
All authors Fractional Hahn differences and fractional Hahn integrals have various applications in fields where discrete fractional calculus plays a significant role, such as in discrete biological modeling and signal processing to handle systems with memory effects. In this study, the existence and uniqueness of solutions for a Riemann-Liouville fractional Hahn integrodifference equation with nonlocal fractional Hahn integral boundary conditions are investigated. To establish these results, we apply the Banach and Schauder fixed-point theorems. Furthermore, the Hyers-Ulam stability of solutions is studied.
Citation: Nichaphat Patanarapeelert, Jiraporn Reunsumrit, Thanin Sitthiwirattham. On nonlinear fractional Hahn integrodifference equations via nonlocal fractional Hahn integral boundary conditions[J]. AIMS Mathematics, 2024, 9(12): 35016-35037. doi: 10.3934/math.20241667
[1] | Nichaphat Patanarapeelert, Thanin Sitthiwiratthame . On nonlocal fractional symmetric Hanh integral boundary value problems for fractional symmetric Hahn integrodifference equation. AIMS Mathematics, 2020, 5(4): 3556-3572. doi: 10.3934/math.2020231 |
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All authors Fractional Hahn differences and fractional Hahn integrals have various applications in fields where discrete fractional calculus plays a significant role, such as in discrete biological modeling and signal processing to handle systems with memory effects. In this study, the existence and uniqueness of solutions for a Riemann-Liouville fractional Hahn integrodifference equation with nonlocal fractional Hahn integral boundary conditions are investigated. To establish these results, we apply the Banach and Schauder fixed-point theorems. Furthermore, the Hyers-Ulam stability of solutions is studied.
Quantum calculus, which is a form of calculus without the traditional concept of limits, deals with a set of non-differentiable functions. Quantum operators are widely utilized in various mathematical fields, including hypergeometric series, complex analysis, orthogonal polynomials, combinatorics, hypergeometric functions, and the calculus of variations. Quantum calculus also has numerous applications in areas such as quantum mechanics and particle physics [1,2,3,4].
In 1949, W. Hahn introduced the Hahn difference operator [5]. This operator is a combination of two well-known operators: the forward difference operator and the Jackson q-difference operator. The Hahn difference operator is defined by
Dq,ωf(t)=f(qt+ω)−f(t)t(q−1)+ω,t≠ω0, |
and Dq,ωf(ω0)=f′(ω0) where ω0:=ω1−q. We note that
Dq,ωf(t)=Δωf(t)whenever q=1,Dq,ωf(t)=Dqf(t)whenever ω=0, |
and Dq,ωf(t)=f′(t)whenever q=1,ω→0. |
The Hahn difference operator has been utilized in the study of families of orthogonal polynomials and in solving certain approximation problems (see [6,7,8]).
The right inverse of the Hahn difference operator was introduced by Aldwoah in 2009 [9,10]. This operator is expressed in terms of the Jackson q-integral, which contains the right inverse of Dq [11], and the Nörlund, which involves the right inverse of Δω [11].
In 2010, Malinowska and Torres [12,13] introduced the Hahn quantum variational calculus. In 2013, Malinowska and Martins [14] extended this work by presenting generalized transversality conditions for the Hahn quantum variational calculus. Subsequently, Hamza and Ahmed [15,16,17] developed the theory of linear Hahn difference equations and a general quantum difference calculus, studied the existence and uniqueness of solutions for initial value problems using the method of successive approximations, and proved Gronwall's and Bernoulli's inequalities in the context of the Hahn difference operator. They also investigated mean value theorems for this calculus. In 2016, Hamza and Makharesh [18] explored the Leibniz rule and Fubini's theorem associated with the Hahn difference operator. That same year, Sitthiwirattham [19] studied nonlocal boundary value problems (BVPs) for nonlinear Hahn difference equations. In 2021, Mac Quarrie et al. [20] proposed the Asymptotic Iteration Method for solving Hahn difference equations. In 2023, Hıra [21] focused on defining and proving fundamental properties of the Hahn Laplace Transform, including linearity, shifting theorems, and convolution theorems.
In 2010, ˇCermˊak and Nechvˊatal [22] introduced the fractional (q,h)-difference operator and the fractional (q,h)-integral for q>1. In 2011, ˇCermˊak, Kisela, and Nechvˊatal [23] presented linear fractional difference equations with discrete Mittag-Leffler functions for q>1. Rahmat [24,25] studied the (q,h)-Laplace transform and some (q,h)-analogues of integral inequalities on discrete time scales for q>1. In 2016, Du, Jai, Erbe, and Peterson [26] presented the monotonicity and convexity for nabla fractional (q,h)-difference for q>0,q≠1. Since fractional Hahn operators require a fractional parameter 0<q<1, the operators previously mentioned are not considered fractional Hahn operators. The fractional Hahn operators have been studied by Brikshavana and Sitthiwirattham [27]. There are research papers that focus on the BVPs for Hahn difference equations, such as [28,29,30,31,32].
Building on the foundation of quantum calculus, the study of fractional Hahn calculus has gained increasing attention due to its ability to generalize classical difference and integral operators. The fractional Hahn difference and integral operators provide a powerful framework for capturing memory effects and non-local behaviors, which are essential in modeling complex systems. In particular, the exploration of boundary value problems within this framework allows for a deeper understanding of dynamic systems governed by fractional discrete processes.
This research focuses on investigating boundary value problems involving fractional Hahn operators, aiming to extend the applicability of fractional Hahn calculus to broader mathematical and physical contexts. Such problems not only enrich the theoretical development of fractional discrete calculus but also pave the way for applications in fields like control theory, population dynamics, and numerical simulations. Specifically, we focus on a nonlocal Riemann-Liouville fractional Hahn integrodifference BVP of the form
Dαq,ωu(t)=λF[t,u(t),(Ψγq,ωu)(t)]+μH[t,u(t),(Υνq,ωu)(t)],t∈ITq,ω,Iβq,ωg1(η)u(η)=ϕ1(u),η∈ITq,ω−{ω0,T},Iβq,ωg2(T)u(T)=ϕ2(u), | (1.1) |
where [ω0,T]q,ω:={qkT+ω[k]q:k∈N0}∪{ω0}; 0<q<1,ω>0;α∈(1,2];β,γ,ν∈(0,1];λ,μ∈R+; F,H∈C([ω0,T]q,ω×R×R,R) and g1,g2∈C([ω0,T]q,ω,R+) are given functions; ϕ1,ϕ2:C([ω0,T]q,ω,R)→R are given functionals, and for φ,ψ∈C([ω0,T]q,ω×[ω0,T]q,ω,[0,∞)), we define operators
(Ψγq,ωu)(t):=(Iγq,ωφu)(t)=1Γq(γ)∫tω0(t−σq,ω(s))γ−1_q,ωφ(t,s)u(s)dq,ωs,(Υνq,ωu)(t):=(Dνq,ωψu)(t)=1Γq(−ν)∫tω0(t−σq,ω(s))−ν−1_q,ωψ(t,s)u(s)dq,ωs. | (1.2) |
Section 2 lays the groundwork by presenting fundamental definitions, properties, and lemmas. In Sections 3 and 4, we delve into the existence analysis and stability analysis of problem (1.1). We employ the powerful Banach fixed-point theorem to prove the existence and uniqueness of solutions, and we use the Schauder fixed-point theorem to establish the existence of at least one solution. To concretize our findings, Section 5 offers illustrative examples.
We establish necessary notation, definitions, and lemmas for the subsequent theorems. Let q∈(0,1), ω>0 and define
[n]q:=1−qn1−q=qn−1+...+q+1and[n]q!:=n∏k=11−qk1−q,n∈R. |
The q-analogue of the power function (a−b)n_q with n∈N0:=[0,1,2,...] is defined by
(a−b)0_q:=1,(a−b)n_q:=n−1∏k=0(a−bqk),a,b∈R. |
The q,ω-analogue of the power function (a−b)n_q,ω with n∈N0:=[0,1,2,...] is defined by
(a−b)0_q,ω:=1,(a−b)n_q,ω:=n−1∏k=0[a−(bqk+ω[k]q)],a,b∈N. |
In general, for α∈R, we define
(a−b)α_q=aα∞∏n=01−(ba)qn1−(ba)qα+n,a≠0, |
(a−b)α_q,ω=(a−ω0)α∞∏n=01−(b−ω0a−ω0)qn1−(b−ω0a−ω0)qα+n=((a−ω0)−(b−ω0))α_q,a≠ω0. |
We note that, aα_q=aα and (a−ω0)α_q,ω=(a−ω0)α and use the notation (0)α_q=(ω0)α_q,ω=0 for α>0. The q-gamma and q-beta functions are defined by
Γq(x):=(1−q)x−1_q(1−q)x−1,x∈R∖{0,−1,−2,...},Bq(x,s):=∫10tx−1(1−qt)s−1_qdqt=Γq(x)Γq(s)Γq(x+s). |
Definition 2.1. For q∈(0,1), ω>0 and f defined on an interval I⊆R that contains ω0:=ω1−q, the Hahn difference of f is defined by
Dq,ωf(t)=f(qt+ω)−f(t)t(q−1)+ωfor t≠ω0, |
and Dq,ωf(ω0)=f′(ω0). Providing that f is differentiable at ω0, we call Dq,ωf the q,ω-derivative of f and say that f is q,ω-differentiable on I.
Remark 2.1. We give some properties for the Hahn difference as follows.
(1) Dq,ω[f(t)+g(t)]=Dq,ωf(t)+Dq,ωg(t),
(2) Dq,ω[αf(t)]=αDq,ωf(t),
(3) Dq,ω[f(t)g(t)]=f(t)Dq,ωg(t)+g(qt+ω)Dq,ωf(t),
(4) Dq,ω[f(t)g(t)]=g(t)Dq,ωf(t)−f(t)Dq,ωg(t)g(t)g(qt+ω).
Letting a,b∈I⊆R with a<ω0<b and [k]q=1−qk1−q,k∈N0:=N∪{0}, we define the q,ω-interval by
Ia,bq,ω=[a,b]q,ω:={qka+ω[k]q:k∈N0}∪{qkb+ω[k]q:k∈N0}∪{ω0}=[a,ω0]q,ω∪[ω0,b]q,ω=(a,b)q,ω∪{a,b}=[a,b)q,ω∪{b}=(a,b]q,ω∪{a},and ITq,ω:=Iω0,Tq,ω=[ω0,T]q,ω. |
Observe that for each s∈[a,b]q,ω, the sequence {σkq,ω(s)}∞k=0={qks+ω[k]q}∞k=0 is uniformly convergent to ω0.
We also define the forward jump operator as σkq,ω(t):=qkt+ω[k]q and the backward jump operator as ρkq,ω(t):=t−ω[k]qqk for k∈N.
Definition 2.2. Let I be any closed interval of R that contains a,b and ω0. Assuming that f:I→R is a given function, we define the q,ω-integral of f from a to b by
∫baf(t)dq,ωt:=∫bω0f(t)dq,ωt−∫aω0f(t)dq,ωt, |
where
∫xω0f(t)dq,ωt:=[x(1−q)−ω]∞∑k=0qkf(xqk+ω[k]q),x∈I. |
Providing that the series converges at x=a and x=b, we call f is q,ω-integrable on [a,b] and the sum to the right hand side of above equation will be called the Jackson-Nörlund sum.
We note that, the actual domain of function f defined on [a,b]q,ω⊂I.
The following lemma presents the fundamental theorem of Hahn calculus.
Lemma 2.1. [9] Let f:I→R be continuous at ω0. Define
F(x):=∫xω0f(t)dq,ωt,x∈I. |
Then, F is continuous at ω0. Furthermore, Dq,ωF(x) exists for every x∈I and
Dq,ωF(x)=f(x). |
Conversely, we have
∫baDq,ωF(t)dq,ωt=F(b)−F(a) for alla,b∈I. |
Lemma 2.2. [19] Let q∈(0,1), ω>0, and f:I→R be continuous at ω0. Then,
∫tω0∫rω0f(s)dq,ωsdq,ωr=∫tω0∫tqs+ωf(s)dq,ωrdq,ωs. |
Lemma 2.3. [19] Let q∈(0,1), and ω>0. Then,
∫tω0dq,ωs=t−ω0 and ∫tω0[t−σq,ω(s)]dq,ωs=(t−ω0)21+q. |
Next, we present the definitions of the fractional Hahn integral and the Riemann-Liouville-type fractional Hahn difference.
Definition 2.3. For α,ω>0,q∈(0,1) and f defined on [ω0,T]q,ω, the fractional Hahn integral is defined by
Iαq,ωf(t):=1Γq(α)∫tω0(t−σq,ω(s))α−1_q,ωf(s)dq,ωs=[t(1−q)−ω]Γq(α)∞∑n=0qn(t−σn+1q,ω(t))α−1_q,ωf(σnq,ω(t)), |
and (I0q,ωf)(t)=f(t).
Definition 2.4. For α,ω>0,q∈(0,1), N−1<α<N,N∈N, and f defined on [ω0,T]q,ω, the fractional Hahn difference of the Riemann-Liouville type of order α is defined by
Dαq,ωf(t):=(DNq,ωIN−αq,ωf)(t)=1Γq(−α)∫tω0(t−σq,ω(s))−α−1_q,ωf(s)dq,ωs. |
The fractional Hahn difference of the Caputo type of order α is defined by
CDαq,ωf(t):=(IN−αq,ωDNq,ωf)(t)=1Γq(N−α)∫tω0(t−σq,ω(s))N−α−1_q,ωDNq,ωf(s)dq,ωs, |
and D0q,ωf(t)=CD0q,ωf(t)=f(t).
Lemma 2.4. [27] Let α>0,q∈(0,1),ω>0 and f:ITq,ω→R. Then,
Iαq,ωDαq,ωf(t)=f(t)+C1(t−ω0)α−1+...+CN(t−ω0)α−N, |
for some Ci∈R,i=N1,N and N−1<α≤N,N∈N.
Lemma 2.5. [27] Let α>0,q∈(0,1),ω>0, and f:ITq,ω→R. Then,
Iαq,ωCDαq,ωf(t)=f(t)+C0+C1(t−ω0)+...+CN−1(t−ω0)N−1, |
for some Ci∈R,i=N0,N−1 and N−1<α≤N,N∈N.
For computational efficiency, we offer these auxiliary results.
Lemma 2.6. [27] Let α,β>0,p,q∈(0,1), and ω>0. Then,
∫tω0(t−σq,ω(s))α−1_q,ω(s−ω0)β_q,ωdq,ωs=(t−ω0)α+βBq(β+1,α),∫tω0∫xω0(t−σp,ω(x))α−1_p,ω(x−σq,ω(s))β−1_q,ωdq,ωsdp,ωx=(t−ω0)α+β[β]qBp(β+1,α). |
To establish a foundation for the analysis of problem (1.1), we present a lemma addressing its linear variant and providing a corresponding solution representation.
Lemma 2.7. Let Ω≠0,α∈(1,2],β∈(0,1], ω>0,p,q∈(0,1),p=qm,m∈N, θ=ω(1−p1−q), and h∈C([ω0,T]q,ω,R) be given function. Then the problem
Dαq,ωu(t)=h(t),Iβq,ωg1(η)u(η)=ϕ1(u),η∈[ω0,T]q,ω−{ω0,T},Iβq,ωg2(T)u(T)=ϕ2(u), | (2.1) |
has the unique solution
u(t)=1Γq(α)∫tω0[t−σq,ω(s)]α−1_q,ωh(s)dq,ωs+(t−ω0)α−1Λ[BηOT−BTOη]+(t−ω0)α−2Λ[ATOη−AηOT], | (2.2) |
where the functionals and the constants are defined by
Λ:=ATBη−AηBT, | (2.3) |
Aη:=1Γq(β)∫ηω0g1(s)(η−σq,ω(s))β−1_q,ω(s−ω0)α−1dq,ωs, | (2.4) |
Bη:=1Γq(β)∫ηω0g1(s)(η−σq,ω(s))β−1_q,ω(s−ω0)α−2dq,ωs, | (2.5) |
AT:=1Γq(β)∫Tω0g2(s)(T−σq,ω(s))β−1_q,ω(s−ω0)α−1dq,ωs, | (2.6) |
BT:=1Γq(β)∫Tω0g2(s)(T−σq,ω(s))β−1_q,ω(s−ω0)α−2dq,ωs, | (2.7) |
Oη[ϕ1,h]:=ϕ1(u(η))−1Γq(β)1Γq(α)∫ηω0∫xω0g1(x)(η−σq,ω(x))β−1_q,ω[x−σq,ω(s)]α−1_q,ω×h(s)dq,ωsdq,ωx, | (2.8) |
OT[ϕ2,h]:=ϕ2(u(T))−1Γq(β)1Γq(α)∫Tω0∫xω0g2(x)(T−σq,ω(x))β−1_q,ω[x−σq,ω(s)]α−1_q,ω×h(s)dq,ωsdq,ωx. | (2.9) |
Proof. Taking the fractional Hahn integral of order α for (2.1), we obtain
u(t)=Iαq,ωh(t)+C1(t−ω0)α−1+C2(t−ω0)α−2=1Γq(α)∫tω0(t−σq,ω(s))α−1_q,ωh(s)dq,ωs+C1(t−ω0)α−1+C2(t−ω0)α−2. | (2.10) |
Multiplying (2.10) by g1(t) and taking the fractional Hahn difference of order β, we obtain
Iβq,ωg1(t)u(t)=1Γq(β)Γq(α)∫tω0∫xω0g1(x)(t−σq,ω(x))β−1_q,ω[x−σq,ω(s)]α−1_q,ωh(s)dq,ωsdq,ωx+1Γq(β)∫tω0g1(s)(t−σq,ω(s))β−1_q,ω[C1(s−ω0)α−1+C2(s−ω0)α−2]dq,ωs. | (2.11) |
Multiplying (2.10) by g2(t) and taking fractional Hahn difference of order β, we obtain
Iβq,ωg2(t)u(t)=1Γq(β)Γq(α)∫tω0∫xω0g2(x)(t−σq,ω(x))β−1_q,ω[x−σq,ω(s)]α−1_q,ωh(s)dq,ωsdq,ωx+1Γq(β)∫tω0g2(s)(t−σq,ω(s))β−1_q,ω[C1(s−ω0)α−1+C2(s−ω0)α−2]dq,ωs. | (2.12) |
Substituting t=η into (2.11) and using the first condition of (2.1), we have
[1Γq(β)∫ηω0g1(s)(η−σq,ω(s))β−1_q,ω(s−ω0)α−1dq,ωs]C1+[1Γq(β)∫ηω0g1(s)(η−σq,ω(s))β−1_q,ω(s−ω0)α−2dq,ωs]C2=AηC1+BηC2=Oη[ϕ1,h]. | (2.13) |
Substituting t=T into (2.12) and using the second condition of (2.1), we have
[1Γq(β)∫Tω0g2(s)(T−σq,ω(s))β−1_qω(s−ω0)α−1dq,ωs]C1+[1Γq(β)∫Tω0g2(s)(T−σq,ω(s))β−1_qω(s−ω0)α−2dq,ωs]C2=ATC1+BTC2=OT[ϕ2,h]. |
To find C1 and C2, we solve the system of Eqs (2.10) and (2.12). Then, we obtain
C1=BηOT−BTOηΛ,C2=ATOη−AηOTΛ, |
where Λ,Aη,AT,Bη,BT,Oη,OT are defined as (2.3)−(2.9), respectively.
Substituting the constants C1,C2 into (2.10), we obtain the solution for 2.1, as shown in Eq (2.2).
To prove the existence of a solution to Eq (1.1), we will employ the well-known Schauder's fixed point theorem.
Lemma 2.8. [33] (Arzelá-Ascoli theorem) A set of functions in C[a,b] with the sup norm, is relatively compact if and only if it is uniformly bounded and equicontinuous on [a,b].
Lemma 2.9. [33] If a set is closed and relatively compact, then it is compact.
Lemma 2.10. [34] (Schauder's fixed point theorem) Let (D,d) be a complete metric space, U be a closed convex subset of D, and T:D→D be the map such that the set Tu:u∈U is relatively compact in D. Then the operator T has at least one fixed point u∗∈U: Tu∗=u∗.
In this section, we prove the existence results for problem (1.1). Let C=C(ITq,ω,R) be a Banach space of all functions u with the norm defined by
‖u‖C=‖u‖+‖Dνq,ωu‖, |
where ‖u‖=maxt∈ITq,ω{|u(t)|} and ‖Dνq,ωu‖=maxt∈ITq,ω{|Dνq,ωu(t)|}.
Lemma 3.1. (C,‖⋅‖C) is Banach space.
Proof. Let {un}∞n=1 be any Cauchy sequence in the space (C,‖⋅‖C). Then ∀ε>0, there exists N>0 such that
‖un−um‖C=‖un−um‖+‖Dνq,ωun−Dνq,ωum‖<ε, |
for n,m>N. Therefore, for any fixed t0∈ITq,ω, the sequence {un(t0)}∞n=1 is a Cauchy sequence in R. In this way, the unique u(t) can be associated for each t∈ITq,ω. This defines (pointwise) a function u on ITq,ω. And can be shown u∈C and um→u with ‖un−um‖+‖Dνq,ωun−Dνq,ωum‖<∞. Letting n→∞, for every t∈ITq,ω, the following inequality holds.
|u(t)−um(t)|≤ε for all m>N. |
This means that um(t) converges to u(t) uniformly on ITq,ω. Since the um are continuous on ITq,ω and the convergence is uniform, the limit function u is continuous on ITq,ω. Hence u∈C and um→u. Next, that ‖un−um‖+‖Dνq,ωun−Dνq,ωum‖<ε will be proven. Consider for t∈ITq,ω
|u|+|Dνq,ωu|=|u(t)−um(t)+um(t)|+|Dνq,ωu−Dνq,ωum+Dνq,ωum|≤|u(t)−um(t)|+|um(t)|+|Dνq,ωu−Dνq,ωum|+|Dνq,ωum|<ε+ε. |
This implies
‖u‖+‖Dνq,ωu‖<∞. |
Hence, (C,‖⋅‖C) is a Banach space.
By Lemma 2.7, replacing h(t) by λF[t,u(t),(Ψγq,ωu)(t)]+μH[t,u(t),(Υνq,ωu)(t)], we define an operator A:C→C by
(Au)(t):=1Γq(α)∫tω0[t−σq,ω(s)]α−1_qω{λF[s,u(s),(Ψγq,ωu)(s)]+μH[s,u(s),(Υνq,ωu)(s)]}dq,ωs−(t−ω0)α−1Λ{BTO∗η(ϕ1,Fu+Hu)−BηO∗T(ϕ2,Fu+Hu)}+(t−ω0)α−2Λ{ATO∗η(ϕ1,Fu+Hu)−AηO∗T(ϕ2,Fu+Hu)}, | (3.1) |
where Λ,Aη,Bη,AT, and BT are defined in (2.3)–(2.7), respectively, and the functionals O∗η[ϕ1,Fu+Hu],O∗T[ϕ2,Fu+Hu] are defined by
O∗η[ϕ1,Fu+Hu]:=ϕ1(u(η))−1Γq(β)Γq(α)∫ηω0∫xω0g1(x)(η−σq,ω(x))β−1_q,ω[x−σq,ω(s)]α−1_q,ω×{λF[s,u(s),(Ψγq,ωu)(s)]+μH[s,u(s),(Υνq,ωu)(s)]}dq,ωsdq,ωx, | (3.2) |
O∗T[ϕ2,Fu+Hu]:=ϕ2(u(T))−1Γq(β)Γq(α)∫Tω0∫xω0g2(x)(T−σq,ω(x))β−1_q,ω[x−σq,ω(s)]α−1_q,ω×{λF[s,u(s),(Ψγq,ωu)(s)]+μH[s,u(s),(Υνq,ωu)(s)]}dq,ωsdq,ωx. | (3.3) |
Obviously, problem (1.1) has solutions if and only if the operator A has fixed points.
Theorem 3.1. Assume that F,H:ITq,ω×R×R→R is continuous, φ,ψ:ITq,ω×ITq,ω→[0,∞) is continuous with φ0=max{φ(t,s):(t,s)∈ITq,ω×ITq,ω} and ψ0=max{ψ(t,s):(t,s)∈ITq,ω×ITq,ω}. In addition, suppose that the following conditions hold:
(H1) There exist constants Mi>0 such that for each t∈ITq,ω and ui,vi∈R,i=1,2,
|F[t,u1,u2]−F[t,v1,v2]|≤M1|u1−v1|+M2|u2−v2|. |
(H2) There exist constants Ni>0 such that for each t∈ITq,ω and ui,vi∈R,i=1,2,
|H[t,u1,u2]−H[t,v1,v2]|≤N1|u1−v1|+N2|u2−v2|. |
(H3) There exist constants ω1,ω2>0 such that for each u,v∈C,
|ϕ1(u)−ϕ1(v)|≤ω1‖u−v‖Cand|ϕ2(u)−ϕ2(v)|≤ω2‖u−v‖C. |
(H4) For each t∈ITq,ω, ˆg1≤g1(t)≤G1 and ˆg2≤g2(t)≤G2.
(H5)X=L[Φ+G1(η−ω0)α+βΘ∗T+G2(T−ω0)α+βΘ∗ηΓq(α+β+1)]+Θ∗Tω1+Θ∗ηω2≤1,
where
L:=λ[M1+M2φ0(T−ω0)γΓq(γ+1)]+μ[N1+N2ψ0(T−ω0)−γΓq(−γ+1)], | (3.4) |
Φ:=(T−ω0)αΓq(α+1)+(T−ω0)α−νΓq(α−ν+1), | (3.5) |
Θ∗η:=Θη+ˉΘη, | (3.6) |
Θ∗T:=ΘT+ˉΘT, | (3.7) |
Θη:=1min|Λ|[max|Bη|(T−ω0)α−1+max|Aη|(T−ω0)α−2], | (3.8) |
ΘT:=1min|Λ|[max|BT|(T−ω0)α−1+max|AT|(T−ω0)α−2], | (3.9) |
ˉΘη:=1min|Λ|[max|Bη|(T−ω0)−ν+α−1Γq(α)Γq(α−ν)+max|Aη|(T−ω0)−ν+α−2Γq(α−1)Γq(α−1−ν)], | (3.10) |
ˉΘT:=1min|Λ|[max|BT|(T−ω0)−ν+α−1Γq(α)Γq(α−ν)+max|AT|(T−ω0)−ν+α−2Γq(α−1)Γq(α−1−ν)]. | (3.11) |
Then, problem (1.1) has a unique solution.
Proof. For each t∈ITq,ω and u,v∈C, we find that
|(Ψγq,ωu)(t)−(Ψγq,ωv)(t)|≤φ0Γq(γ)∫Tω0(T−σq,ω(s))γ−1_q,ω|u(s)−v(s)|dq,ωs≤φ0‖u−v‖Γq(γ)∫Tω0(T−σq,ω(s))γ−1_q,ωdq,ωs=φ0‖u−v‖(T−ω0)γΓq(γ+1). |
Similarly, we have |(Υνq,ωu)(t)−(Υνq,ωv)(t)|≤ψ0(T−ω0)−ν‖u−v‖Γq(−ν+1).
We set
F|u−v|(t):=|F[t,u(t),(Ψγq,ωu)(t)]−F[t,v(t),(Ψγq,ωv)(t)]|H|u−v|(t):=|H[t,u(t),(Υνq,ωu)(t)]−H[t,v(t),(Υνq,ωv)(t)]|. |
Then, we obtain
|O∗η[ϕ1,Fu+Hu,]−O∗η[ϕ1,Fv+Hv]|≤|ϕ1(u(η))−ϕ1(v(η))|+1Γq(β)Γq(α)∫ηω0∫xω0g1(x)(η−σq,ω(x))β−1_q,ω[x−σq,ω(s)]α−1_q,ω×{λF|u−v|(s)+μH|u−v|(s)}dq,ωsdq,ωx≤ω1‖u−v‖C+G1(η−ω0)α+βΓq(α+β+1)(λ[M1+M2φ0(T−ω0)γΓq(γ+1)]+μ[N1+N2ψ0(T−ω0)−νΓq(−ν+1)])‖u−c‖C=ω1‖u−v‖C+LG1(η−ω0)α+βΓq(α+β+1)‖u−c‖C=[ω1+LG1(η−ω0)α+βΓq(α+β+1)]‖u−v‖C. |
Similarly, we obtain
|O∗T[ϕ2,Fu+Hu]−O∗T[ϕ2,Fv+Hv]|≤(ω2+LG2(T−ω0)α+βΓq(α+β+1))‖u−v‖C. |
Next, we find that
|A(u)(t)−A(v)(t)|≤1Γq(α)∫Tω0(T−σq,ω(s))α−1_q,ω{λF|u−v|(s)+μH|u−v|(s)}dq,ωs+(T−ω0)α−1|Λ|[|BT||O∗η[ϕ1,Fu+Hu]−O∗η[ϕ1,Fv+Hv]|+|Bη||O∗T[ϕ2,Fu+Hu]−O∗T[ϕ2,Fv+Hv]|]+(T−ω0)α−2|Λ|[|AT||O∗η[ϕ1,Fu+Hu]−O∗η[ϕ1,Fv+Hv]|+|Aη||O∗T[ϕ2,Fu+Hu−O∗T[ϕ2,Fv+Hv]|]≤[L((T−ω0)αΓq(α+1)+G1(η−ω0)α+βΘT+G2(T−ω0)α+βΘηΓq(α+β+1))+ΘTω1+Θηω2]‖u−v‖C. | (3.12) |
Considering (DνqAu), we have
(Dνq,ωAu)=1Γq(−ν)Γq(α)∫tω0∫xω0(t−σq,ω(x))−ν−1(x−σq,ω(s))α−1_q,ω{λF(s,u(s),(Ψγq,ωu)(s))+μH(s,u(s),(Υνq,ωu)(s))}dq,ωsdq,ωx−{BTO∗η[ϕ1,Fu+Hu]−BηO∗T[ϕ2,Fu+Hu]}×1Γq(−ν)∫tω0(t−σq,ω(s))−ν−1_q,ω(s−ω0)α−1Λdq,ωs+{ATO∗η[ϕ1,Fu+Hu]−BηO∗T[ϕ2,Fu+Hu]}1Γq(−ν)∫tω0(t−σq,ω(s))−ν−1q,ω(s−ω0)α−2Λdq,ωs. | (3.13) |
Hence,
|(Dνq,ωAu)(t)−(Dνq,ωAv)(t)|≤[L((T−ω0)α−νΓq(α−ν+1)+G1(η−ω0)α+β¯ΘT+G2(T−ω0)α+β¯ΘηΓq(α+β+1))+ω1¯ΘT+ω2¯Θη]‖u−v‖C. | (3.14) |
From (3.12) and (3.14), we find that
‖A(u)−A(v)‖C≤ χ‖u−v‖C. |
Thus, the operator A is a contraction. Then, by the Banach contraction mapping principle, A has a fixed point, which is the unique solution for (1.1).
We next show the existence of a solution to (1.1) by the following Schauder's fixed point theorem.
Theorem 3.2. Let us assume that F,H:ITq,ω×R×R→R are continuous functions and ϕ1,ϕ2 : C(ITq,ω,R)→R are given functionals. Let us suppose that the following conditions hold:
(H6) There exist positive constants ˆF,ˆH such that for each t∈ITq,ω and ui∈R,i=1,2,
|F[t,u1,u2]|≤ˆFand|ˆH[t,u1,u2]|≤ˆH. |
(H7) There exist positive constants O1,O2 such that for each u∈C,
|ϕ1(u)|≤O1and|ϕ2(u)|≤O2. |
Then, problem (1.1) has at least one solution on ITq,ω.
Proof. We organize the proof into three steps.
(i) For each t∈ITq,ω and u∈BR, we obtain
|O∗η[ϕ1,Fu+Hu]|≤O1+(λˆF+μˆH)G1Γq(β)Γq(α)∫ηω0∫xω0(η−σq,ω(x))β−1_q,ω[x−σq,ω(s)]α−1_q,ωdq,ωsdq,ωx≤O1+G1(η−ω0)α+β(λˆF+μˆH)Γq(α+β+1). | (3.15) |
Similarly, we have
|O∗T[ϕ2,Fu+Hu]|≤O2+G2(λˆF+μˆH)(T−ω0)α+βΓq(α+β+1). | (3.16) |
From (3.15) and (3.16) and for each t∈ITq,ω, we find that
|(Au)(t)|≤1Γq(α)∫Tω0(T−σq,ω(s))α−1_q,ω|λF[s,u(s),(ψγq,ωu)(s)]+μH[s,u(s),(Υνq,ωu)(s)]|dq,ωs+(T−ω0)α−1|Λ|[|BT||O∗η[ϕ1,Fu+Hu]|+|Bη||O∗T[ϕ2,Fu+Hu]|]+(T−ω0)α−2|Λ|[|AT||O∗η[ϕ1,Fu+Hu]|+|Aη||O∗T[ϕ2,Fu+Hu]|]≤(λˆF+μˆH)[(T−ω0)αΓq(α+1)+G1(η−ω0)α+βΘT+G2(T−ω0)α+βΘηΓq(α+β+1)]+O1ΘT+O2Θη. | (3.17) |
In addition, we obtain
|(Dνq,ωAu)(t)|≤(λˆF+μˆH)[(T−ω0)α−νΓq(α−ν+1)+G1(η−ω0)α+βˉΘT+G2(T−ω0)α+βˉΘηΓq(α+β+1)]+O1ˉΘT+O2ˉΘη. | (3.18) |
From (3.17) and (3.18) we obtain
‖(Au)‖C≤(λˆF+μˆH)[Φ+G1(η−ω0)α+βΘ∗T+G2(T−ω0)α+βΘ∗ηΓq(α+β+1)]+O1Θ∗T+O2Θ∗η<∞, |
which implies that A(BR) is uniformly bounded.
(ii) We show that A(BR) is equicontinuous. for any t1,t2∈ITq,ω with t1<t2, we have
|(Au)(t2)−(Au)(t1)|≤(λˆF+λˆH)Γq(α+1)|(t2−ω0)α−(t1−ω0)α|+|(t2−ω0)α−1−(t1−ω0)α−1||Λ|{|BT||O∗η[ϕ1,Fu+Hu]|+|Bη||O∗T[ϕ2,Fu+Hu]|}+|(t2−ω0)α−2−(t1−ω0)α−2||Λ|{|AT||O∗η[ϕ1,Fu+Hu]|+|Aη||O∗T[ϕ2,Fu+Hu]|} | (3.19) |
and
|(Dνq,ωAu)(t2)−(Dνq,ωAu)(t1)|≤(λˆF+λˆH)Γq(α−ν+1)|(t2−ω0)α−ν−(t1−ω0)α−ν|+Γq(α)|(t2−ω0)α−ν−1−(t1−ω0)α−ν−1||Λ|Γq(α−ν){|BT||O∗η[ϕ1,Fu+Hu]|+|Bη||O∗T[ϕ2,Fu+Hu]|}+Γq(α−1)|(t2−ω0)α−ν−2−(t1−ω0)α−ν−2||Λ|Γq(α−ν−1){|AT||O∗η[ϕ1,Fu+Hu]|+|Aη||O∗T[ϕ2,Fu+Hu]|}. | (3.20) |
The right-hand side of (3.19) and (3.20) tends to zero as t_1\rightarrow t_2 , independently of u , which implies that \mathcal{A}(B_R) is an equicontinuous set. By using the Arzela–Ascoli theorem, the set \mathcal{A}(B_R) is compact.
(iii) Finally, we show that \mathcal{W} = \{u\in C: u = \zeta\mathcal{A} u, 0 < \zeta < 1\} is a bounded set. Let u\in\mathcal{W} . Since |u| = \zeta \|\mathcal{A} u\|\leq \zeta \|\mathcal{A}u\|_\mathcal{C} , and from (i) \|\mathcal{A}u\|_\mathcal{C} is bounded, hence \mathcal{W} is bounded. Then, as in (i), we have
\begin{align*} \big|u(t)\big|\,\leq\; &\zeta\|\mathcal{A}u\|_C\\ \leq\; &\big(\lambda\hat{\mathcal{F}}+\mu\hat{\mathcal{H}}\big) \Big[\Phi+\frac{G_1(\eta-\omega_0)^{\alpha+\beta}\Theta_T^*+G_2(T-\omega_0)^{\alpha+\beta}\Theta_\eta^*}{\Gamma_q(\alpha+\beta+1)}\Big]+\mathcal{O}_1 \Theta^*_T+\mathcal{O}_2 \Theta^*_\eta.\nonumber \end{align*} |
Therefore, \mathcal{W} is bounded.
In this section, we study the Hyers-Ulam stability of system (1.1). Let \varepsilon > 0 and \delta: I_{q, \omega}^T\rightarrow {\mathbb{R}} be a continuous function. Consider
\begin{eqnarray} &&\bigg|D^\alpha_{q,\omega} u(t)-\lambda F\left[ t,u(t),\left( \Psi^\gamma_{q,\omega}u\right)(t)\right] -\mu H\left[ t,u(t),\left( \Upsilon^\nu_{q,\omega}u\right)(t)\right]\bigg| \leq \varepsilon\delta(t), \quad t\in I_{q,\omega}^T,\\ &&{\mathcal{I}}_{q,\omega}^\beta g_1(\eta)u(\eta) = \phi_1(u),\quad\eta \in I_{q,\omega}^T - \left\lbrace \omega_0,T\right\rbrace ,\\ &&{\mathcal{I}}_{q,\omega}^\beta g_2\left(T\right) u\left( T\right) = \phi_2(u). \end{eqnarray} | (4.1) |
Now, we give out the definition of Hyers-Ulam stability of system (1.1).
Definition 4.1. System (1.1) is Hyers-Ulam stable with respect to system (4.1), if there exists A_{F, H} > 0 such that
|\bar{u}-\tilde{u}|\leq\varepsilon A_{F,H} |
for all t\in I_{q, \omega}^T , where \bar{u} is the solution of (4.1) and \tilde{u} is the solution for system (1.1).
Theorem 4.1. Assume that (H_1)-(H_5) hold, and \max_{t \in I_{q, \omega}^T} \delta(t)\leq 1. Then the system (1.1) is Hyers–Ulam stable with respect to system (4.1).
Proof. Let D^\alpha_{q, \omega} \bar{u}(t) = \lambda F\left[ t, \bar{u}(t), \left(\Psi^\gamma_{q, \omega}\bar{u}\right)(t)\right]+\mu H\left[ t, \bar{u}(t), \left(\Upsilon^\nu_{q, \omega}\bar{u}\right)(t)\right]+k(t) . Consider
\begin{eqnarray} &&D^\alpha_{q,\omega} \bar{u}(t) = \lambda F\left[ t,\bar{u}(t),\left( \Psi^\gamma_{q,\omega}\bar{u}\right)(t)\right]+\mu H\left[ t,\bar{u}(t),\left( \Upsilon^\nu_{q,\omega}\bar{u}\right)(t)\right]+k(t), \quad t\in I_{q,\omega}^T,\\ &&{\mathcal{I}}_{q,\omega}^\beta g_1(\eta)u(\eta) = \phi_1(u),\quad\eta \in I_{q,\omega}^T - \left\lbrace \omega_0,T\right\rbrace ,\\ &&{\mathcal{I}}_{q,\omega}^\beta g_2\left(T\right) u\left( T\right) = \phi_2(u). \end{eqnarray} | (4.2) |
Similarly to the system in Theorem 3.1, system (4.2) is equivalent to the following equation in Lemma 2.7.
\begin{align} \bar{u}(t)\,: = \; &\frac{1}{\Gamma_q(\alpha)}\int_{\omega_0}^{t}\Big[t-\sigma_{q,\omega}(s)\Big]_{q\omega}^{\underline{\alpha-1}} \Big\{ \lambda F\left[ s,\bar{u}(s),\left( \Psi^\gamma_{q,\omega}\bar{u}\right)(s)\right]+\mu H\left[ s,\bar{u}(s),\left( \Upsilon^\nu_{q,\omega}\bar{u}\right)(s)\right]+k(s)\Big\}d_{q,\omega}s\\ &- \frac{(t-\omega_0)^{\alpha-1}}{\Lambda}\Big\{\mathcal{B}_T\bar{\mathcal{O}}_\eta^*(\phi_1,F_{\bar{u}}+H_{\bar{u}} +k)-\mathcal{B}_\eta\bar{\mathcal{O}}_T^*(\phi_2,F_{\bar{u}}+H_{\bar{u}}+k)\Big\}\\ &+ \frac{(t-\omega_0)^{\alpha-2}}{\Lambda}\Big\{\mathcal{A}_T\bar{\mathcal{O}}_\eta^*(\phi_1,F_{\bar{u}}+H_{\bar{u}}+k)-\mathcal{A}_\eta\bar{\mathcal{O}}_T^*(\phi_2,F_{\bar{u}}+H_{\bar{u}}+k)\Big\}, \end{align} | (4.3) |
where \Lambda, {\mathcal{A}_\eta}, {\mathcal{B}_\eta}, {\mathcal{A}_T} , and {\mathcal{B}_T} are defined in (2.3)–(2.7), respectively, and the functionals \bar{\mathcal{O}}_\eta^*[\phi_1, F_{\bar{u}}+H_{\bar{u}}+k], \; \bar{\mathcal{O}}_T^*[\phi_2, F_{\bar{u}}+H_{\bar{u}}+k] are defined by
\begin{align} \bar{\mathcal{O}}_\eta^*[\phi_1,F_{\bar{u}}+H_{\bar{u}}+k]\,: = \; &\phi_1(u(\eta))-\frac{1}{\Gamma_q(\beta)\Gamma_q(\alpha)}\int_{\omega_0}^{\eta}\int_{\omega_0}^{x}g_1(x)\big(\eta-\sigma_{q,\omega}(x)\big)_{q,\omega}^{\underline{\beta-1}}\big[x-\sigma_{q,\omega}(s)\big]_{q,\omega}^{\underline{\alpha-1}} \\ &\times \Big\{\lambda F\left[ s,{\bar{u}}(s),\left( \Psi^\gamma_{q,\omega}{\bar{u}}\right)(s)\right] +\mu H\left[ s,{\bar{u}}(s),\left( \Upsilon^\nu_{q,\omega}{\bar{u}}\right)(s)\right]+k(s)\Big\}d_{q,\omega}sd_{q,\omega}x, \end{align} | (4.4) |
\begin{align} \bar{\mathcal{O}}_T^*[\phi_2,F_{\bar{u}}+H_{\bar{u}}+k]\,: = \; & \phi_2(u(T))-\frac{1}{\Gamma_q(\beta)\Gamma_q(\alpha)}\int_{\omega_0}^{T}\int_{\omega_0}^{x}g_2(x)\big(T-\sigma_{q,\omega}(x)\big)_{q,\omega}^{\underline{\beta-1}}\big[x-\sigma_{q,\omega}(s)\big]_{q,\omega}^{\underline{\alpha-1}} \\ & \times\Big\{\lambda F\left[ s,{\bar{u}}(s),\left( \Psi^\gamma_{q,\omega}{\bar{u}}\right)(s)\right] +\mu H\left[ s,{\bar{u}}(s),\left( \Upsilon^\nu_{q,\omega}{\bar{u}}\right)(s)\right]+k(s)\Big\}d_{q,\omega}sd_{q,\omega}x. \end{align} | (4.5) |
Now, we define the operator as
\begin{align} ({\tilde{\mathcal{A}}}u)(t) = ({\mathcal{A}}u)(t)+\mathcal{K}(t), \end{align} | (4.6) |
where
\begin{align} \mathcal{K}(t)\, = \; &\frac{1}{\Gamma_q(\alpha)} \int_{\omega_0}^{t} \Big[t-\sigma_{q,\omega}(s)\Big]_{q,\omega}^{\underline{\alpha-1}}k(s)d_{q,\omega}s+\frac{(t-\omega_0)^{\alpha-1}}{\Lambda}\Big[\mathcal{B}_\eta \mathcal{O}_T[k]-\mathcal{B}_T \mathcal{O}_\eta[k]\Big]\\ &+ \frac{(t-\omega_0)^{\alpha-2}}{\Lambda}\Big[\mathcal{A}_T \mathcal{O}_\eta[k]-\mathcal{A}_\eta \mathcal{O}_T[k]\Big], \end{align} | (4.7) |
where the functionals are defined by
\begin{align} {\mathcal{O}_\eta}[k]\,: = \; &k(\eta)-\frac{1}{\Gamma_q(\beta)}\frac{1}{\Gamma_q(\alpha)} \int_{\omega_0}^{\eta} \int_{\omega_0}^{x}g_1(x)\big(\eta-\sigma_{q,\omega}(x)\big)_{q,\omega}^{\underline{\beta-1}}\big[x-\sigma_{q,\omega}(s)\big]_{q,\omega}^{\underline{\alpha-1}}k(s)d_{q,\omega}s d_{q,\omega}x, \end{align} | (4.8) |
\begin{align} {\mathcal{O}_T}[k]\,: = \; &k(T)-\frac{1}{\Gamma_q(\beta)}\frac{1}{\Gamma_q(\alpha)} \int_{\omega_0}^{T} \int_{\omega_0}^{x}g_2(x)\big(T-\sigma_{q,\omega}(x)\big)_{q,\omega}^{\underline{\beta-1}}\big[x-\sigma_{q,\omega}(s)\big]_{q,\omega}^{\underline{\alpha-1}}k(s)d_{q,\omega}s d_{q,\omega}. \end{align} | (4.9) |
Note that
\begin{align} \|\tilde{\mathcal{A}}u-\tilde{\mathcal{A}}v\| = \|\mathcal{A}u-\mathcal{A}v\|. \end{align} | (4.10) |
Then the existence of a solution of (1.1) implies the existence of a solution to (4.2). It follows from Theorem 3.1 that \tilde{\mathcal{A}} is a contraction. Thus there is a unique fixed point \bar{u} of \tilde{\mathcal{A}} , and \tilde{u} of {\mathcal{A}} .
Since t\in I^T_{q, \omega} and \max_{t \in I_{q, \omega}^T} \delta(t)\leq 1 , we obtain
\begin{align} \|\mathcal{K}\| = \max\limits_{t \in I_{q,\omega}^T} |\mathcal{K}(t)|\leq \varepsilon\hat{\chi}, \end{align} | (4.11) |
where
\begin{align} \hat{\chi} = \Big[\frac{\varphi_0(T-\omega_0)^\gamma}{\Gamma_q(\gamma+1)}+\frac{\psi_0(T-\omega_0)^{-\gamma}}{\Gamma_q(-\gamma+1)}\Big] \Big[\Phi+\frac{G_1(\eta-\omega_0)^{\alpha+\beta}\Theta_T^*+G_2(T-\omega_0)^{\alpha+\beta}\Theta_\eta^*}{\Gamma_q(\alpha+\beta+1)}\Big]+\Theta_T^*+\Theta_\eta^*, \end{align} | (4.12) |
\Phi, \Theta_T^* and \Theta_\eta^* are defined by (3.5)–(3.7), respectively.
Hence, we obtain
\begin{eqnarray} \|\bar{u}-\tilde{u}\| = \|\tilde{\mathcal{A}}\bar{u}-\mathcal{A}\tilde{u}\| = \|{\mathcal{A}}\bar{u}-\mathcal{A}\tilde{u}+\mathcal{K}(t)\| = \|{\mathcal{A}}\bar{u}-\mathcal{A}\tilde{u}\|+\|\mathcal{K}(t)\| = \chi \|\bar{u}-\tilde{u}\|+\varepsilon\hat{\chi}. \end{eqnarray} | (4.13) |
By condition (H_5) , we obtain
\begin{align} \|\bar{u}-\tilde{u}\|\leq \frac{\varepsilon\hat{\chi}}{1-\chi}. \end{align} | (4.14) |
Let A_{F, H} = \frac{\hat{\chi}}{1-\chi} , then
\begin{align} \|\bar{u}-\tilde{u}\|\leq \varepsilon A_{F,H}. \end{align} | (4.15) |
This completes the proof.
To elucidate our results, we present several illustrative examples.
Example 5.1. Consider the following fractional Hahn BVP
\begin{align} D_{\frac{1}{2},\frac{1}{3}}^{\frac{3}{2}} u(t) = \,& \Big[\frac{e^{-[\sin^2(2\pi t)+\pi]}}{100+e^{\cos^2(2\pi t)}}\Big]\frac{|u(t)|+e^{-(5t+\pi)}\big|\Psi_{\frac{1}{2},\frac{1}{3}}^{\frac{1}{4}}u(t)\big|}{1+|u(t)|} \\ & +\Big[\frac{e^{-[\cos^2(2\pi t)+5]}}{(t+10)^2}\Big]\frac{|u(t)|+e^{-(20t+\frac{\pi}{3})}\big|\Upsilon_{\frac{1}{2},\frac{1}{3}}^{\frac{2}{3}}u(t)\big|}{1+|u(t)|},\quad t\in [\frac{2}{3},10]_{\frac{1}{2},\frac{1}{3}} \end{align} | (5.1) |
subject to fractional Hahn integral boundary condition
\begin{align} I_{\frac{1}{2},\frac{1}{3}}^{\frac{1}{2}}\Big(2e+\sin\big(\frac{23}{24}\big)\Big)^2 u\big(\frac{23}{24}\big) = \,& \sum\limits_{i = 0}^{\infty}\frac{C_i|u(t_i)|}{1+|u(t_i)|}\; ,\quad t_i\in \sigma_{\frac{1}{2},\frac{1}{3}}^{i}\big(\frac{23}{24}\big),\\ I_{\frac{1}{2},\frac{1}{3}}^{\frac{1}{2}}\big(2\pi+\cos (10)\big)^2 u(10) = \,& \sum\limits_{i = 0}^{\infty}\frac{D_i|u(t_i)|}{1+|u(t_i)|}\; ,\quad t_i\in \sigma_{\frac{1}{2},\frac{1}{3}}^{i}(10), \end{align} | (5.2) |
where \varphi(t, s) = \frac{e^{-|t-s|}}{(t+2\pi)^5}\; ,\; \psi(t, s) = \frac{e^{-|t-s|}}{(t+2e)^4} and C_i, D_i are given constants, with \frac{1}{2000}\leq\sum\limits_{i = 0}^{\infty}C_i\leq\frac{\pi}{2000} and \frac{1}{1000}\leq\sum\limits_{i = 0}^{\infty}D_i\leq\frac{e}{1000}.
Here \alpha = \frac{3}{2}, \; \beta = \frac{1}{2}, \; \gamma = \frac{1}{4} , q = \frac{1}{2} , \omega = \frac{1}{3}, \; \nu = \frac{2}{3} , \omega_0 = \frac{\omega}{1-q} = \frac{2}{3} , T = 10 , \eta = \sigma_{\frac{1}{2}, \frac{1}{3}}^{5}(10) = \frac{23}{24} , \lambda = e^{-\pi}, \; \mu = e^{-5} , \phi_1 = \sum\limits_{i = 0}^{\infty}\frac{C_i|u(t_i)|}{1+|u(t_i)|} , \phi_2 = \sum\limits_{i = 0}^{\infty}\frac{D_i|u(t_i)|}{1+|u(t_i)|} , g_1(t) = \big(2e+\sin t\big)^2 and g_2(t) = \big(2\pi+\cos t\big)^2 .
\begin{align*} F\big[t,u(t),\big(\Psi_{q,\omega}^{\gamma}\big)(t)\big] = \,& \Big[\frac{e^{-[\sin^2(2\pi t)]}}{100+e^{\cos^2(2\pi t)}}\Big]\frac{|u(t)|+e^{-(5t+\pi)}\big|\Psi_{\frac{1}{2},\frac{1}{3}}^{\frac{1}{4}}u(t)\big|}{1+|u(t)|}, \end{align*} |
and
\begin{align*} H\big[t,u(t),\big(\Upsilon_{q,\omega}^{\nu}\big)(t)\big] = \,& \Big[\frac{e^{-[\cos^2(2\pi t)]}}{(t+10)^2}\Big]\frac{|u(t)|+e^{-(20t+\frac{\pi}{3})}\big|\Upsilon_{\frac{1}{2},\frac{1}{3}}^{\frac{2}{3}}u(t)\big|}{1+|u(t)|}. \end{align*} |
For all t\in I_{\frac{1}{2}, \frac{1}{3}}^{10} and u, v \in \mathbb{R} , we have
\begin{align*} \big|F\big[t,u,\Psi_{q,\omega}^{\gamma}u\big]-F\big[t,v,\Psi_{q,\omega}^{\gamma}v\big]\big|\leq\,& \frac{1}{101}|u-v|+\frac{1}{101e^\pi}\big|\Psi_u^\gamma-\Psi_v^\gamma\big|,\\ \big|H\big[t,u,\Upsilon_{q,\omega}^{\nu}u\big]-H\big[t,v,\Upsilon_{q,\omega}^{\nu}v\big]\big|\leq\,& \frac{1}{100}|u-v|+\frac{1}{100e^{\frac{\pi}{3}}}\big|\Upsilon_u^\gamma-\Upsilon_v^\gamma\big|. \end{align*} |
Thus (H_1) and (H_2) hold with M_1 = 0.0099, M_2 = 0.000428, N_1 = 0.01 and N_2 = 0.00351 .
For all u, v \in C ,
\begin{align*} \big|\phi_1(u)-\phi_1(v)\big|\; \leq\,\; & \frac{\pi}{2000}\|u-v\|_\mathcal{C},\\ \big|\phi_2(u)-\phi_2(v)\big|\; \leq\,\; & \frac{e}{1000}\|u-v\|_\mathcal{C}. \end{align*} |
So (H_3) hold with \omega_1 = \frac{\pi}{2000} = 0.00157 and \omega_2 = \frac{e}{1000} = 0.00272 .
Moreover (H_4) hold with \hat{g}_1 = 19.683, G_1 = 41.429, \hat{g}_2 = 27.912 and G_2 = 53.048 .
After calculating, we find that
\mathcal{A}_\eta = 11.1274,\; \; \; \mathcal{A}_T = 455.939,\; \; \; \mathcal{B}_\eta = 65.1277,\; \; \; \mathcal{B}_T = 83.3932, |
|\Lambda| = 28766.3089,\; \; \; \varphi_0 = 0.0000617,\; \; \; \psi_0 = 0.00072. |
We can show that
\mathcal{L} = 0.0004952,\; \; \; \Theta_T = 0.0140446,\; \; \; \Theta_\eta = 0.00704333,\; \; \; \bar{\theta}_T = 0.0012905, |
\bar{\Theta}_\eta = 0.00130252,\; \; \; \Theta_T^* = 0.01531365,\; \; \; \Theta_\eta^* = 0.0083485. |
So, (H_5) holds with
{\mathcal{X}} \approx 0.027724 < 1. |
Hence, by Theorem 3.1, the BVP (5.1)–(5.2) has a unique solution on I_{\frac{1}{2}, \frac{1}{3}}^{10} .
In view of Theorem 4.1, we have \; \hat{\chi} = 0.049859\; and
A_{F,H} \approx0.051282. |
Therefore, the BVP (5.1)–(5.2) is Hyers-Ulam stable.
For the specific case where \phi_1 = 10, \phi_2 = 20 , and g_1 = g_2 = 30 , we examine the numerical solution to the BVP (5.1)–(5.2) when we let h(t) = \lambda F\left[ t, u(t), \left(\Psi^\gamma_{q, \omega}u\right)(t)\right] +\mu H\left[ t, u(t), \left(\Upsilon^\nu_{q, \omega}u\right)(t)\right] = (t-\omega_0)^\theta . From Figure 1, we obtain the numerical solution graph for \theta = 0, 0.25, 0.5, 0.75, 1, 1.5, 2 . The graph shows that the solution of the equation converges to zero as t\rightarrow \omega_0 .
Example 5.2. Consider the following fractional Hahn BVP
\begin{align} D_{\frac{1}{4},\frac{3}{2}}^{\frac{5}{4}} u(t) = \, \frac{1}{15}\big(t+\frac{2}{5}\big)e^{-(t+5)\big[u(t)+\Psi_{\frac{1}{4},\frac{3}{2}}^{\frac{1}{2}}u(t)\big]}+\frac{1}{5}\big(t+\frac{1}{3}\big)e^{-(t+\pi)\big[u(t)+\Upsilon_{\frac{1}{4},\frac{3}{2}}^{\frac{1}{2}}u(t)\big]}, \; \; t\in I_{\frac{1}{4},\frac{3}{2}}^{15} \end{align} | (5.3) |
with fractional Hahn integral boundary condition
\begin{align} I_{\frac{1}{4},\frac{3}{2}}^{\frac{1}{2}}\big[\pi+\sin\big(\frac{525}{256}\big)\big] u\big(\frac{525}{256}\big) = \,\; & \sum\limits_{i = 0}^{\infty}C_ie^{-|u(t_i)|}, \\ I_{\frac{1}{4},\frac{3}{2}}^{\frac{1}{2}}\big[e+\cos(15)\big] u(15) = \,\; &\sum\limits_{i = 0}^{\infty}D_ie^{-|u(t_i)|}, \quad t_i\in \sigma_{\frac{1}{4},\frac{3}{2}}^i (15), \end{align} | (5.4) |
where C_i and D_i are given constants with \frac{1}{500}\leq \sum_{i = 0}^{\infty} C_i \leq \frac{e}{500} and \frac{1}{1000}\leq\sum_{i = 0}^{\infty}D_i \leq \frac{\pi}{1000}
Here \alpha = \frac{5}{4}, \; \beta = \frac{1}{2}, \; \gamma = \frac{1}{2} , q = \frac{1}{4}, \; \nu = \frac{1}{3} , \omega = \frac{3}{2} , \omega_0 = \frac{\omega}{1-q} = 2 , T = 15 , \eta = \sigma_{\frac{1}{4}, \frac{3}{2}}^{4} = \frac{525}{256} , \lambda = e^{-5} , \mu = e^{-\pi} .
It is clear that \big|F\big[t, u, \Psi_{q, \omega}^{\gamma}u\big]\big|\leq \frac{1}{30} = \hat {F} , \big|H\big[t, u, \Upsilon_{q, \omega}^{\nu}u\big]\big|\leq \frac{1}{15} = \hat {H} for t\in I_{\frac{1}{4}, \frac{3}{2}}^{15} and \big|\phi_1(u)\big| \leq \frac{e}{500} = O_1 , \big|\phi_2(u)\big| \leq \frac{\pi}{1000} = O_2
Hence, (H6) and (H7) hold. Therefore, the BVP (5.3)–(5.4) has at least one solution on I_{\frac{1}{4}, \frac{3}{2}}^{15} by theorem 3.2.
For the specific case where \phi_1 = \frac{e}{500}, \phi_2 = \frac{\pi}{1000} and g_1 = g_2 = 1 , we examine the numerical solution to the BVP (5.3)–(5.4) when we let h(t) = \lambda F\left[ t, u(t), \left(\Psi^\gamma_{q, \omega}u\right)(t)\right] +\mu H\left[ t, u(t), \left(\Upsilon^\nu_{q, \omega}u\right)(t)\right] = (t-\omega_0)^\theta . From Figure 2, we obtain the numerical solution graph for \theta = 0, 0.25, 0.5, 0.75, 1, 1.5, 2 . The graph shows that the solution of the equation converges to zero as t\rightarrow \omega_0 .
We have successfully demonstrated the uniqueness and stability of solutions for the nonlocal Riemann-Liouville fractional Hahn integrodifference BVP through the application of the Banach fixed point theorem. Furthermore, we have established the existence of at least one solution using Schauder's fixed point theorem. Our innovative approach features the integration of two fractional Hahn difference operators and three fractional Hahn integrals.
Nichaphat Patanarapeelert, Jiraporn Reunsumrit, and Thanin Sitthiwirattham: Study conception and design, material preparation, data collection and analysis, and writing the first draft of the manuscript and commenting on previous versions of the manuscript. All the authors read and approved the final manuscript.
This research was funded by National Science, Research and Innovation Fund (NSRF), and King Mongkut's University of Technology North Bangkok with contract no. KMUTNB-FF-67-B-25.
The authors declare no conflicts of interest.
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