Hindawi Publishing Corporation Discrete Dynamics in Nature and Society Volume 2013, Article ID 459161, 7 pages http://dx.doi.org/10.1155/2013/459161
Research Article Existence and Ulam Stability of Solutions for Discrete Fractional Boundary Value Problem
Fulai Chen1 andYongZhou2
1 Department of Mathematics, Xiangnan University, Chenzhou 423000, China 2 School of Mathematics and Computational Science, Xiangtan University, Xiangtan 411005, China
Correspondence should be addressed to Fulai Chen; [email protected]
Received 28 May 2013; Accepted 9 July 2013
Academic Editor: Shurong Sun
Copyright © 2013 F. Chen and Y. Zhou. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
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 operator; here, we choose the Banach contraction mapping principle and the Brower fixed-point theorem. Motivated This paper investigates the existence of solutions for antiperi- by the work of the Ulam stability for fractional differential odicboundaryvalueproblemandtheUlamstabilityfor equations [14], in this paper, we also introduce four types nonlinear fractional difference equations: of the Ulam stability definitions for fractional difference Δ𝛼 𝑥 (𝑡) =𝑓(𝑡+𝛼−1,𝑥(𝑡+𝛼−1)) , equations and study the Ulam-Hyers stable and the Ulam- 𝐶 Hyers-Rassias stable. (1) The rest of the paper is organized as follows. In Section 2, 𝑡∈[0, 𝑏]N , 1<𝛼<2, 0 we introduce some useful preliminaries. In Section 3,we 𝛼 consider the existence of solutions for antiperiodic boundary where Δ 𝐶 is a Caputo fractional difference operator, N𝑎 = value problem of fractional difference equations. In Section 4, {𝑎,𝑎+1,𝑎+2,...}and 𝐼N =𝐼⋂ N𝑎 for any number 𝑎∈𝑅and 𝐼 𝑅, 𝑏 ∈ N𝑎 𝑓: [𝛼−1,𝑏+𝛼] ×𝑅→𝑅 we discuss the Ulam stability for fractional difference equa- each interval of 1,and N𝛼−1 is a continuous function with respect to the second variable. tions. Finally, two examples are given to illustrate our main Accompanied with the development of the theory on results. fractional differential equations, fractional difference equa- tions have also been studied more intensively of late. In 2. Preliminaries particular, some properties and inequalities of the fractional differencecalculusarediscussedin[1–7], the existence and In this section, we introduce preliminary facts which are used asymptotic stability of the solutions for fractional difference throughout this paper. equations are investigated in [8–10], and the boundary value problems of fractional difference equations are considered in Definition 1 (see [3, 4]). Let ] >0.The]th fractional sum of [11–13].Buttherearealotofworkstodointhefuture,andto 𝑓:N𝑎 →𝑅is defined by the best of my knowledge, there is no work on the existence 𝑡−] of solutions for antiperiodic boundary value problem and the −] 1 (]−1) Δ 𝑓 (𝑡) = ∑(𝑡−𝑠−1) 𝑓 (𝑠) ,𝑡∈N , (2) Ulam stability for nonlinear fractional difference equations. Γ (]) 𝑎+] To research the boundary value problem of fractional 𝑠=𝑎 differenceequations,weneedtoselectasuitablefixed-point (]) theorem because of the discrete property of the difference where 𝑡 = Γ(𝑡 + 1)/Γ(𝑡 − ] +1). 2 Discrete Dynamics in Nature and Society
−] In (2), the fractional sum Δ maps functions defined on Then, N𝑎 to functions defined on N𝑎+].AticiandEloe[3]pointed out that this definition is the development of the theory of 𝑡−𝛼 the fractional calculus on time scales. ∑(𝑡−𝑠−1)(𝛼−1) Definition 2 (see [1]). Let 𝜇>0and 𝑛−1<𝜇<𝑛,where𝑛 𝑠=0 denotes a positive integer and 𝑛=⌈𝜇⌉, ⌈⋅⌉ ceiling of number. Γ (𝑡) Γ (𝑡−1) Γ (𝛼+1) = + +⋅⋅⋅+ +Γ(𝛼) Set ] =𝑛−𝜇.The𝜇th fractional Caputo difference operator Γ (𝑡−𝛼+1) Γ (𝑡−𝛼) Γ (2) is defined as 1 Γ (𝑡+1) Γ (𝑡) = ( − ) 𝛼 Γ (𝑡−𝛼+1) Γ (𝑡−𝛼) Δ𝜇 𝑓 (𝑡) =Δ−] (Δ𝑛𝑓 (𝑡)) 𝐶 1 Γ (𝑡) Γ (𝑡−1) + ( − ) 𝑡−] (3) 𝛼 Γ (𝑡−𝛼) Γ (𝑡−𝛼−1) 1 (]−1) 𝑛 = ∑(𝑡−𝑠−1) Δ 𝑓 (𝑠) , ∀𝑡∈𝑁𝑎+], Γ (]) 𝑠=𝑎 1 Γ (𝛼+2) Γ (𝛼+1) +⋅⋅⋅+ ( − )+Γ(𝛼) 𝛼 Γ (2) Γ (1)
𝑛 Γ (𝑡+1) where Δ is the 𝑛th order forward difference operator; the = . 𝜇 𝛼Γ (𝑡−𝛼+1) fractional Caputo like difference Δ 𝐶 maps functions defined on N𝑎 to functions defined on N𝑎−𝜇. (8)
Lemma 3 (see [2, 13]). Assume that 𝜇>0and 𝑓 is defined on N𝑎.Then, The following result is an immediate consequence of Lemma 4.
Corollary 5. One has 𝑛−1 (𝑡−𝑎)(𝑘) Δ−𝜇Δ𝜇 𝑓 (𝑡) =𝑓(𝑡) − ∑ Δ𝑘𝑓 (𝑎) 𝐶 𝑘! 𝑘=0 (4) 𝑏 (𝛼−1) (i) ∑𝑠=0 (𝑏+𝛼−𝑠−1) = Γ(𝑏 + 𝛼 + 1)/𝛼Γ(𝑏, +1) (𝑛−1) =𝑓(𝑡) +𝑐0 +𝑐1𝑡+⋅⋅⋅+𝑐𝑛−1𝑡 , 𝑏−1 (𝛼−2) (ii) ∑𝑠=0 (𝑏+𝛼−𝑠−2) = (1/𝛼)((Γ(𝑏 + 𝛼)/Γ(𝑏 +1))− Γ(𝛼)). where 𝑛 is the smallest integer greater than or equal to 𝜇, 𝑐𝑖 ∈ 𝑅, 𝑖=1,2,...,𝑛−1. 3. Antiperiodic Boundary Value Problem Lemma 4. One has In this section, we consider the following antiperiodic bound- ary value problem: 𝑡−𝛼 Γ (𝑡+1) ∑(𝑡−𝑠−1)(𝛼−1) = . 𝛼Γ (𝑡−𝛼+1) (5) 𝑠=0 Δ𝛼 𝑥 (𝑡)=𝑓(𝑡+𝛼−1,𝑥(𝑡+𝛼−1)) ,𝑡∈[0, 𝑏] ,1<𝛼<2, 𝐶 N0 𝑥 (𝛼−1)+𝑥(𝑏+𝛼)=0, Δ𝑥(𝛼−1)+Δ𝑥(𝑏+𝛼−1)=0, 𝑥>𝑘 𝑥, 𝑘 ∈𝑅 𝑘>−1,𝑥>−1 Proof. For , , ,wehave[1] (9)
Γ (𝑥+1) where Δ is a forward difference operator. Γ 𝑘+1 Γ 𝑥−𝑘+1 𝑏+𝛼 ( ) ( ) Let 𝐵 bethesetofallrealsequences𝑥={𝑥(𝑡)}𝑡=𝛼−1 with (6) norm ‖𝑥‖ = sup𝑡∈[𝛼−1,𝑏+𝛼] |𝑥(𝑡)|.Then,𝐵 is a Banach space. Γ (𝑥+2) Γ (𝑥+1) N𝛼−1 = − ; Γ (𝑘+2) Γ (𝑥−𝑘+1) Γ (𝑘+2) Γ (𝑥−𝑘) Lemma 6. 𝑥:𝑡∈[𝛼−1,𝑏+𝛼] ×𝑅 → 𝑅 Asolution N𝛼−1 is a solution for antiperiodic boundary value problem that is, Δ𝛼 𝑥 (𝑡) =𝑓(𝑡+𝛼−1) ,𝑡∈[0, 𝑏] , 1<𝛼<2, 𝐶 N0 Γ (𝑥+1) 1 Γ (𝑥+2) Γ (𝑥+1) 𝑥 (𝛼−1)+𝑥(𝑏+𝛼)=0, Δ𝑥(𝛼−1)+Δ𝑥(𝑏+𝛼−1)=0, = [ − ]. (7) Γ (𝑥−𝑘+1) 𝑘+1 Γ (𝑥−𝑘+1) Γ (𝑥−𝑘) (10) Discrete Dynamics in Nature and Society 3 if and only if 𝑥(𝑡) is a solution of the the following fractional The following fixed-point theorems are needed to prove Taylor’s difference formula: the existence and uniqueness of solutions for the BVP (9).
1 𝑡−𝛼 Lemma 7 (see [15] (Banach contraction mapping principle)). 𝑥 (𝑡) = ∑(𝑡−𝑠−1)(𝛼−1)𝑓 (𝑠+𝛼−1) A contraction mapping on a complete metric space has exactly Γ (𝛼) 𝑠=0 one fixed point. 1 𝑏 − ∑(𝑏+𝛼−𝑠−1)(𝛼−1)𝑓 (𝑠+𝛼−1) Lemma 8 (see [16] (Brower fixed-point theorem)). Let 𝐹: 2Γ (𝛼) 𝑛 𝑛 𝑠=0 𝐶∈𝑅 →𝐶∈𝑅be a continuous mapping, where 𝐶 is a nonempty, bounded, close, and convex set. Then, 𝐹 has a fixed 𝑏+2𝛼−1−𝑡𝑏−1 + ∑(𝑏+𝛼−𝑠−2)(𝛼−2)𝑓 (𝑠+𝛼−1) , point. 2Γ (𝛼−1) 𝑠=0 Define the operator 𝑡∈[𝛼−1,𝑏+𝛼] . N𝛼−1 (11) (𝑇𝑥)(𝑡) 𝑥(𝑡) [𝛼 − 1, 𝑏 + 𝛼] Proof. Suppose that defined on N𝛼−1 is a 𝑡−𝛼 1 (𝛼−1) solution of (10). Using Lemma 3,forsomeconstants𝑐0,𝑐1 ∈ = ∑(𝑡−𝑠−1) 𝑓 (𝑠+𝛼−1,𝑥(𝑠+𝛼−1)) Γ 𝛼 𝑅,wehave ( ) 𝑠=0
−𝛼 𝑏 𝑥 (𝑡) =Δ 𝑓 (𝑡+𝛼−1) −𝑐0 −𝑐1𝑡 1 − ∑(𝑏+𝛼−𝑠−1)(𝛼−1)𝑓 (𝑠+𝛼−1,𝑥(𝑠+𝛼−1)) 2Γ (𝛼) 1 𝑡−𝛼 𝑠=0 = ∑(𝑡−𝑠−1)(𝛼−1)𝑓 (𝑠+𝛼−1)−𝑐 −𝑐 𝑡, Γ (𝛼) 0 1 (12) 𝑠=0 𝑏+2𝛼−1−𝑡𝑏−1 + ∑(𝑏+𝛼−𝑠−2)(𝛼−2) 𝑡∈[𝛼−1,𝑏+𝛼] . 2Γ (𝛼−1) 𝑠=0 N𝛼−1 ×𝑓(𝑠+𝛼−1,𝑥(𝑠+𝛼−1)) ,𝑡∈[𝛼−1,𝑏+𝛼] . Then, we obtain [3] N𝛼−1 (16) 1 𝑡−𝛼 Δ𝑥 (𝑡) = ∑(𝑡−𝑠−1)(𝛼−2)𝑓 (𝑠+𝛼−1) −𝑐, Γ (𝛼−1) 1 𝑠=0 (13) Obviously, 𝑥(𝑡) is a solution of (9)ifitisafixedpointofthe 𝑡∈[𝛼−1,𝑏+𝛼−1] . operator 𝑇. N𝛼−1 Theorem 9. In view of 𝑥(𝛼−1)+𝑥(𝑏+𝛼)=0and Δ𝑥(𝛼 − 1) + Δ𝑥(𝑏 + Assume that. (𝐻 ) 𝐿>0 |𝑓(𝑡, 𝑥) − 𝛼−1)=0,wehave 1 There exists a constant such that 𝑓(𝑡,𝑦)|≤ 𝐿|𝑥−𝑦| 𝑡∈[𝛼−1,𝑏+𝛼] for each N𝛼−1 and all 1 𝑏 𝑥, 𝑦. ∈𝐵 ∑(𝑏+𝛼−𝑠) −1 (𝛼−1)𝑓 (𝑠+𝛼) −1 −2𝑐 −𝑐 (𝑏+2𝛼) −1 =0, 0 1 Then, the BVP (9) has a unique solution on 𝐵 provided that Γ (𝛼) 𝑠=0
𝑏−1 1 (𝛼−2) 3Γ (𝑏+𝛼+1) 𝑏+𝛼 Γ (𝑏+𝛼) ∑(𝑏+𝛼−𝑠−2) 𝑓 (𝑠+𝛼−1) −2𝑐1 =0. 𝐿< + ( −Γ(𝛼)). Γ (𝛼−1) 𝑠=0 2Γ (𝛼+1) Γ (𝑏+1) 2𝛼Γ (𝛼−1) Γ (𝑏+1) (14) (17)
Then, Proof. Let 𝑥, 𝑦;thenforeach ∈𝐵 𝑡∈[𝛼−1,𝑏+𝛼]N ,we 𝑏 𝛼−1 1 (𝛼−1) have 𝑐0 = ∑(𝑏+𝛼−𝑠−1) 𝑓 (𝑠+𝛼−1) 2Γ (𝛼) 𝑠=0 𝑇𝑥 𝑡 −(𝑇𝑦) 𝑡 𝑏−1 ( )( ) ( ) 𝑏+2𝛼−1 (𝛼−2) − ∑(𝑏+𝛼−𝑠−2) 𝑓 (𝑠+𝛼−1) , (15) 2Γ (𝛼−1) 1 𝑡−𝛼 𝑠=0 ≤ ∑(𝑡−𝑠−1)(𝛼−1) Γ (𝛼) 𝑠=0 1 𝑏−1 𝑐 = ∑(𝑏+𝛼−𝑠−2)(𝛼−2)𝑓 (𝑠+𝛼−1) . 1 × 𝑓 (𝑠+𝛼−1,𝑥(𝑠+𝛼−1)) 2Γ (𝛼−1) 𝑠=0 −𝑓 (𝑠 + 𝛼 −1,𝑦 (𝑠+𝛼−1)) Substituting the values of 𝑐0 and 𝑐1 into (12), we obtain (11). Conversely, if 𝑥(𝑡) is a solution of (11), by a direct 1 𝑏 computation, it follows that the solution given by (11) satisfies + ∑(𝑏+𝛼−𝑠−1)(𝛼−1) (10). The proof is completed. 2Γ (𝛼) 𝑠=0 4 Discrete Dynamics in Nature and Society
× 𝑓 (𝑠+𝛼−1,𝑥(𝑠+𝛼−1)) unique fixed point which is a unique solution of the BVP9 ( ). This completes the proof. −𝑓 (𝑠 + 𝛼 −1,𝑦 (𝑠+𝛼−1)) Theorem 10. Assume that. 𝑏−1 |𝑏+2𝛼−1−𝑡| (𝛼−2) (𝐻2) 𝐾:[𝛼−1,𝑏+𝛼]N + ∑(𝑏+𝛼−𝑠−2) There exists a bounded function 𝛼−1 →𝑅 |𝑓(𝑡, 𝑥)| ≤ 𝐾(𝑡)|𝑥| 𝑥∈𝐵 2Γ (𝛼−1) 𝑠=0 such that for all . Then, the BVP (9) has at least a solution on 𝐵 provided that × 𝑓 (𝑠+𝛼−1,𝑥(𝑠+𝛼−1)) −𝑓 (𝑠 + 𝛼 −1,𝑦 (𝑠+𝛼−1)) 3Γ (𝑏+𝛼+1) 𝑏+𝛼 Γ (𝑏+𝛼) 𝐾∗ < + ( −Γ(𝛼)), 𝐿 𝑡−𝛼 2Γ (𝛼+1) Γ (𝑏+1) 2𝛼Γ (𝛼−1) Γ (𝑏+1) ≤ ∑(𝑡−𝑠−1)(𝛼−1) (21) Γ (𝛼) 𝑠=0 × 𝑥 (𝑠+𝛼−1) −𝑦(𝑠+𝛼−1) ∗ 𝑏 where 𝐾 = max{𝐾(𝑡) : 𝑡 ∈ [𝛼 − 1,N 𝑏+𝛼] }. 𝐿 𝛼−1 + ∑(𝑏+𝛼−𝑠−1)(𝛼−1) 2Γ (𝛼) 𝑠=0 Proof. Let 𝑀>0; define the set × 𝑥 (𝑠+𝛼−1) −𝑦(𝑠+𝛼−1)
𝑏−1 𝐿 |𝑏+2𝛼−1−𝑡| 𝑆={𝑥(𝑡) | [𝛼−1,𝑏+𝛼]N → 𝑅, ‖𝑥‖ ≤𝑀}. (22) + ∑(𝑏+𝛼−𝑠−2)(𝛼−2) 𝛼−1 2Γ (𝛼−1) 𝑠=0 × 𝑥 (𝑠+𝛼−1) −𝑦(𝑠+𝛼−1) To prove this theorem, we only need to show that 𝑇 maps 𝑆 𝑡−𝛼 in 𝑆. 𝐿 𝑥−𝑦 (𝛼−1) 𝐿 𝑥−𝑦 ≤ ∑(𝑡−𝑠−1) + For 𝑥(𝑡) ∈𝑆,wehave Γ (𝛼) 𝑠=0 2Γ (𝛼)
𝑏 (𝛼−1) × ∑(𝑏+𝛼−𝑠−1) |(𝑇𝑥)(𝑡)| 𝑠=0 𝑡−𝛼 𝑏−1 𝐾 (𝑡) (𝛼−1) |𝑏+2𝛼−1−𝑡| ⋅𝐿𝑥−𝑦 (𝛼−2) ≤ ∑(𝑡−𝑠−1) |𝑥 (𝑠+𝛼−1)| + ∑(𝑏+𝛼−𝑠−2) Γ (𝛼) 𝑠=0 2Γ (𝛼−1) 𝑠=0 𝑏 𝐾 (𝑡) (𝛼−1) 𝐿 𝑥−𝑦 Γ (𝑡+1) + ∑(𝑏+𝛼−𝑠−1) |𝑥 (𝑠+𝛼−1)| ≤ ⋅ 2Γ (𝛼) 2Γ (𝛼) 𝛼Γ (𝑡−𝛼+1) 𝑠=0 𝐿 𝑥−𝑦 Γ (𝑏+𝛼+1) (𝑏+2𝛼−1−𝑡) 𝐾 (𝑡) + ⋅ + 2Γ (𝛼) 𝛼Γ (𝑏+1) 2Γ (𝛼−1) 𝑏−1 𝐿 (𝑏+𝛼) 𝑥−𝑦 1 Γ (𝑏+𝛼) (𝛼−2) + ⋅ ( −Γ(𝛼)) × ∑(𝑏+𝛼−𝑠−2) |𝑥 (𝑠+𝛼−1)| 2Γ (𝛼−1) 𝛼 Γ (𝑏+1) 𝑠=0 3Γ (𝑏+𝛼+1) 𝑡−𝛼 ≤[ 𝐾 (𝑡) ‖𝑥‖ (𝛼−1) 2Γ (𝛼+1) Γ (𝑏+1) ≤ ∑(𝑡−𝑠−1) Γ (𝛼) 𝑠=0 𝑏+𝛼 Γ (𝑏+𝛼) + ( −Γ(𝛼)) ]𝐿𝑥−𝑦 . 𝐾 (𝑡) ‖𝑥‖ 𝑏 2𝛼Γ (𝛼−1) Γ (𝑏+1) + ∑(𝑏+𝛼−𝑠−1)(𝛼−1) 2Γ (𝛼) (18) 𝑠=0 According to (17), we obtain (𝑏+2𝛼−1−𝑡) 𝐾 (𝑡) ‖𝑥‖ 𝑏−1 + ∑(𝑏+𝛼−𝑠−2)(𝛼−2) 2Γ 𝛼−1 (𝑇𝑥)(𝑡) −(𝑇𝑦)(𝑡) < 𝑥−𝑦 . (19) ( ) 𝑠=0 Then, 𝐾 (𝑡) ‖𝑥‖ Γ (𝑡+1) 𝐾 (𝑡) ‖𝑥‖ Γ (𝑏+𝛼+1) ≤ ⋅ + ⋅ 𝑇𝑥 − 𝑇𝑦 ≤ 𝑥−𝑦 (20) 2Γ (𝛼) 𝛼Γ (𝑡−𝛼+1) 2Γ (𝛼) 𝛼Γ (𝑏+1) 𝑇 𝐾 (𝑡)(𝑏+𝛼) ‖𝑥‖ 1 Γ (𝑏+𝛼) which implies that is a contraction. Therefore, the Banach + ⋅ ( −Γ(𝛼)) fixed-point theoremLemma ( 7)guaranteesthat𝑇 has a 2Γ (𝛼−1) 𝛼 Γ (𝑏+1) Discrete Dynamics in Nature and Society 5
3Γ (𝑏+𝛼+1) ≤[ (i) |𝑔(𝑡+𝛼−1)|≤𝜀,𝑡∈[0,𝑏]N , 2Γ (𝛼+1) Γ (𝑏+1) 0 𝛼 (ii) Δ 𝑦(𝑡) = 𝑓(𝑡 + 𝛼 − 1, 𝑦(𝑡 + 𝛼 − 1)) + 𝑔(𝑡+𝛼−1),𝑡∈ 𝑏+𝛼 Γ (𝑏+𝛼) 𝐶 + ( −Γ(𝛼))]𝐾(𝑡) ‖𝑥‖ [0, 𝑏]N , 2𝛼Γ (𝛼−1) Γ (𝑏+1) 0 3Γ (𝑏+𝛼+1) similar remark for inequality (25). ≤[ 2Γ (𝛼+1) Γ (𝑏+1) Theorem 15. Assume that (𝐻1) holds. Let 𝑦∈𝐵be a solution 𝑏+𝛼 Γ (𝑏+𝛼) of inequality (24) and let 𝑥∈𝐵be a solution of the following + ( −Γ(𝛼))]𝐾∗𝑀. 2𝛼Γ (𝛼−1) Γ (𝑏+1) boundary value problem:
(23) Δ𝛼 𝑥 (𝑡)=𝑓(𝑡+𝛼−1,𝑥(𝑡+𝛼−1)) ,𝑡∈[0, 𝑏] , 1<𝛼<2, 𝐶 N0 From (21), we have |(𝑇𝑥)(𝑡)|;then, <𝑀 ‖𝑇𝑥‖ ≤𝑀 which 𝑥 (𝛼−1) =𝑦(𝛼−1) ,𝑥(𝑏+𝛼) =𝑦(𝑏+𝛼) . 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 (28) theorem (Lemma 8). This completes the proof. Then, (1) is the Ulam-Hyers stable provided that 4. The Ulam Stability Γ (𝛼+1) Γ (𝑏+1) 𝐿< . Similar to the definitions of the Ulam stability for fractional 2Γ (𝑏+𝛼+1) (29) differential equation14 [ ],weintroducefourtypesoftheUlam stability definitions for fractional difference equation. Proof. By Lemma 3, the solution of the BVP (28)isgivenby Consider (1) and the following inequalities: 𝛼 Δ 𝑦 (𝑡) −𝑓(𝑡+𝛼−1,𝑦(𝑡+𝛼−1)) ≤𝜀, 𝑡∈[0, 𝑏]N , 𝑡 𝐶 0 𝑥 (𝑡) =𝑦(𝛼−1) + (𝑦 (𝑏+𝛼) −𝑦(𝛼−1)) (24) 𝑏+𝛼 Δ𝛼 𝑦 (𝑡) −𝑓(𝑡+𝛼−1,𝑦(𝑡+𝛼−1)) ≤𝜀𝜑(𝑡+𝛼−1) , 𝑡 𝑏 𝐶 − ∑(𝑏+𝛼−𝑠−1)(𝛼−1) (𝑏+𝛼) Γ (𝛼) 𝑡∈[0, 𝑏] . 𝑠=0 N0 (25) 1 𝑡−𝛼 ×𝑓(𝑠+𝛼−1,𝑥(𝑠+𝛼−1)) + ∑(𝑡−𝑠−1)(𝛼−1) Definition 11. Equation (1) is the Ulam-Hyers stable if there Γ (𝛼) 𝑠=0 exists a real number 𝑐𝑓 >0such that for each 𝜀>0and for ×𝑓(𝑠+𝛼−1,𝑥(𝑠+𝛼−1)) ,𝑡∈[𝛼−1,𝑏+𝛼] . each solution 𝑦∈𝐵of inequality (24), there exists a solution N𝛼−1 𝑥∈𝐵of (1)with (30) 𝑦 (𝑡) −𝑥(𝑡) ≤𝑐𝑓𝜀, 𝑡 ∈ [𝛼−1,𝑏+𝛼]N . 𝛼−1 (26) 𝑡∈[𝛼−1,𝑏+𝛼] From inequality (24), for N𝛼−1 ,itfollowsthat Equation (1)isthegeneralizedUlam-Hyersstableifwe 𝜃 (𝜀) 𝑐 𝜀 substitute the function 𝑓 for the constant 𝑓 on inequality 𝑡 + + 𝑦 (𝑡) −𝑦(𝛼−1) − (𝑦 (𝑏+𝛼) −𝑦(𝛼−1)) (26), where 𝜃𝑓(𝜀) ∈ 𝐶(𝑅 ,𝑅 ) and 𝜃𝑓(0) = 0. 𝑏+𝛼
Definition 12. Equation (1) is the Ulam-Hyers-Rassias stable 𝑏 𝑡 (𝛼−1) with respect to 𝜑 if there exists a real number 𝑐𝑓,𝜑 >0such + ∑(𝑏+𝛼−𝑠−1) that for each 𝜀>0and for each solution 𝑦∈𝐵of inequality (𝑏+𝛼) Γ (𝛼) 𝑠=0 (25), there exists a solution 𝑥∈𝐵of (1)with ×𝑓(𝑠+𝛼−1,𝑦(𝑠+𝛼−1)) 𝑦 (𝑡) −𝑥(𝑡) ≤𝑐𝑓,𝜑 𝜀𝜑 (𝑡) ,𝑡∈[𝛼−1,𝑏+𝛼]N . 𝛼−1 (27) 𝑡−𝛼 1 (𝛼−1) Equation (1)isthegeneralizedUlam-Hyers-Rassiasstable − ∑(𝑡−𝑠−1) 𝑓(𝑠+𝛼−1,𝑦(𝑠+𝛼−1)) Γ (𝛼) (31) if we substitute the function 𝜑(𝑡) for the function 𝜀𝜑(𝑡) on 𝑠=0 inequalities (25)and(27). 𝜀 𝑡−𝛼 ≤ ∑(𝑡−𝑠−1)(𝛼−1) Remark 13. If 𝜑 is a constant function in Definition 12,we Γ (𝛼) 𝑠=0 say that the integral equation (25)hasalsotheHyers-Ulam 𝜀 Γ (𝑡+1) stability. = ⋅ Γ (𝛼) 𝛼Γ (𝑡−𝛼+1) Remark 14. Afunction𝑦∈𝐵is a solution of inequality (24)if 𝑔:[𝛼−1,𝑏+𝛼−1] → Γ (𝑏+𝛼+1) and only if there exists a function N𝛼−1 ≤ 𝜀. 𝑅 such that Γ (𝛼+1) Γ (𝑏+1) 6 Discrete Dynamics in Nature and Society 𝑚 𝑡 𝑦−𝑥 Combining (30)and(31), for 𝑡∈[𝛼−1,𝑏+𝛼]N ,wehave Γ (𝑏+𝛼+1) 𝑓 𝛼−1 ≤ 𝜀+ Γ (𝛼+1) Γ (𝑏+1) (𝑏+𝛼) Γ (𝛼) 𝑦 (𝑡) −𝑥(𝑡) 𝑏 (𝛼−1) × ∑(𝑏+𝛼−𝑠−1) 𝑡 𝑠=0 ≤ 𝑦 (𝑡) −𝑦(𝛼−1) − (𝑦 (𝑏+𝛼) −𝑦(𝛼−1)) 𝑏+𝛼 𝑡−𝛼 𝑚𝑓 𝑦−𝑥 𝑡 + ∑(𝑡−𝑠−1)(𝛼−1) + Γ (𝛼) (𝑏+𝛼) Γ (𝛼) 𝑠=0 Γ (𝑏+𝛼+1) 𝑚 𝑡 𝑏 = 𝜀+ 𝑓 ×∑(𝑏+𝛼−𝑠−1)(𝛼−1)𝑓 (𝑠+𝛼−1,𝑥(𝑠+𝛼−1)) Γ (𝛼+1) Γ (𝑏+1) (𝑏+𝛼) Γ (𝛼) 𝑠=0 1 Γ (𝑏+𝛼+1) ⋅ 𝑦−𝑥 1 𝑡−𝛼 − ∑(𝑡−𝑠−1)(𝛼−1) 𝛼 Γ (𝑏+1) Γ (𝛼) 𝑠=0 𝑚 𝑓 Γ (𝑡+1) + ⋅ 𝑦−𝑥 Γ (𝛼) 𝛼Γ (𝑡−𝛼+1) ×𝑓(𝑠+𝛼−1,𝑥(𝑠+𝛼−1)) Γ (𝑏+𝛼+1) 𝑚𝑓𝑡 ≤ 𝜀+ 𝑡 ≤ 𝑦 (𝑡) −𝑦(𝛼−1) − (𝑦 (𝑏+𝛼) −𝑦(𝛼−1)) Γ (𝛼+1) Γ (𝑏+1) (𝑏+𝛼) Γ (𝛼) 𝑏+𝛼 1 Γ (𝑏+𝛼+1) 𝑡 ⋅ 𝑦−𝑥 + 𝛼 Γ (𝑏+1) (𝑏+𝛼) Γ (𝛼) 𝑚 𝑓 Γ (𝑡+1) 𝑏 + ⋅ 𝑦−𝑥 ×∑(𝑏+𝛼−𝑠−1)(𝛼−1)𝑓(𝑠+𝛼−1,𝑦(𝑠+𝛼) −1 ) Γ (𝛼) 𝛼Γ (𝑡−𝛼+1) 𝑠=0 Γ (𝑏+𝛼+1) 2𝑚 Γ (𝑏+𝛼+1) ≤ 𝜀+ 𝑓 1 𝑡−𝛼 Γ (𝛼+1) Γ (𝑏+1) Γ (𝛼+1) Γ (𝑏+1) − ∑(𝑡−𝑠−1)(𝛼−1) Γ (𝛼) 𝑠=0 × 𝑦−𝑥 . (32) ×𝑓(𝑠+𝛼−1,𝑦(𝑠+𝛼−1)) Then, 𝑡 𝑏 Γ (𝑏+𝛼+1) + ∑(𝑏+𝛼−𝑠−1)(𝛼−1) 𝑦−𝑥 ≤ 𝜀 Γ (𝛼+1) Γ (𝑏+1) (𝑏+𝛼) Γ (𝛼) 𝑠=0 (33) 2𝐿Γ (𝑏+𝛼+1) × 𝑓(𝑠+𝛼−1,𝑦(𝑠+𝛼−1)) + 𝑦−𝑥 . Γ (𝛼+1) Γ (𝑏+1) −𝑓 (𝑠+𝛼−1,𝑥(𝑠+𝛼−1)) Applying (29) to the previous inequality yields that 1 𝑡−𝛼 Γ (𝑏+𝛼+1) + ∑(𝑡−𝑠−1)(𝛼−1) 𝑦−𝑥 ≤ 𝜀; Γ (𝛼+1) Γ (𝑏+1) −2𝐿Γ(𝑏+𝛼+1) (34) Γ (𝛼) 𝑠=0 Γ(𝑏 + 𝛼 + 1)/(Γ(𝛼 + 1)Γ(𝑏 + 1) − 2𝐿Γ(𝑏 +𝛼1))>0 × 𝑓(𝑠+𝛼−1,𝑦(𝑠+𝛼−1)) where , thus; (1) is the Ulam-Hyers stable. −𝑓 (𝑠+𝛼−1,𝑥(𝑠+𝛼−1)) Theorem 16. Assume that (𝐻1) and the following condition Γ (𝑏+𝛼+1) 𝑚𝑓𝑡 hold. ≤ 𝜀+ (𝐻 ) 𝜑:[𝛼−1,𝑏+𝛼] →𝑅+ Γ (𝛼+1) Γ (𝑏+1) (𝑏+𝛼) Γ (𝛼) 3 Let N𝛼−1 be an increasing function. There exists a constant 𝜆>0such that 𝑏 (𝛼−1) 1 𝑡−𝛼 × ∑(𝑏+𝛼−𝑠−1) ∑(𝑡−𝑠−1)(𝛼−1)𝜑 (𝑠+𝛼−1) ≤𝜆𝜀𝜑(𝑡+𝛼−1) , 𝑠=0 Γ (𝛼) 𝑠=0 × 𝑦 (𝑠+𝛼−1) −𝑥(𝑠+𝛼−1) 𝑡∈[0, 𝑏] . N0 𝑚 𝑡−𝛼 (35) + 𝑓 ∑(𝑡−𝑠−1)(𝛼−1) 𝑦∈𝐵 𝑥∈𝐵 Γ (𝛼) 𝑠=0 Let be a solution of inequality (25) and let be asolutionofthefollowingboundaryvalueproblem(28).Then, × 𝑦 (𝑠+𝛼−1) −𝑥(𝑠+𝛼−1) (1) is the Ulam-Hyers-Rassias stable provided that (29) holds. Discrete Dynamics in Nature and Society 7
The proof of Theorem 16 is similar to that of Theorem 15, [3] F. M. Atici and P. W. Eloe, “Initial value problems in discrete and we omit it. fractional calculus,” Proceedings of the American Mathematical Society,vol.137,no.3,pp.981–989,2009. 5. Examples [4] F. M. Atici and P.W.Eloe, “Atransform method in discrete frac- tional calculus,” International Journal of Difference Equations, As the applications of our main results, we consider the fol- vol. 2, no. 2, pp. 165–176, 2007. lowing examples. [5]F.M.AtıcıandP.W.Eloe,“Discretefractionalcalculuswith the nabla operator,” Electronic Journal of Qualitative Theory of Example 1. Consider the fractional difference BVP Differential Equations,vol.3,pp.1–12,2009. [6] F. M. Atıcı and S. S¸engul,¨ “Modeling with fractional difference Δ1.5𝑥 (𝑡) =𝜆𝑥(𝑡 + 0.5) ,𝑡∈[0, 10] , equations,” Journal of Mathematical Analysis and Applications, 𝐶 N0 (36) vol. 369, no. 1, pp. 1–9, 2010. 𝑥 (0.5) +𝑥(11.5) =0, Δ𝑥(0.5) +Δ𝑥(10.5) =0, [7] R. A. C. Ferreira, “A discrete fractional Gronwall inequality,” Proceedings of the American Mathematical Society,vol.140,no. 𝑓(𝑡, 𝑥) = 𝜆𝑥(𝑡) 𝑡 ∈ [0.5, 11.5] where for N0.5 and conditions 5, pp. 1605–1612, 2012. (𝐻1) and (𝐻2) are satisfied. Since [8] F. Chen, X. Luo, and Y. Zhou, “Existence results for nonlinear 3Γ (𝑏+𝛼+1) 𝑏+𝛼 Γ (𝑏+𝛼) fractional difference equation,” Advances in Difference Equa- + ( −Γ(𝛼)) tions, vol. 2011, Article ID 713201, 12 pages, 2011. 2Γ (𝛼+1) Γ (𝑏+1) 2𝛼Γ (𝛼−1) Γ (𝑏+1) (37) [9] F. Chen, “Fixed points and asymptotic stability of nonlinear ≈ 47.7268 , fractional difference equations,” Electronic Journal of Qualitative Theory of Differential Equations,vol.39,pp.1–18,2011. inequalities (17)and(21) are satisfied if 𝜆 < 47.7268. [10] F. Chen and Z. Liu, “Asymptotic stability results for nonlinear According to Theorem 9,theBVP(36) has a unique solution. fractional difference equations,” Journal of Applied Mathematics, vol.2012,ArticleID879657,14pages,2012. At the same time, the BVP (36) has at least a solution by Theorem 10. [11] C. S. Goodrich, “Existence of a positive solution to a system of discrete fractional boundary value problems,” Applied Mathe- Example 2. Consider the fractional difference equation matics and Computation,vol.217,no.9,pp.4740–4753,2011. [12] C. S. Goodrich, “On discrete sequential fractional boundary 1.5 value problems,” Journal of Mathematical Analysis and Applica- Δ 𝐶 𝑥 (𝑡) =𝜆𝑥(𝑡+0.5) ,𝑡∈[0, 10]N , (38) 0 tions, vol. 385, no. 1, pp. 111–124, 2012. with the boundary conditions [13] W. Lv, “Existence of solutions for discrete fractional boundary value problems with a 𝑝-Laplacian operator,” Advances in 𝑥 (0.5) =𝑦(0.5) ,𝑥(11.5) =𝑦(11.5) . (39) Difference Equations,vol.2012,article163,2012. [14]J.Wang,L.Lv,andY.Zhou,“Ulamstabilityanddatadepen- Since dence for fractional differential equations with Caputo deriva- Γ (𝛼+1) Γ (𝑏+1) tive,” Electronic Journal of Qualitative Theory of Differential ≈ 0.0176, 2Γ (𝑏+𝛼+1) (40) Equations,vol.63,pp.1–10,2011. [15] A. N. Kolmogorov and S. V.Fomin, Elements of Function Theory if 𝜆 < 0.0176 and the inequality and Functional Analysis,Nauka,Moscow,Russia,1981. [16] R. P. Agarwal, M. Meehan, and D. O’Regan, Fixed Point Theory Δ1.5𝑦 (𝑡) − 𝑓 (𝑡 + 0.5, 𝑦 (𝑡 + 0.5)) ≤𝜀, 𝑡∈[0, 10] , 𝐶 for N0 and Applications,vol.141ofCambridge Tracts in Mathematics, (41) Cambridge University Press, Cambridge, UK, 2001. 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.
References
[1] G. A. Anastassiou, “Nabla discrete fractional calculus and nabla inequalities,” Mathematical and Computer Modelling,vol.51,no. 5-6, pp. 562–571, 2010. [2] T. Abdeljawad, “On Riemann and Caputo fractional differ- ences,” Computers & Mathematics with Applications,vol.62,no. 3, pp. 1602–1611, 2011. Copyright of Discrete Dynamics in Nature & Society is the property of Hindawi Publishing Corporation and its content may not be copied or emailed to multiple sites or posted to a listserv without the copyright holder's express written permission. However, users may print, download, or email articles for individual use.