and a two-dimensional linear Fredholm integral equation. This affirms the utility of fixed point theory in the framework of fractional calculus.



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