# Existence and Stability for a Nonlinear Coupledp-Laplacian System of Fractional Differential Equations

Authors: Merfat Basha; Binxiang Dai; Wadhah Al-Sadi
Journal: Journal of Mathematics
 (2021)

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## Abstract

In this paper, we study the nonlinear coupled system of equations with fractional integral boundary conditions involving the Caputo fractional derivative of ordersθ1andθ2 and Riemann–Liouville derivative of orders ϱ1andϱ2 with the p-Laplacian operator, where n−1<θ1,θ2,ϱ1,ϱ2≤n, and n≥3. With the help of two Green’s functions Gϱ1w,ℑ,Gϱ2w,ℑ, the considered coupled system is changed to an integral system. Since topological degree theory is more applicable in nonlinear dynamical problems, the existence and uniqueness of the suggested coupled system are treated using this technique, and we find appropriate conditions for positive solutions to the proposed problem. Moreover, necessary conditions are highlighted for the Hyer–Ulam stability of the solution for the specified fractional differential problems. To confirm the theoretical analysis, we provide an example at the end.


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## Body

## 1. Introduction

The theoretical development of fractional calculus and its applications is more important to model nonlinear complex problems with the arbitrary fractional order. The subject of fractional differential equations (FDEs) has become an important area in real life because of their ability to model a lot of physical phenomena associated with rapid and concise changes with their significance in science and engineering through the past three decades, such as chemistry, physics, biology, engineering, visco-elasticity, electrotechnical, signal processing, electrochemistry, and controllability (see the details, [1–9], and the reference therein). In the near time, the nonlinear fractional partial differential equations are the most applied research area in which most authors and scientists are focused for their investigation. In this case, the Caputo derivative plays a great role to analyze the specific application of nonlinear PDEs. in [10], the authors have studied the cancer treatment model based on Caputo–Fabrizio fractional derivative. After integrating the model into the Caputo–Fabrizio fractional derivative, they have analyzed the existence of the solution as well. The Caputo–Fabrizio fractional derivative is implemented in [11] for the modeling and characterizing of the alcoholism. By applying the fixed-point theorem, they have studied the existence and uniqueness of the alcoholism model. The spread of the SIQR model is investigated by [12] using the Caputo derivative. They have justified the stability and uniqueness of the nonvirus equilibrium and virus equilibrium point.For this problem, different authors proposed different numerical solution techniques. The analysis with the nonlinear time-fractional HIV/AIDS transmission model is considered in [13], in which the numerical solution is found using the fractional variational iteration method with convergence analysis. The nonlinear garden equation is studied in [14] based on the Atangana–Baleau Caputo derivative. He has highlighted the fixed-point theorem for proving the existence and uniqueness of the garden equation.One of the main difficulties for the solution of the nonlinear fractional PDEs is to analyze the existence theory of solutions. Sufficient conditions for the existence and uniqueness of solutions (EUS) have been obtained by using different nonlinear analysis techniques and fixed-point theorems (for more details, read [15–18]). Also, the boundary value problems with various boundary conditions for many ordinary differential equations are studied [19–23]. However, the theory of boundary value problems for nonlinear FDEs is still not discussed more, and many problems of this theory require to be explored. On the contrary, the investigation of coupled systems of the differential equations is also significant because systems of this kind appear in various applied nature problems (refer [24–28]).The topological degree theory is a useful tool in nonlinear analysis with numerous applications to operatorial equations, optimization theory, fractal theory, and other topics. We will see the following consideration of topological degree theory with boundary conditions based on the Caputo fractional derivative by different authors. Isaia [29] applied the topological degree theory to establish sufficient conditions for the existence of a solution for the following nonlinear integral equations:(1)πw=Zw,πw+∫abQw,ℑ,πℑdℑ,w∈a,b,where Z:a,b×IR⟶IR and Q:a,b×a,b×IR⟶IR are continuous functions. In their study [30], Wang et al. used the topological degree method to obtain some existence conditions of the solution for the following nonlocal Cauchy problem:(2)cDϱπw=Zw,πw,0≤w≤W,π0+hπ=π0,where Dϱc denotes the Caputo fractional derivative with order ϱ∈0,1 and Z:C0,w,ℝ⟶ℝ and π0∈ℝ are continuous. The nonlocal term h:C0,w,ℝ⟶ℝ is a given function. Proceeding on the same fashion, Shah and Khan [31] proved the EUS for a coupled system under the fractional derivatives by using the technique of degree theory given as follows:(3)Dϱ1cπ1w=Z1w,π1w,π2w,w∈0,1,Dϱ2cπ2w=Z2w,π1w,π2w,w∈0,1,α1π10−β1π1θ−δ1π11=ϕ1π1,α2π20−β2π2ϑ−δ2π21=ϕ2π2,where ϱ1,ϱ2,θ,ϑ∈0,1, Z1,Z2∈0,1×R2⟶R, and ϕ1,ϕ2:C0,1,ℝ⟶ℝ are continuous. Khan et al. [32] used the above-mentioned method to study the following coupled system in the sense of Caputo derivatives with p-Laplacian:(4)D0+θ1ϕpD0+ϱ1π1w+Z1w,π2w=0,D0+θ2ϕpD0+ϱ2π2w+Z2w,π1w=0,D0+ϱ1π10=ϕpD0+ϱ1π1w′|w=0=D0+δ1π1w|w=η1=0,π11=Γ2−δ1η11−δ1Jϱ1−δ1ϕqJ0+θ1Z1w,π2w|w=η1,D0+ϱ2π20=ϕpD0+ϱ2π1w′|w=0=D0+δ2π2w|w=η2=0,π21=Γ2−δ2η21−δ2Jϱ2−δ2ϕqJ0+θ2Z1w,π1w|w=η2,where ϱi,θi∈1,2andδi,ηi∈0,1, for i=1and2. The study of positive solutions to boundary value problems for fractional-order differential equations using the topological degree theory technique is rarely available in the literature, so this research field needs further elaboration. Most papers that dealt the topological degree theory with fractional orders belong to 0,1 or 1,2. For the uniqueness and existence analysis of nonlinear fractional differential equations, the case only Caputo fractional derivative is used frequently.Thus, our motivation to this study is developing a sufficient condition for the coupled nonlinear fractional derivative that is based on both Caputo and Riemann–Liouville derivatives. The fractional order in our study is expanded ton−1,n, and we have used a technique of topological degree theory for the analysis of existence and uniqueness of our coupled system defined below. Besides, we have investigated Hyers–Ulam stability to the nonlinear coupled system of fractional-ordered ordinary differential equations with boundary conditions designed by the following:(5)Dθ1cϕpDϱ1Rπ1w=Z1w,π2w,w∈0,1,Dθ2cϕpDϱ2Rπ2w=Z2w,π1w,w∈0,1,ϕpDϱ1Rπ1wi|w=0=0,i=0,1,2,…,n−1,Ik−ϱ1π1w|w=0=0,k=1,2,4,…,n,Dλπ1w|w=1=1Γλ∫0WW−ℑλ−1φπ1ℑdℑ,ϕpDϱ2Rπ2wj|w=0=0,j=0,1,2,…,n−1,Ih−ϱ2π2w|w=0=0,h=1,2,4,…,n,Dσπ2w|w=1=1Γσ∫0WW−ℑσ−1ϖπ2ℑdℑ,where ϱ1,ϱ2,θ1,θ2∈n−1,n,n≥3,1<λ,σ≤2,Dθ1candDθ2c denote the Caputo fractional derivatives, Dϱ1RandDϱ2R are the Riemann–Liouville fractional derivatives, and Z1,Z2:0,1×IR⟶IR are nonlinear functions, and the boundary functions φ,ϖ∈L0,1. ϕp represents the p-Laplacian operator such that ϕpν=ννp−2,andϕq=ϕp−1 denotes the inverse of p-Laplacian, where 1/p+1/q=1. Since it is difficult to find the exact solution of the nonlinear differential equations, stability and uniqueness have played a great role to get the approximate solution for the given nonlinear problems. Therefore, scientists and researchers have given attention to study the various forms of stability to the nonlinear problems in the sense of Ulam and their multiple types in the last few decades. We observe that the concept of Hyers–Ulam stability is fundamental in realistic problems, such as numerical analysis, biology, and economics (see [33–38]).The remaining part of this manuscript is structured as follows. In Section2, we have introduced some basic definitions and lemmas that we need to prove our main results. By using the topological degree theory, the results of existence and uniqueness for the solutions are obtained in Section 3. In Section 4, we investigate the stability of Hyers–Ulam to our proposed coupled system. The theoretical results are demonstrated by providing an example in Section 5, and finally, we have drawn the conclusion in Section 6.


## 2. Preliminaries

In this section, we introduce some basic notions, definitions, and important lemmas which are used in this article.LetΠ=C0,1,IR be a Banach space for all continuous functions π:0,1⟶IR with the norm π=supπw:0≤w≤1. Further, Ω=Π×Π is also a Banach space under the norms π1,π2=π1+π2 and π1,π2=maxπ1,π2. The family of each bounded set of PΩ symbolized by B.Definition 1.
ForZw:0,+∞⟶ℝ, the Caputo fractional derivative of noninteger order ϱ>0 is known by(6)DϱcZw=1Γn−ϱ∫0ww−ℑn−ϱ−1Znℑdℑ,where n−1<ϱ<n, the integral in the right side is pointwise defined on 0,∞, and Zw is a continuous function.Definition 2.
ForZw:0,+∞⟶ℝ, the Riemann–Liouville fractional derivative of noninteger order ϱ>0 is known by(7)DϱRZw=1Γn−ϱddwn∫0ww−ℑn−ϱ−1Zℑdℑ,where n−1<ϱ<n, the integral in the right side is pointwise defined on 0,∞, and Zw is a continuous function.Definition 3.
ForZw:0,+∞⟶ℝ, the Riemann–Liouville fractional integral of order ϱ>0 is defined by(8)IϱZw=1Γϱ∫0ww−ℑϱ−1Zℑdℑ,where the integral on the right side is pointwise defined on 0,+∞ and Γϱ indicates the Gamma function defined as(9)Γϱ=∫0∞e−ℑℑϱ−1dℑ.Lemma 1.
(see [39]). Letϱ>0andZ∈C0,1∩L10,1. Then, the general solution of the fractional differential equationDϱZw=πwis given by(10)Zw=πw+c0+c1w+c2w2+⋯+cn−1wn−1,forci∈ℝ,i=0,1,2,…,n−1.Lemma 2.
(see [2, 8]). Letθ∈n−1,n,Z∈Cn−1, andDθcis the fractional derivative for Caputo, then(11)IθDθcZw=Zw+a1+a2w+a3w2+a4w3+⋯+anwn−1,forai∈ℝandi=1,2,3,4,…,n.Lemma 3.
(see [2, 8]). Letϱ∈n−1,n,Z∈Cn−1, andDϱRis the fractional derivative for Riemman–Liouville, then(12)IϱDϱRZw=Zw+b1wϱ−1+b2wϱ−2+b3wϱ−3+a4wϱ−4+⋯+bnwϱ−n,forbi∈ℝandi=1,2,3,4,…,n.Lemma 4.
(see [22]). Forϱ,ɛ>0, the following relations are satisfying:(13)Dϱξɛ=Γɛ+1Γ1+ɛ−ϱξɛ−ϱ,Iϱξɛ=Γɛ+1Γ1+ɛ+ϱξɛ+ϱ.Definition 4.
The Kuratowski measure of noncompactness is the mapϑ:B⟶0,∞ known as ϑℬ=infd>0:ℬ which admits a finite cover by sets of diameter ≤d, where ℬ∈B.Proposition 1.
(see [40]). The Kuratowski measure ofϑsatisfies the following properties:(a)
The Kuratowski measureϑℬ=0; for a relative compactℬ(b)
ϑis a seminorm, i.e.,ϑμℬ=μϑℬ,μ∈IR, andϑℬ1+ℬ2≤ϑℬ1+ϑℬ2(c)
ℬ1⊂ℬ2impliesϑℬ1≤ϑℬ2;ϑℬ1∪ℬ2=supϑℬ1,ϑℬ2(d)
ϑconvℬ=ϑℬ(e)
ϑℬ¯=ϑℬDefinition 5.
Suppose that the functionℱ:Ψ⟶Π is a continuous and bounded map, where Ψ⊂Π. ℱ is called ϑ−Lipschitz with η≥0, and if ϑℱℬ≤ηϑℬ,∀ℬ⊂Ψ is bounded.
Moreover, ifη<0, then ℱ will be a strict ϑ−contraction.Definition 6.
The functionℱ is called ϑ−condensing, and if ϑℱℬ<ϑℬ,∀ℬ⊂Ψ is bounded with ϑℬ>0.
In other words,ϑℱℬ≥ϑℬ implies ϑℬ=0.
We indicate that the class of eachϑ−condensing mappings ℱ:Ψ⟶Π by CϑΨ and the class of each strict ϑ−contractions ℱ:Ψ⟶Π by ζCϑΨ.We remark thatCϑΨ⊃ζCϑΨ, and every ℱ∈CϑΨ is ϑ−Lipschitz with η=1. As well, we recall that ℱ:Ψ⟶Π is Lipschitz if ∃η>0 such that ℱπ−ℱπ¯≤ηπ−π¯,and∀π,π¯∈Ψ. Also, ℱ is a strict contraction under the condition η<1.Proposition 2.
(see [31]). Letℱ,G:Ψ⟶Πbeϑ−Lipschitz operators with constantsη1andη2, respectively, thenℱ+G:Ψ⟶Πisϑ−Lipschitz with constantsη1+η2.Proposition 3.
(see [41]). The operatorℱ:Ψ⟶Πis compact if and only ifℱisϑ−Lipschitz withη=0.Proposition 4.
(see [31]). The operatorℱ:Ψ⟶Πis Lipschitz with constantηif and only ifℱisϑ−Lipschitz with constantη.Lemma 5.
(see [39]). Letϕpbe a nonlinear p-Laplacian operator.(1)
If1<p≤2,j1j2>0, and j1,j2≥ρ>0, then(14)ϕpj1−ϕpj2≤p−1ρp−2j1−j2.(2)
Ifp>2 and j1,j2≤ρ∗, then(15)ϕpj1−ϕpj2≤p−1ρ∗p−2j1−j2.Theorem 1 (see [29]).
Letℱ:Ω⟶Ωbe aϑ−contraction, andΞ=ω∈Ω:∃,0≤ρ≤1such thatω=ρℱω. IfΞ⊂Ωis a bounded set, there existsr>0such thatΞ⊂ℬr0, then the degreedegI−ρℱ,ℬr0,0=1,∀ρ∈0,1.Thus,ℱ has at least one fixed point, and the set of the fixed points of ℱ lies in ℬr0.The above theorem that we mentioned plays a substantial role in obtaining our main results.


## 3. Main Results

In the current section, we establish some appropriate conditions for proposed coupled system (5).Theorem 2.
LetZ:0,1⟶IRbe aϱ1times’ integrable function. Then, forϱ1∈3,nand positive integern≥4, the solution of the boundary value problem is as follows:(16)Dθ1cϕpDϱ1Rπ1w=Z1w,π2w,w∈0,1,ϕpDϱ1Rπ1wi|w=0=0,i=0,1,2,…,n−1,Ik−ϱ1π1w|w=0=0,k=1,2,4,…,n,Dλπ1w|w=1=1Γλ∫0WW−ℑλ−1φπ1ℑdℑ,is given by(17)π1w=Γϱ1−λ−2wϱ1−3Γϱ1−2Γλ∫0WW−ℑλ−1φπ1ℑdℑ+∫01Gϱ1w,ℑϕq1Γθ1∫0ℑℑ−τθ1−1Z1τ,π2τdτdℑ,whereGϱ1w,ℑis the Green’s function provided by(18)Gϱ1w,ℑ=w−ℑϱ1−1Γϱ1−Γϱ1−λ−2wϱ1−31−ℑϱ1−λ−1Γϱ1−2Γϱ1−λ,0≤ℑ≤w≤1,−Γϱ1−λ−2wϱ1−31−ℑϱ1−λ−1Γϱ1−2Γϱ1−λ,0≤w≤ℑ≤1.Proof.
Applying the integral operatorIθ1 and using Lemma 2 on (16), we get(19)ϕpDϱ1Rπ1w=Iθ1Z1w,π2w+a1+a2w+a3w2+a4w3+⋯+anwn−1.
Using the conditionϕpDϱ1Rπ1wi|w=0=0, for i=0,1,2,…,n−1, in (19), we obtain a1=a2=a3=⋯=an=0, and then, we get(20)ϕpDϱ1Rπ1w=Iθ1Z1w,π2w.
From (20), we have(21)Dϱ1Rπ1w=ϕqIθ1Z1w,π2w.
Applying the operatorIϱ1 and using Lemma 3 in (21), we get(22)π1w=Iϱ1ϕqIθ1Z1w,π2w+b1wϱ1−1+b2wϱ1−2+b3wϱ1−3+b4wϱ−4+⋯+bnwϱ1−n.
Using the conditionIk−ϱ1πw|w=0=0,k=1,2,4,…,n, we get b1=b2=b4=…=bn=0, and then, we obtain(23)π1w=Iϱ1ϕqIθ1Z1w,π2w+b3wϱ1−3.
Using the conditionDλπ1w|w=1=IWλ0φπ1=1/Γλ∫0WW−ℑλ−1φπ1ℑdℑ and Lemma 4 in (22), we get(24)b3=Γϱ1−λ−2Γϱ1−2IWλ0φπ1−Iϱ1−λϕqIθ1Z11,π21.
Putting the value ofb3 in (23), we get(25)π1w=1Γϱ1∫0ww−ℑϱ1−1ϕq1Γθ1∫0ℑℑ−τθ1−1Z1τ,π2τdτdℑ+Γϱ1−λ−2wϱ1−3Γϱ1−2Γλ∫0WW−ℑλ−1φπ1ℑdℑ−Γϱ1−λ−2wϱ1−3Γϱ1−2Γϱ1−λ∫011−ℑϱ1−λ−1ϕq1Γθ1∫0ℑℑ−τθ1−1Z1τ,π2τdτdℑ,which can be written after rearranging as follows:(26)π1w=Γϱ1−λ−2wϱ1−3Γϱ1−2Γλ∫0WW−ℑλ−1φπ1ℑdℑ+∫01Gϱ1w,ℑϕq1Γθ1∫0ℑℑ−τθ1−1Z1τ,π2τdτdℑ,where Gϱ1w,ℑ is the Green’s function defined in (18).In view of Theorem2, the identical coupled system of Hammerstein-kind integral equations to fractional differential equation coupled system (5) is given as follows:(27)π1w=Γϱ1−λ−2wϱ1−3Γϱ1−2Γλ∫0WW−ℑλ−1φπ1ℑdℑ+∫01Gϱ1w,ℑϕq1Γθ1∫0ℑℑ−τθ1−1Z1τ,π2τdτdℑ,π2w=Γϱ2−σ−2wϱ2−3Γϱ2−2Γσ∫0WW−ℑσ−1ϖπ2ℑdℑ+∫01Gϱ2w,ℑϕq1Γθ2∫0ℑℑ−τθ2−1Z2τ,π1τdτdℑ,where Gϱ2w,ℑ is the Green’s function provided by(28)Gϱ2w,ℑ=w−ℑϱ2−1Γϱ2−Γϱ2−σ−2wϱ2−31−ℑϱ2−σ−1Γϱ2−2Γϱ2−σ,0≤ℑ≤w≤1,−Γϱ2−σ−2wϱ2−31−ℑϱ2−σ−1Γϱ2−2Γϱ2−σ,0≤w≤ℑ≤1.FromGϱ1w,ℑ and Gϱ2w,ℑ obviously,(29)maxw∈0,1Gϱ1w,ℑ=Γϱ1−λ−21−ℑϱ1−λ−1Γϱ1−2Γϱ1−λ,maxw∈0,1Gϱ2w,ℑ=Γϱ2−σ−21−ℑϱ2−σ−1Γϱ2−2Γϱ2−σ,ℑ∈0,1.We define the operatorsℱ1:Π1⟶Π1andℱ2:Π2⟶Π2 as(30)ℱ1π1w=Γϱ1−λ−2wϱ1−3Γϱ1−2Γλ∫0WW−ℑλ−1φπ1ℑdℑ,ℱ2π2w=Γϱ2−σ−2wϱ2−3Γϱ2−2Γσ∫0WW−ℑσ−1ϖπ2ℑdℑ,G1,G2:Ω⟶Ω,as(31)G1π2w=∫01Gϱ1w,ℑϕq1Γθ1∫0ℑℑ−τθ1−1Z1τ,π2τdτdℑ,G2π1w=∫01Gϱ2w,ℑϕq1Γθ2∫0ℑℑ−τθ2−1Z2τ,π1τdτdℑ.Therefore, we haveℱπ1,π2=ℱ1,ℱ2π1,π2,Gπ1,π2=G1,G2π1,π2, and Tπ1,π2=ℱπ1,π2+Gπ1,π2. Thus, the equivalent operator equation for the toppled system of Hammerstein-kind integral equations (27) is provided by(32)π1,π2=Tπ1,π2=ℱπ1,π2+Gπ1,π2.Consequently, the solutions of system (27) are the fixed points of operator equation (32).Now, we need to list the following assumptions to complete our results.H1 For ℏ,π1,ℓ,π2∈IR, the nonlocal functions φandϖ satisfy φℏ−φπ1≤Kφℏ−π1andϖℓ−ϖπ2≤Kϖℓ−π2 such that Kφ,Kϖ∈0,1H2 With the positive constants given Cφ,Cϖ,Nφ,Nϖ, and q1∈0,1, the nonlocal functions φandϖ for π1,π2∈IR satisfy the following growth conditions φπ1≤Cφπ1q1+Nφandϖπ2≤Cϖπ2q1+NϖH3 With the presence of constants g,h,NZ1,NZ2, and q2∈0,1, the nonlinear functions Z1andZ2 for π1,π2∈IR satisfy the following growth conditions:(33)Z1w,π2≤ϕpgπ2q2+NZ1,Z2w,π1≤ϕphπ1q2+NZ2.H4 For ℏ,π1,ℓ,π2∈IR, there exists positive constants LZ1andLZ2 such that(34)Z1w,ℓ−Z1w,π2≤LZ1ℓ−π2,Z2w,ℏ−Z2w,π1≤LZ2ℏ−π1.Theorem 3.
Assume thatH1andH2hold true. Then, the operatorℱis Lipschitz and satisfies the following growth condition:(35)ℱπ1,π2≤Cℱπ1,π2q1+Nℱ,∀π1,π2∈Ω.Proof.
By assumptionH1, we get(36)ℱ1π1w−ℱ1π1¯w=Γϱ1−λ−2Γϱ1−2Γλ∫0WW−ℑλ−1φπ1−φπ1¯dℑ≤Γϱ1−λ−2Γϱ1−2Γλ∫0WW−ℑλ−1φπ1−φπ1¯dℑ,which yields(37)ℱ1π1−ℱ1π1¯≤K¯φπ1−π1¯,where(38)K¯φ=KφΓϱ1−λ−2WλΓϱ1−2Γλ+1∈0,1.
To get the growth condition, consider(39)ℱ1π1w=Γϱ1−λ−2Γϱ1−2Γλ∫0WW−ℑλ−1φπ1ℑdℑ≤Γϱ1−λ−2Γϱ1−2Γλ∫0WW−ℑλ−1φπ1ℑdℑ,which means that(40)ℱ1π1c≤Γϱ1−λ−2WλΓϱ1−2Γλ+1Cφπ1cq1+Nφ.
In a similar manner, we have(41)ℱ2π2w=Γϱ2−σ−2Γϱ2−2Γσ∫0WW−ℑσ−1ϖπ2ℑdℑ≤Γϱ2−σ−2Γϱ2−2Γσ∫0WW−ℑσ−1ϖπ2ℑdℑ,which implies that(42)ℱ2π2c≤Γϱ2−σ−2WσΓϱ2−2Γσ+1Cϖπ2cq1+Nϖ.
Now,(43)ℱπ1,π2c≤Γϱ1−λ−2WλΓϱ1−2Γλ+1Cφπ1cq1+Nφ+Γϱ2−σ−2WσΓϱ2−2Γσ+1Cϖπ2cq1+Nϖ≤Γϱ1−λ−2WλΓϱ1−2Γλ+1Cφπ1cq1+Γϱ2−σ−2WσΓϱ2−2Γσ+1Cϖπ2cq1+Γϱ1−λ−2WλNφΓϱ1−2Γλ+1+Γϱ2−σ−2WσNϖΓϱ2−2Γσ+1.
Thus,(44)ℱπ1,π2c≤Cℱπ1q1+π2q1+Nℱ=Cℱπ1,π2q1+Nℱ,where(45)Cℱ=maxΓϱ1−λ−2WλΓϱ1−2Γλ+1Cφ,Γϱ2−σ−2WσΓϱ2−2Γσ+1Cϖ,Nℱ=Γϱ1−λ−2WλNφΓϱ1−2Γλ+1+Γϱ2−σ−2WσNϖΓϱ2−2Γσ+1.Theorem 4.
Suppose thatH3is satisfied. Then, the operatorGis continuous and satisfies the following growth condition:(46)Gπ1,π2≤Λπ1,π2q2+Θ,for allπ1,π2∈Ω,
whereΛ=γg+handΘ=γNZ1+NZ2such that(47)γ=max1Γϱ1+1+Γϱ1−λ−2Γϱ1−2Γϱ1−λ+11Γθ1+1q−1,1Γϱ2+1+Γϱ2−σ−2Γϱ2−2Γϱ2−σ+11Γθ2+1q−1.Proof.
LetBr=π1,π2∈Ω:π1,π2≤r be a bounded set with a sequence π1n,π2n converging to π1,π2 in Br. In order to show that G is continuous, we have to prove that Gπ1n,π2n−Gπ1,π2⟶0 as n⟶∞. Let us choose the following:(48)G1π2n−G1π2w=|∫01Gϱ1w,ℑϕq1Γθ1∫0ℑℑ−τθ1−1Z1τ,π2nτdτdℑ−∫01Gϱ1w,ℑϕq1Γθ1∫0ℑℑ−τθ1−1Z1τ,π2τdτdℑ|≤∫01Gϱ1w,ℑϕq1Γθ1∫0ℑℑ−τθ1−1Z1τ,π2nτdτ−ϕq1Γθ1∫0ℑℑ−τθ1−1Z1τ,π2τdτ|dℑ≤q−1ρ1q−2∫01Gϱ1w,ℑ1Γθ1∫0ℑℑ−τθ1−1Z1τ,π2nτdτ−1Γθ1∫0ℑℑ−τθ1−1Z1τ,π2τdτ|dℑ≤q−1ρ1q−2∫01Gϱ1w,ℑ1Γθ1∫0ℑℑ−τθ1−1Z1τ,π2nτ−Z1τ,π2τdτdℑ.
The continuity ofZ1 implies that Z1τ,π2nτ−Z1τ,π2τ⟶0 as n⟶∞, and then,(49)G1π2nw−G1π2w⟶0,asn⟶∞,and also, we can in the same way prove that(50)G2π1nw−G2π1w≤q−1ρ2q−2∫01Gϱ2w,ℑ1Γθ2∫0ℑℑ−τθ2−1|Z2τ,π1nτ−Z2τ,π1τ|dτdℑ.
The continuity ofZ2 implies that Z2τ,π1nτ−Z2τ,π1τ⟶0 as n⟶∞, and then,(51)G2π1nw−G2π1w⟶0,asn⟶∞.
Thus, from (49) and (51), we have(52)Gπ1n,π2nw−Gπ1,π2w≤q−1ρ1q−2∫01Gϱ1w,ℑ1Γθ1∫0ℑℑ−τθ1−1|Z1τ,π2nτ−Z1τ,π2τ|dτdℑ+q−1ρ2q−2∫01Gϱ2w,ℑ1Γθ2∫0ℑℑ−τθ2−1|Z2τ,π1nτ−Z2τ,π1τ|dτdℑ.
From the continuity ofZ1andZ2 and (52), we have(53)Gπ1n,π2nw−Gπ1,π2w⟶0asn⟶∞.
To calculate (46) for G, using assumption H3 and (29), we obtain(54)G1π2w=∫01Gϱ1w,ℑϕq1Γθ1∫0ℑℑ−τθ1−1Z1τ,π2τdτdℑ≤∫01Gϱ1w,ℑϕq1Γθ1∫0ℑℑ−τθ1−1Z1τ,π2τdτdℑ≤∫01Gϱ1w,ℑϕq1Γθ1∫0ℑℑ−τθ1−1ϕpgπ2q2+NZ1dℑ≤1Γϱ1+1+Γϱ1−λ−2Γϱ1−2Γϱ1−λ+11Γθ1+1q−1gπ2q2+NZ1.
From assumptionH3 and (29), we get(55)G2π1w=∫01Gϱ2w,ℑϕq1Γθ2∫0ℑℑ−τθ2−1Z2τ,π1τdτdℑ≤∫01Gϱ2w,ℑϕq1Γθ2∫0ℑℑ−τθ2−1Z2τ,π1τdτdℑ≤∫01Gϱ2w,ℑϕq1Γθ2∫0ℑℑ−τθ2−1ϕphπ1q2+NZ2dτdℑ.
Then,(56)G2π1w≤∫01Gϱ2w,ℑϕq1Γθ2∫0ℑℑ−τθ2−1ϕphπ1q2+NZ2dτdℑ≤1Γϱ2+1+Γϱ2−σ−2Γϱ2−2Γϱ2−σ+11Γθ2+1q−1hπ1q2+NZ2.
By the help of (54) and (55), we have obtained(57)Gπ1,π2=G1π2+G2π1≤γgπ2q2+NZ1+γhπ1q2+NZ2≤γg+hπ2q2+π1q2+γNZ1+NZ2=Λπ1,π2q2+Θ.Theorem 5.
The operatorG:Ω⟶Ωisϑ−Lipschitz with constant zero and is compact.Proof.
Take a bounded setE and a sequence π1n,π2n such that E⊂Br⊆Ω. Then, using (46), we have(58)Gπ1n,π2n≤Λπ1,π2+Θ,for allπ1,π2∈Ω,which means that G is bounded. Now, for all π1n,π2n∈E, we have, for 0≤w1<w2≤1,(59)G1π2nw1−G1π2nw2=|∫01Gϱ1w1,ℑϕq1Γθ1∫0ℑℑ−τθ1−1Z1τ,π2τdτdℑ−∫01Gϱ1w2,ℑϕq1Γθ1∫0ℑℑ−τθ1−1Z1τ,π2τdτdℑ|≤∫01Gϱ1w1,ℑ−Gϱ1w2,ℑϕq1Γθ1∫0ℑℑ−τθ1−1ϕpgπ1q2+NZ1dτdℑ.
Hence, it follows that(60)G1π2nw1−G1π2nw2≤w1ϱ1−w2ϱ1Γϱ1+1+w1ϱ1−3−w2ϱ1−3Γϱ1−2Γϱ1−λ+11Γθ1+1q−1×gπ1q2+NZ1.
Similarly, we have(61)G2π1nw1−G2π1nw2≤w1ϱ2−w2ϱ2Γϱ2+1+w1ϱ2−3−w2ϱ2−3Γϱ2−2Γϱ2−σ+11Γθ2+1q−1×hπ2q2+NZ2.
Both the right sides of (55) and (61) tend to be zero as w1⟶w2. Therefore, the operators G1andG2 are equicontinuous, and hence, G=G1,G2 is equicontinuous on E. Thus, GE is compact by the theorem of Arzela–Ascoli. Moreover, through Proposition 3, G is ϑ−Lipschitz with constant zero.Theorem 6.
Suppose thatH1–H3are satisfied withΛ+Cℱ≤1. Then, the toppled system (5) has at least one solutionπ1,π2∈Ω. Furthermore, the set of solutions of (5) is bounded inΩ.Proof.
With the help of Theorem3, ℱ is ϑ−Lipschitz with constant 0≤Cℱ<1, and G is ϑ−Lipschitz with constant zero by Theorem 5. Thus, T is strictly ϑ−condensing with constant η by Proposition 2. Now, let us set that we have to show that B is bounded in Ω. In fact,(62)π1,π2=κTπ1,π2≤Tπ1,π2≤ℱπ1,π2+Gπ1,π2≤Cℱπ1,π2q1+Nℱ+Λπ1,π2q2+Θ=Cℱ+Λπ1,π2q3+Nℱ+Θ,whereq3=maxq1,q2.
Obviously,π1,π2 is bounded. If not correct, take π1,π2=S such that S⟶∞ and q3∈0,1. Consequently,(63)1≤Cℱ+Λπ1,π2q3π1,π2+Nℱ+Θπ1,π2,1≤Cℱ+ΛSq3S+Nℱ+ΘS,1≤Cℱ+ΛS1−q3+Nℱ+ΘS⟶0asS⟶∞,which is a contradiction. So, B is bounded. Thus, by Theorem 7, we conclude that T has at least one fixed point and that is a solution of system (5), and the set of solutions is bounded in Ω.Theorem 7.
SupposethatH1–H4hold andχ<1, where(64)χ=KφΓϱ1−λ−2WλΓϱ1−2Γλ+1+KϖΓϱ2−σ−2WσΓϱ2−2Γσ+1+q−1ρ1q−2LZ11Γϱ1+1+Γϱ1−λ−2Γϱ1−2Γϱ1−λ+11Γθ1+1+q−1ρ2q−2LZ21Γϱ2+1+Γϱ2−σ−2Γϱ2−2Γϱ2−σ+11Γθ2+1.Then, toppled system (5) has a unique solution.Proof.
Letπ1,π2andπ1¯,π2¯∈Ω are two solutions, then(65)Tπ1,π2−Tπ1¯,π2¯=ℱπ1,π2+Gπ1,π2−ℱπ1¯,π2¯+Gπ1¯,π2¯≤ℱπ1,π2−ℱπ1¯,π2¯+Gπ1,π2−Gπ1¯,π2¯,and after simplification, we obtain(66)Tπ1,π2−Tπ1¯,π2¯≤KφΓϱ1−λ−2WλΓϱ1−2Γλ+1+KϖΓϱ2−σ−2WσΓϱ2−2Γσ+1+q−1ρ1q−2LZ11Γϱ1+1+Γϱ1−λ−2Γϱ1−2Γϱ1−λ+11Γθ1+1+q−1ρ2q−2LZ21Γϱ2+1+Γϱ2−σ−2Γϱ2−2Γϱ2−σ+11Γθ2+1×π1,π2−π1¯,π2¯,which implies that(67)Tπ1,π2−Tπ1¯,π2¯≤χπ1,π2−π1¯,π2¯.
Thus, the operatorT is a contraction as χ<1, and by the Banach fixed-point theorem, T has a unique fixed point, and then, considered toppled system (5) has a unique solution.


## 4. Hyers–Ulam Stability

In this section, we investigate the stability of Hyers–Ulam for the suggested toppled system.Definition 7.
We say that the toppled system of Hammerstein-kind integral equations (27) is Hyers–Ulam stable if there exists positive constants a,b,c,andd such that, for each ξ1,ξ2>0 and any solution π1∗,π2∗ of the system(68)|π1∗w−Γϱ1−λ−2wϱ1−3Γϱ1−2Γλ∫0WW−ℑλ−1φπ1∗ℑdℑ+∫01Gϱ1w,ℑϕq1Γθ1∫0ℑℑ−τθ1−1Z1τ,π2∗τdτdℑ|≤ξ1,|π2∗w−Γϱ2−σ−2wϱ2−3Γϱ2−2Γσ∫0WW−ℑσ−1ϖπ2∗ℑdℑ+∫01Gϱ2w,ℑϕq1Γθ2∫0ℑℑ−τθ2−1Z2τ,π1∗τdτdℑ|≤ξ2,there exists π1,π2 which is the unique solution of (27) satisfying that(69)π1w−π1∗w≤aξ1+bξ2,π2w−π2∗w≤cξ1+dξ2.Theorem 8.
The toppled system (5) is Hyers–Ulam stable under hypothesesH1–H4.Proof.
With the help of Definition7 and Theorem 7, suppose that π1,π2 to be the correct solution and the pair π1∗,π2∗ be the other solution of system (27). Then, we have, from the first equation of (27),(70)π1w−π1∗w=|Γϱ1−λ−2wϱ1−3Γϱ1−2Γλ∫0WW−ℑλ−1φπ1ℑdℑ+∫01Gϱ1w,ℑϕq1Γθ1∫0ℑℑ−τθ1−1Z1τ,π2τdτdℑ−Γϱ1−λ−2Γϱ1−2Γλ∫0WW−ℑλ−1φπ1∗ℑdℑ−∫01Gϱ1w,ℑϕq1Γθ1∫0ℑℑ−τθ1−1Z1τ,π2∗τdτdℑ|≤Γϱ1−λ−2Γϱ1−2Γλ∫0WW−ℑλ−1φπ1−φπ1∗dℑ+∫01Gϱ1w,ℑϕq1Γθ1∫0ℑℑ−τθ1−1Z1τ,π2τdτ−ϕq1Γθ1∫0ℑℑ−τθ1−1Z1τ,π2∗τdτ|dℑ≤Γϱ1−λ−2Γϱ1−2Γλ∫0WW−ℑλ−1φπ1−φπ1∗dℑ+q−1ρ1q−2∫01Gϱ1w,ℑ1Γθ1∫0ℑℑ−τθ1−1×Z1τ,π2τ−Z1τ,π2∗τdτdℑ.
Then,(71)π1w−π1∗w≤Γϱ1−λ−2WλKφΓϱ1−2Γλ+1π1w−π1∗w+q−1ρ1q−2LZ11Γϱ1+1+Γϱ1−λ−2Γϱ1−2Γϱ1−λ+11Γθ1+1×π2w−π2∗w≤aξ1+bξ2,where(72)a=Γϱ1−λ−2WλKφΓϱ1−2Γλ+1,b=q−1ρ1q−2LZ11Γϱ1+1+Γϱ1−λ−2Γϱ1−2Γϱ1−λ+11Γθ1+1.
Similarly, we get(73)π2w−π2∗w≤cξ1+dξ2,where(74)c=Γϱ2−σ−2WσKϖΓϱ2−2Γσ+1,d=q−1ρ2q−2LZ21Γϱ2+1+Γϱ2−σ−2Γϱ2−2Γϱ2−σ+11Γθ2+1.
Hence, by (71) and (73), integral equations’ toppled system (27) is Hyers–Ulam stable. Thus, proposed toppled system (5) is Hyers–Ulam stable.


## 5. Illustrative Example

In this section, we introduce an application of our results, which were proved in Sections3 and 4.Example 1.
Consider the following toppled fractional system with thep-Laplacian operator and integral boundary conditions for n = 5:(75)D14/3cϕ4D13/3Rπ1w=−21w12+110cosπ2w,w∈0,1,D14/3cϕ4D13/3Rπ2w=3215+110sinπ1w,w∈0,1,ϕ4D13/3Rπ1wi|w=0=0,i=0,1,2,3,4,Ik−13/3π1w|w=0=0,k=1,2,4,5,D3/2π1w|w=1=1Γ3/2∫011−ℑ1/2cosπ16dℑ,ϕ4D13/3Rπ2wj|w=0=0,j=0,1,2,3,4,Ih−13/3π2w|w=0=0,h=1,2,4,5,D3/2π2w|w=1=1Γ3/2∫011−ℑ1/2cosπ26dℑ,where θ1=θ2=14/3,p=4,ϱ1=ϱ2=13/3,andλ=σ=3/2. Then, we obtain Kφ=Kϖ=1/6 and LZ1=LZ2=1/10. Via simple calculation and taking ρ1=ρ2=1/2, we get χ=0.1767<1. Hence, by Theorem 7, toppled system (75) has a unique solution. With comparable fashion, it is easy to verify the fulfillment of the conditions of Theorem 6. Likewise, the conditions of Theorem 8 can be easily confirmed, and consequently, the solution of system (75) is Hyers–Ulam stable.


## 6. Conclusion

In this study, we analyzed the stability and uniqueness solution of Caputo and Riemann–Liouville fractional derivatives with fractional ordersn−1<θ1,θ2,ϱ1,ϱ2≤n, and n≥3. By using the topological degree theory, we have proved sufficient conditions for the EUS of the coupled system of fractional differential equations with integral boundary conditions involving the p-Laplacian operator. Also, we have found appropriate conditions for Hyers–Ulam stability of the solution for the considered system. At the end, we have provided an example that supported our results as we have done in Section 5 to confirm the theoretical analysis.



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