# Existence and Stability Results for Caputo-Type Sequential Fractional Differential Equations with New Kind of Boundary Conditions

Authors: Muath Awadalla; Murugesan Manigandan
Journal: Mathematical Problems in Engineering
 (2022)

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## Abstract

In this paper, we present the existence and the stability results for a nonlinear coupled system of sequential fractional differential equations supplemented with a new kind of coupled boundary conditions. Existence and uniqueness results are established by using Schaefer’s fixed point theorem and Banach’s contraction mapping principle. We examine the stability of the solutions involved in the Hyers–Ulam type. A few examples are presented to illustrate the main results.


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## Body

## 1. Introduction

In recent years, it has been demonstrated that the definition of fractional calculus is more suitable for describing historical dependence processes than the local limit definitions of ordinary or partial differential equations, and it has garnered increasing attention in a variety of fields [1–4]. Fractional calculus have arisen as a significant topic of research due to the multiple applications of their techniques in scientific and technical disciplines, such as monetary economics, ecological economics, aerodynamics, marketing, mathematical physics, mathematical biology, aeronautics, and financial mathematics [5–11]. Numerous scholars have become interested in this area of mathematical analysis as a result of its popularity and have contributed to its numerous facets. The boundary value problems of fractional order, in particular, garnered much attention. See [12–19] for the most recent results on FDEs with multipoint and integral boundary conditions.Numerous authors have also examined coupled systems of differential equations of fractional order. These systems naturally exist in a wide variety of real-world circumstances [19–22]. A series of papers [19, 20, 23–27] and the sources listed therein contain some recent results on this subject. More recently, in [28], the authors discussed the existence and uniqueness of coupled system of nonlinear FDEs with a new kind of coupled boundary conditions specified by(1)CDϖut=ft,ut,vt,t∈ℱ=0,T,CDϖut=gt,ut,vt,t∈ℱ=0,T,u+v0=−u+vT,∫ξηu−vsds=A,where DχC is the Caputo fractional derivatives (CFDs) of order χ∈ϖ,ξ, ϖ,ξ∈0,1, f,g:0,T×ℝ2⟶ℝ are continuous functions, and A is a non-negative constant. The authors proved the existence and uniqueness results with the aid of standard fixed point to obtain their results. Recently, Subramanian et al. [29] studied the existence of positive solutions for nonlinear coupled system of fractional differential equations complemented with boundary conditions.In [19], authors studied the existence for a the system of nonlinear coupled Caputo-type SFDEs and inclusions subject to multipoint and fractional integral boundary conditions of the form(2)DϖC+φDϖ−1Cpζ=ϑ1ζ,pζ,qζ,ζ∈ℱ=0,T,DϖC+φDϖ−1Cqζ=ϑ2ζ,pζ,qζ,ζ∈ℱ=0,T,p+q0=−p+qT,∑i=0mxip−qξi+μ∫0ηη−sδ−1Γδp−qsds=Υ,and(3)CDϖ+φCDϖ−1pζ∈ϑ1ζ,pζ,qζ,ζ∈ℱ=0,T,CDϖ+φCDϖ−1qζ∈ϑ2ζ,pζ,qζ,ζ∈ℱ=0,T,p+q0=−p+qT,∑i=0mxip−qξi+μ∫0ηη−sδ−1Γδp−qsds=Υ,where ς∈ϖ,ξ, ϖ,ξ∈0,1, ϑ1,ϑ2:0,T×ℝ2⟶ℝ are continuous functions, Uℝ is the collection of nonempty subsets of ℝ, and Υ is positive constant. The author investigation is mainly based on the theorems of Schaefer’s, Banach’s, Covitz–Nadler, and nonlinear alternatives for kakutani.In this article, we consider the following system of nonlinear sequential fractional differential equations,(4)CDϖ+φCDϖ−1pζ=ϑ1ζ,pζ,qζ,ζ∈ℱ=0,1,CDϖ+φCDϖ−1qζ=ϑ2ζ,pζ,qζ,ζ∈ℱ=0,1,p+q0=0,p+q0=−p+q1,μ∫0ηη−sδ−1Γδp−qsds=Υ,where CDϖ and DξC are the Caputo fractional derivatives (CFDs) of orders ϖ,ξ∈1,2, ϑ1,ϑ2:0,1×ℝ2⟶ℝ are continuous functions, and Υ is a non-negative constant. The boundary conditions (BCs) introduced in the problem (4) can be interpreted as the sum of the unknown functions p and q at the interval 0,1 equals zero, while the contribution of sum of the unknown functions on an arbitrary domain 0,η of the given interval 0,1 remains constant. Existence and uniqueness of the above mentioned nonlinear system is investigated. Unlike in [28], the main results of this article are entirely different. Because we consider the problems in the context of sequential fractional differential equations, unique techniques based on Schaefer’s and Banach’s contraction mapping principle are employed. In addition, the Ulam–Hyers stability technique for the problem (4) is also investigated, while it was not considered in [28, 30–33].The rest of the paper is organized as follows: in Section2, we recall some basic definitions from fractional calculus and present an auxiliary result, which plays a pivotal role in transforming system into equivalent integral equations. In Section 3, the existence results for the problem at hand are proved via the standard fixed point theorems. In Section 4, we present certain criteria under which the proposed problem is Ulam-Hyers stable. Furthermore, as an application, two examples are given to shows the applicability of the obtained results.


## 2. Preliminaries

In this section we present some definitions and auxiliary results that will be used to prove our main theorems (see [13, 14, 21, 22, 33, 34]).The space of Lebesgue measurable functionsf:b,c⟶ℝ,fXaq<∞, where a∈ℝ, 1≤q<∞ and(5)fXaq=∫bcθafθqdθθ1/q,1≤q<∞.Definition 1.
The generalized Riemann–Liouville fractional integral (GRLFI) of orderτ>0 and ρ>0, of a function f∈Xaqb,c, for all −∞<b<ζ<c<∞, is defined as(6)ρℐb+τfζ=ρ1−τΓτ∫bζθρ−1fθζρ−θρ1−τdθ,and(7)ρℐc−τfζ=ρ1−τΓτ∫ζcθρ−1fθθρ−ζρ1−τdθ,for ζ∈b,c are called the left and right sided generalized Riemann–Liouville fractional integral (GRLFI) of order τ, respectively. The operators ρℐb+τf and ρℐc−τf are defined for f∈Xaqb,c.Remark 1.
The above definition for generalized Riemann–Liouville fractional integral (GRLFI)s reduce to the RLFIs forρ⟶1.(8)ℐb+τfζ=1Γτ∫bζζ−θτ−1fθdθ,and(9)ℐc−τfζ=1Γτ∫ζcθ−ζτ−1fθdθ.Definition 2.
The RL fractional derivative of orderτ>0, n−1<τ<n, n∈ℕ, is defined as(10)D0+τfζ=1Γn−τζdζn∫ζ−θn−τ−1fθdθ,where the function fζ has absolutely continuous derivative up to order n−1.Definition 3.
The Caputo fractional derivative (CFD) of orderτ for a function f:0,∞⟶ℝ can be written as(11)DτCfζ=D0τ+fζ−∑j=0n−1ζjj!fj0,ζ>0,n−1<τ<n.Remark 2.
Iffζ∈Cn0,∞, then,(12)DτCfζ=1Γn−τ∫0ζfnθζ−θτ+1−ndθ=ℐn−τfnζ,ζ>0,n−1<τ<n.
The following auxiliary lemma, which concerns the linear variant of problem (4) plays a key role in the sequel.Lemma 1.
Letϑ^1,ϑ^2∈C0,1 and p,q∈ACℱ. The solution of the linear system of FDEs(13)CDϖ+φCDϖ−1pζ=ϑ^1ζ,ζ∈ℱ≔0,1,CDξ+φCDξ−1qζ=ϑ^2ζ,ζ∈ℱ≔0,1,p+q0=0,p+q0=−p+q1,μ∫0ηη−sδ−1Γδp−qsds=Υ,is given by(14)pζ=1−e−φζ2φ−1ℬ1∫01e−φ1−s∫0ss−mϖ−2Γϖ−1ϑ^1mdmds+∫01e−φ1−s∫0ss−mξ−2Γξ−1ϑ^2mdmds+1ℬ2Υ−μ∫0ηη−sδ−1Γδ∫0se−φs−m∫0mm−aϖ−2Γϖ−1ϑ^1adadmds+μ∫0ηη−sδ−1Γδ∫0se−φs−m∫0mm−aξ−2Γξ−1ϑ^2adadmds+∫0ζe−φζ−s∫0ss−mϖ−2Γϖ−1ϑ^1mdmds,(15)qζ=1−e−φζ2φ−1ℬ1∫01e−φ1−s∫0ss−mϖ−2Γϖ−1ϑ^1mdmds+∫01e−φ1−s∫0ss−mξ−2Γξ−1ϑ^2mdmds−1ℬ2Υ−μ∫0ηη−sδ−1Γδ∫0se−φs−m∫0mm−aϖ−2Γϖ−1ϑ^1adadmds+μ∫0ηη−sδ−1Γδ∫0se−φs−m∫0mm−aξ−2Γξ−1ϑ^2adadmds+∫0ζe−φζ−s∫0ss−mξ−2Γξ−1ϑ^2mdmds,(16)ℬ1=1+1−e−φφ,ℬ2=μ∫0ηη−sδ−1Γδ1−e−φsφds.Proof.
Solving,(17)pζ=c0e−φζ+c11−e−φζφ+∫0ζe−φζ−s∫0ss−mϖ−2Γϖ−1ϑ^1mdmds,qζ=d0e−φζ+d11−e−φζφ+∫0ζe−φζ−s∫0ss−mξ−2Γξ−1ϑ^2mdmds.
Using the conditions under whichp+q0=0 in (17), we find that c0=0 and d0=0 and using the boundary conditions p+q0=−p+q1, and μ∫0ηη−sδ−1/Γδp−qsds=Υ in (17), we obtain(18)c1=12−1ℬ1∫01e−φ1−s∫0ss−mϖ−2Γϖ−1ϑ^1mdmds+∫01e−φ1−s∫0ss−mξ−2Γξ−1ϑ^2mdmds+1ℬ2Υ−μ∫0ηη−sδ−1Γδ∫0se−φs−m∫0mm−aϖ−2Γϖ−1ϑ^1adadmds+μ∫0ηη−sδ−1Γδ∫0se−φs−m∫0mm−aξ−2Γξ−1ϑ^2adadmds,d1=12−1ℬ1∫01e−φ1−s∫0ss−mϖ−2Γϖ−1ϑ^1mdmds+∫01e−φ1−s∫0ss−mξ−2Γξ−1ϑ^2mdmds−1ℬ2Υ−μ∫0ηη−sδ−1Γδ∫0se−φs−m∫0mm−aϖ−2Γϖ−1ϑ^1adadmds+μ∫0ηη−sδ−1Γδ∫0se−φs−m∫0mm−aξ−2Γξ−1ϑ^2adadmds.
Substituting the values ofc1 and d1 in (17) leads to the solution (14) and (15). The proof is complete.


## 3. Existence Results

Denote the Banach space byP=E0,1,ℝ×E0,1,ℝ endowed with the norm p,q=supζ∈0,1pζ+supζ∈0,1qζ, for p,q∈P. In view of Lemma 1, we introduce the operator Π:P⟶P associated with the problem (4) as follows(19)Πp,qζ=Π1p,qζ,Π2p,qζ,where(20)Π1p,qζ=1−e−φζ2φ−1ℬ1∫01e−φ1−s∫0ss−mϖ−2Γϖ−1ϑ1mpm,qmdmds+∫01e−φ1−s∫0ss−mξ−2Γξ−1ϑ2mpm,qmdmds+1ℬ2Υ−μ∫0ηη−sδ−1Γδ∫0se−φs−m∫0mm−aϖ−2Γϖ−1ϑ1a,pa,qadadmds+μ∫0ηη−sδ−1Γδ∫0se−φs−m∫0mm−aξ−2Γξ−1ϑ2a,pa,qadadmds+∫0ζe−φζ−s∫0ss−mϖ−2Γϖ−1ϑ1mpm,qmdmds,(21)Π2p,qζ=1−e−φζ2φ−1ℬ1∫01e−φ1−s∫0ss−mϖ−2Γϖ−1ϑ1mpm,qmdmds+∫01e−φ1−s∫0ss−mξ−2Γξ−1ϑ2mpm,qmdmds−1ℬ2Υ−μ∫0ηη−sδ−1Γδ∫0se−φs−m∫0mm−aϖ−2Γϖ−1ϑ1a,pa,qadadmds+μ∫0ηη−sδ−1Γδ∫0se−φs−m∫0mm−aξ−2Γξ−1ϑ2a,pa,qadadmds+∫0ζe−φζ−s∫0ss−mξ−2Γξ−1ϑ2mpm,qmdmds.Next, the following assumptions will be used to demonstrate the paper’s results.Letϑ1,ϑ2:0,1×ℝ2⟶ℝ be continuous functions.W1 There a exist continuous non-negative function βi,κi∈E0,1,ℝ+,i=1,2,3 such that(22)ϑ1ζ,p,q≤β1ζ+β2ζp+β3ζqfor allζ,p,q∈0,1×ℝ2,ϑ2ζ,p,q≤κ1ζ+κ2ζp+κ3ζqfor allζ,p,q∈0,1×ℝ2.W2 There exist non-negative constants Z1,Z2, Q1 and Q2 such that ∀ζ∈0,1pi,qi∈ℝ,i=1,2.(23)ϑ1ζ,p1,q1−ϑ1ζ,p2,q2≤Z1p1−p2+Z2q1−q2,for allζ∈0,1,ϑ2ζ,p1,q1−ϑ2ζ,p2,q2≤Q1p1−p2+Q2q1−q2,for allζ∈0,1.To facilitate the computations, we introduce the notation:(24)Ω1=1−e−φζ2φ1ℬ11−e−φφΓϖ+1ℬ2ηϖ+δ−1φ2ΓδΓϖηφ+e−φη−1,(25)Ω2=1−e−φζ2φ1ℬ11−e−φφΓξ+1ℬ2ηξ+δ−1φ2ΓδΓξηφ+e−φη−1,and(26)Φ=min1−β22Ω1+ζϖ−1φΓϖ1−e−φζ+κ22Ω2+ζξ−1φΓξ1−e−φζ1−β32Ω1+ζϖ−1φΓϖ1−e−φζ+κ32Ω2+ζξ−1φΓξ1−e−φζ.In this part, we prove the existence of a solution to the BVPs via fixed point theorem of Schaefer’s.Theorem 1.
Assume thatW1 holds. In addition, the assumption is that(27)β22Ω1+ζϖ−1φΓϖ1−e−φζ+κ22Ω2+ζξ−1φΓξ1−e−φζ<1,β32Ω1+ζϖ−1φΓϖ1−e−φζ+κ32Ω2+ζξ−1φΓξ1−e−φζ<1,where Ωii=1,2 are defined by (24) and (25). Then, the problem has at least one solution on 0,1.Proof.
In the first part, we demonstrate that the operatorΠ:P⟶P is completely continuous. The continuity of the operator Π comes from the continuity of the function of Ω1 and Ω2. Following that, we show that the Π operator is continuously bounded. Now let ψr¯⊂P be bounded. Then, there exist non-negative ℒϑ1 and ℒϑ2 constants such that(28)ϑ1ζ,pζ,qζ≤ℒϑ1,ϑ2ζ,pζ,qζ≤ℒϑ2.
So, for anyp,q∈ψr¯,ζ∈0,1, we get(29)Π1p,qζ≤ℒϑ1Ω1+ζϖ−1φΓϖ1−e−φζ+ℒϑ2Ω2+Υℬ2,Π2p,qζ≤ℒϑ1Ω1+ℒϑ2Ω2+ζξ−1φΓξ1−e−φζ+Υℬ2.
Thus,(30)Πp,q=Π1p,q+Π2p,q≤ℒϑ12Ω1+ζϖ−1φΓϖ1−e−φζ+ℒϑ22Ω2+ζξ−1φΓξ1−e−φζ+2Υℬ2.
Hence, it follows from the above inequality that the operatorΠ is uniformly bounded.
In order to show thatΠ maps bounded sets into equicontinuous sets of P, let ζ1,ζ2∈0,ζ, ζ1<ζ2, and p,q∈ψr¯. Then,(31)Π1p,qζ2−Π1p,qζ1≤|e−φζ2−e−φζ12−1ℬ1∫01e−φ1−s∫0ss−mϖ−2Γϖ−1ϑ1m,pm,qmdmds+∫01e−φ1−s∫0ss−mξ−2Γξ−1ϑ2m,pm,qmdmds+1ℬ2Υ−μ∫0ηη−sδ−1Γδ∫0se−φs−m∫0mm−aϖ−2Γϖ−1ϑ1a,pa,qadadmds+μ∫0ηη−sδ−1Γδ∫0se−φs−m∫0mm−aξ−2Γξ−1ϑ2a,pa,qadadmds+∫0ζ1e−φζ2−s−e−φζ1−s∫0ss−aϖ−2Γϖ−1ℒϑ1dads+∫ζ1ζ2e−φζ2−s∫0ss−aϖ−2Γϖ−1ℒϑ1dads|,similarly,(32)Π2p,qζ2−Π2p,qζ1≤|e−φζ2−e−φζ12−1ℬ1∫01e−φ1−s∫0ss−mϖ−2Γϖ−1ϑ1m,pm,qmdmds+∫01e−φ1−s∫0ss−mξ−2Γξ−1ϑ2m,pm,qmdmds−1ℬ2Υ−μ∫0ηη−sδ−1Γδ∫0se−φs−m∫0mm−aϖ−2Γϖ−1ϑ1a,pa,qadadmds+μ∫0ηη−sδ−1Γδ∫0se−φs−m∫0mm−aξ−2Γξ−1ϑ2a,pa,qadadmds+∫0ζ1e−φζ2−s−e−φζ1−s∫0ss−aξ−2Γξ−1ℒϑ2dads+∫ζ1ζ2e−φζ2−s∫0ss−aξ−2Γξ−1ℒϑ2dads|.
Note that the right-hand sides of the above inequalities tend to zero asζ1⟶ζ2 and are independent of p,q∈ψr¯. Thus, it follows by the Arzela–Ascoli theorem that the operator Π:E⟶E is completely continuous. Next, we consider the set Θ=p,q∈P|p,q=χΠp,q,0<χ<1. For any ζ∈0,1, we have(33)pζ=χΠ1p,qζ,qζ=χΠ2p,qζ.
UsingΩii=1,2 given by (24) and (25), we find that(34)pζ=χΠ1p,qζ≤β1+β2|=p+β3+qΩ1+ζϖ−1φΓϖ1−e−φζ+κ1+κ2|=p+κ3+qΩ2+Υℬ2,qζ=χΠ2p,qζ≤β1+β2|=p+β3+qΩ1+κ1+κ2|=p+κ3+qΩ2+ζξ−1φΓξ1−e−φζ+Υℬ2.
In consequence, we get(35)p+q≤β12Ω1+ζϖ−1φΓϖ1−e−φζ+κ12Ω2+ζξ−1φΓξ1−e−φζ+2Υℬ2+β22Ω1+ζϖ−1φΓϖ1−e−φζ+κ22Ω2+ζξ−1φΓξ1−e−φζp+β32Ω1+ζϖ−1φΓϖ1−e−φζ+κ32Ω2+ζξ−1φΓξ1−e−φζq.
Thus, by the condition (27), we obtain(36)p,q≤β12Ω1+ζϖ−1/φΓϖ1−e−φζ+κ12Ω2+ζϖ−1/φΓϖ1−e−φζ+2Υ/ℬ2Φ,which shows that p,q is bounded for ζ∈0,1. The set Θ is bounded. Therefore, the fixed point theorem of Schaefer’s applies and hence the operator Π has at least one fixed point, which is a solution for the problem (4).
Ifβ2ζ=β3ζ≡0 and κ2ζ=κ3ζ≡0, then, the statement of Theorem 1 takes the following special form.Remark 3.
There exist positive functionsβ1,κ1∈C0,1,ℝ+ and ϑ1,ϑ2:0,1×ℝ2⟶ℝ which are continuous functions such that(37)ϑ1ζ,p,q≤β1ζ,ϑ2ζ,p,q≤κ1ζfor allζ,p,q∈0,1×ℝ2.
Then, the problem (4) has at least one solution on 0,1.Remark 4.
According to the assumptions of Theorem1, if βiζ=δi,κiζ=εi,i=1,2,3, are non-negative constants, then, the conditions on the functions ϑ1 and ϑ2 take the form:
W1 There are real constants δi,εi>0,i=1,2,3, such that(38)ϑ1ζ,p,q≤δ1+δ2p+δ3qfor allζ,p,q∈0,1×ℝ2,ϑ2ζ,p,q≤ε1+ε2p+ε3qfor allζ,p,q∈0,1×ℝ2,and (27) becomes(39)β22Ω1+ζϖ−1φΓϖ1−e−φζ+κ22Ω2+ζξ−1φΓξ1−e−φζ<1,β32Ω1+ζϖ−1φΓϖ1−e−φζ+κ32Ω2+ζξ−1φΓξ1−e−φζ<1.Theorem 2.
Assume thatW2 holds and that(40)Z2Ω1+ζϖ−1φΓϖ1−e−φζ+Q2Ω2+ζξ−1φΓξ1−e−φζ<1,where Z=maxZ1,Z2, Q=maxQ1,Q2 and Ωi,i=1,2 are defined by (24) and (25). Then, the problem (4) has a unique solution on 0,1.Proof.
Consider the operatorΠ:P⟶P defined by (19) and let ℬς=p,q∈P:p,q≤ς, fix(41)ς>ℳ12Ω1+ζϖ−1/φΓϖ1−e−φζ+ℳ22Ω2+ζξ−1/φΓξ1−e−φζ1−Z2Ω1+ζϖ−1/φΓϖ1−e−φζ+Q2Ω2+ζξ−1/φΓξ1−e−φζ,where ℳ1=supζ∈0,1ϑ1ζ,0,0 and ℳ2=supζ∈0,1ϑ2ζ,0,0. Then, we show that Πℬς⊂ℬς, and we have(42)Π1p,qζ≤ZΩ1+ζϖ−1φΓϖ1−e−φζ+QΩ2p+q+ℳ1Ω1+ζϖ−1φΓϖ1−e−φζ+ℳ2Ω2,which to taking the norm for ζ∈0,1 leads to(43)Π1p,q≤ZΩ1+ζϖ−1φΓϖ1−e−φζ+QΩ2p+q+ℳ1Ω1+ζϖ−1φΓϖ1−e−φζ+ℳ2Ω2,when the norm for ζ∈0,1. In the same way, for p,q∈ℬς, one can obtain(44)Π2p,q≤QΩ2+ζξ−1φΓξ1−e−φζ+ZΩ1p+q+ℳ2Ω2+ζξ−1φΓξ1−e−φζ+ℳ1Ω1.
Therefore, for anyp,q∈ℬς, we have(45)Πp,q=Π1p,q+Π2p,q≤Z2Ω1+ζϖ−1φΓϖ1−e−φζ+Q2Ω2+ζϖ−1φΓϖ1−e−φζp+q+ℳ12Ω1+ζϖ−1φΓϖ1−e−φζ+ℳ22Ω2+ζξ−1φΓξ1−e−φζ<ς,which shows that Π maps ℬς into itself. In order to demonstrate that the Π operator is a contraction, let p1,q1,p2,q2∈P,ζ∈0,1 in view of W2, and we obtain(46)Π1p1,q1ζ−Π1p2,q2ζ≤ZΩ1+ζϖ−1φΓϖ1−e−φζ+QΩ2p+q,Π2p1,q1ζ−Π2p2,q2ζ≤QΩ2+ζξ−1φΓξ1−e−φζ+ZΩ1p+q.
Clearly, the preceding inequalities imply that(47)Πp1,q1−Πp2,q2=Π1p1,q1−Π2p2,q2+Π2p1,q1−Π2p2,q2,≤Z2Ω1+ζϖ−1φΓϖ1−e−φζ+Q2Ω2+ζξ−1φΓξ1−e−φζp1−p2,q1−q2,which, in view of (40), implies that Π is a contraction mapping. Hence, Π has a unique fixed point by Banach’s contraction mapping principle. This, in turn, shows the problems (4) has a unique solution on 0,1. The proof is completed.


## 4. Stability Results

This section is devoted to the investigation of Hyers–Ulam stability for our proposed system. Consider the following inequality:(48)CDϖ+φCDϖ−1pζ−ϑ1ζ,pζ,qζ≤ε1,ζ∈0,1,CDϖ+φCDϖ−1pζ−ϑ2ζ,pζ,qζ≤ε2,ζ∈0,1.where ε1,ε2 are given two positive numbers.Definition 4.
Problem (4) is Hyers–Ulam stable if there exist ℳi>0,i=1,2,3,4 such that for given ε1,ε2>0 and for each solution p,q∈E0,1×ℝ2,ℝ of problem (4) with(49)pζ−p∗ζ≤ℳ1ε1+ℳ2ε2,ζ∈0,1,qζ−q∗ζ≤ℳ3ε1+ℳ4ε2,ζ∈0,1.Remark 5.
p,q is a solution of inequality (48) if there exist functions Qi∈0,1,ℝ,i=1,2 which depend upon p,q, respectively, such that(50)Q1ζ≤ε1,Q2ζ≤ε2,ζ∈0,1,(51)CDϖ+φCDϖ−1pζ=ϑ1ζ,pζ,qζ+Q1ζ,ζ∈0,1,DξC+φDξ−1Cqζ=ϑ2ζ,pζ,qζ+Q2ζ,ζ∈0,1.Remark 6.
Ifp,q represent a solution of inequality (48), then, p,q is a solution of following inequality:(52)pζ−p∗ζ≤ℳ1ε1+ℳ2ε2,ζ∈0,1,qζ−q∗ζ≤ℳ3ε1+ℳ4ε2,ζ∈0,1.
∀p,q∈E0,1,ℝ of inequality.Theorem 3.
Assume thatW2 holds. Then, the problem (4) is Ulam–Hyers stable.Proof.
PetE0,1,ℝ×E0,1,ℝ be the solution to (4) that satisfies (19) and (20). Let p,q be any solution that meets the condition (50):(53)CDϖ+φCDϖ−1pζ=ϑ1ζ,pζ,qζ+ϑ1p,qζ,ζ∈0,1,DξC+φDξ−1Cqζ=ϑ2ζ,pζ,qζ+ϑ2p,qζ,ζ∈0,1,so,(54)pζ=p∗ζ+1−e−φζ2φ−1ℬ1∫01e−φ1−s∫0ss−mϖ−2Γϖ−1ϑ1p,qmdmds+∫01e−φ1−s∫0ss−mξ−2Γξ−1ϑ2p,qmdmds+1ℬ2Υ−μ∫0ηη−sδ−1Γδ∫0se−φs−m∫0mm−aϖ−2Γϖ−1ϑ1p,qadadmds+μ∫0ηη−sδ−1Γδ∫0se−φs−m∫0mm−aξ−2Γξ−1ϑp,qadadmds+∫0ζe−φζ−s∫0ss−mϖ−2Γϖ−1ϑ1p,qmdmds.
It follows that(55)pζ−p∗ζ≤1−e−φζ2φ−1ℬ1∫01e−φ1−s∫0ss−mϖ−2Γϖ−1ε1dmds+∫01e−φ1−s∫0ss−mξ−2Γξ−1ε2dmds+1ℬ2Υ−μ∫0ηη−sδ−1Γδ∫0se−φs−m∫0mm−aϖ−2Γϖ−1ε1dadmds+μ∫0ηη−sδ−1Γδ∫0se−φs−m∫0mm−aϖ−2Γϖ−1ε2dadmds+∫0ζe−φζ−s∫0ss−mϖ−2Γϖ−1ε1dmds,≤Ω1ε1+Ω2ε2.
Similarly, we obtain(56)qζ−q∗ζ≤1−e−φζ2φ−1ℬ1∫01e−φ1−s∫0ss−mϖ−2Γϖ−1ε1dmds+∫01e−φ1−s∫0ss−mξ−2Γξ−1ε2dmds+1ℬ2Υ−μ∫0ηη−sδ−1Γδ∫0se−φs−m∫0mm−aϖ−2Γϖ−1ε1dadmds+μ∫0ηη−sδ−1Γδ∫0se−φs−m∫0mm−aξ−2Γξ−1ε2dadmds+∫0ζe−φζ−s∫0ss−mξ−2Γξ−1ε2dmds,≤Ω1ε1+Ω2ε2,where Ω1 and Ω2 are defined in (24) and (40), respectively. Hence, the problem (4) is U-H stable.


## 5. Numerical Examples

In this section, we introduce two numerical examples to support our existence results.Example 1.
Consider the following system:(57)CDϖ+φCDϖ−1pζ=ϑ1ζ,pζ,qζ,ζ∈ℱ=0,1,where φ=3/2,ϖ=5/3,ξ=7/4,η=1/10,δ=3/2,μ=1,andΥ=2,andϑ1ζ,pζ,qζ, and ϑ2ζ,pζ,qζ will be fixed later.
Using the above data, we getΩ1=0.4025220540 and Ω2=0.3953485857, where Ω1 and Ω2 are, respectively, given by (24) and (25). Have(58)ϑ1ζ,pζ,qζ=e−ζ225+ζ2pζ+sinq+cosζ,ϑ2ζ,pζ,qζ=19+ζ2sinp+q2+e−ζ.
Next,ϑ1 and ϑ2 are continuous and fulfill the hypothesis W1 with(59)β1ζ=e−ζcosζ225+ζ2,β2ζ=ζe−ζ225+ζ2,β3ζ=e−ζ225+ζ2,κ2=e−ζ9+ζ2,κ1=19+ζ2,andκ3=129+ζ2.
Also,(60)β22Ω1+ζϖ−1φΓϖ1−e−φζ+κ22Ω2+ζξ−1φΓξ1−e−φζ≈0.0537931248,β32Ω1+ζϖ−1φΓϖ1−e−φζ+κ32Ω2+ζξ−1φΓξ1−e−φζ≈0.0142934345.
Thus, by Theorem1, ∃ a solution to the system (57) on 0,1.Example 2.
Consider the following functions:(61)ϑ1ζ,pζ,qζ=1401+ζ2p1+p+tan−1q,ϑ2ζ,pζ,qζ=1400+ζ22tan−1p+sinq.
Observe thatp and q are continuous and satisfy the condition ℳ2 with Z1=Z2=1/40=Z and Q1=1/10,Q2=1/20 and so, Q=1/15. Also,(62)Z2Ω1+ζϖ−1φΓϖ1−e−φζ+Q2Ω2+ζξ−1φΓξ1−e−φζ≈0.2754435781.
Thus, all the conditions of Theorem2 are satisfied, that is, the problem has a unique solution in 0,1 the problem (57).


## 6. Conclusion

In this paper, we have introduced a new kind of boundary conditions which deal with the sum of unknown functions at the boundary points and on an arbitrary segment of the given domain. Equipped with these conditions, we have solved a nonlinear coupled system of Caputo sequential fractional differential equations. The existence and uniqueness results obtained for the given problem are new, and we also study the Hyers–Ulam stability. Our results are not only in the given configuration but also yield some new results by specializing the parameters involved in the problems at hand. For example, our results correspond to those for new coupled integral boundary conditions of the form:(63)p+q0=0,p+q0=−p+q′1,μ∫0ηη−sδ−1Γδp−qsds=Υ,where, as δ=1, the integral boundary conditions reduce to the usual form correspond to the nonlocal integral boundary conditions of the form:(64)p+q0=0,p+q0=−p+qT,μ∫0ηp−qsds=Υ,Furthermore, the methods employed in this paper can be used to solve the systems involving time-delay systems/inclusions, a time-delay system/inclusions with finite delay, discrete boundary conditions, and multiterm fractional differential equations complemented with the boundary conditions considered in the problem (4).



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