# An Existence Study on the Fractional Coupled Nonlinearq-Difference Systems via Quantum Operators along with Ulam–Hyers and Ulam–Hyers–Rassias Stability

Authors: Shahram Rezapour; Chatthai Thaiprayoon; Sina Etemad; Weerawat Sudsutad; Chernet Tuge Deressa; Akbar Zada
Journal: Journal of Function Spaces
 (2022)

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## Abstract

In this paper, we study the existence of solutions and their uniqueness and different kinds of Ulam–Hyers stability for a new class of nonlinear Caputo quantum boundary value problems. Also, we investigate such properties for the relevant generalized coupledq-system involving fractional quantum operators. By using the Banach contraction principle and Leray-Schauder’s fixed–point theorem, we prove the existence and uniqueness of solutions for the suggested fractional quantum problems. The Ulam–Hyers stability of solutions in different forms are studied. Finally, some examples are provided for both q-problem and coupled q-system to show the validity of the main results.


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## Body

## 1. Introduction

Fractional calculus is one of the most important fields in applied mathematics. In recent years, many researchers have studied different branches of this theory and conducted numerous analyses analytically and numerically. Particularly, in recent decades, we can see some papers on the applications of fixed-point theorems to prove the existence of solutions of fractional boundary value problems [1–4]. Because of the quick developments in fractional calculus, many mathematicians discussed on the theory of q-calculus that is an equivalent of traditional calculus without defining the concept of limit, and also the parameter q refers to quantum. This theory was originally developed by Jackson [5, 6], and it includes many practical aspects in the fields of hypergeometric series, theory of relativity, particle physics, discrete mathematics, quantum mechanics, combinatorics, and complex analysis. For a fundamental introduction of the basic notions of q-calculus, one can refer to [7–9]. In the early years, for finding positive solutions of given q-difference equations in the nonlinear settings, we lead you to study a work published by both El-Shahed and Al-Askar [10] and also a manuscript by Graef and Kong [11].So later, various mathematicalq-difference fractional models of IVPs and BVPs have been presented in which different methods like the lower-upper solutions technique, fixed-point results, and iterative methods have been implemented. For instance, we see q-intego-equation on time scales in [12], q-delay equations in [13], q-integro-equations under the q-integral conditions in [14], singular q-equations in [15], q-sequential symmetric BVPs in [16], q-difference equations having p-Laplacian in [17], four-point q-BVP with different orders in [18], oscillation on q-difference inclusions in [19], etc.Here, we apply similar techniques to discuss the existence property of solutions for givenq-integro-sum-difference FBVPs depending on the quantum operators. This shows an application of fixed-point theory in relation to q-difference theory. This specifies the main contribution of the present reseach.In 2014, Ahmad et al. [20] studied a q-sequential equation in the nonlinear case via four-point q-integral conditions given by
(1)DqC0+k1DqC0+k2+σur=Gr,ur,r∈0,1,q∈0,1,u0=e1Iq0+s−1ub1,u1=e2Iq0+s−1ub2,so that k1,k2∈0,1, b1,b2∈0,1, s>2, and σ,e1,e2∈ℝ. As well as, G:0,1×ℝ⟶ℝ is continuous, and Iq0+s−1 indicates the q-RL-integral. These mathematicians extracted different qualitative aspects of solutions for the above q-FBVP by means of the classical methods which are available in fixed-point theory.In 2015, Etemad et al. [21] focused on the new four-point three-term q-difference FBVP
(2)DqC0+ρur=Gr,ur,DqC0+1ur,0<q<1,c1u0+d1DqC0+1u0=b1Iq0+αuk1=b1∫0k1k1−qzα−1Γqαuzdqz,c2u1+d2DqC0+1u1=b2Iq0+αuk2=b2∫0k2k2−qzα−1Γqαuzdqz,where 0≤r≤1, 1<ρ≤2, α∈0,2, c1,c2,d1,d2,b1,b2∈ℝ, and k1,k2∈0,1 with k1<k2.In 2019, two mathematicians named Ntouyas and Samei [22] devoted their attention to investigate the existence property about the multiterm q-integro-difference FBVP
(3)DqC0+ρur=Gr,ur,ℏ1ur,ℏ2ur,DqC0+ρ1ur,DqC0+ρ2ur,⋯,DqC0+ρmur,u0+b1u1=0,u′0+b2u′1=0where r∈0,1, q∈0,1, ρ∈1,2, ρi∈0,1 with i=1,2,⋯,m, b1,b2≠−1, ℏj are formulated as
(4)ℏjur=∫0rvjr,zuzdqz,for j=1,2 and G:0,1×ℝm+3⟶ℝ is continuous with respect to all variables [22].In 2020, Phuong et al. [23] formulated a novel extended configuration of the Caputo q-multi-integro-difference equation with two nonlinearity under q-multi-order-integrals conditions
(5)mDqC0+ρ−m+1Iq0+k1−m+2Iq0+k2ur=b1Iq0+k3G1r,ur+b2Iq0+k4G2r,ur,u0=0,nIq0+p1u1+n+1Iq0+p2u1+n+2Iq0+p3u1=0,where r∈0,1, ρ∈1,2, k1,k2,k3,k4∈0,1, p1,p2,p3,m,n>0, and b1,b2∈ℝ≥0.In this paper, inspired by aboveq-problems, we analyze a structure of the nonlinear Caputo quantum difference fractional boundary problem (or q-CFBVP) in the form
(6)DqC0+ςμr=Gr,μr,JqR0+ωμr≔Gμr,r∈O≔0,1,q∈0,1,μ0+μζ=∑j=1kαjJqR0+σjμ1,DqC0+ϱμ0+DqC0+ϱμζ=∑j=1kβjJqR0+σjμ1,DqC0+2μ0+DqC0+2μζ=∑j=1kγjJqR0+ϱDqC0+2μ1,where ς∈2,3, ϱ∈1,2, ζ∈0,1, αj,βj,γj∈ℝ>0, ω,σj>0 for j=1,2,⋯,k, and G:O×ℝ2⟶ℝ are continuous. As the same way, the operators DqC0+· and Iq0+· denote the q-Caputo derivative and the q-RL integral, respectively. In the direction of the above problem, we consider a coupled system of nonlinear q-CFBVPs with the same q-boundary conditions. In other words, the mentioned fractional q-system is organized as
(7)DqC0+ς1μr=G1r,ϑr,JqR0+ω1ϑr:Uϑr,r∈O,q∈0,1,DqC0+ς2ϑr=G2r,μr,JqR0+ω2μr≔Vμr,μ0+μζ=∑j=1kαjJqR0+σjμ1,ϑ0+ϑζ=∑j=1kϕjJqR0+δjϑ1,DqC0+ϱμ0+DqC0+ϱμζ=∑j=1kβjJqR0+σjμ1,DqC0+ρϑ0+DqC0+ρϑζ=∑j=1kφjJqR0+δjϑ1,DqC0+2μ0+DqC0+2μζ=∑j=1kγjJqR0+σjDqC0+2μ1,DqC0+2ϑ0+DqC0+2ϑζ=∑j=1kηjJqR0+δjDqC0+2ϑ1,where ς1,ς2∈2,3, ϱ,ρ∈1,2, ζ∈0,1, αj,βj,γj,ϕj,φj,ηj∈ℝ>0, ω1,ω2,σj,δj>0 for j=1,2,⋯,k, and G1,G2:O×ℝ2⟶ℝ are continuous.In other words, we extend ourq-CFBVP to a coupled q-difference system and derive the existence and stability results on such a generalized coupled q-CFBVP system. In fact, a large number of researchers have devoted their concentration to the discussion on various categories of Ulam-Hyers stabilities for standard systems of FDEs (or refer to [24, 25]), while a few articles can be found in the literature in which the researchers developed the relevant existence and stability theory in relation to nonlinear fractional q-difference systems.The present work is assembled as follows: In Section2, we state some basic materials required to prove our theoretical results. In both Section 3 and Section 4, several criteria and conditions are presented for the desired uniqueness-existence results, along with different classes of stabilities in relation to the proposed q-CFBVPs (6) and (7), respectively, with the help of some known fixed–point theorems. A simulative example, to represent the consistency of our results, is given with each suggested q-CFBVP in the relevant section. We give Section 6 to the presentation of the conclusion of this research work.


## 2. Preliminaries

The basic notions ofq-calculus are collected in this section by assuming q∈0,1. The q-analogue of a1−a2k is given by
(8)a1−a20=1,a1−a2k=∏j=0k−1a1−a2qj,a1,a2∈ℝ,k∈ℕ0≔0,1,2,⋯Rajkovic et al. [26]. Now, if k=ς∈ℝ, then
(9)a1−a2ς=a1ς∏k=0∞1−a2/a1qk1−a2/a1qς+k,a1≠0.On the other side, by takinga2=0, we have a1ς=a1ς [26]. A q-number a1q for a1∈ℝ is defined by
(10)a1q=1−qa11−q=qa1−1+⋯+q+1.Accordingly, the Gamma function in the quantum settings is defined by(11)Γqr=1−qr−11−qr−1,r∈ℝ\ℤ−∪0,and Γqr+1=rqΓqr [5, 26].Definition 1 (see [27]).
Theq-difference-derivative of the given function μ is defined by
(12)Dq0+μr=ddrqμr=μr−μqr1−qr,where Dq0+μ0=limr⟶0Dq0+μr.Clearly, we haveDq0+kμr=Dq0+Dq0+k−1μr for all k∈ℕ and Dq0+0μr=μr [27].Definition 2 (see [27]).
Theq-integral of the supposed function μ∈C0,m2,ℝ is defined as
(13)Iq0+μr=∫0rμvdqv=r1−q∑j=0∞μrqjqj,if the series is absolutely convergent.Similarly,Iq0+kμr=Iq0+Iq0+k−1μr for all k≥1 and Iq0+0μr=μr [27].Definition 3 (see [27]).
By lettinga1∈0,a2, the definite q-integral of the given function μ∈C0,a2,ℝ is defined by
(14)∫a1a2μvdqv=Iq0+μa2−Iq0+μa1=∫0a2μvdqv−∫0a1μvdqv=1−q∑j=0∞a2μa2qj−a1μa1qjqj,if the series exists.By consideringμ as a continuous function at r=0, then Iq0+Dq0+μr=μr−μ0 [27]. Furthermore, Dq0+Iq0+μr=μr for all r.Definition 4 (see [11, 28]).
Theςth-RL-q-integral of μ∈Cℝ0,+∞ is defined by
(15)IqR0+ςμr=1Γqς∫0rr−qvς−1μvdqv,ς>0,μr,ς=0,if integral exists.One can simply see that theq-semi-group property satisfies as IqR0+ς1IqR0+ς2μr=IqR0+ς1+ς2μr for ς1,ς2≥0 [28]. Also, for ζ>−1, we have
(16)IqR0+ςrζ=Γqζ+1Γqζ+ς+1rζ+ς,IqR0+ς1r=1Γqς+1rς,r>0.Definition 5 (see [11, 28]).
Letℓ−1<ς<ℓ, i.e., ℓ=ς+1. The ςth-Caputo q-derivative of μ∈Cℝℓ0,+∞ is defined as
(17)DqC0+ςμr=1Γqℓ−ς∫0rr−qvℓ−ς−1qD0+ℓμvdqv,if the integral exists.Note that forζ>−1, we have
(18)DqC0+ςrι=Γqι+1Γqι−ς+1rι−ς,DqC0+ς1r=0,r>0.Lemma 6 (see [10]).
Letℓ−1<ς<ℓ. Then,
(19)IqC0+ςDqC0+ςμr=μr−∑j=0ℓ−1rjΓqj+1Dq0+jμ0.By Lemma6, the general series solution of q-difference FDE DqC0+ςμr=0 is given as μr=c~0+c~1r+c~2r2+⋯+c~ℓ−1rℓ−1 with c~0,⋯,c~ℓ−1∈ℝ and ℓ=ς+1 [10]. In this case, we get
(20)IqR0+ςDqC0+ςμr=μr+c~0+c~1r+c~2r2+⋯+c~ℓ−1rℓ−1.


## 3. Analysis of the Cap-q-Difference FBVP (6)

LetA=CℝO be the space of all real-valued continuous functions on O=0,1. Clearly, A is a Banach space under the norm μA=Supr∈Oμr for all members μ∈A. In the first step, we provide the following fundamental lemma which presents a characterization of the structure of solutions for the proposed Cap-q-difference FBVP (6)Remark 7.
For convenience, we consider the following nonzero constants:(21)W1=2−∑j=1kαjΓqσj+1,W2=ζ−∑j=1kαjΓqσj+2,W3=ζ2−∑j=1kαj1+qΓqσj+3,(22)W4=−∑j=1kβjΓqσj+1,W5=−∑j=1kβjΓqσj+2,W6=2ζ2−ρΓq3−ρ−∑j=1kβj1+qΓqσj+3,(23)W7=21+q−∑j=1kγj1+qΓqσj+1,W8=W2W4−W1W5,W9=W3W4−W1W6,(24)W10=W8−W2W4,W11=W3W8−W2W9.Lemma 8.
Letϕ∗∈A, ς∈2,3, ρ∈1,2, ζ∈0,1, αj,βj,γj∈ℝ>0, and σj>0 for j=1,2,⋯,k. The solution of the linear Cap-q-difference FBVP
(25)DqC0+ςμr=ϕ∗r,r∈O,q∈0,1,μ0+μζ=∑j=1kαjJqR0+σjμ1,DqC0+ϱμ0+DqC0+ϱμζ=∑j=1kβjJqR0+σjμ1,DqC0+2μ0+DqC0+2μζ=∑j=1kγjJqR0+σjDqC0+2μ1is given by
(26)μr=∫0rr−qvς−1Γqςϕ∗vdqv−Θ1rW1W8∫0ζζ−qvς−1Γqςϕ∗vdqv+Θ2rW8∫0ζζ−qvς−ρ−1Γqς−ρϕ∗vdqv−Θ3rW1W7W8∫0ζζ−qvς−3Γqς−2ϕ∗vdqv+Θ1rW1W8∑j=1kαj∫011−qvς+σj−1Γqς+σjϕ∗vdqv−Θ2rW8∑j=1kβj∫011−qvς+σj−1Γqς+σjϕ∗vdqv+Θ3rW1W7W8∑j=1kγj∫011−qvς+σj−3Γqς+σj−2ϕ∗vdqv,where
(27)Θ1r=rW1W4+W10,Θ2r=rW1−W2,Θ3r=r2W1W8−rW1W9−W11,and Wi are defined in (24).Proof.
Letμ satisfies the linear Cap-q-difference FBVP (25). Then DqC0+ςμr=ϕ∗r. By virtue of ς∈2,3 and taking ςth-RL-q-integral, we have
(28)μr=1Γqς∫0rr−qvς−1ϕ∗vdqv+c~0+c~1r+c~2r2,where c~0,c~1,c~2∈ℝ are unknown coefficients that we have to explore them. It is immediately computed that
(29)DqC0+2μr=1Γqς−2∫0rr−qvς−3ϕ∗vdqv+c~21+q,(30)DqC0+ϱμr=1Γqς−ϱ∫0rr−qvς−ϱ−1ϕ∗vdqv+c~22Γq3−ϱr2−ϱ,(31)IqR0+σjμr=1Γqς+σj∫0rr−qvς+σj−1ϕ∗vdqv+c~01Γqσj+1rσj+c~11Γqσj+2rσj+1+c~21+qΓqσj+3rσj+2,(32)IqR0+σjDqC0+2μr=1Γqς+σj−2∫0rr−qvς+σj−3ϕ∗vdqv+c~21+qΓqσj+1rσj.
By considering the constantsW1,⋯,W11 given by (24) and by virtue the given boundary conditions implemented on (29)–(32) and by some straightforward computations, we obtain the following coefficients
(33)c~0=W2W8∑j=1kβj∫011−qvς+σj−1Γqς+σjϕ∗vdqv−W2W8∫0ζζ−qvς−ρ−1Γqς−ρϕ∗vdqv+W10W1W8∑j=1kαj∫011−qvς+σj−1Γqς+σjϕ∗vdqv−W10W1W8∫0ζζ−qvς−1Γqςϕ∗vdqv+W11W1W7W8∫0ζζ−qvς−3Γqς−2ϕ∗vdqv−W11W1W7W8∑j=1kγj∫011−qvς+σj−3Γqς+σj−2ϕ∗vdqv,(34)c~1=W4W8∑j=1kαj∫011−qvς+σj−1Γqς+σjϕ∗vdqv−W4W8∫0ζζ−qvς−1Γqςϕ∗vdqv+W1W8∫0ζζ−qvς−ρ−1Γqς−ρϕ∗vdqv−W1W8∑j=1kβj∫011−qvς+σj−1Γqς+σjϕ∗vdqv+W9W7W8∫0ζζ−qvς−3Γqς−2ϕ∗vdqv−W9W7W8∑j=1kγj∫011−qvς+σj−3Γqς+σj−2ϕ∗vdqv,(35)c~2=1W7∑j=1kγj∫011−qvς+σj−3Γqς+σj−2ϕ∗vdqv−1W7∫0ζζ−qvς−3Γqς−2ϕ∗vdqv.
By inserting (33), (34), and (35) into (28), we derive equation (26) which is the same desired q-integral solution of the linear Cap-q-difference FBVP (25). The proof is completed.Now, consider the following estimates:(36)Supr∈OΘ1r≤Supr∈OrW1W4+W10≤W1W4+W10≔Θ1∗>0,Supr∈OΘ2r≤Supr∈OrW1+W2≤W1+W2≔Θ2∗>0,Supr∈OΘ3r≤Supr∈Or2W1W8+rW1W9+W11≤W1W8+W1W9+W11≔Θ3∗>0.In this paper, for convenience in computation, we set(37)IqR0+ςGμvr=1Γqς∫0rr−qvς−1Gμυdqv.According to Lemma8, we define the operator F:A⟶A as
(38)Fμr=IqR0+ςGμvr+Θ1rW1W8−IqR0+ςGμvζ+∑j=1kαjIqR0+ς+σjGμv1+Θ2rW8IqR0+ς−ϱGμvζ−∑j=1kβjIqR0+ς+σjGμv1+Θ3rW1W7W8−IqR0+ς−2Gμvζ+∑j=1kγjIqR0+ς+σj−2Gμv1.Notice that the Cap-q-difference FBVP (6) has solutions if and only if F has fixed points.To simplify the computations, we set the following notation and the constants(39)Λ=1Γqς+1+Θ1∗W1W8ζςΓqς+1+∑j=1kαjΓqς+σj+1+Θ2∗W8ζς−ϱΓqς−ρ+1+∑j=1kβjΓqς+σj+1+Θ3∗W1W7W8ζς−2Γqς−1+∑j=1kγjΓqς+σj−1.


### 3.1. Uniqueness Result

The uniqueness result for the Cap-q-difference FBVP (6) is proved by using the Banach’s fixed-point theorem.Theorem 9.
Assume thatG∈CO×ℝ2,ℝ satisfies the following assumptions.
(H1) There are L1, L2>0 such that
(40)Gr,u1,v1−Gr,u2,v2≤L1u1−u2+L2v1−v2,for every ui, vi∈ℝ, i=1,2, and r∈O.
If(41)L1+L2Γqω+1Λ<1,where Λ is given in (39), and then the Cap-q-difference FBVP (6) has a unique solution μ in A.Proof.
We convert the Cap-q-difference FBVP (6) into μ=Fμ, where F is defined by (38). By the Banach’s contraction principle, we shall guarantee that F has exactly one fixed point.
At first, we define a bounded, closed convex subsetBY1≔μ∈A:μA≤Y1≠∅ with
(42)Y1≥ΛG1−L1+L2/Γqω+1Λ,where Λ is defined by (39).
Letsupr∈OGr,0,0≔G<∞. The proof will be completed in two steps:
Step 1.FBY1⊂BY1.
Letμ∈BY1 and r∈O. Estimate
(43)Fμr≤IqR0+ςGμvr+Θ1rW1W8IqR0+ςGμvζ+∑j=1kαjIqR0+ς+σjGμv1+Θ2rW8IqR0+ς−ϱGμvζ+∑j=1kβjIqR0+ς+σjGμv1+Θ3rW1W7W8IqR0+ς−2Gμvζ+∑j=1kγjIqR0+ς+σj−2Gμv1.
By using the property of integral (16), we get
(44)IqR0+ωμvr=1Γqω∫0rr−qvω−1μvdqv≤rωμAΓqω+1.
From the assumptionsH1 and (44), we can estimate
(45)Gμr≤gr,μr,IqR0+ςμr−gr,0,0+gr,0,0,0≤L1μr+L2Iqr0+ςμr+G≤L1+L2Γqω+1μA+G.
From (45) and by the property of integral (16), we obtain
(46)IqR0+ςGμvr≤L1+L2Γqω+1μA+GrςΓqς+1,(47)IqR0+ςGμvζ≤L1+L2Γqω+1μA+GζςΓqς+1,(48)IqR0+ς−ϱGμvζ≤L1+L2Γqω+1μA+Gζς−ϱΓqς−ϱ+1,(49)IqR0+ς−2Gμvζ≤L1+L2Γqω+1μA+Gζς−2Γqς−1,(50)IqR0+ς+σjGμv1≤L1+L2Γqω+1μA+G1Γqς+σj+1,(51)IqR0+ς+σj−2Gμv1≤L1+L2Γqω+1μA+G1Γqς+σj−1.
Substituting (46)–(51) into (43), we obtain
(52)Fμr≤L1+L2Γqω+1μA+GrςΓqς+1+Θ1rW1W8ζςΓqς+1+∑j=1kαjΓqς+σj+1+Θ2rW8ζς−ϱΓqς−ϱ+1+∑j=1kβjΓqς+σj+1+Θ3rW1W7W8ζς−2Γqς−1+∑j=1kγjΓqς+σj−1.
Then,(53)Fμr≤L1+L2Γqω+1μA+GΛ,which implies that FμA≤Y1. Thus, FBY1⊂BY1.
Step 2.F:A⟶A is a contraction.
Letμ, ϑ∈A. For each r∈O, we have
(54)Fμr−Fϑr≤Θ1rW1W8IqR0+ςGμv−Gϑvζ+∑j=1kαjIqR0+ς+σjGμv−Gϑv1+Θ2rW8IqR0+ς−ϱGϑv−Gϑvζ+∑j=1kβjIqR0+ς+σjGμv−Gϑv1+Θ3rW1W7W8IqR0+ς−2Gμv−Gϑvζ+∑j=1kγjIqR0+ς+σj−2Gμv−Gϑv1+IqR0+ςGμv−Gϑvr.
ByH1, it follows that
(55)Gμv−Gϑv≤gr,μr,IqR0+ςμr−gr,ϑr,IqR0+ςϑr≤L1+L2Γqω+1μ−ϑA.
Hence, by inserting (55) into (54) and using the property of integral (16), we get
(56)Fμr−Fϑr≤L1+L2Γqω+1μ−ϑArςΓqς+1+Θ1rW1W8ζςΓqς+1+∑j=1kαjΓqς+σj+1+Θ2rW8ζς−ϱΓqς−ϱ+1+∑j=1kβjΓqς+σj+1+Θ3rW1W7W8ζς−2Γqς−1+∑j=1kγjΓqς+σj−1,which implies that Fμ−FϑA≤L1+L2/Γqω+1Λμ−ϑA.
In view of (41), L1+L2/Γqω+1Λ<1, and we conclude that F is a contraction. Hence, in accordance with the Banach’s contraction principle, the Cap-q-difference FBVP (6) has a unique solution μ∈A.


### 3.2. Existence Result

The second result is based on the Leray-Schauder’s nonlinear alternative theorem.Lemma 10 (Leray-Schauder’s nonlinear alternative theorem [29]).
LetM be a Banach space, C be its closed convex subset, and X be an open set in C such that 0∈X. Let G:X¯⟶C be a continuous and compact function. Then either (i) there is μ∈X¯ such that μ=Gμ or (ii) there are μ∈∂X and ϱ∈0,1 such that μ=ϱGμ.Theorem 11.
LetG∈CO×ℝ2,ℝ satisfies the following assumptions:
(H2) There is continuous nondecreasing functions Y: ℝ+⟶ℝ+, p1, p2∈CJ,ℝ+ such that
(57)Gr,u,v≤p1rYu+p2rv,∀r,u,v∈O×ℝ2,where p¯i=supr∈Jpir, i=1,2.
(H3) There is M∗>0 such that
(58)1−Λp¯2/Γqω+1M∗Λ1p¯1YM∗>1.Then the Cap-q-difference FBVP (6) has at least one solution μ in A.Proof.
ConsiderF as (38). In the first step, we will prove that F corresponds bounded sets (balls) to bounded ones in A. For each positive real constant Y2, BY2≔μ∈A:μ≤Y2 is a bounded set (ball) in A. Let μ∈BY2. We have
(59)Fμr≤IqR0+ςGμvr+Θ1rW1W8IqR0+ςGμvζ+∑j=1kαjIqR0+ς+σjGμv1+Θ2rW8IqR0+ς−ϱGμvζ+∑j=1kβjIqR0+ς+σjGμv1+Θ3rW1W7W8IqR0+ς−2Gμvζ+∑j=1kγjIqR0+ς+σj−2Gμv1.
From (H2) and (44) in Theorem 9, we obtain
(60)Gr,μr,IqR0+ςμr≤p1rYμ+p2rIqR0+ςμr≤p¯1YY2+p¯2Y2Γqω+1≔g¯.
By the same process in Theorem9, we can estimate
(61)FμrA≤Λg¯.
Further, it will be investigated thatF corresponds bounded sets to equicontinuous sets of A.
Letr1, r2∈O with r1<r2 and μ∈BY2, where BY2 is a bounded set in A. Also, we obtain
(62)Fμr2−Fμr1≤IqR0+ςGμvr2−IqR0+ςGμvr1+Θ1r2−Θ1r1W1W8IqR0+ςGμvζ+∑j=1kαjIqR0+ς+σjGμv1+Θ2r2−Θ2r1W8IqR0+ς−ϱGμvζ+∑j=1kβjIqR0+ς+σjGμv1+Θ3r2−Θ3r1W1W7W8IqR0+ς−2Gμvζ+∑j=1kγjIqR0+ς+σj−2Gμv1≤1Γqς∫r1r2r2−qvς−1Gμvdqv+1Γqς∫0r1r2−qvς−1−r1−qvς−1Gμvdqv+Θ1r2−Θ1r1W1W8IqR0+ςGμvζ+∑j=1kαjIqR0+ς+σjGμv1+Θ2r2−Θ2r1W8IqR0+ς−ϱGμvζ+∑j=1kβjIqR0+ς+σjGμv1+Θ3r2−Θ3r1W1W7W8IqR0+ς−2Gμvζ+∑j=1kγjIqR0+ς+σj−2Gμv1≤g¯Γqς∫r1r2r2−qvς−1dqv+∫0r1r2−qvς−1−r1−qvς−1dqv+Θ1r2−Θ1r1W1W8IqR0+ςGμvζ+∑j=1kαjIqR0+ς+σjGμv1+Θ2r2−Θ2r1W8IqR0+ς−ρGμvζ+∑j=1kβjIqR0+ς+σjGμv1+Θ3r2−Θ3r1W1W7W8IqR0+ς−2Gμvζ+∑j=1kγjIqR0+ς+σj−2Gμv1.
Obviously, the above inequality goes to zero asr2−r1⟶0, independent of μ∈BY2. Hence, by helping the Arzelá-Ascoli theorem, F:A⟶A is completely continuous.
Now, we prove that there is an open setD⊂A such that μ≠κFμ for κ∈0,1 and xμ∈∂D.
Letμ∈A satisfies μ=κFμ for each κ∈0,1. So, for r∈O, by following the calculations applied in proving the boundedness of F, we have
(63)μr=κFμr≤Λp¯1μA+p¯2μAΓqω+1.
It yields that(64)μA≤p¯1ΛYμA+p¯2ΛμAΓqω+1.
Consequently, we obtain(65)Γqω+1−p¯2ΛμAp¯1ΛΓqω+1YμA≤1.
FromH3, there is M∗>0 such that μA≠M∗. Let
(66)D≔μ∈A:μ≤M∗+1,U=D∪BY2.
Notice thatF:U¯⟶A is completely continuous. For the sake of the choice of D, ∄x∈∂D such that μ=κFμ for some κ∈0,1. Therefore, by Lemma 10, we find out that F has the fixed point μ∈U¯ which implies that the Cap-q-difference FBVP (6) has at least one solution on O.


### 3.3. On the Stability Property for (6)

Stability analysis is one of the most important parts of each research in the field of existence of solution of fractional boundary value problems. For instances, we can mention to such a stability analysis in some newly published works including [24, 25, 30–32]. In this subsection, we introduce some concepts of stabilities for the Cap-q-difference FBVP (6). These definitions were extracted from [33].Letϵ>0, G:O×ℝ2⟶ℝ be continuous and θ:O⟶ℝ+ be a nondecreasing mapping. Assume that
(67)DqC0+ςμr−Gr,μr,IqR0+ωμr≤ε,(68)DqC0+ςμr−Gr,μr,IqR0+ωμr≤θr,(69)DqC0+ςμr−Gr,μr,IqR0+ωμr≤εθr.Definition 12.
The Cap-q-difference FBVP (6) is called Ulam-Hyers stable if ∃CG∈ℝ+ s.t. ∀ε>0 and every solution μ∈U of (67), a solution κ∈U of (6) exists s.t.
(70)μr−κr≤CGϵ,r∈O.Definition 13.
The Cap-q-difference FBVP (6) is called generalized Ulam-Hyers stable if ∃P∈Cℝ+,ℝ+, P0=0 s.t. ∀μ∈U fulfilling (67), a solution κ∈U of (6) exists s.t.
(71)μr−κr≤Pε,r∈O.Definition 14.
The Cap-q-difference FBVP (6) is Ulam-Hyers-Rassias stable w.r.t. θ if ∃Cθ∈ℝ+ s.t. ∀ε>0 and every solution μ∈U of (69), ∃ a solution κ∈U of (6) s.t.
(72)μr−κr≤Cθθrε,r∈O.Definition 15.
The Cap-q-difference FBVP (6) is termed generalized Ulam-Hyers-Rassias stable w.r.t. θ if ∃Cθ∈ℝ+ s.t. for every solution μ∈U of (68), ∃ a solution κ∈U of (6) s.t.
(73)μr−κr≤Cθθr,r∈O.Remark 16.
μ∈U is a solution of (67) if ∃ως∈U (dependent on μ) s.t.
(74)b1DqC0+ςμr=Gr,μr,IqR0+ωμr+ωςr,r∈O,b2ωςr≤ε.Lemma 17.
Ifμ∈U satisfies (67), then
(75)μr−λr≤Λε,where Λ is given as in (39) and λr is introduced in the proof.Proof.
Letμ satisfie (67). By (b1) of Remark 16, there is ως∈U (dependent on μ) such that
(76)DqC0+ςμr=Gr,μr,JqR0+ωμr+ωςr,r∈O,q∈0,1,μ0+μζ=∑j=1kαjJqR0+σjμ1,DqC0+ϱμ0+DqC0+ϱμζ=∑j=1kβjJqR0+σjμ1,DqC0+2μ0+DqC0+2μζ=∑j=1kγjJqR0+σjDqC0+2μ1.
Then, the solution of (76) is given as
(77)μr=IqR0+ςGμvr+Θ1rW1W8−IqR0+ςGμvζ+∑j=1kαjIqR0+ς+σjGμv1+Θ2rW8IqR0+ς−ϱGμvζ−∑j=1kβjIqR0+ς+σjGμv1+Θ3rW1W7W8−IqR0+ς−2Gμvζ+∑j=1kγjIqR0+ς+σj−2Gμv1+IqR0+ςωςr+Θ1rW1W8IqR0+ςωςζ+∑j=1kαjIqR0+ς+σjως1+Θ2rW8IqR0+ς−ϱωςζ−∑j=1kβjIqR0+ς+σjως1+Θ3rW1W7W8−IqR0+ς−2ωςζ+∑j=1kγjIqR0+ς+σj−2ως1.
For convenience, considerλr for the terms that are independent of ωςr. That is,
(78)λr=IqR0+ςGμvr+Θ1rW1W8−IqR0+ςGμvζ+∑j=1kαjIqR0+ς+σjGμv1+Θ2rW8IqR0+ς−ϱGμvζ−∑j=1kβjIqR0+ς+σjGμv1+Θ3rW1W7W8−IqR0+ς−2Gμvζ+∑j=1kγjIqR0+ς+σj−2Gμv1.
Therefore, (77) can be rewritten and by using (b2) of Remark 16, we have
(79)μr−λr≤IqR0+ςωςr+Θ1rW1W8IqR0+ςωςζ+∑j=1kαjIqR0+ς+σjως1+Θ2rW8IqR0+ς−ϱωςζ+∑j=1kβjIqR0+ς+σjως1+Θ3rW1W7W8IqR0+ς−2ωςζ+∑j=1kγjIqR0+ς+σj−2ως1≤Λε.
This inequality completes the proof.Theorem 18.
LetH1 and
(80)L1+L2Γqω+1Λ<1,to be held. Then, the Cap-q-difference FBVP (6) is both Ulam-Hyers and generalized Ulam-Hyers stable.Proof.
Letμ∈U satisfies (67) and κ fulfills the Cap-q-difference FBVP (6) given as
(81)DqC0+ςκr=Gr,κr,JqR0+ωκr,r∈O,q∈0,1,κ0+κζ=∑j=1kαjJqR0+σjκ1,DqC0+ϱκ0+DqC0+ϱκζ=∑j=1kβjJqR0+σjκ1,DqC0+2κ0+DqC0+2κζ=∑j=1kγjJqR0+σjDqC0+2κ1.
By the previous lemma, let(82)μr−κr≤μr−λr+λr−κr.
By using Lemma17 in (82), we have
(83)μr−κr≤Λϵ+L1+L2Γqω+1Λμr−κr.
Forr∈O, we have
(84)μ−κU≤Λϵ+L1+L2Γqω+1Λμ−κU.
After simplification, we get(85)μ−κU≤Λ1−L1+L2/Γqω+1Λϵ.
Thus(86)μr−κr≤CGϵ,where
(87)CG=Λ1−L1+L2/Γqω+1Λ.
Thus, the Cap-q-difference FBVP (6) is Ulam-Hyers stable.
In the sequel, the functionPϵ=CGϵ implies that the Cap-q-difference FBVP (6) is generalized Ulam-Hyers stable and P0=0.
Now, we add another condition.
(A1) Consider an increasing map πς∈CO,ℝ+. Then, there is ξπς>0 such that
(88)IqR0+ςπςr≤ξπςπςr.Remark 19.
Under the hypotheses (H1) and (A1) and (80), the Cap-q-difference FBVP (6) is the Ulam-Hyers-Rassias and generalized Ulam-Hyers-Rassias stable.


## 3.1. Uniqueness Result

The uniqueness result for the Cap-q-difference FBVP (6) is proved by using the Banach’s fixed-point theorem.Theorem 9.
Assume thatG∈CO×ℝ2,ℝ satisfies the following assumptions.
(H1) There are L1, L2>0 such that
(40)Gr,u1,v1−Gr,u2,v2≤L1u1−u2+L2v1−v2,for every ui, vi∈ℝ, i=1,2, and r∈O.
If(41)L1+L2Γqω+1Λ<1,where Λ is given in (39), and then the Cap-q-difference FBVP (6) has a unique solution μ in A.Proof.
We convert the Cap-q-difference FBVP (6) into μ=Fμ, where F is defined by (38). By the Banach’s contraction principle, we shall guarantee that F has exactly one fixed point.
At first, we define a bounded, closed convex subsetBY1≔μ∈A:μA≤Y1≠∅ with
(42)Y1≥ΛG1−L1+L2/Γqω+1Λ,where Λ is defined by (39).
Letsupr∈OGr,0,0≔G<∞. The proof will be completed in two steps:
Step 1.FBY1⊂BY1.
Letμ∈BY1 and r∈O. Estimate
(43)Fμr≤IqR0+ςGμvr+Θ1rW1W8IqR0+ςGμvζ+∑j=1kαjIqR0+ς+σjGμv1+Θ2rW8IqR0+ς−ϱGμvζ+∑j=1kβjIqR0+ς+σjGμv1+Θ3rW1W7W8IqR0+ς−2Gμvζ+∑j=1kγjIqR0+ς+σj−2Gμv1.
By using the property of integral (16), we get
(44)IqR0+ωμvr=1Γqω∫0rr−qvω−1μvdqv≤rωμAΓqω+1.
From the assumptionsH1 and (44), we can estimate
(45)Gμr≤gr,μr,IqR0+ςμr−gr,0,0+gr,0,0,0≤L1μr+L2Iqr0+ςμr+G≤L1+L2Γqω+1μA+G.
From (45) and by the property of integral (16), we obtain
(46)IqR0+ςGμvr≤L1+L2Γqω+1μA+GrςΓqς+1,(47)IqR0+ςGμvζ≤L1+L2Γqω+1μA+GζςΓqς+1,(48)IqR0+ς−ϱGμvζ≤L1+L2Γqω+1μA+Gζς−ϱΓqς−ϱ+1,(49)IqR0+ς−2Gμvζ≤L1+L2Γqω+1μA+Gζς−2Γqς−1,(50)IqR0+ς+σjGμv1≤L1+L2Γqω+1μA+G1Γqς+σj+1,(51)IqR0+ς+σj−2Gμv1≤L1+L2Γqω+1μA+G1Γqς+σj−1.
Substituting (46)–(51) into (43), we obtain
(52)Fμr≤L1+L2Γqω+1μA+GrςΓqς+1+Θ1rW1W8ζςΓqς+1+∑j=1kαjΓqς+σj+1+Θ2rW8ζς−ϱΓqς−ϱ+1+∑j=1kβjΓqς+σj+1+Θ3rW1W7W8ζς−2Γqς−1+∑j=1kγjΓqς+σj−1.
Then,(53)Fμr≤L1+L2Γqω+1μA+GΛ,which implies that FμA≤Y1. Thus, FBY1⊂BY1.
Step 2.F:A⟶A is a contraction.
Letμ, ϑ∈A. For each r∈O, we have
(54)Fμr−Fϑr≤Θ1rW1W8IqR0+ςGμv−Gϑvζ+∑j=1kαjIqR0+ς+σjGμv−Gϑv1+Θ2rW8IqR0+ς−ϱGϑv−Gϑvζ+∑j=1kβjIqR0+ς+σjGμv−Gϑv1+Θ3rW1W7W8IqR0+ς−2Gμv−Gϑvζ+∑j=1kγjIqR0+ς+σj−2Gμv−Gϑv1+IqR0+ςGμv−Gϑvr.
ByH1, it follows that
(55)Gμv−Gϑv≤gr,μr,IqR0+ςμr−gr,ϑr,IqR0+ςϑr≤L1+L2Γqω+1μ−ϑA.
Hence, by inserting (55) into (54) and using the property of integral (16), we get
(56)Fμr−Fϑr≤L1+L2Γqω+1μ−ϑArςΓqς+1+Θ1rW1W8ζςΓqς+1+∑j=1kαjΓqς+σj+1+Θ2rW8ζς−ϱΓqς−ϱ+1+∑j=1kβjΓqς+σj+1+Θ3rW1W7W8ζς−2Γqς−1+∑j=1kγjΓqς+σj−1,which implies that Fμ−FϑA≤L1+L2/Γqω+1Λμ−ϑA.
In view of (41), L1+L2/Γqω+1Λ<1, and we conclude that F is a contraction. Hence, in accordance with the Banach’s contraction principle, the Cap-q-difference FBVP (6) has a unique solution μ∈A.


## 3.2. Existence Result

The second result is based on the Leray-Schauder’s nonlinear alternative theorem.Lemma 10 (Leray-Schauder’s nonlinear alternative theorem [29]).
LetM be a Banach space, C be its closed convex subset, and X be an open set in C such that 0∈X. Let G:X¯⟶C be a continuous and compact function. Then either (i) there is μ∈X¯ such that μ=Gμ or (ii) there are μ∈∂X and ϱ∈0,1 such that μ=ϱGμ.Theorem 11.
LetG∈CO×ℝ2,ℝ satisfies the following assumptions:
(H2) There is continuous nondecreasing functions Y: ℝ+⟶ℝ+, p1, p2∈CJ,ℝ+ such that
(57)Gr,u,v≤p1rYu+p2rv,∀r,u,v∈O×ℝ2,where p¯i=supr∈Jpir, i=1,2.
(H3) There is M∗>0 such that
(58)1−Λp¯2/Γqω+1M∗Λ1p¯1YM∗>1.Then the Cap-q-difference FBVP (6) has at least one solution μ in A.Proof.
ConsiderF as (38). In the first step, we will prove that F corresponds bounded sets (balls) to bounded ones in A. For each positive real constant Y2, BY2≔μ∈A:μ≤Y2 is a bounded set (ball) in A. Let μ∈BY2. We have
(59)Fμr≤IqR0+ςGμvr+Θ1rW1W8IqR0+ςGμvζ+∑j=1kαjIqR0+ς+σjGμv1+Θ2rW8IqR0+ς−ϱGμvζ+∑j=1kβjIqR0+ς+σjGμv1+Θ3rW1W7W8IqR0+ς−2Gμvζ+∑j=1kγjIqR0+ς+σj−2Gμv1.
From (H2) and (44) in Theorem 9, we obtain
(60)Gr,μr,IqR0+ςμr≤p1rYμ+p2rIqR0+ςμr≤p¯1YY2+p¯2Y2Γqω+1≔g¯.
By the same process in Theorem9, we can estimate
(61)FμrA≤Λg¯.
Further, it will be investigated thatF corresponds bounded sets to equicontinuous sets of A.
Letr1, r2∈O with r1<r2 and μ∈BY2, where BY2 is a bounded set in A. Also, we obtain
(62)Fμr2−Fμr1≤IqR0+ςGμvr2−IqR0+ςGμvr1+Θ1r2−Θ1r1W1W8IqR0+ςGμvζ+∑j=1kαjIqR0+ς+σjGμv1+Θ2r2−Θ2r1W8IqR0+ς−ϱGμvζ+∑j=1kβjIqR0+ς+σjGμv1+Θ3r2−Θ3r1W1W7W8IqR0+ς−2Gμvζ+∑j=1kγjIqR0+ς+σj−2Gμv1≤1Γqς∫r1r2r2−qvς−1Gμvdqv+1Γqς∫0r1r2−qvς−1−r1−qvς−1Gμvdqv+Θ1r2−Θ1r1W1W8IqR0+ςGμvζ+∑j=1kαjIqR0+ς+σjGμv1+Θ2r2−Θ2r1W8IqR0+ς−ϱGμvζ+∑j=1kβjIqR0+ς+σjGμv1+Θ3r2−Θ3r1W1W7W8IqR0+ς−2Gμvζ+∑j=1kγjIqR0+ς+σj−2Gμv1≤g¯Γqς∫r1r2r2−qvς−1dqv+∫0r1r2−qvς−1−r1−qvς−1dqv+Θ1r2−Θ1r1W1W8IqR0+ςGμvζ+∑j=1kαjIqR0+ς+σjGμv1+Θ2r2−Θ2r1W8IqR0+ς−ρGμvζ+∑j=1kβjIqR0+ς+σjGμv1+Θ3r2−Θ3r1W1W7W8IqR0+ς−2Gμvζ+∑j=1kγjIqR0+ς+σj−2Gμv1.
Obviously, the above inequality goes to zero asr2−r1⟶0, independent of μ∈BY2. Hence, by helping the Arzelá-Ascoli theorem, F:A⟶A is completely continuous.
Now, we prove that there is an open setD⊂A such that μ≠κFμ for κ∈0,1 and xμ∈∂D.
Letμ∈A satisfies μ=κFμ for each κ∈0,1. So, for r∈O, by following the calculations applied in proving the boundedness of F, we have
(63)μr=κFμr≤Λp¯1μA+p¯2μAΓqω+1.
It yields that(64)μA≤p¯1ΛYμA+p¯2ΛμAΓqω+1.
Consequently, we obtain(65)Γqω+1−p¯2ΛμAp¯1ΛΓqω+1YμA≤1.
FromH3, there is M∗>0 such that μA≠M∗. Let
(66)D≔μ∈A:μ≤M∗+1,U=D∪BY2.
Notice thatF:U¯⟶A is completely continuous. For the sake of the choice of D, ∄x∈∂D such that μ=κFμ for some κ∈0,1. Therefore, by Lemma 10, we find out that F has the fixed point μ∈U¯ which implies that the Cap-q-difference FBVP (6) has at least one solution on O.


## 3.3. On the Stability Property for (6)

Stability analysis is one of the most important parts of each research in the field of existence of solution of fractional boundary value problems. For instances, we can mention to such a stability analysis in some newly published works including [24, 25, 30–32]. In this subsection, we introduce some concepts of stabilities for the Cap-q-difference FBVP (6). These definitions were extracted from [33].Letϵ>0, G:O×ℝ2⟶ℝ be continuous and θ:O⟶ℝ+ be a nondecreasing mapping. Assume that
(67)DqC0+ςμr−Gr,μr,IqR0+ωμr≤ε,(68)DqC0+ςμr−Gr,μr,IqR0+ωμr≤θr,(69)DqC0+ςμr−Gr,μr,IqR0+ωμr≤εθr.Definition 12.
The Cap-q-difference FBVP (6) is called Ulam-Hyers stable if ∃CG∈ℝ+ s.t. ∀ε>0 and every solution μ∈U of (67), a solution κ∈U of (6) exists s.t.
(70)μr−κr≤CGϵ,r∈O.Definition 13.
The Cap-q-difference FBVP (6) is called generalized Ulam-Hyers stable if ∃P∈Cℝ+,ℝ+, P0=0 s.t. ∀μ∈U fulfilling (67), a solution κ∈U of (6) exists s.t.
(71)μr−κr≤Pε,r∈O.Definition 14.
The Cap-q-difference FBVP (6) is Ulam-Hyers-Rassias stable w.r.t. θ if ∃Cθ∈ℝ+ s.t. ∀ε>0 and every solution μ∈U of (69), ∃ a solution κ∈U of (6) s.t.
(72)μr−κr≤Cθθrε,r∈O.Definition 15.
The Cap-q-difference FBVP (6) is termed generalized Ulam-Hyers-Rassias stable w.r.t. θ if ∃Cθ∈ℝ+ s.t. for every solution μ∈U of (68), ∃ a solution κ∈U of (6) s.t.
(73)μr−κr≤Cθθr,r∈O.Remark 16.
μ∈U is a solution of (67) if ∃ως∈U (dependent on μ) s.t.
(74)b1DqC0+ςμr=Gr,μr,IqR0+ωμr+ωςr,r∈O,b2ωςr≤ε.Lemma 17.
Ifμ∈U satisfies (67), then
(75)μr−λr≤Λε,where Λ is given as in (39) and λr is introduced in the proof.Proof.
Letμ satisfie (67). By (b1) of Remark 16, there is ως∈U (dependent on μ) such that
(76)DqC0+ςμr=Gr,μr,JqR0+ωμr+ωςr,r∈O,q∈0,1,μ0+μζ=∑j=1kαjJqR0+σjμ1,DqC0+ϱμ0+DqC0+ϱμζ=∑j=1kβjJqR0+σjμ1,DqC0+2μ0+DqC0+2μζ=∑j=1kγjJqR0+σjDqC0+2μ1.
Then, the solution of (76) is given as
(77)μr=IqR0+ςGμvr+Θ1rW1W8−IqR0+ςGμvζ+∑j=1kαjIqR0+ς+σjGμv1+Θ2rW8IqR0+ς−ϱGμvζ−∑j=1kβjIqR0+ς+σjGμv1+Θ3rW1W7W8−IqR0+ς−2Gμvζ+∑j=1kγjIqR0+ς+σj−2Gμv1+IqR0+ςωςr+Θ1rW1W8IqR0+ςωςζ+∑j=1kαjIqR0+ς+σjως1+Θ2rW8IqR0+ς−ϱωςζ−∑j=1kβjIqR0+ς+σjως1+Θ3rW1W7W8−IqR0+ς−2ωςζ+∑j=1kγjIqR0+ς+σj−2ως1.
For convenience, considerλr for the terms that are independent of ωςr. That is,
(78)λr=IqR0+ςGμvr+Θ1rW1W8−IqR0+ςGμvζ+∑j=1kαjIqR0+ς+σjGμv1+Θ2rW8IqR0+ς−ϱGμvζ−∑j=1kβjIqR0+ς+σjGμv1+Θ3rW1W7W8−IqR0+ς−2Gμvζ+∑j=1kγjIqR0+ς+σj−2Gμv1.
Therefore, (77) can be rewritten and by using (b2) of Remark 16, we have
(79)μr−λr≤IqR0+ςωςr+Θ1rW1W8IqR0+ςωςζ+∑j=1kαjIqR0+ς+σjως1+Θ2rW8IqR0+ς−ϱωςζ+∑j=1kβjIqR0+ς+σjως1+Θ3rW1W7W8IqR0+ς−2ωςζ+∑j=1kγjIqR0+ς+σj−2ως1≤Λε.
This inequality completes the proof.Theorem 18.
LetH1 and
(80)L1+L2Γqω+1Λ<1,to be held. Then, the Cap-q-difference FBVP (6) is both Ulam-Hyers and generalized Ulam-Hyers stable.Proof.
Letμ∈U satisfies (67) and κ fulfills the Cap-q-difference FBVP (6) given as
(81)DqC0+ςκr=Gr,κr,JqR0+ωκr,r∈O,q∈0,1,κ0+κζ=∑j=1kαjJqR0+σjκ1,DqC0+ϱκ0+DqC0+ϱκζ=∑j=1kβjJqR0+σjκ1,DqC0+2κ0+DqC0+2κζ=∑j=1kγjJqR0+σjDqC0+2κ1.
By the previous lemma, let(82)μr−κr≤μr−λr+λr−κr.
By using Lemma17 in (82), we have
(83)μr−κr≤Λϵ+L1+L2Γqω+1Λμr−κr.
Forr∈O, we have
(84)μ−κU≤Λϵ+L1+L2Γqω+1Λμ−κU.
After simplification, we get(85)μ−κU≤Λ1−L1+L2/Γqω+1Λϵ.
Thus(86)μr−κr≤CGϵ,where
(87)CG=Λ1−L1+L2/Γqω+1Λ.
Thus, the Cap-q-difference FBVP (6) is Ulam-Hyers stable.
In the sequel, the functionPϵ=CGϵ implies that the Cap-q-difference FBVP (6) is generalized Ulam-Hyers stable and P0=0.
Now, we add another condition.
(A1) Consider an increasing map πς∈CO,ℝ+. Then, there is ξπς>0 such that
(88)IqR0+ςπςr≤ξπςπςr.Remark 19.
Under the hypotheses (H1) and (A1) and (80), the Cap-q-difference FBVP (6) is the Ulam-Hyers-Rassias and generalized Ulam-Hyers-Rassias stable.


## 4. Analysis of the Cap-q-Difference System (7)

Here, we continue to discuss the existence and uniqueness results for the proposed system (7). In view of the assumptions of Section 3 for the Banach space A, the norm considered on the product space A×A is μ,ϑA×A=μA+ϑA which implies that A×A,μ,ϑA×A is a Banach space.Remark 20.
For convenience, and based on the given parameters in (7), we have nonzero constants:
(89)W¯1=2−∑j=1kϕjΓqδj+1,W¯2=ζ−∑j=1kϕjΓqδj+2,W¯3=ζ2−∑j=1kϕj1+qΓqδj+3,W¯4=−∑j=1kφjΓqδj+1,W¯5=−∑j=1kφjΓqδj+2,W¯6=2ζ2−ρΓq3−ρ−∑j=1kφj1+qΓqδj+3,W¯7=21+q−∑j=1kηj1+qΓqδj+1,W¯8=W¯2W¯4−W¯1W¯5,W¯9=W¯3W¯4−W¯1W¯6,W¯10=W¯8−W¯2W¯4,W¯11=W¯3W¯8−W¯2W¯9,Θ¯1r=rW¯1W¯4+W¯10,Θ¯2r=rW¯1−W¯2,Θ¯3r=r2W¯1W¯8−rW¯1W¯9−W¯11.
Keeping in mind Lemma8, consider the operator S:A×A⟶A×A as
(90)Sx,yr≔S1μ,ϑr,S2μ,ϑr,where
(91)S1μ,ϑr=IqR0+ς1Uϑvr+Θ1rW1W8−IqR0+ς1Uϑvζ+∑j=1kαjIqR0+ς1+σjUϑv1+Θ2rW8IqR0+ς1−ϱUϑvζ−∑j=1kβjIqR0+ς1+σjUϑv1+Θ3rW1W7W8−IqR0+ς1−2Uϑvζ+∑j=1kγjIqR0+ς1+σj−2Uϑv1,S2μ,ϑr=IqR0+ς2Vμvr+Θ¯1rW¯1W¯8−IqR0+ς2Vμvζ+∑j=1kϕjIqR0+ς2+δjVμv1+Θ¯2rW¯8IqR0+ς2−ρVμvζ−∑j=1kφjIqR0+ς2+δjVμv1+Θ¯3rW¯1W¯7W¯8−IqR0+ς2−2Vμvζ+∑j=1kηjIqR0+ς2+δj−2Vμv1.Before proceeding, consider the following estimates(92)Supr∈OΘ¯1r≔Θ¯1∗,Supr∈OΘ2r≔Θ¯2∗,Supr∈OΘ3r≔Θ¯3∗.To simplify, we also set the following notation and the constants(93)Λ1=1Γqς1+1+Θ1∗W1W8ζς1Γqς1+1+∑j=1kαjΓqς1+σj+1+Θ2∗W8ζς1−ϱΓqς1−ϱ+1+∑j=1kβjΓqς1+σj+1+Θ3∗W1W7W8ζς1−2Γqς1−1+∑j=1kγjΓqς1+σj−1,Λ2=1Γqς2+1+Θ¯1∗W¯1W¯8ζς2Γqς2+1+∑j=1kϕjΓqς2+δj+1+Θ¯2∗W¯8ζς2−ρΓqς2−ρ+1+∑j=1kφjΓqς2+δj+1+Θ¯3∗W¯1W¯7W¯8ζς2−2Γqς2−1+∑j=1kηjΓqς2+δj−1.


### 4.1. Uniqueness Result

In this step, we shall establish the existence of a unique solution to the coupled system of nonlinearq-CFBVPs (7), by the Banach’s contraction principle.Theorem 21.
LetG1, G2:O×ℝ2⟶ℝ be continuous. Assume that
(H4) There exist positive constants Li,Ki,i=1,2 such that for each r∈0,1 and ui, vi, u¯i, v¯i∈ℝ, and for i=1,2(94)G1r,u1,v1−G1r,u2,v2≤L1u1−u2+L2v1−v2,G2r,u¯1,v¯1−G2r,u¯2,v¯2≤K1u¯1−u¯2+K2v¯1−v¯2.
Then the coupled system of nonlinearq-CFBVPs (7) has a solution on O provided that
(95)Ω≔maxL1+L2Γqω1+1Λ1,K1+K2Γqω2+1Λ2<1.Proof.
We transform the coupled system of nonlinearq-CFBVPs (7) into a fixed-point problem μ,ϑr=Sμ,ϑr, where S is an operator as (90).
Letsupr∈OG1r,0,0≔MU<∞ and supr∈OG2r,0,0≔MV<∞. Next, we set BY3≔μ,ϑ∈A×A:μ,ϑA×A≤Y3 with
(96)Y3≥MUΛ1+MVΛ21−Ω.
Note thatBY3 is a bounded convex closed set in A.
Step 1.SBY3⊂BY3.
For eachμ,ϑ∈BY3 and r∈O, and by using the condition H4 and (44), we have
(97)Uϑr≤G1r,ϑr,qRI0+ω1ϑr−G1r,0,0+G1r,0,0≤L1ϑr+L2IqR0+ω1ϑr+MU≤L1+L2Γqω1+1ϑA+MU,Vμr≤G2r,μr,IqR0+ω2μr−G2r,0,0+G2r,0,0≤K1+K2Γqω2+1μA+MV.
Then, we get(98)S1μ,ϑr≤IqR0+ς1Uϑvr+Θ1rW1W8IqR0+ς1Uϑvζ+∑j=1kαjIqR0+ς1+σjUϑv1+Θ2rW8IqR0+ς1−ϱUϑvζ+∑j=1kβjIqR0+ς1+σjUϑv1+Θ3rW1W7W8IqR0+ς1−2Uϑvζ+∑j=1kγjIqR0+ς1+σj−2Uϑv1,≤rς1Γqς1+1+Θ1∗W1W8ζς1Γqς1+1+∑j=1kαjΓqς1+σj+1+Θ2∗W8ζς1−ϱΓqς1−ϱ+1+∑j=1kβjΓqς1+σj+1+Θ3∗W1W7W8ζς1−2Γqς1−1+∑j=1kγjΓqς1+σj−1×L1+L2Γqω1+1ϑA+MU.
Hence(99)S1μ,ϑA≤L1+L2Γqω1+1Λ1ϑA+MUΛ1.
Similarly, we find that(100)S2μ,ϑA≤K1+K2Γqω2+1Λ2μA+MVΛ2.
Consequently, we have(101)Sμ,ϑA×A≤L1+L2Γqω1+1Λ1ϑA+MUΛ1+K1+K2Γqω2+1Λ2μA+MVΛ2≤ΩY3+MUΛ1+MVΛ2≤Y3,which implies that SBY3⊂BY3.
Step 2. We show thatS:A×A⟶A×A is a contraction.
Using conditionH4, for any μ1,ϑ1, μ2,ϑ2∈A×A and for each r∈O, we have
(102)S1μ1,ϑ1r−S1μ2,ϑ2r≤Θ1rW1W8IqR0+ς1Uϑ1v−Uϑ2vζ+∑j=1kαjIqR0+ς1+σjUϑ1v−Uϑ2v1+Θ2rW8IqR0+ς1−ϱUϑ1v−Uϑ2vζ+∑j=1kβjIqR0+ς1+σjUϑ1v−Uϑ2v1+Θ3rW1W7W8IqR0+ς1−2Uϑ1v−Uϑ2vζ+∑j=1kγjIqR0+ς1+σj−2Uϑ1v−Uϑ2v1+IqR0+ς1Uϑ1v−Uϑ2vr≤rς1Γqς1+1+Θ1∗W1W8ζς1Γqς1+1+∑j=1kαjΓqς1+σj+1+Θ2∗W8ζς1−ϱΓqς1−ϱ+1+∑j=1kβjΓqς1+σj+1+Θ3∗W1W7W8ζς1−2Γqς1−1+∑j=1kγjΓqς1+σj−1×L1+L2Γqω1+1ϑ1−ϑ2A,and therefore
(103)S1μ1,ϑ1−S1μ2,ϑ2A≤L1+L2Γqω1+1Λ1ϑ1−ϑ2A.
Similarly, we get(104)S2μ1,ϑ1−S2μ2,ϑ2A≤K1+K2Γqω2+1Λ2μ1−μ2A.
From (103) and (104), it yields
(105)Sμ1,ϑ1−Sμ2,ϑ2A×A≤Ωϑ1−ϑ2A+μ1−μ2A.
AsΩ<1, by (95), the operator S is a contraction. The Banach’s contraction principle implies the existence of unique solution for the coupled system of nonlinear q-CFBVPs (7) on 0.1.


### 4.2. Existence Result

We get help from Lemma10 to complete the main result of this subsection.Theorem 22.
LetG1, G2:O×ℝ2⟶ℝ be continuous. Assume that
(H4) There exist nonnegative continuous maps xir,yir∈CO,ℝ+∪0, for i=1,2,3 such that
(106)G1r,u,v≤x1r+x2ru+x3rv,r,u,v∈O,ℝ2,G2r,u¯,v¯≤y1r+y2ru¯+y3rv¯,r,u¯,v¯∈O,ℝ2,with xi∗=supr∈Oxit and yi∗=supr∈Oyit.Then the coupled system of nonlinearq-CFBVPs (7) has at least one solution on O.Proof.
Here, the process of the proof will be continued during four steps as follows.
Step 1.S is continuous.
Letμn and ϑn be two sequences such that μn⟶μ and ϑn⟶ϑ in A . Then for each r∈O, we get
(107)S1μn,ϑnr−S1μ,ϑr≤Θ1rW1W8IqR0+ς1Uϑnv−Uϑvζ+∑j=1kαjIqR0+ς1+σjUϑnv−Uϑv1+Θ2rW8IqR0+ς1−ϱUϑnv−Uϑvζ+∑j=1kβjIqR0+ς1+σjUϑnv−Uϑv1+Θ3rW1W7W8IqR0+ς1−2Uϑnv−Uϑvζ+∑j=1kγjIqR0+ς1+σj−2Uϑnv−Uϑv1,+IqR0+ς1Uϑnv−Uϑvr≤rς1Γqς1+1+Θ1∗W1W8ζς1Γqς1+1+∑j=1kαjΓqς1+σj+1+Θ2∗W8ζς1−ϱΓqς1−ϱ+1+∑j=1kβjΓqς1+σj+1+Θ3∗W1W7W8ζς1−2Γqς1−1+∑j=1kγjΓqς1+σj−1Uϑn−UϑA,and therefore
(108)S1μn,ϑn−S1μ,ϑA≤Λ1Uϑn−UϑA.
Similarly, we get(109)S2μn,ϑn−S2μ,ϑA≤Λ2Vμn−VμA.
From (108) and (109), it yields
(110)Sμn,ϑn−Sμ,ϑA×A≤Λ1Uϑn−UϑA+Λ2Vμn−VμA.
Since the continuity ofG1 and G2 imply that of Uϑ, Vμ, so we have Uϑn−UϑA⟶0 and Vμn−VμA⟶0 as n⟶∞; and S is continuous.
Step 2.S is uniformly bounded.
We prove that forY4>0, there exists NS>0 such that for every μ,ϑ∈BY4, where
(111)BY4=μ,ϑ∈A×A:x,yA×A<Y4,we get Sμ,ϑA×A≤NS.
Using the conditionH5 and (16), we have
(112)Uϑr=G1r,ϑr,qRI0+ω1ϑr≤x1t+x2tϑr+x3tIqR0+ω1ϑr≤x1∗+x2∗+x3∗Γqω1+1ϑA,Vμr=G2r,μr,IqR0+ω2μr≤y1∗+y2∗+y3∗Γqω2+1μA.
Then, we get(113)S1μ,ϑr≤IqR0+ς1Uϑvr+Θ1rW1W8IqR0+ς1Uϑvζ+∑j=1kαjIqR0+ς1+σjUϑv1+Θ2rW8IqR0+ς1−ρUϑvζ+∑j=1kβjIqR0+ς1+σjUϑv1+Θ3rW1W7W8IqR0+ς1−2Uϑvζ+∑j=1kγjIqR0+ς1+σj−2Uϑv1≤rς1Γqς1+1+Θ1∗W1W8ζς1Γqς1+1+∑j=1kαjΓqς1+σj+1+Θ2∗W8ζς1−ϱΓqς1−ϱ+1+∑j=1kβjΓqς1+σj+1+Θ3∗W1W7W8ζς1−2Γqς1−1+∑j=1kγjΓqς1+σj−1×x1∗+x2∗+x3∗Γqω1+1ϑA.
Hence(114)S1μ,ϑA≤Λ1x1∗+x2∗+x3∗Γqω1+1ϑA.
Similarly, we find that(115)S2μ,ϑA≤K1+K2Γqω2+1Λ2μA+MVΛ2.
Consequently, we have(116)Sμ,ϑA×A≤Λ1x1∗+x2∗+x3∗Γqω1+1ϑA+Λ2y1∗+y2∗+y3∗Γqω2+1μA≔NS.
Then,S is uniformly bounded.
Step 3.S maps bounded sets into equi-continuous sets of A. Let r1,r2∈O such that r1<r2 and μ,ϑ∈BY4 where BY4 is defined as in Step 2. Then we have
(117)S1μ,ϑr2−S1μ,ϑr1≤IqR0+ς1Uϑvr2−IqR0+ς1Uϑvr1+Θ1r2−Θ1r2W1W8IqR0+ς1Uϑvζ+∑j=1kαjIqR0+ς1+σjUϑv1+Θ2r2−Θ2r1W8IqR0+ς1−ϱUϑvζ+∑j=1kβjIqR0+ς1+σjUϑv1+Θ3r2−Θ3r1W1W7W8IqR0+ς1−2Uϑvζ+∑j=1kγjIqR0+ς1+σj−2Uϑv1≤1Γqς1x1∗+x2∗+x3∗Γqω1+1ϑA∫r1r2r2−qvς1−1dqv+∫0r1r2−qvς1−1−r1−qvς1−1dqv+Θ1r2−Θ1r2W1W8IqR0+ς1Uϑvζ+∑j=1kαjIqR0+ς1+σjUϑv1+Θ2r2−Θ2r1W8IqR0+ς1−ϱUϑvζ+∑j=1kβjIqR0+ς1+σjUϑv1+Θ3r2−Θ3r1W1W7W8IqR0+ς1−2Uϑvζ+∑j=1kγjIqR0+ς1+σj−2Uϑv1,S2μ,ϑr2−S2μ,ϑr1≤1Γqς2y1∗+y2∗+y3∗Γqω2+1μA∫r1r2r2−qvς2−1dqv+∫0r1r2−qvς2−1−r1−qvς2−1dqv+Θ¯1r2−Θ¯1r1W¯1W¯8IqR0+ς2Vμvζ+∑j=1kϕjIqR0+ς2+δjVμv1+Θ¯2r2−Θ¯2r1W¯8IqR0+ς2−ρVμvζ+∑j=1kφjIqR0+ς2+δjVμv1+Θ¯3r2−Θ¯3r1W¯1W¯7W¯8IqR0+ς2−2Vμvζ+∑j=1kηjIqR0+ς2+δj−2Vμv1,which implies that
(118)Sμ,ϑr2−Sμ,ϑr1≤1Γqς1x1∗+x2∗+x3∗Γqω1+1ϑA∫r1r2r2−qvς1−1dqv+∫0r1r2−qvς1−1−r1−qvς1−1dqv+1Γqς2y1∗+y2∗+y3∗Γqω2+1μA∫r1r2r2−qvς2−1dqv+∫0r1r2−qvς2−1−r1−qvς2−1dqv+Θ1r2−Θ1r2W1W8IqR0+ς1Uϑvζ+∑j=1kαjIqR0+ς1+σjUϑv1+Θ¯1r2−Θ¯1r1W¯1W¯8IqR0+ς2Vμvζ+∑j=1kϕjIqR0+ς2+δjVμv1+Θ2r2−Θ2r1W8IqR0+ς1−ϱUϑvζ+∑j=1kβjIqR0+ς1+σjUϑv1+Θ¯2r2−Θ¯2r1W¯8IqR0+ς2−ρVμvζ+∑j=1kφjIqR0+ς2+δjVμv1+Θ3r2−Θ3r1W1W7W8IqR0+ς1−2Uϑvζ+∑j=1kγjIqR0+ς1+σj−2Uϑv1+Θ¯3r2−Θ¯3r1W¯1W¯7W¯8IqR0+ς2−2Vμvζ+∑j=1kηjIqR0+ς2+δj−2Vμv1.
The right-hand side tends to 0 asr2⟶r1, which is independent of μ,ϑ∈BY4. By helping the Arzelá-Ascoli theorem, S:A⟶A is completely continuous.
Step 4. The setB=μ,ϑ∈A×A:μ,ϑ=κSμ,ϑ,κ∈0,1 is bounded.
Letμ,ϑ∈B. Then μ,ϑ=κSμ,ϑ for some κ∈0,1. Thus, for any r∈O, by using the computations of Step 2, we have
(119)Sμ,ϑrA×A≤NS.
This means thatB is bounded. Consequently, by Lemma 10, S has a fixed point and so a solution to the coupled system of nonlinear q-CFBVPs (7).


## 4.1. Uniqueness Result

In this step, we shall establish the existence of a unique solution to the coupled system of nonlinearq-CFBVPs (7), by the Banach’s contraction principle.Theorem 21.
LetG1, G2:O×ℝ2⟶ℝ be continuous. Assume that
(H4) There exist positive constants Li,Ki,i=1,2 such that for each r∈0,1 and ui, vi, u¯i, v¯i∈ℝ, and for i=1,2(94)G1r,u1,v1−G1r,u2,v2≤L1u1−u2+L2v1−v2,G2r,u¯1,v¯1−G2r,u¯2,v¯2≤K1u¯1−u¯2+K2v¯1−v¯2.
Then the coupled system of nonlinearq-CFBVPs (7) has a solution on O provided that
(95)Ω≔maxL1+L2Γqω1+1Λ1,K1+K2Γqω2+1Λ2<1.Proof.
We transform the coupled system of nonlinearq-CFBVPs (7) into a fixed-point problem μ,ϑr=Sμ,ϑr, where S is an operator as (90).
Letsupr∈OG1r,0,0≔MU<∞ and supr∈OG2r,0,0≔MV<∞. Next, we set BY3≔μ,ϑ∈A×A:μ,ϑA×A≤Y3 with
(96)Y3≥MUΛ1+MVΛ21−Ω.
Note thatBY3 is a bounded convex closed set in A.
Step 1.SBY3⊂BY3.
For eachμ,ϑ∈BY3 and r∈O, and by using the condition H4 and (44), we have
(97)Uϑr≤G1r,ϑr,qRI0+ω1ϑr−G1r,0,0+G1r,0,0≤L1ϑr+L2IqR0+ω1ϑr+MU≤L1+L2Γqω1+1ϑA+MU,Vμr≤G2r,μr,IqR0+ω2μr−G2r,0,0+G2r,0,0≤K1+K2Γqω2+1μA+MV.
Then, we get(98)S1μ,ϑr≤IqR0+ς1Uϑvr+Θ1rW1W8IqR0+ς1Uϑvζ+∑j=1kαjIqR0+ς1+σjUϑv1+Θ2rW8IqR0+ς1−ϱUϑvζ+∑j=1kβjIqR0+ς1+σjUϑv1+Θ3rW1W7W8IqR0+ς1−2Uϑvζ+∑j=1kγjIqR0+ς1+σj−2Uϑv1,≤rς1Γqς1+1+Θ1∗W1W8ζς1Γqς1+1+∑j=1kαjΓqς1+σj+1+Θ2∗W8ζς1−ϱΓqς1−ϱ+1+∑j=1kβjΓqς1+σj+1+Θ3∗W1W7W8ζς1−2Γqς1−1+∑j=1kγjΓqς1+σj−1×L1+L2Γqω1+1ϑA+MU.
Hence(99)S1μ,ϑA≤L1+L2Γqω1+1Λ1ϑA+MUΛ1.
Similarly, we find that(100)S2μ,ϑA≤K1+K2Γqω2+1Λ2μA+MVΛ2.
Consequently, we have(101)Sμ,ϑA×A≤L1+L2Γqω1+1Λ1ϑA+MUΛ1+K1+K2Γqω2+1Λ2μA+MVΛ2≤ΩY3+MUΛ1+MVΛ2≤Y3,which implies that SBY3⊂BY3.
Step 2. We show thatS:A×A⟶A×A is a contraction.
Using conditionH4, for any μ1,ϑ1, μ2,ϑ2∈A×A and for each r∈O, we have
(102)S1μ1,ϑ1r−S1μ2,ϑ2r≤Θ1rW1W8IqR0+ς1Uϑ1v−Uϑ2vζ+∑j=1kαjIqR0+ς1+σjUϑ1v−Uϑ2v1+Θ2rW8IqR0+ς1−ϱUϑ1v−Uϑ2vζ+∑j=1kβjIqR0+ς1+σjUϑ1v−Uϑ2v1+Θ3rW1W7W8IqR0+ς1−2Uϑ1v−Uϑ2vζ+∑j=1kγjIqR0+ς1+σj−2Uϑ1v−Uϑ2v1+IqR0+ς1Uϑ1v−Uϑ2vr≤rς1Γqς1+1+Θ1∗W1W8ζς1Γqς1+1+∑j=1kαjΓqς1+σj+1+Θ2∗W8ζς1−ϱΓqς1−ϱ+1+∑j=1kβjΓqς1+σj+1+Θ3∗W1W7W8ζς1−2Γqς1−1+∑j=1kγjΓqς1+σj−1×L1+L2Γqω1+1ϑ1−ϑ2A,and therefore
(103)S1μ1,ϑ1−S1μ2,ϑ2A≤L1+L2Γqω1+1Λ1ϑ1−ϑ2A.
Similarly, we get(104)S2μ1,ϑ1−S2μ2,ϑ2A≤K1+K2Γqω2+1Λ2μ1−μ2A.
From (103) and (104), it yields
(105)Sμ1,ϑ1−Sμ2,ϑ2A×A≤Ωϑ1−ϑ2A+μ1−μ2A.
AsΩ<1, by (95), the operator S is a contraction. The Banach’s contraction principle implies the existence of unique solution for the coupled system of nonlinear q-CFBVPs (7) on 0.1.


## 4.2. Existence Result

We get help from Lemma10 to complete the main result of this subsection.Theorem 22.
LetG1, G2:O×ℝ2⟶ℝ be continuous. Assume that
(H4) There exist nonnegative continuous maps xir,yir∈CO,ℝ+∪0, for i=1,2,3 such that
(106)G1r,u,v≤x1r+x2ru+x3rv,r,u,v∈O,ℝ2,G2r,u¯,v¯≤y1r+y2ru¯+y3rv¯,r,u¯,v¯∈O,ℝ2,with xi∗=supr∈Oxit and yi∗=supr∈Oyit.Then the coupled system of nonlinearq-CFBVPs (7) has at least one solution on O.Proof.
Here, the process of the proof will be continued during four steps as follows.
Step 1.S is continuous.
Letμn and ϑn be two sequences such that μn⟶μ and ϑn⟶ϑ in A . Then for each r∈O, we get
(107)S1μn,ϑnr−S1μ,ϑr≤Θ1rW1W8IqR0+ς1Uϑnv−Uϑvζ+∑j=1kαjIqR0+ς1+σjUϑnv−Uϑv1+Θ2rW8IqR0+ς1−ϱUϑnv−Uϑvζ+∑j=1kβjIqR0+ς1+σjUϑnv−Uϑv1+Θ3rW1W7W8IqR0+ς1−2Uϑnv−Uϑvζ+∑j=1kγjIqR0+ς1+σj−2Uϑnv−Uϑv1,+IqR0+ς1Uϑnv−Uϑvr≤rς1Γqς1+1+Θ1∗W1W8ζς1Γqς1+1+∑j=1kαjΓqς1+σj+1+Θ2∗W8ζς1−ϱΓqς1−ϱ+1+∑j=1kβjΓqς1+σj+1+Θ3∗W1W7W8ζς1−2Γqς1−1+∑j=1kγjΓqς1+σj−1Uϑn−UϑA,and therefore
(108)S1μn,ϑn−S1μ,ϑA≤Λ1Uϑn−UϑA.
Similarly, we get(109)S2μn,ϑn−S2μ,ϑA≤Λ2Vμn−VμA.
From (108) and (109), it yields
(110)Sμn,ϑn−Sμ,ϑA×A≤Λ1Uϑn−UϑA+Λ2Vμn−VμA.
Since the continuity ofG1 and G2 imply that of Uϑ, Vμ, so we have Uϑn−UϑA⟶0 and Vμn−VμA⟶0 as n⟶∞; and S is continuous.
Step 2.S is uniformly bounded.
We prove that forY4>0, there exists NS>0 such that for every μ,ϑ∈BY4, where
(111)BY4=μ,ϑ∈A×A:x,yA×A<Y4,we get Sμ,ϑA×A≤NS.
Using the conditionH5 and (16), we have
(112)Uϑr=G1r,ϑr,qRI0+ω1ϑr≤x1t+x2tϑr+x3tIqR0+ω1ϑr≤x1∗+x2∗+x3∗Γqω1+1ϑA,Vμr=G2r,μr,IqR0+ω2μr≤y1∗+y2∗+y3∗Γqω2+1μA.
Then, we get(113)S1μ,ϑr≤IqR0+ς1Uϑvr+Θ1rW1W8IqR0+ς1Uϑvζ+∑j=1kαjIqR0+ς1+σjUϑv1+Θ2rW8IqR0+ς1−ρUϑvζ+∑j=1kβjIqR0+ς1+σjUϑv1+Θ3rW1W7W8IqR0+ς1−2Uϑvζ+∑j=1kγjIqR0+ς1+σj−2Uϑv1≤rς1Γqς1+1+Θ1∗W1W8ζς1Γqς1+1+∑j=1kαjΓqς1+σj+1+Θ2∗W8ζς1−ϱΓqς1−ϱ+1+∑j=1kβjΓqς1+σj+1+Θ3∗W1W7W8ζς1−2Γqς1−1+∑j=1kγjΓqς1+σj−1×x1∗+x2∗+x3∗Γqω1+1ϑA.
Hence(114)S1μ,ϑA≤Λ1x1∗+x2∗+x3∗Γqω1+1ϑA.
Similarly, we find that(115)S2μ,ϑA≤K1+K2Γqω2+1Λ2μA+MVΛ2.
Consequently, we have(116)Sμ,ϑA×A≤Λ1x1∗+x2∗+x3∗Γqω1+1ϑA+Λ2y1∗+y2∗+y3∗Γqω2+1μA≔NS.
Then,S is uniformly bounded.
Step 3.S maps bounded sets into equi-continuous sets of A. Let r1,r2∈O such that r1<r2 and μ,ϑ∈BY4 where BY4 is defined as in Step 2. Then we have
(117)S1μ,ϑr2−S1μ,ϑr1≤IqR0+ς1Uϑvr2−IqR0+ς1Uϑvr1+Θ1r2−Θ1r2W1W8IqR0+ς1Uϑvζ+∑j=1kαjIqR0+ς1+σjUϑv1+Θ2r2−Θ2r1W8IqR0+ς1−ϱUϑvζ+∑j=1kβjIqR0+ς1+σjUϑv1+Θ3r2−Θ3r1W1W7W8IqR0+ς1−2Uϑvζ+∑j=1kγjIqR0+ς1+σj−2Uϑv1≤1Γqς1x1∗+x2∗+x3∗Γqω1+1ϑA∫r1r2r2−qvς1−1dqv+∫0r1r2−qvς1−1−r1−qvς1−1dqv+Θ1r2−Θ1r2W1W8IqR0+ς1Uϑvζ+∑j=1kαjIqR0+ς1+σjUϑv1+Θ2r2−Θ2r1W8IqR0+ς1−ϱUϑvζ+∑j=1kβjIqR0+ς1+σjUϑv1+Θ3r2−Θ3r1W1W7W8IqR0+ς1−2Uϑvζ+∑j=1kγjIqR0+ς1+σj−2Uϑv1,S2μ,ϑr2−S2μ,ϑr1≤1Γqς2y1∗+y2∗+y3∗Γqω2+1μA∫r1r2r2−qvς2−1dqv+∫0r1r2−qvς2−1−r1−qvς2−1dqv+Θ¯1r2−Θ¯1r1W¯1W¯8IqR0+ς2Vμvζ+∑j=1kϕjIqR0+ς2+δjVμv1+Θ¯2r2−Θ¯2r1W¯8IqR0+ς2−ρVμvζ+∑j=1kφjIqR0+ς2+δjVμv1+Θ¯3r2−Θ¯3r1W¯1W¯7W¯8IqR0+ς2−2Vμvζ+∑j=1kηjIqR0+ς2+δj−2Vμv1,which implies that
(118)Sμ,ϑr2−Sμ,ϑr1≤1Γqς1x1∗+x2∗+x3∗Γqω1+1ϑA∫r1r2r2−qvς1−1dqv+∫0r1r2−qvς1−1−r1−qvς1−1dqv+1Γqς2y1∗+y2∗+y3∗Γqω2+1μA∫r1r2r2−qvς2−1dqv+∫0r1r2−qvς2−1−r1−qvς2−1dqv+Θ1r2−Θ1r2W1W8IqR0+ς1Uϑvζ+∑j=1kαjIqR0+ς1+σjUϑv1+Θ¯1r2−Θ¯1r1W¯1W¯8IqR0+ς2Vμvζ+∑j=1kϕjIqR0+ς2+δjVμv1+Θ2r2−Θ2r1W8IqR0+ς1−ϱUϑvζ+∑j=1kβjIqR0+ς1+σjUϑv1+Θ¯2r2−Θ¯2r1W¯8IqR0+ς2−ρVμvζ+∑j=1kφjIqR0+ς2+δjVμv1+Θ3r2−Θ3r1W1W7W8IqR0+ς1−2Uϑvζ+∑j=1kγjIqR0+ς1+σj−2Uϑv1+Θ¯3r2−Θ¯3r1W¯1W¯7W¯8IqR0+ς2−2Vμvζ+∑j=1kηjIqR0+ς2+δj−2Vμv1.
The right-hand side tends to 0 asr2⟶r1, which is independent of μ,ϑ∈BY4. By helping the Arzelá-Ascoli theorem, S:A⟶A is completely continuous.
Step 4. The setB=μ,ϑ∈A×A:μ,ϑ=κSμ,ϑ,κ∈0,1 is bounded.
Letμ,ϑ∈B. Then μ,ϑ=κSμ,ϑ for some κ∈0,1. Thus, for any r∈O, by using the computations of Step 2, we have
(119)Sμ,ϑrA×A≤NS.
This means thatB is bounded. Consequently, by Lemma 10, S has a fixed point and so a solution to the coupled system of nonlinear q-CFBVPs (7).


## 5. Numerical Examples

In this section, we provide some illustrative examples of the exactness and applicability of our main results.Example 1.
i Consider the Cap-q-difference FBVP of the form
(120)D0.8C0+2.5μr=Gr,μr,J0.8R0+3.8μr,r∈O,q∈0,1,μ0+μ0.4=∑j=1212−4j10J0.8R0+3j/10μ1,D0.8C0+1.2μ0+D0.8C0+1.2μ0.4=∑j=122j+310J0.8R0+3j/10μ1,D0.8C0+2μ0+D0.8C0+2μ0.4=∑j=1212−5j10J0.8R0+3j/10D0.8C0+2μ1.Hereς=2.5, q=0.8, ω=3.8, ζ=0.4, ϱ=1.2, αj=2j+3/10, βj=12−5j/10, γj=12−4j/10, σj=3j/10, and j=1,2. From the given data, we obtain W1≈0.676686276≠0, W7≈1.814092676≠0, and W8≈1.431872331≠0. We consider the functions
(121)Gr,μr,I0.8R0+3.8μr=4r−1re2r+4+9cosπ/32er+6·μrμr+3+10sinπ/62r+32+2e3r·I0.8R0+3.8μrI0.8R0+3.8μr+2.Forui, vi∈ℝ, and r∈O, we can find that
(122)Gr,u1,v1−Gr,u2,v2≤38u1−u2+511v1−v2.The assumptionH1 is satisfied under the values L1=3/8 and L2=5/11. Thus,
(123)L1+L2Γqω+1Λ≈0.8324696807<1.All assumptions of Theorem9 are valid. Then the Cap-q-difference FBVP (120) has a unique solution on 0,1. Moreover,
(124)CG=Λ1−L1+L2/Γqω+1Λ≈11.85782552>0.By the conclusions of Theorem18, the Cap-q-difference FBVP (120) is both Ulam–Hyers and also generalized Ulam-Hyers stable on 0,1. (ii) Set Gr,μr,I0.8R0+3.8μr=rλ.By using the property of integral (16) and setting λ=2.8, the implicit solution of the Cap-q-difference FBVP (120) is given by
(125)μr=Γqλ+1Γqλ+ς+1rλ+ς+Θ1rW1W8−Γqλ+1Γqλ+ς+1ζλ+ς+∑j=1kαjΓqλ+1Γqλ+ς+σj+1+Θ2rW8Γqλ+1Γqλ+ς−ρ+1ζλ+ς−ϱ−∑j=1kβjΓqλ+1Γqλ+ς+σj+1+Θ3rW1W7W8−Γqλ+1Γqλ+ς−1ζλ+ς−2+∑j=1kγjΓqλ+1Γqλ+ς+σj−1.Figure1 displays the solution of the Cap-q-difference FBVP (120) involving various values of ς=2.78,2.80,⋯,2.90 and q=0.56,0.60,⋯,0.80.Figure 1
The exact solutionμr of (120) for r∈0,1.Example 2.
Consider the coupled system of nonlinear Cap-q-difference FBVP under the conditions
(126)D0.7C0+2.8μr=G1r,ϑr,I0.7R0+1.7ϑr,r∈O,D0.7C0+2.9ϑr=G2r,μr,I0.7R0+2.3μr,μ0+μ0.3=∑j=124j10I0.7R0+5j−3/10μ1,ϑ0+ϑ0.3=∑j=1k12−5j10I0.7R0+3j−1/10ϑ1,D0.7C0+1.8μ0+D0.7C0+1.8μ0.3=∑j=127−2j10I0.7R0+5j−3/10μ1,D0.7C0+1.4ϑ0+D0.7C0+1.4ϑ0.3=∑j=1210−4j10I0.7R0+3j−1/10ϑ1,D0.7C0+2μ0+D0.7C0+2μ0.3=∑j=1210−3j10I0.7R0+5j−3/100.7CD0+2μ1,D0.7C0+2ϑ0+D0.7C0+2ϑ0.3=∑j=128−3j10I0.7R0+3j−1/100.7CD0+2ϑ1.Hereς1=2.8, ς2=2.9, q=0.7, ω1=1.7, ω2=2.3, ζ=0.3, ρ=1.8, ρ=1.4, αj=4j/10, βj=7−2j/10, γj=10−3j/10, ϕj=12−5j/10, φj=10−4j/10, ηj=8−3j/10, σj=5j−3/10, δj=3j−1/10, and j=1,2. From all the given data, we obtain W1≈0.705064917≠0, W7≈1.385967560≠0, W8≈1.029770834≠0, W¯1≈1.026846802≠0, W¯7≈2.110974612≠0, and W¯8≈1.174518052≠0. We consider the functions
(127)G1r,ϑr,I0.7R0+1.7ϑr=3r2−2r+1+r+1sin2r+6·ϑrϑr+3+2cosr3r+42·I0.7R0+1.7ϑrI0.7R0+1.7ϑr+1,G2r,μr,I0.7R0+2.3μr=re2r−3r+2r+sinr3er+4·μrμr+1+rln2r+1+3·I0.7R0+2.3μrI0.7R0+2.3μr+2.Forui, vi, u¯i, v¯i∈ℝ, and r∈O, we can find that
(128)G1r,u1,v1−Gr,u2,v2≤19u1−u2+18v1−v2,G2r,u¯1,v¯1−Gr,u¯2,v¯2≤37u¯1−u¯2+13v¯1−v¯2.The assumptionH4 is satisfied with L1=1/9, L2=1/9, K1=3/7, and K2=1/3. Hence, L1+L2/Γqω1+1Λ1≈0.6937912556<1 and K1+K2/Γqω2+1Λ1≈0.8947974715<1. All assumptions of Theorem 21 are satisfied. Then the coupled system of nonlinear Cap-q-difference FBVPs (126) has a unique solution on 0,1.


## 6. Conclusion

In this paper, a new category of nonlinear Caputo quantum boundary problems and its relevant generalized coupledq-system involving fractional quantum operators was discussed. We presented new q-difference equations and system in which we dealt with q-integro-sum-difference bundary conditions. Some qualitative aspects of solutions such as the existence, uniqueness, and different classes of stabilities of Ulam-Hyers type were investigated for both given q-Cap-difference problems. The results were examined with some examples. As a new idea in the next papers, we aim to extend our method for similar generalized coupled systems under the newly introduced generalized p,q-operators (postquantum operators).



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