# Some Existence and Stability Criteria to a Generalized FBVP Having Fractional Compositep-Laplacian Operator

Authors: Sh. Rezapour; S. T. M. Thabet; M. M. Matar; J. Alzabut; S. Etemad
Journal: Journal of Function Spaces
 (2021)

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## Abstract

In this paper, we consider a generalized Caputo boundary value problem of fractional differential equation with compositep-Laplacian operator. Boundary value conditions of this problem are of three-point integral type. First, we obtain Green’s function in relation to the proposed fractional boundary value problem and then for establishing the existence and uniqueness results, we use topological degree theory and Banach contraction principle. Further, we consider a stability analysis of Ulam-Hyers and Ulam-Hyers-Rassias type. To examine the validity of theoretical results, we provide an illustrative example.


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## Body

## 1. Introduction

Fractional calculus, as a generalization of classical ordinary calculus to integrodifferential operators in the noninteger settings, has attracted considerable interest in recent years and has grown rapidly since its introduction. Fractional calculus is now broadly used in several fields such as biology, fluid dynamics, viscoelastic theory, neural networks, and epidemic models; see for instance [1, 2] and references therein.By using fixed point techniques, a large number of researchers studied the existence-uniqueness properties of solutions for different classes of differential equations in the fractional settings. In 2016, Ntouyas et al. [3] studied two fractional boundary value problems (FBVPs) with three-point boundary conditions and derived the existence results by using the fixed point notion. Similarly, in [4], Boutiara et al. used fixed point theorems to prove the existence results for another FBVP with three-point boundary conditions in the context of the Caputo-Hadamard and Hadamard operators. More recently, Derbazi et al. [5] designed a new FBVP by applying the generalized ψ-operators and proved their desired results via monotone iterative techniques.As you know, every numerical method must be accurate in order to give desired results which are acceptable for different applications. For this purpose, the analysis of the stability is needed. Various types of stability involving exponential, Lyapunov, and Mittag-Leffler have been studied for different types of problems. The abovementioned types of stability have been improved for many differential equations in both linear and nonlinear fractional cases and their related systems over the last few years. However, the stability of some nonlinear systems undergo unavoidable deficiencies which appear due to the need of predefining Lyapunov function. This is often considered as an uneasy task.In [6, 7], Ulam and Hyers have initiated the concept of Ulam-Hyers stability. In addition, this notion has been considered for nonlinear fractional differential equations and their related systems. For instance, Abdo et al. [8] investigated the stability criteria for ψ-Hilfer fractional integrodifferential equations and in the same time, Zada et al. [9] derived similar results for impulsive integrodifferential equations with Riemann-Liouville boundary conditions. In [10], Kheiryan and Rezapour considered a new multisingular FBVP and checked its Hyers-Ulam stability.On the other hand, some properties of solutions of FBVPs including the uniqueness, existence, and stability notions have been investigated with the help of various techniques such as topological degree theory (T-degree theory) and fixed point theory. In this paper, we will apply the existing concepts in T-degree theory, as well as there are a large number of nonlinear mathematical models in engineering and the scientific fields to investigate and analyze dynamical systems. One of the most important nonlinear operators frequently used is the classicalp-Laplacian operator. Models with p-Laplacian operators are often used to simulate practical problems such as tides caused by celestial gravity and elastic deformation of beams. Such extensive applications attract the attention of many researchers to study mathematical models having p-Laplacian operators.Specifically, Ma et al. [11] defined a new multipoint FBVP with p-Laplacian operator and derived the existence and iteration of monotone positive solutions for the given system. Next, Matar et al. [12] studied another p-Laplacian FBVP having Caputo-katugampula fractional derivatives recently. For more details about p-Laplacian fractional boundary value problems, we refer to [13, 14].Also, to see the importance of existing techniques in T-degree theory, we can point out to a paper published by Shah and Khan [15] on the existence-uniqueness results to a coupled system of FBVPs. Further, Sher et al. [16] implemented a qualitative analysis on a multiterm delay FBVP with the help of the same technique in T-degree theory.In 2017, Ali et al. [17] studied a coupled fractional structure of a system involving two differential equations with non-integer boundary conditions of integral type which takes the form
(1)D0+r1v1t=ϕ1t,v2t,C≤t≤1,D0+r1v1t=ϕ2t,v1t,C≤t≤1,v10=0,v11=1Γα∫0KK−sα−1pv1sds,v20=0,v21=1Γβ∫0KK−sβ−1pv2sds,whereDC0+r1·,DC0+r2· stands for the Caputo derivative of orders 1<r1<2 and 1<r2<2, respectively, and ϕ1,ϕ2:0,1×ℝ2⟶ℝ is continuous functions along with p,q∈L0,1 which satisfy some certain linear growth conditions. By setting certain particular conditions, they derived their desired existence results using some techniques in T-degree theory. The authors also investigated the Hyers-Ulam stability for the proposed problem.In [18], Khan et al. studied the existence of solutions and their uniqueness for the proposed coupled fractional structure of a FBVP having the nonlinear operator of p-Laplacian type and integral boundary conditions given by
(2)D0+r1∗φpDC0+r1v1t=ϱ1t,v1,v2t,C0≤t≤1,D0+r2∗φpDC0+r2v1t=ϱ2t,v1,v2t,C0≤t≤1,φpDC0+r1v10=v1′0=0,v10=c11Γα−1∫0KK−sα−2v1sds,φpDC0+r2v20=v2′0=0,v20=c21Γβ−1∫0Kη−sβ−2v2sds,where DC0+rj∗· and DC0+rj· for j=1,2 denote the Caputo derivative of orders 0<rj∗<1 and 1<rj<2. Additionally, ϱ1,ϱ2:0,1×ℝ2⟶ℝ is continuous, and α,β≥1,−1≤c1,c2≤0 and φpθ=θp−2θ stand for the p-Laplacian operator such that 1/q+1/p=1. The authors established their desired theorems using the techniques attributed to Leray-Schauder and Banach. Further, the Hyers-Ulam stability was investigated.In [19] and by means of T-degree theory, Shah and Hussain established sufficient conditions for investigation of the existence of solutions and their stability on the following nonlinear FBVP
(3)DC0+r=Θt,μt,2<r<4,μ0=ζt,μ′0=μ′′0=0,μ1=vμη,where DC0+r· represents the Caputo derivative of order r. Further, Θ:0,1×ℝ⟶ℝ and ζ:0,1⟶ℝ are regarded to be continuous and μ,η∈0,1.By considering the existing literatures, we see that all differential equations having ap-Laplacian operator with three-point integral boundary conditions are not well explored by T-degree theory, and even the boundary conditions of integral type cover a wide range of applications which have direct contributions in the theory of fluid mechanics, optimization, and viscoelasticity.Inspired and motivated by the above fractional systems, we focus on the existence of solutions and establish four classes of Hyres-Ulam stability of a generalized FBVP havingp-Laplacian operator with 3-point integral boundary conditions given by
(4)DCβ,δφpDCχ,ρxt=ℏ1t,xtt∈t0,K,t0≥0,xt0+μ1xK=σ1∫t0Kℏ2sds,μ1≠−1,x′t0+μ2x′K=σ2∫t0Kℏ3sds,μ2≠−t0Kρ−1,DCχ,ρxt0=0,DCχ,ρxK=υDCχ,ρxη,η∈t0,K,so that DCβ,δ and DCχ,ρ are generalized derivatives in the sense of Caputo, of order β,χ∈ (1,2) and δ,ρ>0. Along with these, μ1,μ2,υ,σ1,σ2∈ℝ with υ≠Kδ−t0δ/ηδ−t0δq−1 and also ℏ2,ℏ3:t0,K⟶ℝ and ℏ1:t0,K×ℝ⟶ℝ are assumed to be continuous. We emphasize that the proposed FBVP (4) has a novel structure and is designed for the first time in the context of a generalized fractional settings along with the p-Laplacian operator.


## 2. Auxiliary Preliminaries

The main purpose of this section is to collect some important definitions, primitive lemmas, and theoretical results of generalized fractional integrals and derivatives which are applicable in this paper.ByCt0,K, we mean the category of all continuous real functions defined on t0,K which is simply proved that it is a Banach space along with x=supt∈t0,Kxt. Moreover, AClt0,K,ℝ=x:t0,K⟶ℝ:xl−1t∈ACt0,K,ℝ stands for the space of absolutely continuous functions on t0,K having real values up to l−1-derivative. Thus, in this regard, we define
(5)ACδlt0,K,ℝ=x:t0,K⟶ℝ:δl−1xt∈ACt0,K,ℝ,δ=1tρ−1ddt,as a category of functions having absolutely continuous δl−1-derivatives, and a norm is defined by
(6)xCδl=∑k=0l−1δkxtC,so that δk=δδ⋯δ⏞k−times.Definition 1. (see [20]).
Let0<a,b<+∞, q>0 and x∈Xcpa,b, where Xcpa,b is the space of all Lebesgue measurable complex functions. The integral operator given by
(7)Iρa+qxt=ρ1−qΓq∫a+tsρ−1tρ−sρq−1xsds,ρ>0,t>a+is named as the generalized Riemann-Liouville integral such that the R.H.S. integral is finite-valued.Definition 2. (see [20]).
The generalized derivative in sense of Caputo for a given functionf∈ACδla,b of order l−1<q<l with l=q+1 is defined by
(8)Da+q,ρCxt=Iρa+l−qδlft=ρq−l+1Γl−q∫a+tsρ−1tρ−sρl−q−1δlfsds,ρ>0,t>a+
In particular, ifq=l∈ℕ,
(9)Da+q,ρCxt=δlfs.Lemma 3. (see [20]).
Letf∈ACδla,b. Then, for every l−1<q<l,
(10)ρIa+qCDa+q,ρxt=xt−∑k=0l−1δkfaρkk!tρ−aρk.Moreover, for0<q<1, (10) becomes
(11)ρIa+qCDa+q,ρxt=xt−xa.Now, we will present a definition of Kuratowski’s measure of noncompactnessξ· which is constructed by
(12)ξD=infε>0:D=⋃i=1nDiandDiamDi≤εfori=1,⋯,n,where DiamDi=supx−y:x,y∈Di and D are a bounded subset of the Banach space Ct0,K. It is clear that 0≤ξD≤DiamD<+∞ [21].Definition 4. (see [21]).
LetG:V⟶W be bounded and continuous with V⊂W. Then, G will be ξ-Lipschitz if ∃ς≥0 so that
(13)ξGA<ςξA,∀boundedA⊂V.
As well asG is named as strict ξ-contraction when ς<1 holds.Definition 5. (see [21]).
A functionG is ξ-condensing if
(14)ξGA<ξA,∀A⊂Vbounded,withξA>0.
So,ξGA≥ξA gives ξA=0. Also, G:V⟶W is Lipschitz for ς>0 such that
(15)Gμ−Gμ′≤ςμ−μ′forallμ,μ′∈V.
Ifς<1, in this case G is called a strict contraction.Proposition 6. (see [21]).
G is ξ-Lipschitz with constant ς=0iffG:V⟶W is compact.Proposition 7. (see [21]).
A functionGisξ-Lipschitz with constant ςif and only if G:V⟶W is Lipschitz with Lipschitz constant ς.Theorem 8. (see [22]).
LetG:Ct0,K⟶Ct0,K be a ξ-condensing and
(16)M=u∈Ct0,K:θ∈0,1existssothatu=θGu.
IfM is a bounded subset contained in Ct0,K, i.e., a constant r>0 exists with M⊂Kr0, then degI−θG,Kr0,0=1 for all θ∈0,1. Therefore, G has a fixed point, and the set FIXG belongs to Kr0.Lemma 9. (see [13]).
Considerφp as an operator in the p-Laplacian settings.
(p1) For 1<p≤2, if Y1,Y2>0 and Y1,∣Y2∣≥λ>0, then
(17)φpY1−φpY2≤p−1λp−2Y1−Y2.
(p2) For p>2, if Y1,Y2≤λ∗, then
(18)∣φpY1−φpY2∣≤p−1λ∗p−2Y1−Y2.


## 3. Main Analytical Results

This important section is divided into some subsections. In each part, we shall study desired theorems about different specifications of solutions to the proposedp-Laplacian FBVP (4).


### 3.1. Existence-Uniqueness Results

We here present the first result which yields the solution of the proposedp-Laplacian FBVP (4) in the equivalent format of integral equations.Theorem 10.
Letℏ1,ℏ2,ℏ3∈Ct0,K, K>t0≥0, χ,β∈1,2, and δ,ρ,υ>0. The p-Laplacian FBVP with given integral composite conditions
(19)DCβ,δφpDCχ,ρxt=ℏ1t,xt0+μ1xK=σ1∫t0Kgsds,x′t0+μ2x′K=σ2∫t0Kℏ3sds,DCχ,ρxt0=0,DCχ,ρxK=υDCχ,ρxη,has a solution given by
(20)xt=∫t0KH1t,ϖφq∫t0KH2ϖ,sℏ1sds+σ1ℏ2ϖ1+μ1+σ2ℏ3ϖt0ρ−1+μ2Kρ−1tρ−t0ρρ−μ11+μ1Kρ−t0ρρdϖ,such that H1t,ϖ and H2t,s stand for Green’s functions defined as follows:
(21)H1t,ϖ=ρ1−χΓχtρ−ϖρχ−1ϖρ−1−μ11+μ1ρ1−χΓχKρ−ϖρχ−1ϖρ−1+μ2ρ2−χt0ρ−1+μ2Kρ−1Γχ−1×μ11+μ1Kρ−t0ρρ−tρ−t0ρρKρ−ϖρχ−2ϖKρ−1,−μ11+μ1ρ1−χΓχKρ−ϖρχ−1ϖρ−1+μ2ρ2−χt0ρ−1+μ2Kρ−1Γχ−1t0≤ϖ≤t≤K,μ11+μ1Kρ−t0ρρ−tρ−t0ρρKρ−ϖρχ−2ϖKρ−1,t0≤t≤ϖ≤K,(22)H2t,s=δ1−βΓβtδ−sδβ−1sδ−1−δ1−βtδ−t0δKδ−t0δ−υ1/q−1ηδ−t0δΓβKδ−sδβ−1sδ−1+δ1−βυ1/q−1tδ−t0δKδ−t0δ−υ1/q−1ηδ−t0δΓβηδ−sδβ−1sδ−1,t0≤s≤η≤t,≤K,δ1−βΓβtδ−sδβ−1sδ−1−δ1−βtδ−t0δKδ−sδβ−1sδ−1Kδ−t0δ−υ1/q−1ηδ−t0δΓβ,t0≤η≤s≤t,≤K,−δ1−βtδ−t0δKδ−t0δ−υ1/q−1ηδ−t0δΓβKδ−sδβ−1sδ−1,t0≤η≤t≤s.≤K.Proof.
By utilizing the integral operatorIβδ on (19), the following relation is produced
(23)φpDCχ,ρxt=Iβδℏ1t−a1−a2tδ−t0δδ,a1,a2∈ℝ.By taking the inverseφq of φp on above equation, we have
(24)DCχ,ρxt=φqδ1−βΓβ∫t0ttδ−sδβ−1sδ−1ℏ1sds−a1−a2tδ−t0δδ.Using the boundary conditionsDCχ,ρxt0=0 and DCχ,ρxK=υDCχ,ρxη, we have a1=0 and
(25)φqδ1−βΓβ∫t0KKδ−sδβ−1sδ−1ℏ1sds−a2Kδ−t0δδ=υφqδ1−βΓβ∫t0ηηδ−sδβ−1sδ−1ℏ1sds−a2ηδ−t0δδ=φqυ1/q−1δ1−βΓβ∫t0ηηδ−sδβ−1sδ−1ℏ1sds−a2υ1q−1ηδ−t0δδwhich is obtained by using the property of p-Laplacian operator. Therefore,
(26)a2Kδ−t0δδ−υ1q−1ηδ−t0δδ=δ1−βΓβ∫t0KKδ−sδβ−1sδ−1ℏ1sds−υ1/q−1δ1−βΓβ∫t0ηηδ−sδβ−1sδ−1ℏ1sds,which implies that
(27)a2=δ2−βKδ−t0δ−υ1/q−1ηδ−t0δΓβ∫t0KKδ−sδβ−1sδ−1ℏ1sds−υ1/q−1δ2−βKδ−t0δ−υ1/q−1ηδ−t0δΓβ∫t0ηηδ−sδβ−1sδ−1ℏ1sds.According to (24) and the definition of H2t,s, we get
(28)DCχ,ρxt=φq∫t0KH2t,sℏ1sds.Applying the integral operatorIχρ, we shall write
(29)xt=Iχρφq∫t0KH2t,sℏ1sds−b1−b2tρ−t0ρρ.By virtue of the boundary conditionxt0+μ1xK=σ1∫t0Kℏ2sds, we have
(30)−b1+μ1Iχ,ρφq∫t0KH2t,sℏ1sds−b1−b2Kρ−t0ρρ=σ1∫t0Kℏ2sds,which implies
(31)b1=μ11+μ1ρ1−χΓχ∫t0KKρ−ϖρχ−1ϖρ−1φq∫t0KH2ϖ,sℏ1sdsdϖ−μ11+μ1Kρ−t0ρρb2−σ11+μ1∫t0Kℏ2sds.Therefore, we obtain(32)x′t=ρ2−χΓχ−1∫t0ttρ−ϖρχ−2tϖρ−1φq∫t0KH2ϖ,sℏsdsdϖ−b2tρ−1.Using the boundary conditionx′t0+μ2x′K=σ2∫t0Kℏ3sds,(33)−b2t0ρ−1+μ2ρ2−χΓχ−1∫t0KKρ−ϖρχ−2ϖKρ−1φq∫t0KH2ϖ,sℏ1sdsdϖ−μ2b2Kρ−1=σ2∫t0Kℏ3sds.Then,(34)b2=μ2t0ρ−1+μ2Kρ−1ρ2−χΓχ−1∫t0KKρ−ϖρχ−2ϖKρ−1φq∫t0KH2ϖ,sℏ1sdsdϖ−σ2t0ρ−1+μ2Kρ−1∫t0Kℏ3sds.Therefore,(35)xt=ρ1−χΓχ∫t0ttρ−ϖρχ−1ϖρ−1φq∫t0KH2ϖ,sℏ1sdsdϖ−μ11+μ1ρ1−χΓχ∫t0KKρ−ϖρχ−1ϖρ−1φq∫t0KH2ϖ,sℏ1sdsdϖ+μ2ρ2−χt0ρ−1+μ2Kρ−1Γχ−1μ11+μ1Kρ−t0ρρ−tρ−t0ρρ×∫t0KKρ−ϖρχ−2ϖKρ−1φq∫t0KH2ϖ,sℏ1sdsdϖ+σ11+μ1∫t0Kℏ2ϖdϖ+σ2t0ρ−1+μ2Kρ−1tρ−t0ρρ−μ11+μ1Kρ−t0ρρ∫t0Kℏ3ϖdϖ=∫t0KH1t,ϖφq∫t0KH2ϖ,sℏ1sds+σ1ℏ2ϖ1+μ1+σ2hϖt0ρ−1+μ2Kρ−1tρ−t0ρρ−μ11+μ1Kρ−t0ρρdϖ,where H1t,ϖ is defined in (18). This ends the proof.In this part, we intend to state and prove our required existence-uniqueness theorems. To achieve such an intention and in view of Theorem10, the solution of the suggested p-Laplacian FBVP (4) is equivalent to a fixed point xt of the self–map Ξ:Ct0,K⟶Ct0,K which is formulated as
(36)Ξxt=∫t0KH1t,ϖφq∫t0KH2ϖ,sℏ1s,xsds+σ1ℏ2ϖ1+μ1+σ2ℏ3ϖt0ρ−1+μ2Kρ−1tρ−t0ρρ−μ11+μ1Kρ−t0ρρdϖ,where H1t,ϖ and H2ϖ,s are represented by (21) and (22), respectively. In the sequel, we utilize the following notations:
(37)Δ1=Kρ−t0ρρχμ11+μ1+11Γχ+1+μ2Kρ−1Γχt0ρ−1+μ2Kρ−1,Δ2=Kδ−t0δδβ1Γβ+1+Kδ−t0δKδ−t0δ−υ1/q−1ηδ−t0δKδ−t0δδβ1Γβ+1+ηδ−t0δδβυ1/q−1Γβ+1,Δ3=σ11+μ1ℏ2∗+σ2ℏ3∗t0ρ−1+μ2Kρ−1Kρ−t0ρρμ11+μ1+1,and ω1∗=supt∈t0,Kω1t, ω2∗=supt∈t0,Kω2t, Ω1=ω1∗Δ1Δ2q−1+Δ3, Ω2=ω2∗Δ1Δ2q−1, ÿ2∗=supt∈t0,K∣ℏ2t∣, and ℏ3∗=supt∈t0,Kℏ3t.Theorem 11.
Let (HP1): the functionsω1,ω2∈Ct0,K exist so that ℏ1t,x≤φpω1t+ω2txt for any x∈Ct0,K and t∈t0,K.
Then,Ξ:Ct0,K⟶Ct0,K is continuous, and also the growth condition Ξx≤Ω1+Ω2x holds.Proof.
Define a setBε=x∈Ct0,K:x≤ε having the boundedness property. In order to prove the continuity of Ξ, we consider xn as a sequence converging to x in Bε. Then, Lemma 9 yields
(38)Ξxnt−Ξxt=∫t0KH1t,ϖφq∫t0KH2ϖ,sℏ1s,xnsdsdϖ−∫t0KH1t,ϖφq∫t0KH2ϖ,sℏ1s,xsdsdϖ≤∫t0KH1t,ϖφq∫t0KH2ϖ,sℏ1s,xnsds−φq∫t0KH2ϖ,sℏ1s,xsdsdϖ≤q−1λq−2∫t0KH1t,ϖ∫t0KH2ϖ,sℏ1s,xns−ℏ1s,xsdsdϖ.
According to Lebesgue’s dominated convergence theorem and the continuity of the functionℏ1, we get Ξxn−Ξx⟶0 when n⟶∞. Hence, Ξ is continuous.Now, about the growth condition, by (HP1), we obtain(39)Ξxt≤∫t0KH1t,ϖφq∫t0KH2ϖ,sℏ1s,xsds+σ1ℏ2ϖ1+μ1+σ2ℏ3ϖt0ρ−1+μ2Kρ−1tρ−t0ρρ−μ11+μ1Kρ−t0ρρdϖ≤∫t0KH1t,ϖφq∫t0KH2ϖ,sφpω1s+ω2sxsds+σ1ℏ2ϖ1+μ1+σ2ℏ3ϖt0ρ−1+μ2Kρ−1tρ−t0ρρ−μ11+μ1Kρ−t0ρρdϖ≤∫t0KH1t,ϖΔ2q−1ω1∗+ω2∗x+σ1ℏ2ϖ1+μ1+σ2ℏ3ϖt0ρ−1+μ2Kρ−1tρ−t0ρρ−μ11+μ1Kρ−t0ρρdϖ≤Δ1Δ2q−1ω1∗+ω2∗x+Δ3≤ω1∗Δ1Δ2q−1+Δ3+ω2∗Δ1Δ2q−1x≤Ω1+Ω2x.Thus,Ξx≤Ω1+Ω2x and this complete the argument.Theorem 12.
Under hypothesis (HP1), the single-valued operatorΞ:Ct0,K⟶Ct0,K is ξ-Lipschitz with the constant zero and is compact.Proof.
In view of Theorem11, Ξ is bounded. In the subsequent step, we show that Ξ is an equicontinuous operator. Then, by the hypothesis (HP1), for any x∈Bε and t1,t2∈t0,K subject to t1<t2, we have
(40)Ξxt2−Ξxt1≤∫t0KH1t2,ϖφq∫t0KH2ϖ,sℏ1s,xsds+σ2ℏ3ϖt0ρ−1+μ2Kρ−1t2ρ−t0ρρdϖ−∫t0KH1t1,ϖφq∫t0KH2ϖ,sℏ1s,xsds+σ2ℏ3ϖt0ρ−1+μ2Kρ−1t1ρ−t0ρρdϖ≤∫t0KH1t2,ϖ−H1t1,ϖφq∫t0KH2ϖ,sφpω1s+ω2sxsds+∣σ2ℏ3ϖ∣∣t0ρ−1+μ2Kρ−1∣t2ρ−t0ρρ−t1ρ−t0ρρdϖ.
Clearly, the R.H.S. of (40) goes to zero by taking t2⟶t1, and so ΞBε is equicontinuous. Therefore, by virtue of the well-known Arzelá-Ascoli theorem, ΞBε is compact, and thus Proposition 6 gives a result stating this fact that Ξ is ξ-Lipschitz with the constant zero.Theorem 13.
Under the following hypothesis, i.e.,
(HP2) A real constantℓ exists so that for any μ1,μ2∈Ct0,K and t∈t0,K,
(41)ℏ1t,μ1−ℏ1t,μ2≤ℓμ1t−μ2t.The generalized p-Laplacian FBVP (4) has a unique solution such that
(42)ℓq−1λq−2Δ1Δ2<1.Proof.
ConsiderΞ as defined in (36). Then by Lemma 9, we obtain
(43)Ξxt−Ξyt≤∫t0KH1t,ϖφq∫t0KH2ϖ,sℏ1s,xsdsdϖ−∫t0KH1t,ϖφq∫t0KH2ϖ,sℏ1s,ysdsdϖ≤∫t0KH1t,ϖφq∫t0KH2ϖ,sℏ1s,xsds−φq∫t0KH2ϖ,sℏ1s,ysdsdϖ≤q−1λq−2∫t0KH1t,ϖ∫t0KH2ϖ,sℏ1s,xs−ℏÿ1s,ysdsdϖ≤ℓq−1λq−2∫t0KH1t,ϖ∫t0KH2ϖ,sxs−ysdsdϖ≤ℓq−1λq−2∫t0KH1t,ϖΔ2x−ydϖ≤ℓq−1λq−2Δ1Δ2x−y,for x,y∈Ct0,K. So, Ξx−Ξy≤ℓq−1λq−2Δ1Δ2x−y. Hence, in view of the well-known contraction principle due to Banach, we follow that Ξ admits a fixed point uniquely. Thus, the generalized p-Laplacian FBVP (4) involves a solution uniquely.Theorem 14.
If hypotheses (HP1) and (HP2) hold, then the generalizedp-Laplacian FBVP (4) has a solution such that Ω2<1. Moreover, the set containing solutions of the generalized p-Laplacian FBVP (4) is bounded.Proof.
According to Theorem13, Ξ is Lipschitz and by Proposition 7, Ξ is ξ-Lipschitz which yields that Ξ is ξ-condensing. With the aid of Theorem 8, we need to prove that
(44)W=x∈Ct0,K:θ∈0,1existssothatx=θΞx,is bounded. For this regard, we suppose that x∈W for some θ∈0,1 and for each t∈t0,K. Then, from the growth condition of Ξ derived in Theorem 11, we may write
(45)x=θΞx≤Ω1+Ω2x.
Hence,(46)x≤Ω11−Ω2,which yields that W is a bounded set contained in Ct0,K. By Theorem 8, one can understand that Ξ involves at least a fixed point which confirms the existence of at least a solution for the proposed generalized p-Laplacian FBVP (4), and hence W consisting of solutions of the mentioned FBVP (4) is a bounded subset of Ct0,K. This ends the proof.☐


### 3.2. Analysis of the Stability

In this part, we discuss on four kinds of stability for the generalizedp-Laplacian FBVP (4) as follows [6, 7].Definition 15.
The generalizedp-Laplacian FBVP (4) is called Ulam-Hyers stable if there is a real number Cÿ1>0 such that for every κ>0 and every solution x~∈Ct0,K of the inequality
(47)Dβ,δCφpCDχ,ρx~t−ℏ1t,x~t≤κ,
there is a unique solutionx∈Ct0,K of (4) such that
(48)x−x~≤Cÿ1κ,t∈t0,K.Definition 16.
The generalizedp-Laplacian FBVP (4) is called the generalized Ulam-Hyers stable with respect to ρ∈Cℝ>0,ℝ>0 with ρ0=0, if for each approximate solution x~∈Ct0,K of inequality (47), there is a unique solution x∈Ct0,K of (4) so that
(49)x−x~≤ρκ,t∈t0,K.Definition 17.
The generalizedp-Laplacian FBVP (4) is called Ulam-Hyers-Rassias stable with respect to ψ∈Ct0,K if there is a real number Kℏ1,ψ>0 such that for every κ>0 and approximate solution x~∈Ct0,K of the inequality
(50)Dβ,δCφpDχ,ρCx~t−ℏ1t,x~t≤ψtκ,there is a unique solution x∈Ct0,K of (4) so that
(51)x−x~≤Kÿ1,ψκψt,t∈t0,K.Definition 18.
The generalizedp-Laplacian FBVP (4) is called the generalized Ulam-Hyers-Rassias stable with respect to ψ∈Ct0,K, if there is a real number Kÿ1,ψ>0 such that for each approximate solution x~∈Ct0,K of inequality (50), there is a unique solution x∈Ct0,K of (4) such that
(52)x−x~≤Kÿ1,ψψt,t∈t0,K.Remark 19.
The functionx~∈Ct0,K is a solution of (47) if and only if there exists a function Ψ∈Ct0,K such that
Ψt≤κ for t∈t0,KDβ,δCφpCDχ,ρx~t=ℏ1t,x~t+Ψt for t∈t0,KRemark 20.
The functionx~∈Ct0,K is a solution of (50) if and only if there exists a function Φ∈Ct0,K such that
Φt≤κψt for t∈t0,KDβ,δCφpDCχ,ρx~t=ℏ1t,x~t+Φt for t∈t0,KTheorem 21.
If the hypothesis (HP2) and the inequality (42) are valid, then the unique solution of the generalized p-Laplacian FBVP (4) is Ulam-Hyers stable and is the generalized Ulam-Hyers stable.Proof.
Setκ>0 and let x~∈Ct0,K be the approximate solution of (47) and x∈Ct0,K be the unique solution of the approximate generalized p-Laplacian FBVP
(53)DCβ,δφpDCχ,ρx~t=ℏ1t,x~t+Ψt,t∈t0,K,t0≥0,x~t0+μ1x~K=σ1∫t0Kℏ2sds,μ1≠−1,x~′t0+μ2x~′K=σ2∫t0Kℏ3sds,μ2≠−t0Kρ−1,DCχ,ρx~t0=0,DCχ,ρx~K=υDCχ,ρx~η,η∈t0,K,with υ≠Kδ−t0δ/ηδ−t0δq−1. According to Theorem 10, we get
(54)x~t=∫t0KH1t,ϖφq∫t0KH2ϖ,sℏs,x~s+Ψtds+σ1ℏϖ1+μ1+σ2ℏ3ϖt0ρ−1+μ2Kρ−1tρ−t0ρρ−μ11+μ1Kρ−t0ρρdϖ,where H1t,ϖ and H2t,s are defined by (21) and (22), respectively. Hence, from Theorem 13, we estimate
(55)xt−x~t≤∫t0KH1t,ϖφq∫t0KH2ϖ,sℏ1s,xsdsdϖ−∫t0vH1t,ϖφq∫t0KH2ϖ,sℏ1s,x~s+Ψtdsdϖ≤∫t0KH1t,ϖφq∫t0KH2ϖ,sℏ1s,xsds−φq∫t0KH2ϖ,sℏ1s,x~sdsdϖ+∫t0KH1t,ϖφq∫t0KH2ϖ,sΨtdsdϖ≤ℓq−1λq−2Δ1Δ2x−x~+Δ1Δ2κq−1.
Thus,(56)x−x~≤Δ1Δ2κq−11−ℓq−1λq−2Δ1Δ2=Cÿ1κ,
whereCÿ1≔Δ1Δ2q−1κq−2/1−ℓq−1λq−2Δ1Δ2. This shows that the generalized p-Laplacian FBVP (4) is Ulam-Hyers stable. Along with this, if ∥x−x~∥≤ρκ so that ρ0=0, then the solution related to the generalized p-Laplacian FBVP (4) is the generalized Ulam-Hyers stable, and the proof is completed.Theorem 22.
Let the hypothesis (HP2) and (42) are valid, and there exists an increasing function ψt∈Ct0,K; there exists λψ>0 such that ∫t0Kψsds≤λψψt,∀t∈t0,K. Then, the unique solution of the generalized p-Laplacian FBVP (4) is Ulam-Hyers-Rassias stable and thus is the generalized Ulam-Hyers-Rassias stable.Proof.
Considerκ>0 and let x~∈Ct0,K be the approximate solution of (50) and x∈Ct0,K be the unique solution of the generalized p-Laplacian FBVP (4). By remark 20, we have
(57)DCβ,δφpDCχ,ρx~t=ℏ1t,x~t+Φt,t∈t0,K,t0≥0,x~t0+μ1x~K=σ1∫t0Kℏ2sds,μ1≠−1,x~′t0+μ2x~′K=σ2∫t0Kℏ3sds,μ2≠−t0Kρ−1,DCχ,ρx~t0=0,DCχ,ρx~K=υDCχ,ρx~η,η∈t0,K,
withυ≠Kδ−t0δ/ηδ−t0δq−1. In view of Theorem 10, we have
(58)x~t=∫t0KH1t,ϖφq∫t0KH2ϖ,sℏ1s,x~s+Φsds+σ1ℏ2ϖ1+μ1+σ2ℏ3ϖt0ρ−1+μ2Kρ−1tρ−t0ρρ−μ11+μ1Kρ−t0ρρdϖ,where H1t,ϖ and H2t,s are defined by (21) and (22), respectively. Hence, we can immediately estimate that
(59)x−x~≤Δ1Δ2κλψψtq−11−ℓq−1λq−2Δ1Δ2=Kℏ,ψκψt,
whereKÿ1,ψ≔Δ1Δ2q−1λψq−1κψtq−2/1−ℓq−1λq−2Δ1Δ2. This proves that the generalized p-Laplacian FBVP (4) is Ulam-Hyers-Rassias stable. Furthermore, if κ=1, then the solution of generalized p-Laplacian FBVP (4) is the generalized Ulam-Hyers-Rassias stable, and the proof is completed.


## 3.1. Existence-Uniqueness Results

We here present the first result which yields the solution of the proposedp-Laplacian FBVP (4) in the equivalent format of integral equations.Theorem 10.
Letℏ1,ℏ2,ℏ3∈Ct0,K, K>t0≥0, χ,β∈1,2, and δ,ρ,υ>0. The p-Laplacian FBVP with given integral composite conditions
(19)DCβ,δφpDCχ,ρxt=ℏ1t,xt0+μ1xK=σ1∫t0Kgsds,x′t0+μ2x′K=σ2∫t0Kℏ3sds,DCχ,ρxt0=0,DCχ,ρxK=υDCχ,ρxη,has a solution given by
(20)xt=∫t0KH1t,ϖφq∫t0KH2ϖ,sℏ1sds+σ1ℏ2ϖ1+μ1+σ2ℏ3ϖt0ρ−1+μ2Kρ−1tρ−t0ρρ−μ11+μ1Kρ−t0ρρdϖ,such that H1t,ϖ and H2t,s stand for Green’s functions defined as follows:
(21)H1t,ϖ=ρ1−χΓχtρ−ϖρχ−1ϖρ−1−μ11+μ1ρ1−χΓχKρ−ϖρχ−1ϖρ−1+μ2ρ2−χt0ρ−1+μ2Kρ−1Γχ−1×μ11+μ1Kρ−t0ρρ−tρ−t0ρρKρ−ϖρχ−2ϖKρ−1,−μ11+μ1ρ1−χΓχKρ−ϖρχ−1ϖρ−1+μ2ρ2−χt0ρ−1+μ2Kρ−1Γχ−1t0≤ϖ≤t≤K,μ11+μ1Kρ−t0ρρ−tρ−t0ρρKρ−ϖρχ−2ϖKρ−1,t0≤t≤ϖ≤K,(22)H2t,s=δ1−βΓβtδ−sδβ−1sδ−1−δ1−βtδ−t0δKδ−t0δ−υ1/q−1ηδ−t0δΓβKδ−sδβ−1sδ−1+δ1−βυ1/q−1tδ−t0δKδ−t0δ−υ1/q−1ηδ−t0δΓβηδ−sδβ−1sδ−1,t0≤s≤η≤t,≤K,δ1−βΓβtδ−sδβ−1sδ−1−δ1−βtδ−t0δKδ−sδβ−1sδ−1Kδ−t0δ−υ1/q−1ηδ−t0δΓβ,t0≤η≤s≤t,≤K,−δ1−βtδ−t0δKδ−t0δ−υ1/q−1ηδ−t0δΓβKδ−sδβ−1sδ−1,t0≤η≤t≤s.≤K.Proof.
By utilizing the integral operatorIβδ on (19), the following relation is produced
(23)φpDCχ,ρxt=Iβδℏ1t−a1−a2tδ−t0δδ,a1,a2∈ℝ.By taking the inverseφq of φp on above equation, we have
(24)DCχ,ρxt=φqδ1−βΓβ∫t0ttδ−sδβ−1sδ−1ℏ1sds−a1−a2tδ−t0δδ.Using the boundary conditionsDCχ,ρxt0=0 and DCχ,ρxK=υDCχ,ρxη, we have a1=0 and
(25)φqδ1−βΓβ∫t0KKδ−sδβ−1sδ−1ℏ1sds−a2Kδ−t0δδ=υφqδ1−βΓβ∫t0ηηδ−sδβ−1sδ−1ℏ1sds−a2ηδ−t0δδ=φqυ1/q−1δ1−βΓβ∫t0ηηδ−sδβ−1sδ−1ℏ1sds−a2υ1q−1ηδ−t0δδwhich is obtained by using the property of p-Laplacian operator. Therefore,
(26)a2Kδ−t0δδ−υ1q−1ηδ−t0δδ=δ1−βΓβ∫t0KKδ−sδβ−1sδ−1ℏ1sds−υ1/q−1δ1−βΓβ∫t0ηηδ−sδβ−1sδ−1ℏ1sds,which implies that
(27)a2=δ2−βKδ−t0δ−υ1/q−1ηδ−t0δΓβ∫t0KKδ−sδβ−1sδ−1ℏ1sds−υ1/q−1δ2−βKδ−t0δ−υ1/q−1ηδ−t0δΓβ∫t0ηηδ−sδβ−1sδ−1ℏ1sds.According to (24) and the definition of H2t,s, we get
(28)DCχ,ρxt=φq∫t0KH2t,sℏ1sds.Applying the integral operatorIχρ, we shall write
(29)xt=Iχρφq∫t0KH2t,sℏ1sds−b1−b2tρ−t0ρρ.By virtue of the boundary conditionxt0+μ1xK=σ1∫t0Kℏ2sds, we have
(30)−b1+μ1Iχ,ρφq∫t0KH2t,sℏ1sds−b1−b2Kρ−t0ρρ=σ1∫t0Kℏ2sds,which implies
(31)b1=μ11+μ1ρ1−χΓχ∫t0KKρ−ϖρχ−1ϖρ−1φq∫t0KH2ϖ,sℏ1sdsdϖ−μ11+μ1Kρ−t0ρρb2−σ11+μ1∫t0Kℏ2sds.Therefore, we obtain(32)x′t=ρ2−χΓχ−1∫t0ttρ−ϖρχ−2tϖρ−1φq∫t0KH2ϖ,sℏsdsdϖ−b2tρ−1.Using the boundary conditionx′t0+μ2x′K=σ2∫t0Kℏ3sds,(33)−b2t0ρ−1+μ2ρ2−χΓχ−1∫t0KKρ−ϖρχ−2ϖKρ−1φq∫t0KH2ϖ,sℏ1sdsdϖ−μ2b2Kρ−1=σ2∫t0Kℏ3sds.Then,(34)b2=μ2t0ρ−1+μ2Kρ−1ρ2−χΓχ−1∫t0KKρ−ϖρχ−2ϖKρ−1φq∫t0KH2ϖ,sℏ1sdsdϖ−σ2t0ρ−1+μ2Kρ−1∫t0Kℏ3sds.Therefore,(35)xt=ρ1−χΓχ∫t0ttρ−ϖρχ−1ϖρ−1φq∫t0KH2ϖ,sℏ1sdsdϖ−μ11+μ1ρ1−χΓχ∫t0KKρ−ϖρχ−1ϖρ−1φq∫t0KH2ϖ,sℏ1sdsdϖ+μ2ρ2−χt0ρ−1+μ2Kρ−1Γχ−1μ11+μ1Kρ−t0ρρ−tρ−t0ρρ×∫t0KKρ−ϖρχ−2ϖKρ−1φq∫t0KH2ϖ,sℏ1sdsdϖ+σ11+μ1∫t0Kℏ2ϖdϖ+σ2t0ρ−1+μ2Kρ−1tρ−t0ρρ−μ11+μ1Kρ−t0ρρ∫t0Kℏ3ϖdϖ=∫t0KH1t,ϖφq∫t0KH2ϖ,sℏ1sds+σ1ℏ2ϖ1+μ1+σ2hϖt0ρ−1+μ2Kρ−1tρ−t0ρρ−μ11+μ1Kρ−t0ρρdϖ,where H1t,ϖ is defined in (18). This ends the proof.In this part, we intend to state and prove our required existence-uniqueness theorems. To achieve such an intention and in view of Theorem10, the solution of the suggested p-Laplacian FBVP (4) is equivalent to a fixed point xt of the self–map Ξ:Ct0,K⟶Ct0,K which is formulated as
(36)Ξxt=∫t0KH1t,ϖφq∫t0KH2ϖ,sℏ1s,xsds+σ1ℏ2ϖ1+μ1+σ2ℏ3ϖt0ρ−1+μ2Kρ−1tρ−t0ρρ−μ11+μ1Kρ−t0ρρdϖ,where H1t,ϖ and H2ϖ,s are represented by (21) and (22), respectively. In the sequel, we utilize the following notations:
(37)Δ1=Kρ−t0ρρχμ11+μ1+11Γχ+1+μ2Kρ−1Γχt0ρ−1+μ2Kρ−1,Δ2=Kδ−t0δδβ1Γβ+1+Kδ−t0δKδ−t0δ−υ1/q−1ηδ−t0δKδ−t0δδβ1Γβ+1+ηδ−t0δδβυ1/q−1Γβ+1,Δ3=σ11+μ1ℏ2∗+σ2ℏ3∗t0ρ−1+μ2Kρ−1Kρ−t0ρρμ11+μ1+1,and ω1∗=supt∈t0,Kω1t, ω2∗=supt∈t0,Kω2t, Ω1=ω1∗Δ1Δ2q−1+Δ3, Ω2=ω2∗Δ1Δ2q−1, ÿ2∗=supt∈t0,K∣ℏ2t∣, and ℏ3∗=supt∈t0,Kℏ3t.Theorem 11.
Let (HP1): the functionsω1,ω2∈Ct0,K exist so that ℏ1t,x≤φpω1t+ω2txt for any x∈Ct0,K and t∈t0,K.
Then,Ξ:Ct0,K⟶Ct0,K is continuous, and also the growth condition Ξx≤Ω1+Ω2x holds.Proof.
Define a setBε=x∈Ct0,K:x≤ε having the boundedness property. In order to prove the continuity of Ξ, we consider xn as a sequence converging to x in Bε. Then, Lemma 9 yields
(38)Ξxnt−Ξxt=∫t0KH1t,ϖφq∫t0KH2ϖ,sℏ1s,xnsdsdϖ−∫t0KH1t,ϖφq∫t0KH2ϖ,sℏ1s,xsdsdϖ≤∫t0KH1t,ϖφq∫t0KH2ϖ,sℏ1s,xnsds−φq∫t0KH2ϖ,sℏ1s,xsdsdϖ≤q−1λq−2∫t0KH1t,ϖ∫t0KH2ϖ,sℏ1s,xns−ℏ1s,xsdsdϖ.
According to Lebesgue’s dominated convergence theorem and the continuity of the functionℏ1, we get Ξxn−Ξx⟶0 when n⟶∞. Hence, Ξ is continuous.Now, about the growth condition, by (HP1), we obtain(39)Ξxt≤∫t0KH1t,ϖφq∫t0KH2ϖ,sℏ1s,xsds+σ1ℏ2ϖ1+μ1+σ2ℏ3ϖt0ρ−1+μ2Kρ−1tρ−t0ρρ−μ11+μ1Kρ−t0ρρdϖ≤∫t0KH1t,ϖφq∫t0KH2ϖ,sφpω1s+ω2sxsds+σ1ℏ2ϖ1+μ1+σ2ℏ3ϖt0ρ−1+μ2Kρ−1tρ−t0ρρ−μ11+μ1Kρ−t0ρρdϖ≤∫t0KH1t,ϖΔ2q−1ω1∗+ω2∗x+σ1ℏ2ϖ1+μ1+σ2ℏ3ϖt0ρ−1+μ2Kρ−1tρ−t0ρρ−μ11+μ1Kρ−t0ρρdϖ≤Δ1Δ2q−1ω1∗+ω2∗x+Δ3≤ω1∗Δ1Δ2q−1+Δ3+ω2∗Δ1Δ2q−1x≤Ω1+Ω2x.Thus,Ξx≤Ω1+Ω2x and this complete the argument.Theorem 12.
Under hypothesis (HP1), the single-valued operatorΞ:Ct0,K⟶Ct0,K is ξ-Lipschitz with the constant zero and is compact.Proof.
In view of Theorem11, Ξ is bounded. In the subsequent step, we show that Ξ is an equicontinuous operator. Then, by the hypothesis (HP1), for any x∈Bε and t1,t2∈t0,K subject to t1<t2, we have
(40)Ξxt2−Ξxt1≤∫t0KH1t2,ϖφq∫t0KH2ϖ,sℏ1s,xsds+σ2ℏ3ϖt0ρ−1+μ2Kρ−1t2ρ−t0ρρdϖ−∫t0KH1t1,ϖφq∫t0KH2ϖ,sℏ1s,xsds+σ2ℏ3ϖt0ρ−1+μ2Kρ−1t1ρ−t0ρρdϖ≤∫t0KH1t2,ϖ−H1t1,ϖφq∫t0KH2ϖ,sφpω1s+ω2sxsds+∣σ2ℏ3ϖ∣∣t0ρ−1+μ2Kρ−1∣t2ρ−t0ρρ−t1ρ−t0ρρdϖ.
Clearly, the R.H.S. of (40) goes to zero by taking t2⟶t1, and so ΞBε is equicontinuous. Therefore, by virtue of the well-known Arzelá-Ascoli theorem, ΞBε is compact, and thus Proposition 6 gives a result stating this fact that Ξ is ξ-Lipschitz with the constant zero.Theorem 13.
Under the following hypothesis, i.e.,
(HP2) A real constantℓ exists so that for any μ1,μ2∈Ct0,K and t∈t0,K,
(41)ℏ1t,μ1−ℏ1t,μ2≤ℓμ1t−μ2t.The generalized p-Laplacian FBVP (4) has a unique solution such that
(42)ℓq−1λq−2Δ1Δ2<1.Proof.
ConsiderΞ as defined in (36). Then by Lemma 9, we obtain
(43)Ξxt−Ξyt≤∫t0KH1t,ϖφq∫t0KH2ϖ,sℏ1s,xsdsdϖ−∫t0KH1t,ϖφq∫t0KH2ϖ,sℏ1s,ysdsdϖ≤∫t0KH1t,ϖφq∫t0KH2ϖ,sℏ1s,xsds−φq∫t0KH2ϖ,sℏ1s,ysdsdϖ≤q−1λq−2∫t0KH1t,ϖ∫t0KH2ϖ,sℏ1s,xs−ℏÿ1s,ysdsdϖ≤ℓq−1λq−2∫t0KH1t,ϖ∫t0KH2ϖ,sxs−ysdsdϖ≤ℓq−1λq−2∫t0KH1t,ϖΔ2x−ydϖ≤ℓq−1λq−2Δ1Δ2x−y,for x,y∈Ct0,K. So, Ξx−Ξy≤ℓq−1λq−2Δ1Δ2x−y. Hence, in view of the well-known contraction principle due to Banach, we follow that Ξ admits a fixed point uniquely. Thus, the generalized p-Laplacian FBVP (4) involves a solution uniquely.Theorem 14.
If hypotheses (HP1) and (HP2) hold, then the generalizedp-Laplacian FBVP (4) has a solution such that Ω2<1. Moreover, the set containing solutions of the generalized p-Laplacian FBVP (4) is bounded.Proof.
According to Theorem13, Ξ is Lipschitz and by Proposition 7, Ξ is ξ-Lipschitz which yields that Ξ is ξ-condensing. With the aid of Theorem 8, we need to prove that
(44)W=x∈Ct0,K:θ∈0,1existssothatx=θΞx,is bounded. For this regard, we suppose that x∈W for some θ∈0,1 and for each t∈t0,K. Then, from the growth condition of Ξ derived in Theorem 11, we may write
(45)x=θΞx≤Ω1+Ω2x.
Hence,(46)x≤Ω11−Ω2,which yields that W is a bounded set contained in Ct0,K. By Theorem 8, one can understand that Ξ involves at least a fixed point which confirms the existence of at least a solution for the proposed generalized p-Laplacian FBVP (4), and hence W consisting of solutions of the mentioned FBVP (4) is a bounded subset of Ct0,K. This ends the proof.☐


## 3.2. Analysis of the Stability

In this part, we discuss on four kinds of stability for the generalizedp-Laplacian FBVP (4) as follows [6, 7].Definition 15.
The generalizedp-Laplacian FBVP (4) is called Ulam-Hyers stable if there is a real number Cÿ1>0 such that for every κ>0 and every solution x~∈Ct0,K of the inequality
(47)Dβ,δCφpCDχ,ρx~t−ℏ1t,x~t≤κ,
there is a unique solutionx∈Ct0,K of (4) such that
(48)x−x~≤Cÿ1κ,t∈t0,K.Definition 16.
The generalizedp-Laplacian FBVP (4) is called the generalized Ulam-Hyers stable with respect to ρ∈Cℝ>0,ℝ>0 with ρ0=0, if for each approximate solution x~∈Ct0,K of inequality (47), there is a unique solution x∈Ct0,K of (4) so that
(49)x−x~≤ρκ,t∈t0,K.Definition 17.
The generalizedp-Laplacian FBVP (4) is called Ulam-Hyers-Rassias stable with respect to ψ∈Ct0,K if there is a real number Kℏ1,ψ>0 such that for every κ>0 and approximate solution x~∈Ct0,K of the inequality
(50)Dβ,δCφpDχ,ρCx~t−ℏ1t,x~t≤ψtκ,there is a unique solution x∈Ct0,K of (4) so that
(51)x−x~≤Kÿ1,ψκψt,t∈t0,K.Definition 18.
The generalizedp-Laplacian FBVP (4) is called the generalized Ulam-Hyers-Rassias stable with respect to ψ∈Ct0,K, if there is a real number Kÿ1,ψ>0 such that for each approximate solution x~∈Ct0,K of inequality (50), there is a unique solution x∈Ct0,K of (4) such that
(52)x−x~≤Kÿ1,ψψt,t∈t0,K.Remark 19.
The functionx~∈Ct0,K is a solution of (47) if and only if there exists a function Ψ∈Ct0,K such that
Ψt≤κ for t∈t0,KDβ,δCφpCDχ,ρx~t=ℏ1t,x~t+Ψt for t∈t0,KRemark 20.
The functionx~∈Ct0,K is a solution of (50) if and only if there exists a function Φ∈Ct0,K such that
Φt≤κψt for t∈t0,KDβ,δCφpDCχ,ρx~t=ℏ1t,x~t+Φt for t∈t0,KTheorem 21.
If the hypothesis (HP2) and the inequality (42) are valid, then the unique solution of the generalized p-Laplacian FBVP (4) is Ulam-Hyers stable and is the generalized Ulam-Hyers stable.Proof.
Setκ>0 and let x~∈Ct0,K be the approximate solution of (47) and x∈Ct0,K be the unique solution of the approximate generalized p-Laplacian FBVP
(53)DCβ,δφpDCχ,ρx~t=ℏ1t,x~t+Ψt,t∈t0,K,t0≥0,x~t0+μ1x~K=σ1∫t0Kℏ2sds,μ1≠−1,x~′t0+μ2x~′K=σ2∫t0Kℏ3sds,μ2≠−t0Kρ−1,DCχ,ρx~t0=0,DCχ,ρx~K=υDCχ,ρx~η,η∈t0,K,with υ≠Kδ−t0δ/ηδ−t0δq−1. According to Theorem 10, we get
(54)x~t=∫t0KH1t,ϖφq∫t0KH2ϖ,sℏs,x~s+Ψtds+σ1ℏϖ1+μ1+σ2ℏ3ϖt0ρ−1+μ2Kρ−1tρ−t0ρρ−μ11+μ1Kρ−t0ρρdϖ,where H1t,ϖ and H2t,s are defined by (21) and (22), respectively. Hence, from Theorem 13, we estimate
(55)xt−x~t≤∫t0KH1t,ϖφq∫t0KH2ϖ,sℏ1s,xsdsdϖ−∫t0vH1t,ϖφq∫t0KH2ϖ,sℏ1s,x~s+Ψtdsdϖ≤∫t0KH1t,ϖφq∫t0KH2ϖ,sℏ1s,xsds−φq∫t0KH2ϖ,sℏ1s,x~sdsdϖ+∫t0KH1t,ϖφq∫t0KH2ϖ,sΨtdsdϖ≤ℓq−1λq−2Δ1Δ2x−x~+Δ1Δ2κq−1.
Thus,(56)x−x~≤Δ1Δ2κq−11−ℓq−1λq−2Δ1Δ2=Cÿ1κ,
whereCÿ1≔Δ1Δ2q−1κq−2/1−ℓq−1λq−2Δ1Δ2. This shows that the generalized p-Laplacian FBVP (4) is Ulam-Hyers stable. Along with this, if ∥x−x~∥≤ρκ so that ρ0=0, then the solution related to the generalized p-Laplacian FBVP (4) is the generalized Ulam-Hyers stable, and the proof is completed.Theorem 22.
Let the hypothesis (HP2) and (42) are valid, and there exists an increasing function ψt∈Ct0,K; there exists λψ>0 such that ∫t0Kψsds≤λψψt,∀t∈t0,K. Then, the unique solution of the generalized p-Laplacian FBVP (4) is Ulam-Hyers-Rassias stable and thus is the generalized Ulam-Hyers-Rassias stable.Proof.
Considerκ>0 and let x~∈Ct0,K be the approximate solution of (50) and x∈Ct0,K be the unique solution of the generalized p-Laplacian FBVP (4). By remark 20, we have
(57)DCβ,δφpDCχ,ρx~t=ℏ1t,x~t+Φt,t∈t0,K,t0≥0,x~t0+μ1x~K=σ1∫t0Kℏ2sds,μ1≠−1,x~′t0+μ2x~′K=σ2∫t0Kℏ3sds,μ2≠−t0Kρ−1,DCχ,ρx~t0=0,DCχ,ρx~K=υDCχ,ρx~η,η∈t0,K,
withυ≠Kδ−t0δ/ηδ−t0δq−1. In view of Theorem 10, we have
(58)x~t=∫t0KH1t,ϖφq∫t0KH2ϖ,sℏ1s,x~s+Φsds+σ1ℏ2ϖ1+μ1+σ2ℏ3ϖt0ρ−1+μ2Kρ−1tρ−t0ρρ−μ11+μ1Kρ−t0ρρdϖ,where H1t,ϖ and H2t,s are defined by (21) and (22), respectively. Hence, we can immediately estimate that
(59)x−x~≤Δ1Δ2κλψψtq−11−ℓq−1λq−2Δ1Δ2=Kℏ,ψκψt,
whereKÿ1,ψ≔Δ1Δ2q−1λψq−1κψtq−2/1−ℓq−1λq−2Δ1Δ2. This proves that the generalized p-Laplacian FBVP (4) is Ulam-Hyers-Rassias stable. Furthermore, if κ=1, then the solution of generalized p-Laplacian FBVP (4) is the generalized Ulam-Hyers-Rassias stable, and the proof is completed.


## 4. Example

As an application to validate the theoretical results, an illustrative example is given here.Example 23.
Regarding to the given FBVP (4), we provide a special structure of the generalized problem having the fractional composite p-Laplacian as
(60)DC1.08,0.5φ54DC1.09,0.7xt=6sinxt12000sinxt+12000+0.18et,t∈0,1x0+0.1x1=0.8∫012e3s+1ds,x′0+0.2x′1=0.9∫01cossds,DC1.09,0.7x0=0,DC1.09,0.7x1=0.3DC1.09,0.7x0.25,
in which the following parameters are consideredβ=1.08, χ=1.09, δ=0.5, ρ=0.7, μ1=0.1, μ2=0.2, υ=0.3, σ1=0.8, σ2=0.9, η=0.25, λ=1.5, p=5/4, q=5, t0=0, K=1, and φpθ=θp−2θ. In addition to these, the continuous functions ÿ1, ÿ2, and ÿ3 are introduced by
(61)ℏ1t,xt=6sinxt12000sinxt+12000+0.18etd,ℏ2t=2e3t+1,ℏ3t=cost,for t∈0,1. By utilizing some of above data, we get Δ1≃=3.0899 and Δ2≃6.4094. On the other side, for any x,x^∈ℝ, we can write
(62)ℏ1t,xt−ℏ1t,x^t≤6sinxt12000sinxt+12000−6sinx^t12000sinx^t+12000≤612000sinxtsinxt+1−sinx^tsinx^t+1≤612000sinxt−sinx^t≤612000xt−x^t,where ℓ=6/12000=0.0005 is obtained. Then, since
(63)ℓq−1λq−2Δ1Δ2≃0.1336797342<1,thus the conditions of Theorem 13 are satisfied, and so the generalized composite p-Laplacian FBVP (60) has a unique solution. On the other side, since all hypotheses of Theorems 21 and 22 hold, we find out that the given generalized composite p-Laplacian FBVP (60) is Ulam-Hyers and Ulam-Hyers-Rassias stable and thus is stable of their generalized type.


## 5. Conclusion

Qualitative analysis such as the investigation of the existence, uniqueness, and stability of fractional differential equations is an important and useful task. In this paper, we studied a generalized fractional composite differential equation withp-Laplacian operator equipped with three-point integral boundary value conditions. We used the classical results for this purpose and obtained the relevant Green’s function. The existence and uniqueness of solutions were established by means of topological degree theory and Banach contraction principle. Besides, four types of stability in the sense of Ulam-Hyers, Ulam-Hyers-Rassias, and their generalized versions were analyzed. Finally, we provided an illustrative example to validate our results. In the next researches, one can study these qualitative behaviors of solutions for different generalizations of fractional p-Laplacian boundary value problems by means of generalized operators with nonsingular kernels such as Caputo-Fabrizio operators or Atangana-Baleanu operators.



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*Source: 9554076-2021-10-25.xml*