# Existence and Ulam Stability of Solutions for Discrete Fractional Boundary Value Problem

Authors: Fulai Chen; Yong Zhou
Journal: Discrete Dynamics in Nature and Society
 (2013)

---

## Abstract

We discuss the existence of solutions for antiperiodic boundary value problem and the Ulam stability for nonlinear fractional difference equations. Two examples are also provided to illustrate our main results.


---

## Body

## 1. Introduction

This paper investigates the existence of solutions for antiperiodic boundary value problem and the Ulam stability for nonlinear fractional difference equations:(1)ΔCαx(t)=f(t+α-1,x(t+α-1)),t∈[0,b]ℕ0,1<α<2,
where ΔCα is a Caputo fractional difference operator, ℕa={a,a+1,a+2,…} and Iℕa=I⋂‍ℕa for any number a∈R and each interval I of R,b∈ℕ1, and f:[α-1,b+α]ℕα-1×R→R is a continuous function with respect to the second variable.Accompanied with the development of the theory on fractional differential equations, fractional difference equations have also been studied more intensively of late. In particular, some properties and inequalities of the fractional difference calculus are discussed in [1–7], the existence and asymptotic stability of the solutions for fractional difference equations are investigated in [8–10], and the boundary value problems of fractional difference equations are considered in [11–13]. But there are a lot of works to do in the future, and to the best of my knowledge, there is no work on the existence of solutions for antiperiodic boundary value problem and the Ulam stability for nonlinear fractional difference equations.To research the boundary value problem of fractional difference equations, we need to select a suitable fixed-point theorem because of the discrete property of the difference operator; here, we choose the Banach contraction mapping principle and the Brower fixed-point theorem. Motivated by the work of the Ulam stability for fractional differential equations [14], in this paper, we also introduce four types of the Ulam stability definitions for fractional difference equations and study the Ulam-Hyers stable and the Ulam-Hyers-Rassias stable.The rest of the paper is organized as follows. In Section2, we introduce some useful preliminaries. In Section 3, we consider the existence of solutions for antiperiodic boundary value problem of fractional difference equations. In Section 4, we discuss the Ulam stability for fractional difference equations. Finally, two examples are given to illustrate our main results.


## 2. Preliminaries

In this section, we introduce preliminary facts which are used throughout this paper.Definition 1 (see [3, 4]).
Letν>0. The νth fractional sum of f:ℕa→R is defined by
(2)Δ-νf(t)=1Γ(ν)∑s=at-ν‍(t-s-1)(ν-1)f(s),t∈ℕa+ν,
where t(ν)=Γ(t+1)/Γ(t-ν+1).In (2), the fractional sum Δ-ν maps functions defined on ℕa to functions defined on ℕa+ν. Atici and Eloe [3] pointed out that this definition is the development of the theory of the fractional calculus on time scales.Definition 2 (see [1]).
Letμ>0 and n-1<μ<n, where n denotes a positive integer and n=⌈μ⌉, ⌈·⌉ ceiling of number. Set ν=n-μ. The μth fractional Caputo difference operator is defined as
(3)ΔCμf(t)=Δ-ν(Δnf(t))=1Γ(ν)∑s=at-ν‍(t-s-1)(ν-1)Δnf(s),∀t∈Na+ν,
where Δn is the nth order forward difference operator; the fractional Caputo like difference ΔCμ maps functions defined on ℕa to functions defined on ℕa-μ.Lemma 3 (see [2, 13]).
Assume thatμ>0 and f is defined on ℕa. Then,
(4)Δ-μΔCμf(t)=f(t)-∑k=0n-1‍(t-a)(k)k!Δkf(a)=f(t)+c0+c1t+⋯+cn-1t(n-1),
where n is the smallest integer greater than or equal to μ, ci∈R,i=1,2,…,n-1.Lemma 4.
One has(5)∑s=0t-α‍(t-s-1)(α-1)=Γ(t+1)αΓ(t-α+1).Proof.
Forx>k,  x,k∈R,   k>-1,x>-1, we have [1]
(6)Γ(x+1)Γ(k+1)Γ(x-k+1)=Γ(x+2)Γ(k+2)Γ(x-k+1)-Γ(x+1)Γ(k+2)Γ(x-k);
that is,
(7)Γ(x+1)Γ(x-k+1)=1k+1[Γ(x+2)Γ(x-k+1)-Γ(x+1)Γ(x-k)].
Then,
(8)∑s=0t-α‍(t-s-1)(α-1)=Γ(t)Γ(t-α+1)+Γ(t-1)Γ(t-α)+⋯+Γ(α+1)Γ(2)+Γ(α)=1α(Γ(t+1)Γ(t-α+1)-Γ(t)Γ(t-α))+1α(Γ(t)Γ(t-α)-Γ(t-1)Γ(t-α-1))+⋯+1α(Γ(α+2)Γ(2)-Γ(α+1)Γ(1))+Γ(α)=Γ(t+1)αΓ(t-α+1).
The following result is an immediate consequence of Lemma 4.Corollary 5.
One has(i)
∑s=0b(b+α-s-1)(α-1)=Γ(b+α+1)/αΓ(b+1),(ii)
∑s=0b-1(b+α-s-2)(α-2)=(1/α)((Γ(b+α)/Γ(b+1))-Γ(α)).


## 3. Antiperiodic Boundary Value Problem

In this section, we consider the following antiperiodic boundary value problem:(9)ΔCαx(t)=f(t+α-1,x(t+α-1)),t∈[0,b]ℕ0,1<α<2,x(α-1)+x(b+α)=0,Δx(α-1)+Δx(b+α-1)=0,
where Δ is a forward difference operator.LetB be the set of all real sequences x={x(t)}t=α-1b+α with norm ∥x∥=supt∈[α-1,b+α]ℕα-1|x(t)|. Then, B is a Banach space.Lemma 6.
A solutionx:t∈[α-1,b+α]ℕα-1×R→R is a solution for antiperiodic boundary value problem
(10)ΔCαx(t)=f(t+α-1),t∈[0,b]ℕ0,1<α<2,x(α-1)+x(b+α)=0,Δx(α-1)+Δx(b+α-1)=0,
if and only if x(t) is a solution of the the following fractional Taylor’s difference formula:
(11)x(t)=1Γ(α)∑s=0t-α‍(t-s-1)(α-1)f(s+α-1)-12Γ(α)∑s=0b‍(b+α-s-1)(α-1)f(s+α-1)+b+2α-1-t2Γ(α-1)∑s=0b-1‍(b+α-s-2)(α-2)f(s+α-1),t∈[α-1,b+α]ℕα-1.Proof.
Suppose thatx(t) defined on [α-1,b+α]ℕα-1 is a solution of (10). Using Lemma 3, for some constants c0,c1∈R, we have
(12)x(t)=Δ-αf(t+α-1)-c0-c1t=1Γ(α)∑s=0t-α‍(t-s-1)(α-1)f(s+α-1)-c0-c1t,t∈[α-1,b+α]ℕα-1.
Then, we obtain [3]
(13)Δx(t)=1Γ(α-1)∑s=0t-α‍(t-s-1)(α-2)f(s+α-1)-c1,t∈[α-1,b+α-1]ℕα-1.
In view ofx(α-1)+x(b+α)=0 and Δx(α-1)+Δx(b+α-1)=0, we have
(14)1Γ(α)∑s=0b‍(b+α-s-1)(α-1)f(s+α-1)-2c0-c1(b+2α-1)=0,1Γ(α-1)∑s=0b-1(b+α-s-2)(α-2)f(s+α-1)-2c1=0.
Then,
(15)c0=12Γ(α)∑s=0b‍(b+α-s-1)(α-1)f(s+α-1)-b+2α-12Γ(α-1)∑s=0b-1‍(b+α-s-2)(α-2)f(s+α-1),c1=12Γ(α-1)∑s=0b-1‍(b+α-s-2)(α-2)f(s+α-1).
Substituting the values of c0 and c1 into (12), we obtain (11).
Conversely, ifx(t) is a solution of (11), by a direct computation, it follows that the solution given by (11) satisfies (10). The proof is completed.The following fixed-point theorems are needed to prove the existence and uniqueness of solutions for the BVP (9).Lemma 7 (see [15] (Banach contraction mapping principle)).
A contraction mapping on a complete metric space has exactly one fixed point.Lemma 8 (see [16] (Brower fixed-point theorem)).
LetF:C∈Rn→C∈Rn be a continuous mapping, where C is a nonempty, bounded, close, and convex set. Then, F has a fixed point.Define the operator(16)(Tx)(t)=1Γ(α)∑s=0t-α‍(t-s-1)(α-1)f(s+α-1,x(s+α-1))-12Γ(α)∑s=0b‍(b+α-s-1)(α-1)f(s+α-1,x(s+α-1))+b+2α-1-t2Γ(α-1)∑s=0b-1‍(b+α-s-2)(α-2)×f(s+α-1,x(s+α-1)),t∈[α-1,b+α]ℕα-1.
Obviously, x(t) is a solution of (9) if it is a fixed point of the operator T.Theorem 9.
Assume that.
(
H
1
) There exists a constant L>0 such that |f(t,x)-f(t,y)|≤L|x-y|  for each t∈[α-1,b+α]ℕα-1 and all x,y∈B.
Then, the BVP (9) has a unique solution on B provided that
(17)L<3Γ(b+α+1)2Γ(α+1)Γ(b+1)+b+α2αΓ(α-1)(Γ(b+α)Γ(b+1)-Γ(α)).Proof.
Letx,y∈B; then for each t∈[α-1,b+α]ℕα-1, we have
(18)|(Tx)(t)-(Ty)(t)|≤1Γ(α)∑s=0t-α‍(t-s-1)(α-1)×|f(s+α-1,x(s+α-1))-f(s+α-1,y(s+α-1))|+12Γ(α)∑s=0b‍(b+α-s-1)(α-1)×|f(s+α-1,x(s+α-1))-f(s+α-1,y(s+α-1))|+|b+2α-1-t|2Γ(α-1)∑s=0b-1‍(b+α-s-2)(α-2)×|f(s+α-1,x(s+α-1))-f(s+α-1,y(s+α-1))|≤LΓ(α)∑s=0t-α‍(t-s-1)(α-1)×|x(s+α-1)-y(s+α-1)|+L2Γ(α)∑s=0b(b+α-s-1)(α-1)×|x(s+α-1)-y(s+α-1)|+L|b+2α-1-t|2Γ(α-1)∑s=0b-1‍(b+α-s-2)(α-2)×|x(s+α-1)-y(s+α-1)|≤L∥x-y∥Γ(α)∑s=0t-α‍(t-s-1)(α-1)+L∥x-y∥2Γ(α)×∑s=0b‍(b+α-s-1)(α-1)+|b+2α-1-t|·L∥x-y∥2Γ(α-1)∑s=0b-1‍(b+α-s-2)(α-2)≤L∥x-y∥2Γ(α)·Γ(t+1)αΓ(t-α+1)+L∥x-y∥2Γ(α)·Γ(b+α+1)αΓ(b+1)+L(b+α)∥x-y∥2Γ(α-1)·1α(Γ(b+α)Γ(b+1)-Γ(α))≤[3Γ(b+α+1)2Γ(α+1)Γ(b+1)+b+α2αΓ(α-1)(Γ(b+α)Γ(b+1)-Γ(α))]L∥x-y∥.
According to (17), we obtain
(19)|(Tx)(t)-(Ty)(t)|<∥x-y∥.
Then,
(20)∥Tx-Ty∥≤∥x-y∥
which implies that T is a contraction. Therefore, the Banach fixed-point theorem (Lemma 7) guarantees that T has a unique fixed point which is a unique solution of the BVP (9). This completes the proof.Theorem 10.
Assume that.
(
H
2
) There exists a bounded function K:[α-1,b+α]ℕα-1→R such that |f(t,x)|≤K(t)|x| for all x∈B.
Then, the BVP (9) has at least a solution on B provided that
(21)K*<3Γ(b+α+1)2Γ(α+1)Γ(b+1)+b+α2αΓ(α-1)(Γ(b+α)Γ(b+1)-Γ(α)),
where  K*=max{K(t):t∈[α-1,b+α]ℕα-1}.Proof.
LetM>0; define the set
(22)S={x(t)∣[α-1,b+α]ℕα-1⟶R,∥x∥≤M}.
To prove this theorem, we only need to show that T maps S in S.
Forx(t)∈S, we have
(23)|(Tx)(t)|≤K(t)Γ(α)∑s=0t-α‍(t-s-1)(α-1)|x(s+α-1)|+K(t)2Γ(α)∑s=0b‍(b+α-s-1)(α-1)|x(s+α-1)|+(b+2α-1-t)K(t)2Γ(α-1)×∑s=0b-1‍(b+α-s-2)(α-2)|x(s+α-1)|≤K(t)∥x∥Γ(α)∑s=0t-α‍(t-s-1)(α-1)+K(t)∥x∥2Γ(α)∑s=0b‍(b+α-s-1)(α-1)+(b+2α-1-t)K(t)∥x∥2Γ(α-1)∑s=0b-1‍(b+α-s-2)(α-2)≤K(t)∥x∥2Γ(α)·Γ(t+1)αΓ(t-α+1)+K(t)∥x∥2Γ(α)·Γ(b+α+1)αΓ(b+1)+K(t)(b+α)∥x∥2Γ(α-1)·1α(Γ(b+α)Γ(b+1)-Γ(α))≤[3Γ(b+α+1)2Γ(α+1)Γ(b+1)+b+α2αΓ(α-1)(Γ(b+α)Γ(b+1)-Γ(α))]K(t)∥x∥≤[3Γ(b+α+1)2Γ(α+1)Γ(b+1)+b+α2αΓ(α-1)(Γ(b+α)Γ(b+1)-Γ(α))]K*M.
From (21), we have |(Tx)(t)|<M; then, ∥Tx∥≤M which implies that T maps S in S. T has at least a fixed point which is a solution of the BVP (9) according to Brower fixed-point theorem (Lemma 8). This completes the proof.


## 4. The Ulam Stability

Similar to the definitions of the Ulam stability for fractional differential equation [14], we introduce four types of the Ulam stability definitions for fractional difference equation.Consider (1) and the following inequalities:
(24)|ΔCαy(t)-f(t+α-1,y(t+α-1))|≤ɛ,t∈[0,b]ℕ0,(25)|ΔCαy(t)-f(t+α-1,y(t+α-1))|≤ɛφ(t+α-1),t∈[0,b]ℕ0.Definition 11.
Equation (1) is the Ulam-Hyers stable if there exists a real number cf>0 such that for each ɛ>0 and for each solution y∈B of inequality (24), there exists a solution x∈B of (1) with
(26)|y(t)-x(t)|≤cfɛ,t∈[α-1,b+α]ℕα-1.
Equation (1) is the generalized Ulam-Hyers stable if we substitute the function θf(ɛ) for the constant cfɛ on inequality (26), where θf(ɛ)∈C(R+,R+) and θf(0)=0.Definition 12.
Equation (1) is the Ulam-Hyers-Rassias stable with respect to φ if there exists a real number cf,φ>0 such that for each ɛ>0 and for each solution y∈B of inequality (25), there exists a solution x∈B of (1) with
(27)|y(t)-x(t)|≤cf,φɛφ(t),t∈[α-1,b+α]ℕα-1.
Equation (1) is the generalized Ulam-Hyers-Rassias stable if we substitute the function φ(t) for the function ɛφ(t) on inequalities (25) and (27).Remark 13.
Ifφ is a constant function in Definition 12, we say that the integral equation (25) has also the Hyers-Ulam stability.Remark 14.
A functiony∈B is a solution of inequality (24) if and only if there exists a function g:[α-1,b+α-1]ℕα-1→R such that (i)
|g(t+α-1)|≤ɛ,t∈[0,b]ℕ0,(ii)
ΔCαy(t)=f(t+α-1,y(t+α-1))+g(t+α-1),t∈[0,b]ℕ0,similar remark for inequality (25).Theorem 15.
Assume that(H1) holds. Let y∈B be a solution of inequality (24) and let x∈B be a solution of the following boundary value problem:
(28)ΔCαx(t)=f(t+α-1,x(t+α-1)),t∈[0,b]ℕ0,1<α<2,x(α-1)=y(α-1),x(b+α)=y(b+α).
Then, (1) is the Ulam-Hyers stable provided that
(29)L<Γ(α+1)Γ(b+1)2Γ(b+α+1).Proof.
By Lemma3, the solution of the BVP (28) is given by
(30)x(t)=y(α-1)+tb+α(y(b+α)-y(α-1))-t(b+α)Γ(α)∑s=0b‍(b+α-s-1)(α-1)×f(s+α-1,x(s+α-1))+1Γ(α)∑s=0t-α‍(t-s-1)(α-1)×f(s+α-1,x(s+α-1)),t∈[α-1,b+α]ℕα-1.
From inequality (24), for t∈[α-1,b+α]ℕα-1, it follows that
(31)|∑s=0t-α‍∑s=0t-α‍y(t)-y(α-1)-tb+α(y(b+α)-y(α-1))+t(b+α)Γ(α)∑s=0b‍(b+α-s-1)(α-1)×f(s+α-1,y(s+α-1))-1Γ(α)∑s=0t-α‍(t-s-1)(α-1)f(s+α-1,y(s+α-1))|≤ɛΓ(α)∑s=0t-α‍(t-s-1)(α-1)=ɛΓ(α)·Γ(t+1)αΓ(t-α+1)≤Γ(b+α+1)Γ(α+1)Γ(b+1)ɛ.
Combining (30) and (31), for t∈[α-1,b+α]ℕα-1, we have
(32)|y(t)-x(t)|≤|y(t)-y(α-1)-tb+α(y(b+α)-y(α-1))+t(b+α)Γ(α)×∑s=0b‍(b+α-s-1)(α-1)f(s+α-1,x(s+α-1))-1Γ(α)∑s=0t-α‍(t-s-1)(α-1)×f(s+α-1,x(s+α-1))tb+α|≤|y(t)-y(α-1)-tb+α(y(b+α)-y(α-1))+t(b+α)Γ(α)×∑s=0b‍(b+α-s-1)(α-1)f(s+α-1,y(s+α-1))-1Γ(α)∑s=0t-α‍(t-s-1)(α-1)×f(s+α-1,y(s+α-1))tb+α|+t(b+α)Γ(α)∑s=0b‍(b+α-s-1)(α-1)×|f(s+α-1,y(s+α-1))-f(s+α-1,x(s+α-1))|+1Γ(α)∑s=0t-α‍(t-s-1)(α-1)×|f(s+α-1,y(s+α-1))-f(s+α-1,x(s+α-1))|≤Γ(b+α+1)Γ(α+1)Γ(b+1)ɛ+mft(b+α)Γ(α)×∑s=0b‍(b+α-s-1)(α-1)×|y(s+α-1)-x(s+α-1)|+mfΓ(α)∑s=0t-α‍(t-s-1)(α-1)×|y(s+α-1)-x(s+α-1)|≤Γ(b+α+1)Γ(α+1)Γ(b+1)ɛ+mft∥y-x∥(b+α)Γ(α)×∑s=0b‍(b+α-s-1)(α-1)+mf∥y-x∥Γ(α)∑s=0t-α(t-s-1)(α-1)‍=Γ(b+α+1)Γ(α+1)Γ(b+1)ɛ+mft(b+α)Γ(α)·1αΓ(b+α+1)Γ(b+1)∥y-x∥+mfΓ(α)·Γ(t+1)αΓ(t-α+1)∥y-x∥≤Γ(b+α+1)Γ(α+1)Γ(b+1)ɛ+mft(b+α)Γ(α)·1αΓ(b+α+1)Γ(b+1)∥y-x∥+mfΓ(α)·Γ(t+1)αΓ(t-α+1)∥y-x∥≤Γ(b+α+1)Γ(α+1)Γ(b+1)ɛ+2mfΓ(b+α+1)Γ(α+1)Γ(b+1)×∥y-x∥.
Then,
(33)∥y-x∥≤Γ(b+α+1)Γ(α+1)Γ(b+1)ɛ+2LΓ(b+α+1)Γ(α+1)Γ(b+1)∥y-x∥.
Applying (29) to the previous inequality yields that
(34)∥y-x∥≤Γ(b+α+1)Γ(α+1)Γ(b+1)-2LΓ(b+α+1)ɛ;
where Γ(b+α+1)/(Γ(α+1)Γ(b+1)-2LΓ(b+α+1))>0, thus; (1) is the Ulam-Hyers stable.Theorem 16.
Assume that(H1) and the following condition hold.
(
H
3
) Let φ:[α-1,b+α]ℕα-1→R+ be an increasing function. There exists a constant λ>0 such that
(35)1Γ(α)∑s=0t-α‍(t-s-1)(α-1)φ(s+α-1)≤λɛφ(t+α-1),t∈[0,b]ℕ0.
Lety∈B be a solution of inequality (25) and let x∈B be a solution of the following boundary value problem (28). Then, (1) is the Ulam-Hyers-Rassias stable  provided that (29) holds.The proof of Theorem16 is similar to that of Theorem 15, and we omit it.


## 5. Examples

As the applications of our main results, we consider the following examples.Example 1.
Consider the fractional difference BVP(36)ΔC1.5x(t)=λx(t+0.5),t∈[0,10]ℕ0,x(0.5)+x(11.5)=0,Δx(0.5)+Δx(10.5)=0,
where f(t,x)=λx(t) for t∈[0.5,11.5]ℕ0.5 and conditions (H1) and (H2) are satisfied. Since
(37)3Γ(b+α+1)2Γ(α+1)Γ(b+1)+b+α2αΓ(α-1)(Γ(b+α)Γ(b+1)-Γ(α))≈47.7268,
inequalities (17) and (21) are satisfied if λ<47.7268. According to Theorem 9, the BVP (36) has a unique solution. At the same time, the BVP (36) has at least a solution by Theorem 10.Example 2.
Consider the fractional difference equation(38)ΔC1.5x(t)=λx(t+0.5),t∈[0,10]ℕ0,
with the boundary conditions
(39)x(0.5)=y(0.5),x(11.5)=y(11.5).
Since
(40)Γ(α+1)Γ(b+1)2Γ(b+α+1)≈0.0176,
if λ<0.0176 and the inequality
(41)|ΔC1.5y(t)-f(t+0.5,y(t+0.5))|≤ɛ,fort∈[0,10]ℕ0,
hold, then (38) is the Ulam-Hyers stable by Theorem 15.



---

*Source: 459161-2013-08-20.xml*