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

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.

1. Introduction

This paper investigates the existence of solutions for antiperiodic boundary value problem and the Ulam stability for nonlinear fractional difference equations: Δ^𝛼_𝐶𝑥(𝑡)((,[]=𝑓𝑡+𝛼−1,𝑥𝑡+𝛼−1))𝑡∈0,𝑏_ℕ0,1<𝛼<2, (1) where Δ^𝛼_𝐶 is a Caputo fractional difference operator, ℕ_𝑎={𝑎,𝑎+1,𝑎+2,…} and 𝐼_ℕ𝑎=𝐼⋂ℕ_𝑎 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 Section 2, 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]). Let 𝜈>0. The 𝜈th fractional sum of 𝑓∶ℕ_𝑎→𝑅 is defined by Δ^−𝜈𝑓(𝑡)=1Γ(𝜈)^𝑡−𝜈_𝑠=𝑎(𝑡−𝑠−1)^(𝜈−1)𝑓(𝑠),𝑡∈ℕ_𝑎+𝜈, (2) where 𝑡^(𝜈)=Γ(𝑡+1)/Γ(𝑡−𝜈+1).

In (2), the fractional sum Δ^−𝜈 maps functions defined on ℕ_𝑎 to functions defined on ℕ_𝑎+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]). Let 𝜇>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 Δ^𝜇_𝐶𝑓(𝑡)=Δ^−𝜈Δ^𝑛𝑓(𝑡)=1Γ(𝜈)^𝑡−𝜈_𝑠=𝑎(𝑡−𝑠−1)^(𝜈−1)Δ^𝑛𝑓(𝑠),∀𝑡∈𝑁_𝑎+𝜈, (3) where Δ^𝑛 is the 𝑛th order forward difference operator; the fractional Caputo like difference Δ^𝜇_𝐶 maps functions defined on ℕ_𝑎 to functions defined on ℕ_𝑎−𝜇.

Lemma 3 (see [2, 13]). Assume that 𝜇>0 and 𝑓 is defined on ℕ_𝑎. Then, Δ^−𝜇Δ^𝜇_𝐶𝑓(𝑡)=𝑓(𝑡)−^𝑛−1_𝑘=0(𝑡−𝑎)^(𝑘)Δ𝑘!^𝑘𝑓(𝑎)=𝑓(𝑡)+𝑐₀+𝑐₁𝑡+⋯+𝑐_𝑛−1𝑡^(𝑛−1), (4) where 𝑛 is the smallest integer greater than or equal to 𝜇, 𝑐_𝑖∈𝑅,𝑖=1,2,…,𝑛−1.

Lemma 4. One has ^𝑡−𝛼_𝑠=0(𝑡−𝑠−1)^(𝛼−1)=Γ(𝑡+1)().𝛼Γ𝑡−𝛼+1 (5)

Proof. For 𝑥>𝑘,  𝑥,𝑘∈𝑅,   𝑘>−1,𝑥>−1, we have [1] Γ(𝑥+1)Γ()Γ()=Γ()𝑘+1𝑥−𝑘+1𝑥+2Γ()Γ()−Γ()𝑘+2𝑥−𝑘+1𝑥+1Γ()Γ();𝑘+2𝑥−𝑘 (6) that is, Γ(𝑥+1)Γ()=1𝑥−𝑘+1Γ()𝑘+1𝑥+2Γ()−Γ()𝑥−𝑘+1𝑥+1Γ().𝑥−𝑘 (7) Then, ^𝑡−𝛼_𝑠=0(𝑡−𝑠−1)^(𝛼−1)=Γ(𝑡)Γ()+Γ()𝑡−𝛼+1𝑡−1Γ()Γ()𝑡−𝛼+⋯+𝛼+1Γ(2)(𝛼)=1+Γ𝛼Γ(𝑡+1)Γ()−Γ(𝑡)𝑡−𝛼+1Γ()+1𝑡−𝛼𝛼Γ(𝑡)Γ()−Γ()𝑡−𝛼𝑡−1Γ()1𝑡−𝛼−1+⋯+𝛼Γ(𝛼+2)Γ(2)−Γ(𝛼+1)Γ(1)(𝛼)=Γ()+Γ𝑡+1().𝛼Γ𝑡−𝛼+1 (8) The following result is an immediate consequence of Lemma 4.

Corollary 5. One has (i)∑^𝑏_𝑠=0(𝑏+𝛼−𝑠−1)^(𝛼−1)=Γ(𝑏+𝛼+1)/𝛼Γ(𝑏+1), (ii)∑^𝑏−1_𝑠=0(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: Δ^𝛼_𝐶𝑥(𝑡)=𝑓(𝑡+𝛼−1,𝑥(𝑡+𝛼−1)),𝑡∈[0,𝑏]_ℕ0𝑥()()()(),1<𝛼<2,𝛼−1+𝑥𝑏+𝛼=0,Δ𝑥𝛼−1+Δ𝑥𝑏+𝛼−1=0, (9) where Δ is a forward difference operator.

Let 𝐵 be the set of all real sequences 𝑥={𝑥(𝑡)}^𝑏+𝛼_{𝑡=𝛼−1} with norm ‖𝑥‖=sup_{𝑡∈[𝛼−1,𝑏+𝛼]ℕ𝛼−1}|𝑥(𝑡)|. Then, 𝐵 is a Banach space.

Lemma 6. A solution 𝑥∶𝑡∈[𝛼−1,𝑏+𝛼]_ℕ𝛼−1×𝑅→𝑅 is a solution for antiperiodic boundary value problem Δ^𝛼_𝐶𝑥(𝑡)=𝑓(𝑡+𝛼−1),𝑡∈[0,𝑏]_ℕ0𝑥()()()(),1<𝛼<2,𝛼−1+𝑥𝑏+𝛼=0,Δ𝑥𝛼−1+Δ𝑥𝑏+𝛼−1=0, (10) if and only if 𝑥(𝑡) is a solution of the the following fractional Taylor’s difference formula: 𝑥(𝑡)=1Γ(𝛼)^𝑡−𝛼_𝑠=0(𝑡−𝑠−1)^(𝛼−1)𝑓()−1𝑠+𝛼−12Γ(𝛼)^𝑏_𝑠=0(𝑏+𝛼−𝑠−1)^(𝛼−1)𝑓()+𝑠+𝛼−1𝑏+2𝛼−1−𝑡2Γ(𝛼−1)^𝑏−1_𝑠=0(𝑏+𝛼−𝑠−2)^(𝛼−2)𝑓(),[]𝑠+𝛼−1𝑡∈𝛼−1,𝑏+𝛼_ℕ𝛼−1. (11)

Proof. Suppose that 𝑥(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 𝑥(𝑡)=Δ^−𝛼𝑓(𝑡+𝛼−1)−𝑐₀−𝑐₁𝑡=1Γ(𝛼)^𝑡−𝛼_𝑠=0(𝑡−𝑠−1)^(𝛼−1)𝑓(𝑠+𝛼−1)−𝑐₀−𝑐₁[]𝑡,𝑡∈𝛼−1,𝑏+𝛼_ℕ𝛼−1. (12)
Then, we obtain [3] (𝑡)=1Δ𝑥Γ(𝛼−1)^𝑡−𝛼_𝑠=0(𝑡−𝑠−1)^(𝛼−2)𝑓(𝑠+𝛼−1)−𝑐₁,𝑡∈[𝛼−1,𝑏+𝛼−1]_ℕ𝛼−1. (13)
In view of 𝑥(𝛼−1)+𝑥(𝑏+𝛼)=0 and Δ𝑥(𝛼−1)+Δ𝑥(𝑏+𝛼−1)=0, we have 1Γ(𝛼)^𝑏_𝑠=0(𝑏+𝛼−𝑠−1)^(𝛼−1)𝑓(𝑠+𝛼−1)−2𝑐₀−𝑐₁()1𝑏+2𝛼−1=0,Γ(𝛼−1)^𝑏−1_𝑠=0(𝑏+𝛼−𝑠−2)^(𝛼−2)𝑓(𝑠+𝛼−1)−2𝑐₁=0. (14) Then, 𝑐₀=12Γ(𝛼)^𝑏_𝑠=0(𝑏+𝛼−𝑠−1)^(𝛼−1)𝑓()−𝑠+𝛼−1𝑏+2𝛼−12Γ(𝛼−1)^𝑏−1_𝑠=0(𝑏+𝛼−𝑠−2)^(𝛼−2)𝑓(),𝑐𝑠+𝛼−1₁=12Γ(𝛼−1)^𝑏−1_𝑠=0(𝑏+𝛼−𝑠−2)^(𝛼−2)𝑓(𝑠+𝛼−1). (15) Substituting the values of 𝑐₀ and 𝑐₁ into (12), we obtain (11).
Conversely, if 𝑥(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)). Let 𝐹∶𝐶∈𝑅^𝑛→𝐶∈𝑅^𝑛 be a continuous mapping, where 𝐶 is a nonempty, bounded, close, and convex set. Then, 𝐹 has a fixed point.

Define the operator (𝑡)=1𝑇𝑥)(Γ(𝛼)^𝑡−𝛼_𝑠=0(𝑡−𝑠−1)^(𝛼−1)𝑓((−1𝑠+𝛼−1,𝑥𝑠+𝛼−1))2Γ(𝛼)^𝑏_𝑠=0(𝑏+𝛼−𝑠−1)^(𝛼−1)𝑓((+𝑠+𝛼−1,𝑥𝑠+𝛼−1))𝑏+2𝛼−1−𝑡2Γ(𝛼−1)^𝑏−1_𝑠=0(𝑏+𝛼−𝑠−2)^(𝛼−2)×𝑓(𝑠+𝛼−1,𝑥(𝑠+𝛼−1)),𝑡∈[𝛼−1,𝑏+𝛼]_ℕ𝛼−1. (16) Obviously, 𝑥(𝑡) is a solution of (9) if it is a fixed point of the operator 𝑇.

Theorem 9. Assume that.
(𝐻₁) 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 ()𝐿<3Γ𝑏+𝛼+1()Γ()+2Γ𝛼+1𝑏+1𝑏+𝛼()Γ()2𝛼Γ𝛼−1𝑏+𝛼Γ()(𝛼).𝑏+1−Γ (17)

Proof. Let 𝑥,𝑦∈𝐵; then for each 𝑡∈[𝛼−1,𝑏+𝛼]_ℕ𝛼−1, we have 𝑡)−(𝑡≤1(𝑇𝑥)(𝑇𝑦)Γ(𝛼)^𝑡−𝛼_𝑠=0(𝑡−𝑠−1)^(𝛼−1)×𝑓(((+1𝑠+𝛼−1,𝑥𝑠+𝛼−1))−𝑓𝑠+𝛼−1,𝑦𝑠+𝛼−1)2Γ(𝛼)^𝑏_𝑠=0(𝑏+𝛼−𝑠−1)^(𝛼−1)×𝑓((()+||𝑠+𝛼−1,𝑥𝑠+𝛼−1))−𝑓𝑠+𝛼−1,𝑦𝑠+𝛼−1𝑏+2𝛼−1−𝑡2Γ(𝛼−1)^𝑏−1_𝑠=0(𝑏+𝛼−𝑠−2)^(𝛼−2)×𝑓((()≤𝐿𝑠+𝛼−1,𝑥𝑠+𝛼−1))−𝑓𝑠+𝛼−1,𝑦𝑠+𝛼−1Γ(𝛼)^𝑡−𝛼_𝑠=0(𝑡−𝑠−1)^(𝛼−1)×𝑥()(+𝐿𝑠+𝛼−1−𝑦𝑠+𝛼−1)2Γ(𝛼)^𝑏_𝑠=0(𝑏+𝛼−𝑠−1)^(𝛼−1)×𝑥()(+𝐿||𝑠+𝛼−1−𝑦𝑠+𝛼−1)𝑏+2𝛼−1−𝑡2Γ(𝛼−1)^𝑏−1_𝑠=0(𝑏+𝛼−𝑠−2)^(𝛼−2)×𝑥()(≤𝐿𝑠+𝛼−1−𝑦𝑠+𝛼−1)𝑥−𝑦Γ(𝛼)^𝑡−𝛼_𝑠=0(𝑡−𝑠−1)^(𝛼−1)+𝐿𝑥−𝑦(𝛼)×2Γ_𝑏𝑠=0(𝑏+𝛼−𝑠−1)^(𝛼−1)+|𝑏+2𝛼−1−𝑡|⋅𝐿𝑥−𝑦2Γ(𝛼−1)^𝑏−1_𝑠=0(𝑏+𝛼−𝑠−2)^(𝛼−2)≤𝐿𝑥−𝑦(𝛼)⋅Γ()2Γ𝑡+1()+𝐿𝛼Γ𝑡−𝛼+1𝑥−𝑦(𝛼)⋅Γ()2Γ𝑏+𝛼+1()+𝐿(𝛼Γ𝑏+1𝑏+𝛼)𝑥−𝑦()⋅12Γ𝛼−1𝛼Γ(𝑏+𝛼)Γ()(𝛼)≤()𝑏+1−Γ3Γ𝑏+𝛼+1()Γ()+2Γ𝛼+1𝑏+1𝑏+𝛼()Γ()2𝛼Γ𝛼−1𝑏+𝛼Γ()(𝛼)𝐿.𝑏+1−Γ𝑥−𝑦 (18)
According to (17), we obtain (𝑇𝑥)(𝑡)−𝑇𝑦(𝑡)<𝑥−𝑦. (19) Then, 𝑇𝑥−𝑇𝑦≤𝑥−𝑦 (20) 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.
(𝐻₂) 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 𝐾^∗<3Γ(𝑏+𝛼+1)()Γ()+2Γ𝛼+1𝑏+1𝑏+𝛼()Γ()2𝛼Γ𝛼−1𝑏+𝛼Γ()(𝛼),𝑏+1−Γ (21) where 𝐾^∗=max{𝐾(𝑡)∶𝑡∈[𝛼−1,𝑏+𝛼]_ℕ𝛼−1}.

Proof. Let 𝑀>0; define the set 𝑆=𝑥(𝑡)∣[𝛼−1,𝑏+𝛼]_ℕ𝛼−1⟶𝑅,‖𝑥‖≤𝑀. (22) To prove this theorem, we only need to show that 𝑇 maps 𝑆 in 𝑆.
For 𝑥(𝑡)∈𝑆, we have 𝑡≤𝐾(𝑡)|(𝑇𝑥)()|Γ(𝛼)^𝑡−𝛼_𝑠=0(𝑡−𝑠−1)^(𝛼−1)|𝑥(+𝐾(𝑡)𝑠+𝛼−1)|2Γ(𝛼)^𝑏_𝑠=0(𝑏+𝛼−𝑠−1)^(𝛼−1)|𝑥(+()𝐾(𝑡)𝑠+𝛼−1)|𝑏+2𝛼−1−𝑡()×2Γ𝛼−1_{𝑏−1𝑠=0}(𝑏+𝛼−𝑠−2)^(𝛼−2)|𝑥(≤𝐾(𝑡𝑥‖𝑠+𝛼−1)|)‖Γ(𝛼)^𝑡−𝛼_𝑠=0(𝑡−𝑠−1)^(𝛼−1)+𝐾(𝑡)‖𝑥‖2Γ(𝛼)^𝑏_𝑠=0(𝑏+𝛼−𝑠−1)^(𝛼−1)+(𝑏+2𝛼−1−𝑡)𝐾(𝑡)‖𝑥‖2Γ(𝛼−1)^𝑏−1_𝑠=0(𝑏+𝛼−𝑠−2)^(𝛼−2)≤𝐾(𝑡)‖𝑥‖(𝛼)⋅Γ()2Γ𝑡+1()+𝐾(𝑡𝑥‖𝛼Γ𝑡−𝛼+1)‖(𝛼)⋅Γ()2Γ𝑏+𝛼+1()+𝐾(𝑡𝑥‖𝛼Γ𝑏+1)(𝑏+𝛼)‖()⋅12Γ𝛼−1𝛼Γ(𝑏+𝛼)Γ()(𝛼)≤()𝑏+1−Γ3Γ𝑏+𝛼+1()Γ()+2Γ𝛼+1𝑏+1𝑏+𝛼()Γ()2𝛼Γ𝛼−1𝑏+𝛼Γ()(𝛼)𝐾(𝑡𝑥‖≤()𝑏+1−Γ)‖3Γ𝑏+𝛼+1()Γ()+2Γ𝛼+1𝑏+1𝑏+𝛼()Γ()2𝛼Γ𝛼−1𝑏+𝛼Γ()(𝛼)𝐾𝑏+1−Γ^∗𝑀. (23) 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: Δ^𝛼_𝐶𝑦(𝑡)−𝑓𝑡+𝛼−1,𝑦(𝑡+𝛼−1)≤𝜀,𝑡∈[0,𝑏]_ℕ0,Δ^𝛼_𝐶𝑦(𝑡)()(),[]−𝑓𝑡+𝛼−1,𝑦𝑡+𝛼−1≤𝜀𝜑𝑡+𝛼−1𝑡∈0,𝑏_ℕ0. (24)(25)

Definition 11. Equation (1) is the Ulam-Hyers stable if there exists a real number 𝑐_𝑓>0 such that for each 𝜀>0 and for each solution 𝑦∈𝐵 of inequality (24), there exists a solution 𝑥∈𝐵 of (1) with 𝑦(𝑡)−𝑥(𝑡)≤𝑐_𝑓𝜀,𝑡∈[𝛼−1,𝑏+𝛼]_ℕ𝛼−1. (26) Equation (1) is the generalized Ulam-Hyers stable if we substitute the function 𝜃_𝑓(𝜀) for the constant 𝑐_𝑓𝜀 on inequality (26), where 𝜃_𝑓(𝜀)∈𝐶(𝑅⁺,𝑅⁺) and 𝜃_𝑓(0)=0.

Definition 12. Equation (1) is the Ulam-Hyers-Rassias stable with respect to 𝜑 if there exists a real number 𝑐_𝑓,𝜑>0 such that for each 𝜀>0 and for each solution 𝑦∈𝐵 of inequality (25), there exists a solution 𝑥∈𝐵 of (1) with 𝑦(𝑡)−𝑥(𝑡)≤𝑐_𝑓,𝜑𝜀𝜑(𝑡),𝑡∈[𝛼−1,𝑏+𝛼]_ℕ𝛼−1. (27) Equation (1) is the generalized Ulam-Hyers-Rassias stable if we substitute the function 𝜑(𝑡) for the function 𝜀𝜑(𝑡) on inequalities (25) and (27).

Remark 13. If 𝜑 is a constant function in Definition 12, we say that the integral equation (25) has also the Hyers-Ulam stability.

Remark 14. A function 𝑦∈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)|≤𝜀,𝑡∈[0,𝑏]_ℕ0, (ii)Δ^𝛼_𝐶𝑦(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: Δ^𝛼_𝐶𝑥(𝑡)=𝑓(𝑡+𝛼−1,𝑥(𝑡+𝛼−1)),𝑡∈[0,𝑏]_ℕ0𝑥()()()().,1<𝛼<2,𝛼−1=𝑦𝛼−1,𝑥𝑏+𝛼=𝑦𝑏+𝛼 (28) Then, (1) is the Ulam-Hyers stable provided that Γ()Γ()𝐿<𝛼+1𝑏+1().2Γ𝑏+𝛼+1 (29)

Proof. By Lemma 3, the solution of the BVP (28) is given by 𝑥(𝑡)()+𝑡=𝑦𝛼−1𝑦()()−𝑡𝑏+𝛼𝑏+𝛼−𝑦𝛼−1(𝑏+𝛼)Γ(𝛼)^𝑏_𝑠=0(𝑏+𝛼−𝑠−1)^(𝛼−1)((+1×𝑓𝑠+𝛼−1,𝑥𝑠+𝛼−1))Γ(𝛼)^𝑡−𝛼_𝑠=0(𝑡−𝑠−1)^(𝛼−1)×𝑓(𝑠+𝛼−1,𝑥(𝑠+𝛼−1)),𝑡∈[𝛼−1,𝑏+𝛼]_ℕ𝛼−1. (30) From inequality (24), for 𝑡∈[𝛼−1,𝑏+𝛼]_ℕ𝛼−1, it follows that 𝑦(𝑡)()−𝑡−𝑦𝛼−1𝑦()(+𝑡𝑏+𝛼𝑏+𝛼−𝑦𝛼−1)(𝑏+𝛼)Γ(𝛼)^𝑏_𝑠=0(𝑏+𝛼−𝑠−1)^(𝛼−1)()−1×𝑓𝑠+𝛼−1,𝑦𝑠+𝛼−1Γ(𝛼)^𝑡−𝛼_𝑠=0(𝑡−𝑠−1)^(𝛼−1)𝑓()≤𝜀𝑠+𝛼−1,𝑦𝑠+𝛼−1Γ(𝛼)^𝑡−𝛼_𝑠=0(𝑡−𝑠−1)^(𝛼−1)=𝜀Γ(𝛼)⋅Γ(𝑡+1)()≤Γ()𝛼Γ𝑡−𝛼+1𝑏+𝛼+1Γ(𝛼+1)Γ(𝑏+1)𝜀. (31) Combining (30) and (31), for 𝑡∈[𝛼−1,𝑏+𝛼]_ℕ𝛼−1, we have 𝑦(𝑡)(𝑡≤𝑦(𝑡)()−𝑡−𝑥)−𝑦𝛼−1𝑦()()+𝑡𝑏+𝛼𝑏+𝛼−𝑦𝛼−1()Γ(𝛼)×𝑏+𝛼_𝑏𝑠=0(𝑏+𝛼−𝑠−1)^(𝛼−1)𝑓((−1𝑠+𝛼−1,𝑥𝑠+𝛼−1))Γ(𝛼)^𝑡−𝛼_𝑠=0(𝑡−𝑠−1)^(𝛼−1)((≤𝑦(𝑡)()−𝑡×𝑓𝑠+𝛼−1,𝑥𝑠+𝛼−1))−𝑦𝛼−1𝑦()()+𝑡𝑏+𝛼𝑏+𝛼−𝑦𝛼−1()Γ(𝛼)×𝑏+𝛼_𝑏𝑠=0(𝑏+𝛼−𝑠−1)^(𝛼−1)𝑓()−1𝑠+𝛼−1,𝑦𝑠+𝛼−1Γ(𝛼)^𝑡−𝛼_𝑠=0(𝑡−𝑠−1)^(𝛼−1)()+𝑡×𝑓𝑠+𝛼−1,𝑦𝑠+𝛼−1(𝑏+𝛼)Γ(𝛼)^𝑏_𝑠=0(𝑏+𝛼−𝑠−1)^(𝛼−1)×𝑓()((+1𝑠+𝛼−1,𝑦𝑠+𝛼−1−𝑓𝑠+𝛼−1,𝑥𝑠+𝛼−1))Γ(𝛼)^𝑡−𝛼_𝑠=0(𝑡−𝑠−1)^(𝛼−1)×𝑓()((≤Γ()𝑠+𝛼−1,𝑦𝑠+𝛼−1−𝑓𝑠+𝛼−1,𝑥𝑠+𝛼−1))𝑏+𝛼+1Γ()Γ()𝑚𝛼+1𝑏+1𝜀+^𝑓𝑡()Γ(𝛼)×𝑏+𝛼_𝑏𝑠=0(𝑏+𝛼−𝑠−1)^(𝛼−1)×𝑦()(+𝑚𝑠+𝛼−1−𝑥𝑠+𝛼−1)_𝑓Γ(𝛼)^𝑡−𝛼_𝑠=0(𝑡−𝑠−1)^(𝛼−1)×𝑦()(≤Γ()𝑠+𝛼−1−𝑥𝑠+𝛼−1)𝑏+𝛼+1Γ()Γ()𝑚𝛼+1𝑏+1𝜀+^𝑓𝑡𝑦−𝑥()Γ(𝛼)×𝑏+𝛼_𝑏𝑠=0(𝑏+𝛼−𝑠−1)^(𝛼−1)+𝑚_𝑓𝑦−𝑥Γ(𝛼)^𝑡−𝛼_𝑠=0(𝑡−𝑠−1)^(𝛼−1)=Γ(𝑏+𝛼+1)Γ()Γ()𝑚𝛼+1𝑏+1𝜀+^𝑓𝑡()Γ(𝛼)⋅1𝑏+𝛼𝛼Γ(𝑏+𝛼+1)Γ()+𝑚𝑏+1𝑦−𝑥_𝑓Γ(𝛼)⋅Γ(𝑡+1)()≤Γ()𝛼Γ𝑡−𝛼+1𝑦−𝑥𝑏+𝛼+1Γ()Γ()𝑚𝛼+1𝑏+1𝜀+^𝑓𝑡()Γ(𝛼)⋅1𝑏+𝛼𝛼Γ(𝑏+𝛼+1)Γ()+𝑚𝑏+1𝑦−𝑥_𝑓Γ(𝛼)⋅Γ(𝑡+1)()≤Γ()𝛼Γ𝑡−𝛼+1𝑦−𝑥𝑏+𝛼+1Γ(𝛼+1)Γ(𝑏+1)𝜀+2𝑚_𝑓Γ(𝑏+𝛼+1)Γ()Γ()×.𝛼+1𝑏+1𝑦−𝑥 (32) Then, ≤Γ()𝑦−𝑥𝑏+𝛼+1Γ()Γ()𝜀+()𝛼+1𝑏+12𝐿Γ𝑏+𝛼+1Γ()Γ().𝛼+1𝑏+1𝑦−𝑥 (33) Applying (29) to the previous inequality yields that ≤Γ()𝑦−𝑥𝑏+𝛼+1Γ(𝛼+1)Γ(𝑏+1)−2𝐿Γ(𝑏+𝛼+1)𝜀; (34) 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.
(𝐻₃) Let 𝜑∶[𝛼−1,𝑏+𝛼]_ℕ𝛼−1→𝑅⁺ be an increasing function. There exists a constant 𝜆>0 such that 1Γ(𝛼)^𝑡−𝛼_𝑠=0(𝑡−𝑠−1)^(𝛼−1)𝜑()(),[]𝑠+𝛼−1≤𝜆𝜀𝜑𝑡+𝛼−1𝑡∈0,𝑏_ℕ0. (35)
Let 𝑦∈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 Theorem 16 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 Δ_𝐶^1.5𝑥(𝑡)=𝜆𝑥(𝑡+0.5),𝑡∈[0,10]_ℕ0,𝑥(0.5)+𝑥(11.5)=0,Δ𝑥(0.5)+Δ𝑥(10.5)=0, (36) where 𝑓(𝑡,𝑥)=𝜆𝑥(𝑡) for 𝑡∈[0.5,11.5]_ℕ0.5 and conditions (𝐻₁) and (𝐻₂) are satisfied. Since 3Γ(𝑏+𝛼+1)()Γ()+2Γ𝛼+1𝑏+1𝑏+𝛼()Γ()2𝛼Γ𝛼−1𝑏+𝛼Γ()(𝛼),𝑏+1−Γ≈47.7268 (37) 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 Δ_𝐶^1.5𝑥(𝑡)=𝜆𝑥(𝑡+0.5),𝑡∈[0,10]_ℕ0, (38) with the boundary conditions 𝑥(0.5)=𝑦(0.5),𝑥(11.5)=𝑦(11.5). (39) Since Γ(𝛼+1)Γ(𝑏+1)2Γ(𝑏+𝛼+1)≈0.0176, (40) if 𝜆<0.0176 and the inequality Δ_𝐶^1.5𝑦(𝑡)−𝑓𝑡+0.5,𝑦(𝑡+0.5)≤𝜀,for𝑡∈[0,10]_ℕ0, (41) hold, then (38) is the Ulam-Hyers stable by Theorem 15.

Acknowledgments

This work was supported in part by the Natural Science Foundation of China (10971173), the National Natural Science Foundation of Hunan Province under Grant 13JJ3120, and the Construct Program of the Key Discipline in Hunan Province.