# On Coupledp-Laplacian Fractional Differential Equations with Nonlinear Boundary Conditions

Authors: Aziz Khan; Yongjin Li; Kamal Shah; Tahir Saeed Khan
Journal: Complexity
 (2017)

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## Abstract

This paper is related to the existence and uniqueness of solutions to a coupled system of fractional differential equations (FDEs) with nonlinearp-Laplacian operator by using fractional integral boundary conditions with nonlinear term and also to checking the Hyers-Ulam stability for the proposed problem. The functions involved in the proposed coupled system are continuous and satisfy certain growth conditions. By using topological degree theory some conditions are established which ensure the existence and uniqueness of solution to the proposed problem. Further, certain conditions are developed corresponding to Hyers-Ulam type stability for the positive solution of the considered coupled system of FDEs. Also, from applications point of view, we give an example.


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## Body

## 1. Introduction

Due to high profile accuracy and usability, FDEs become an area of interest for various fields of scientists and mathematicians. In last few years, some physical phenomenons were described through FDEs and compared with integer order differential equations which have better results, that is why researchers of different areas have paid great attention to study FDEs. The applications of FDEs can be studied in several disciplines including aerodynamics, engineering, electrical circuits, plasma physics, chemical reaction design, turbulent filtration in porous media, and signal and image processing; for further details we refer to [1–5].Nonlinear operators have vital roles in differential equations; one of the most important operators used in FDEs is the classical nonlinearp-Laplacian operator, which is defined as (1)1p+1q=1,ϕps=sp-2s,p≥1,ϕqτ=ϕp-1.For further details and applications of the nonlinear p-Laplacian operators, reader should study [6].Researchers study different aspects of FDEs involvingp-Laplacian operators like the existence theory, which has been extensively investigated by using classical fixed point theory. The mentioned theory has been investigated very well for the aforesaid equations of ordinary and partial fractional differential equations. Since p-Laplacian operators have been greatly applied in the mathematical modeling of large numbers of real world phenomenons devoted to physics, mechanics, dynamical systems, electrodynamics, and so forth, therefore researchers paid much attention to study such type of differential equation dealing with p-Laplacian operators from different aspects including existence theory, multiplicity results, and stability analysis. For instance, Lu et al. [7] discussed Sturm-Liouville boundary value problems (BVP) of FDEs with p-Laplacian operator for existence of two or three positive solutions by using fixed point theory. By applying Leggett-William fixed point theorem, the mentioned author studied the following problem: (2)DβϕpDαμx+Fx,μx=0,0<x<1,ȷμ0-ημ′0=0,γμ1+δμ′1=0,Dαμ0=0and also he provided proper example, where Dα and Dβ denote standard Caputo fractional derivatives with 1<α≤2, 0<β≤1, 1/p+1/q=1, ϕp(τ)=|s|p-2s, p>1, ϕq(τ)=ϕp-1, ρ=ȷγ+ȷδ+ηγ>0, ȷ,η,δ,γ≥0, and F is continuous.Hu and Zhang [8] have investigated a nonlinear FDE with p-Laplacian operator for existence of solution as given by (3)DβϕpDαμx+Fx,μx,Dαμx=0,x∈0,1,Dαμ0=0=Dαμ1,where 0<α,β<1, 1<α+β<2, Dα and Dβ represent standard Caputo fractional derivatives, F is continuous and 1/p+1/q=1, ϕp(τ)=|s|p-2s, p>1, and ϕq(τ)=ϕp-1. Zhi et al. [9] have investigated the existence of positive solutions for the nonlocal BVP of FDEs with p-Laplacian operator and illustrated the problem with expressive example. The corresponding problem is given by(4)ϕpDαμx′′=Fx,μx,Dβμx,x∈0,1,μxx=0=μ′′xx=0=0,μ1=∫01gτμτdτ,ϕpDαμ0′=ξ1ϕpDαμȷ1′,ϕpDαμ1=ξ2ϕpDαμȷ2,where ϕp is p-Laplacian operator and 2<α≤3, 1<β<α-1<2, 0<ȷ1≤ȷ2<1, 0≤ξ1,ξ2<1, and Dα expresses Caputo derivative of order α. For further study about the existence theory and multiplicity results of p-Laplacian operator involved in differential equations, see [10–14].Using classical fixed point theory needs strong conditions to establish conditions for existence and uniqueness of solutions to FDEs and therefore restrict the applicability to certain classes of FDEs and their systems. To relax the criteria degree theory plays excellent roles for the existence of solutions to FDEs and their systems. Various degree theories including Brouwer and Leray-Schauder were established to deal with the existence theory of differential equations. An important degree theory known as topological degree theory which was introduced by Stamova [15] and later on extended by Isaia [16] has been used to establish existence theory for solutions to nonlinear differential and integral equations. The mentioned method is called prior-estimate method which needs no compactness of the operator and relaxes much the condition for existence and uniqueness of solutions to differential and integral equations. Recently, the aforesaid degree theory has been applied to investigate certain classes of FDEs with boundary conditions; see [17–19].In recent years another aspect of FDEs which has greatly attracted the attentions of researchers is devoted to the stability analysis of the mentioned equations. Stability analysis plays significant roles in the optimization and numerical analysis of the aforesaid equations. Different kinds of stability have been studied for fractional differential equations including exponential, Mittag-Leffler, and Lyapunov stabilities; see [15, 20, 21]. An important stability was pointed out by Ulam [22], in 1940, which was formally introduced by Hyers [23], in 1941. The aforesaid stability has now been considered in many papers for classical differential equations; see [24–26]. For instance, Urs [27] has investigated the Hyers-Ulam stability for the following coupled periodic BVPs given as (5)μ′′x-F1x,μx=F2x,vx,x∈0,T,v′′x-F1x,vx=F2x,μx,μxx=0=μxx=T,vxx=0=vxx=T.The Hyers-Ulam stability has been investigated for certain FDEs with boundary and initial conditions; see [28–30]. In many situations, Lyapunov type stability and its investigation are very difficult and time-consuming for certain nonlinear fractional differential equations. This is due to the predefined Lyapunov function which is often very difficult to construct for FDEs. Therefore, Hyers-Ulam type stability plays important roles in such a situation. Inspired from the above-mentioned work, in this paper, we study a coupled system of FDEs with nonlinearp-Laplacian operator by using topological degree theory. Further, we also investigated some conditions for the Hyers-Ulam stability of the solution to the proposed problem. The proposed problem is given by (6)Dβ1ϕpDα1μx+F1x,vx=0,x∈0,1,Dβ2ϕpDα2vx+F2x,μx=0,x∈0,1,Dα1μxx=0=0=μxx=0=μ′′xx=0,μ′xx=1=η1IPγ1ψ1μ=η1Γγ1∫0PP-τγ1-1ψ1μτdτ,Dα2vxx=0=vxx=0=v′′xx=0=0,v′xx=1=η2IPγ2ψ2v=η2Γγ2∫0PP-τγ2-1ψ2vτdτ,where 2<αi<3, 0<βi<1, P, ηi,γi>0, ψ1,ψ2∈L[0,1], and Dαi and Dβi where i=1,2 stand for Caputo fractional derivative, ϕp(κ)=κp-2κ is p-Laplacian operator, where 1/p+1/q=1, ϕq denotes inverse of p-Laplacian, and Fi:[0,1]×[0,∞)→(0,∞), i=1,2, are continuous functions. Here, we remark that applying degree method to deal with existence and uniqueness and to find conditions for Hyers-Ulam stability to a coupled system of FDEs with p-Laplacian operator has not been investigated properly to the best of our knowledge. Therefore thanks to the coincidence degree theory and nonlinear functional analysis greatly developed by Deimling [31], we establish necessary and sufficient conditions for existence and uniqueness as well as for Hyers-Ulam stability corresponding to the aforementioned problem considered by us. We also demonstrate our result through expressive example.


## 2. Axillary Results

Here we recall some special definitions, theorems, and Hyers-Ulam stability results from the literature [1–4] which have important applications throughout this paper.Definition 1.
The integral with fractional orderα>0 of Riemann-Liouville type is defined for the function F as (7)I0αFx=1Γα∫0xx-τα-1Fτdτ,provided that the integral on the right converges pointwise on (0,∞).Definition 2.
The derivative with fractional orderα>0 of Caputo type is defined for the function F as (8)D0αFx=1Γm-α∫0xx-τm-α-1Fmτdτ,where m=[α]+1, [α] is the integer part of α such that the integral on the right converges pointwise on (0,∞).Lemma 3.
Letα>0 and μ∈C(0,1)∩L1(0,1), and then the general solution of FDE (9)D0αμx=yxis given by (10)μx=I0αyx+A0+A1x+A2x2+⋯+Am-1xm-1,for some Ai∈R,  i=0,1,2,…,m-1 and m is smallest integer such that m≥α.LetL be the space of all continuous functions μ:[0,1]→R endowed with a norm supx∈[0,1]{|μ(x)|:μ∈C[0,1]} which is obviously a Banach space. Then the product space denoted by L=L1×L2 under the norms μ,v=μ+v is also a Banach space which will be used throughout this paper. For the coincidence degree theory and nonlinear functional analysis, we recall the following definitions which can be traced in [15, 16, 31] as follows.Definition 4.
Let the class of all bounded set ofP(L) be denoted by ℵ. Then the mapping ȷ:ℵ→(0,∞) for Kuratowski measure of noncompactness is defined as (11)ȷħ=infd>0:ħisthefinitecoverforsetsofdiameter≤d,where ħ∈ℵ.Proposition 5.
The following are the characteristics of Kuratowski measureȷ: (1)
For relative compactħ, then Kuratowski measure ȷ(ħ)=0.(2)
For seminormȷ, ȷ(κħ)=κȷ(ħ), κ∈R, and ȷ(ħ1+ħ2)≤ȷ(ħ1)+ȷ(ħ2).(3)
ħ1⊂ħ2 yields ȷ(ħ1); ȷ(ħ1∪ħ2)=sup{ȷ(ħ1),ȷ(ħ2)}.(4)
ȷ(conv⁡ħ)=ȷ(ħ).(5)
ȷ(ħ¯)=ȷ(ħ).Definition 6.
Assume thatφ:ϑ→U is bounded and continuous mapping such that ϑ⊂U. Then φ is a ȷ-Lipschitz, where ζ≥0 such that (12)ȷφħ≤ζȷħforallboundedħ⊂ϑ.Then φ is called strict ȷ-contraction under the condition ζ<1.Definition 7.
The functionφ is ȷ-condensing if (13)ȷφħ<ȷħ,forallboundedħ⊂ϑsuchthatȷħ>0.Therefore ȷ(φ(ħ))≥ȷ(ħ) yields ȷ(ħ)=0.Further we haveφ:ϑ→U which is Lipschitz for ζ>0, such that (14)φυ-φυ¯≤ζυ-υ¯,forallυ,υ¯∈ϑ.The condition ζ<1 yields that φ is a strict contraction.Proposition 8.
The mappingφ is ȷ-Lipschitz with constant ζ=0 if and only if φ:ϑ→U, which is said to be compact.Proposition 9.
The operatorφ is ȷ-Lipschitz for some constant ζ if and only if φ:ϑ→U, which is Lipschitz with constant ζ.Theorem 10.
Letφ:L→L be a ȷ-contraction and(15)G=z∈L:thereexist0≤λ≤1suchthatz=λφz.Under the conditions G⊂L is bounded for r>0 and G⊂ħr(0), with degree (16)degI-λφ,ħr0,0=1,foreveryλ∈0,1.Then, φ has at least one fixed point.Lemma 11 (see [10]).
Letϕp be a p-Laplacian operator. Then (i)
if1<p≤2, κ1κ2>0, and κ1,κ2≥m>0, then (17)ϕpκ1-ϕpκ2≤p-1mp-2κ1-κ2;(ii)
ifp>2 and κ1,κ2≤M, then (18)ϕpκ1-ϕpκ2≤p-1Mp-2κ1-κ2.Definition 12.
LetT:L→L. Then the operator equation given by (19)Tμx=μx,x∈0,1is called Hyers-Ulam stable if, for any ξ>0, the inequality given as (20)μ-Tμ≤ξ,x∈0,1has a unique fixed point say μ∗ with constant D>0 such that μ-μ∗≤Dξ, for all x∈[0,1] holds.In view of Definition12, we give the following definition.Definition 13.
The system of Hammerstein type integral equations(21)μx=∫01Ωα1x,τϕqIβ1F1τ,vτdτ+η1xΓγ1∫0PP-τγ1-1ψ1μτdτ,vx=∫01Ωα2x,τϕqIβ2F2τ,μτdτ+η2xΓγ2∫0PP-τγ2-1ψ2vτdτis called Hyers-Ulam stable such that, for Di>0 (i=1,2,3,4) and for all ξ1,ξ2>0 and for every solution (μ∗,v∗) to the system(22)μ∗x-∫01Ωα1x,τϕqIβ1F1τ,vτdτ+η1xΓγ1∫0PP-τγ1-1ψ1μτdτ≤ξ1,v∗x-∫01Ωα2x,τϕqIβ2F2τ,μτdτ+η2xΓγ2∫0PP-τγ2-1ψ2vτdτ≤ξ2,there exists a unique solution (λ1,λ2) of (21) satisfying the following system of inequalities:(23)λ1-μ∗≤D1ξ1+D2ξ2,x∈0,1,λ2-v∗≤D3ξ1+D4ξ2,x∈0,1.


## 3. Some Data Dependence Assumptions

To proceed further, let the following hypothesis hold:(A1)
The nonlocal functionsψ1 and ψ2 where ω,μ,ν,υ∈R satisfy the following:(24)ψ1ω-ψ1μ≤Kψ1ω-μ,ψ2ν-ψ2υ≤Kψ2ν-υ,whereKψ1,Kψ2∈[0,1).(A2)
To satisfy the following growth conditions by the constantsCψ1,Cψ2,Mψ1,Mψ2>0 and q1∈[0,1) for μ,v∈R with the nonlocal functions ψ1 and ψ2 we have(25)ψ1μ≤Cψ1μq1+Mψ1,ψ2v≤Cψ2vq1+Mψ2.(A3)
The functionsF1 and F2 satisfy the following growth conditions under the constants a,b,M1∗,M2∗, p1∈(0,1]: (26)F1x,μ≤aμp1+MF1∗,F2x,υ≤bυp1+MF2∗.(A4)
There exist real valued constantsLF1 and LF2, and for all μ,υ,ω1,ω2∈R, (27)F1x,υ-ψ1x,ω1≤LF1υ-ω1,F2x,μ-ψ2x,ω2≤LF1μ-ω2.


## 4. Main Results

Theorem 14.
LetF1(x)∈C[0,1] be integrable function for FDEs and with integral boundary conditions; then the solution of(28)Dβ1ϕpDα1μx+F1x=0,0<β1≤1,2<α1≤3,x∈0,1,Dα1μxx=0=μxx=0=μ′′xx=0=0,μ′xx=1=η1ITγ1ψ1μ=η1Γγ1∫0PP-τγ1-1ψ1μτdτis provided by(29)μx=∫01Ωα1x,τϕqIβ1F1τdτ+η1xΓγ1∫0qPP-τγ1-1ψ1μτdτ,where Ωα1(x,τ) is Green’s function, given by(30)Ωα1x,τ=x1-τα1-2Γα1-1-x-τα1-1Γα1,0≤τ≤x≤1,x1-τα1-2Γα1-1,0≤x≤τ≤1.Proof.
Applying the operatorIβ1 on (28) and using Lemma 3, we get from problem (28) the following equivalent integral form as(31)ϕpDα1μx=-Iβ1F1x+A0.By using conditions Dα1μ(x)x=0=0, we get A0=0. From (31), we have(32)Dα1μx=-ϕqIβ1F1x.Applying the operator Iα1 on (32) and using Lemma 3 again, we get from problem (32) the following equivalent integral form given by(33)μx=-Iα1ϕqIβ1F1x+A1+A2x+A3x2.By using the condition μ(x)|x=0=μ′′(x)|x=0=0 in (33), we obtain A1=A3=0. Also in view of condition μ′(x)|x=1=η1ITγψ1(μ) in (33), we get(34)A2=1Γα1-1∫011-τα1-2ϕqIβ1F1xdτ+η1Γγ1∫0PP-τγ1-1ψ1μτdτ.By substituting the values of A1, A2, and A3 in (33), we get the following integral equation:(35)μx=1Γα1-1∫01x1-τα1-2-1Γα1∫0xx-τα1-1ϕqIβ1F1τdτ+η1xΓγ1∫0PP-τγ1-1ψ1μτdτ=∫01Ωα1x,τϕqIβ1F1τdτ+η1xΓγ1∫0PP-τγ1-1ψ1μτdτ.According to Theorem14, the equivalent system of Hammerstein type integral equations corresponding to coupled system (6) is given by(36)μx=∫01Ωα1x,τϕqIβ1F1τ,vτdτ+η1xΓγ1∫0PP-τγ1-1ψ1μτdτ,υx=∫01Ωα2x,τϕqIβ2F2τ,μτdτ+η2xΓγ2∫0PP-τγ2-1ψ2υτdτ,where Ωα2(x,τ) is defined as(37)Ωα2x,τ=x1-τα2-2Γα2-1-x-τα2-1Γα2,0≤τ≤x≤1,x1-τα2-2Γα2-1,0≤x≤τ≤1.From Ωα1(x,τ) and Ωα2(x,τ) clearly,(38)maxx∈JΩα1x,τ=1-τα1-2Γα1-1,maxt∈JΩα2x,τ=1-τα2-2Γα2-1.Further, we define the operators φ1:L1→L1 and φ2:L2→L2 by(39)φ1μx=η1xΓγ1∫0PP-τγ1-1ψ1μτdτ,x∈0,1,φ2υx=η2xΓγ2∫0PP-τγ2-1ψ2υτdτ,x∈0,1,and Υi:L→L for (i=1,2) by(40)Υ1υx=∫01Ωα1x,τϕqIβ1F1τ,vτdτ,x∈0,1Υ2μx=∫01Ωα2x,τϕqIβ2F2τ,μτdτ,x∈0,1.Hence we have φ(μ,v)=(φ1,φ2)(μ,v), Υ(μ,v)=(Υ1,Υ2)(μ,v), and T(μ,v)=φ(μ,v)+Υ(μ,v). Therefore the operator equation of Hammerstein type integral equations (36) is given by(41)μ,v=Tμ,v=φμ,v+Υμ,v.Thus the solution of Hammerstein type equation (36) is the fixed points of operator equation (41).Theorem 15.
In view of hypotheses(A1) and (A4), the operator φ is ȷ-Lipschitz and satisfies the growth condition given by(42)φμ,υ≤Cφμ,υq1+Mφ,∀μ,υ∈L.Proof.
From condition(A1) and using x≤1, we get(43)φ1μx-φ1μ¯x=η1Γγ1∫0PP-τγ1-1ψ1μ-ψ1μ¯dτ≤η1Γγ1∫0PP-τγ1-1ψ1μ-ψ1μ¯dτ,which implies that |φ1(μ)(x)-φ2(μ¯)(x)|≤K¯ψ1μ-μ¯, where K¯ψ1=Kψ1Pγ1/Γ(γ1+1)∈[0,1). To obtain the growth condition we have(44)φ1μx=η1Γγ1∫0PP-τγ1-1ψ1μτdτ≤η1Γγ1∫0PP-τγ1-1ψ1μτdτ.Upon simplification, we get from (44) (45)φ1μ≤η1Pγ1Γγ1+1Cψ1μq1+Mψ1.Similarly, we get(46)φ2υ≤η2Pγ2Γγ2+1Cψ2υq1+Mψ2.Now(47)φμ,υ≤η1Pγ1Γγ1+1Cψ1μq1+Mψ1+η2Pγ2Γγ2+1Cψ2υq1+Mψ2≤η1Pγ1Γγ1+1Cψ1μq1+η2Pγ2Γγ2+1Cψ2υq1+η1Pγ1Mψ1Γγ1+1+η2Pγ2Mψ2Γγ2+1≤Cφμq1+υq1+Mφ=Cφμ,υq1+Mφ, where(48)maxη1Pγ1Γγ1+1Cψ1,η2Pγ2Γγ2+1Cψ2=Cφ,Mφ=η1Pγ1Mψ1Γγ1+1+η2Pγ2Mψ2Γγ2+1.Theorem 16.
In view of hypothesis(A3), the operator Υ is continuous and satisfies the growth condition given by(49)Υμ,υp1≤Ϝμ,υp1+Ξ,where Ϝ=℘(a+b), ℘=max{(q-1m1q-2/Γβ1+1)(1/Γα1+1),((q-1)m2q-2/Γ(β2+1))(1/Γ(α2+1))}, and Ξ=℘(M1∗+M2∗) for each (μ,υ)∈L.Proof.
Consider bounded setBr={(μ,υ)∈L:μ,υ≤r} with sequence (μn,υn) converging to (μ,υ) in Br. We have to show that Υ(μn,υn)-Υ(μ,υ)→0 as n→∞. Therefore, we have to consider(50)Υ1υnx-Υυx≤xΓα1-1∫011-τα1-2ϕqIβF1τ,υnτ-ϕqIβF1τ,υτdτ+1Γα1∫0xx-τα1-1ϕqIβF1τ,υnτ-ϕqIβF1τ,υτdτ≤q-1m1q-21Γα1-1∫011-τα1-2-1Γα1∫0xx-τα1-1×IβF1τ,υnτ-IβF1τ,υτdτ.Due to continuity of F1, one has F1(τ,υn(τ))→F1(τ,υ(τ)) as n→∞. Thus in view of Lebesgue dominated convergent theorem, we have |IβF1(x,υn(x))-IβF1(x,υ(x))|→0 as n→∞. Thus |Υ1(υn(x))-Υ(υ(x))|→0,asn→∞. Hence Υ1 is continuous. Now for growth condition (49), we have(51)Υ1υx=∫01Ωα1x,τϕqIβ1F1τ,υτdτ≤q-1m1q-2∫01Ωα1x,τIβ1F1τ,υτdτ,Υ1υp1≤q-1m1q-2Γβ1+11Γα1+1aυp1+M1∗,Υ2μx=∫01Ωα2x,τϕqIβ2F2τ,μτds≤q-1m2q-2∫01Ωα2x,τIβ2F1τ,υτds,Υ2μp1≤q-1m2q-2Γβ2+11Γα2+1bμp1+M2∗.From, (51), we obtain the following result(52)Υμ,υp1=Υ1υp1+Υ2μp1≤℘aυp1+M1∗+℘bμp1+M2∗≤℘a+bυp1+μp1+℘M1∗+M2∗=Ϝμ,υp1+Ξ.Theorem 17.
The operatorΥ:L→L is compact and ȷ-Lipschitz with constant zero.Proof.
Consider a sequence{Fn} in bounded set D such that D is subset of Br; in view of (49), we see(53)Υμn,υnp1≤Ϝμn,υnp1+Ξ,foreveryμ,υ∈L.Thus Υ is bounded for (μn,υn)∈D. Let us, for any x1,x2∈[0,1], consider(54)Υυnx1-Υυnx2=∫01Ωα1x1,τϕqIβ1F1τ,υnτdτ-∫01Ωα1x2,τϕqIβ1F1τ,υnτdτ≤q-1m1q-2∫01Ω1x1,τ-∫01Ω1x2,τIβ1F1τ,υnτdτ≤q-1m1q-2Γβ1+1x1-x2Γα1-1∫011-τα1-2dτ+1Γα1∫0x1x1-τα1-1dτ-∫0x2x2-τα1-1dτaυnp1+M1∗.By simplification, we get the following result:(55)Υ1υnx1-Υ1υnx2≤q-1m1q-2Γβ1+1x1-x2Γα1+x1α1-x2α1Γα1+1aμnp1+M1∗.And also in same fashion, one has(56)Υ2μnx1-Υ2μnx2≤q-1m2q-2Γβ2+1x1-x2Γα2+x1α2-x2α2Γα2+1bυp2+M2∗.Therefore if x1→x2, then the right hand side of both sides of (55) and (56) tends to zero. Thus Υ1,Υ2 are equicontinuous, and therefore Υ=(Υ1,Υ2) is equicontinuous on D. Hence, thanks to the Arzelá-Ascoli theorem, Υ(D) is compact. Also, by proposition Υ is ȷ-Lipschitz with constant zero.Remark 18.
The results of Theorems15, 16, and 17 are also hold for using q1=p1=1.Theorem 19.
In view of assumptions(A1)–(A4) such that Cφ+Ϝ≤1, then the coupled system (6) has at least one solution (μ,υ)∈L together with condition that the set of the solutions B is bounded in L.Proof.
φ and Υ are continuous and ȷ-Lipschitz with constant Cφ and 0, respectively. Thanks to Theorem 10, T is strict ȷ-contraction, and then consider a set (57)B=μ,υ∈L:λ∈0,1∣μ,υ=λTμ,υ.To prove that B is bounded, let(58)μ,υ=λTμ,υ≤λφμ,υ+λΥμ,υ≤Cφμ,υ+Mφ+Ϝμ,υ+Ξ=Cφ+Ϝμ,υ+Mφ+Ξ.Therefore, the set of solutions B is bounded.Theorem 20.
Assume that hypotheses(A1)–(A4) hold. Then system (6) has a unique solution if and only if Θ<1, where(59)Θ=η1Kψ1Pγ1Γγ1+1+η2Kψ2Tγ2Γγ2+1+q-1m1q-2LF1Γβ1+1Γα1+q-1m2q-2LF2Γβ2+1Γα2.Proof.
Let(μ,υ) and (μ¯,υ¯)∈L be two solutions; then(60)Tμ,υ-Tμ¯,υ¯≤φμ,υ-φμ¯,υ¯+Υμ,υ-Υμ¯,υ¯;therefore by simplification (60), we get the following result:(61)Tμ,υ-Tμ¯,υ¯≤η1Kψ1Pγ1Γγ1+1+η2Kψ2Tγ2Γγ2+1+q-1m1q-2LF1Γβ1+1Γα1+q-1m2q-2LF2Γβ2+1Γα2μ,υ-μ¯,υ¯≤Θμ,υ-μ¯,υ¯. Thus the operator T is a contraction. Hence the uniqueness of solution to system (6) follows due to the Banach fixed point theorem.


## 5. Hyers-Ulam Stability of Coupled System

In the present section, we derive the Hyers-Ulam type stability for the solution of the considered problem.Theorem 21.
By assumptions(A1)–(A4), system (6) is Hyers-Ulam stable.Proof.
By Theorem20 and Definition 13, let (λ1,λ2) be the exact solution, and let (μ∗,υ∗) be any other solution of system (36). Then from equation of (36) we have(62)λ1x-μ∗x≤∫01Ωα1x,τϕqIβ1F1τ,yτ-F1τ,υ∗τdτ+η1xΓγ1∫0PP-τγ1-1ψ1x-ψ1μ∗dτ(63)impliesthatλ1-μ∗≤q-1m1q-2LF1Γβ1+1Γα1x-υ∗+η1Pγ1Kψ1Γγ1+1y-μ∗≤D1ξ1+D2ξ2,where D1=(q-1)m1q-2LF1/Γ(β1+1)Γ(α1)andD2=η1Pγ1Kψ1/Γ(γ1+1). By the same process we also get that(64)λ2-υ∗≤D3ξ1+D4ξ2,where D3=(q-1)m1q-2LF2/Γ(β2+1)Γ(α2) and D4=η2Pγ2Kψ2/Γ(γ2+1).
Hence in view of (62) and (64), the system of integral equation (36) is Hyers-Ulam stable, and consequently, the solution of system (6) is Hyers-Ulam stable.


## 6. Illustrative Examples

Example 1.
Consider the following coupled system of FDEs withp-Laplacian operator and fractional order differential and integral boundary conditions of the type(65)D1/2ϕpD5/2μx+x99+19sin⁡νx=0,(66)D1/2ϕpD5/2υx+x100+3sin⁡μx+75=0,D5/2μxx=0=0=μxx=0=μ′′xx=0,μ′xx=1=13Γ3/2∫011-τ1/2cos⁡μdτ,D5/2υxx=0=0=υxx=0=υ′′xx=0,υ′xx=1=13Γ3/2∫011-τ1/2cosυdτ,where x∈[0,1], P=1, p=3, αi=5/2, βi=1/2, ηi=1/3, γi=3/2, where i=1,2,  Kψ1,Kψ2=1/5, Mi=2, and LFi=1/6. By simple calculation we obtain the following:(67)1/31/5Γ3/2+1+1/31/5Γ3/2+1+3/2-12-1/21/6Γ1/2+1Γ5/2+3/2-12-1/21/6Γ1/2+1Γ5/2<1.Therefor from Theorem 20, we concluded that (65) has unique solution. Similarly, the conditions of Theorem 21 can be verified easily. Thus the solution of system (65) is Hyers-Ulam stable.


## 7. Conclusion

We have investigated sufficient conditions for existence and uniqueness of solutions to a coupled system of nonlinear FDEs with fractional integral boundary conditions with nonlinearp-Laplacian operator by using topological degree method. Further, we have established some adequate conditions for the Hyers-Ulam stability to the proposed problem.



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