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

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Research Article Existence and Ulam Stability of Solutions for Discrete Fractional Boundary Value Problem 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 Δ () =(+−1,(+−1)) , equations and study the Ulam-Hyers stable and the Ulam- Hyers-Rassias stable. (1) The rest of the paper is organized as follows. In Section 2, ∈[0, ]N , 1<<2, 0 we introduce some useful preliminaries. In Section 3,we consider the existence of solutions for antiperiodic boundary where Δ is a Caputo fractional difference operator, N = value problem of fractional difference equations. In Section 4, {,+1,+2,...}and N =⋂ N for any number ∈and , ∈ N : [−1,+] ×→ we discuss the Ulam stability for fractional difference equa- each interval of 1,and N−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 :N →is defined by the best of my knowledge, there is no work on the existence −] of solutions for antiperiodic boundary value problem and the −] 1 (]−1) Δ () = ∑(−−1) () ,∈N , (2) Ulam stability for nonlinear fractional difference equations. Γ (]) +] To research the boundary value problem of fractional = differenceequations,weneedtoselectasuitablefixed-point (]) theorem because of the discrete property of the difference where = Γ( + 1)/Γ( − ] +1). 2 Discrete Dynamics in Nature and Society −] In (2), the fractional sum Δ maps functions defined on Then, N to functions defined on N+].AticiandEloe[3]pointed out that this definition is the development of the theory of − the fractional calculus on time scales. ∑(−−1)(−1) Definition 2 (see [1]). Let >0and −1<<,where =0 denotes a positive integer and =⌈⌉, ⌈⋅⌉ ceiling of number. Γ () Γ (−1) Γ (+1) = + +⋅⋅⋅+ +Γ() Set ] =−.Theth fractional Caputo difference operator Γ (−+1) Γ (−) Γ (2) is defined as 1 Γ (+1) Γ () = ( − ) Γ (−+1) Γ (−) Δ () =Δ−] (Δ ()) 1 Γ () Γ (−1) + ( − ) −] (3) Γ (−) Γ (−−1) 1 (]−1) = ∑(−−1) Δ () , ∀∈+], Γ (]) = 1 Γ (+2) Γ (+1) +⋅⋅⋅+ ( − )+Γ() Γ (2) Γ (1) Γ (+1) where Δ is the th order forward difference operator; the = . Γ (−+1) fractional Caputo like difference Δ maps functions defined on N to functions defined on N−. (8) Lemma 3 (see [2, 13]). Assume that >0and is defined on N.Then, The following result is an immediate consequence of Lemma 4. Corollary 5. One has −1 (−)() Δ−Δ () =() − ∑ Δ () ! =0 (4) (−1) (i) ∑=0 (+−−1) = Γ( + + 1)/Γ(, +1) (−1) =() +0 +1+⋅⋅⋅+−1 , −1 (−2) (ii) ∑=0 (+−−2) = (1/)((Γ( + )/Γ( +1))− Γ()). where is the smallest integer greater than or equal to , ∈ , =1,2,...,−1. 3. Antiperiodic Boundary Value Problem Lemma 4. One has In this section, we consider the following antiperiodic bound- ary value problem: − Γ (+1) ∑(−−1)(−1) = . Γ (−+1) (5) =0 Δ ()=(+−1,(+−1)) ,∈[0, ] ,1<<2, N0 (−1)+(+)=0, Δ(−1)+Δ(+−1)=0, > , ∈ >−1,>−1 Proof. For , , ,wehave[1] (9) Γ (+1) where Δ is a forward difference operator. Γ +1 Γ −+1 + ( ) ( ) Let bethesetofallrealsequences={()}=−1 with (6) norm ‖‖ = sup∈[−1,+] |()|.Then, is a Banach space. Γ (+2) Γ (+1) N−1 = − ; Γ (+2) Γ (−+1) Γ (+2) Γ (−) Lemma 6. :∈[−1,+] × → Asolution N−1 is a solution for antiperiodic boundary value problem that is, Δ () =(+−1) ,∈[0, ] , 1<<2, N0 Γ (+1) 1 Γ (+2) Γ (+1) (−1)+(+)=0, Δ(−1)+Δ(+−1)=0, = [ − ]. (7) Γ (−+1) +1 Γ (−+1) Γ (−) (10) Discrete Dynamics in Nature and Society 3 if and only if () 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 − Lemma 7 (see [15] (Banach contraction mapping principle)). () = ∑(−−1)(−1) (+−1) A contraction mapping on a complete metric space has exactly Γ () =0 one fixed point. 1 − ∑(+−−1)(−1) (+−1) Lemma 8 (see [16] (Brower fixed-point theorem)). Let : 2Γ () =0 ∈ →∈be a continuous mapping, where is a nonempty, bounded, close, and convex set. Then, has a fixed +2−1−−1 + ∑(+−−2)(−2) (+−1) , point. 2Γ (−1) =0 Define the operator ∈[−1,+] . N−1 (11) ()() () [ − 1, + ] Proof. Suppose that defined on N−1 is a − 1 (−1) solution of (10). Using Lemma 3,forsomeconstants0,1 ∈ = ∑(−−1) (+−1,(+−1)) Γ ,wehave ( ) =0 − () =Δ (+−1) −0 −1 1 − ∑(+−−1)(−1) (+−1,(+−1)) 2Γ () 1 − =0 = ∑(−−1)(−1) (+−1)− − , Γ () 0 1 (12) =0 +2−1−−1 + ∑(+−−2)(−2) ∈[−1,+] . 2Γ (−1) =0 N−1 ×(+−1,(+−1)) ,∈[−1,+] . Then, we obtain [3] N−1 (16) 1 − Δ () = ∑(−−1)(−2) (+−1) −, Γ (−1) 1 =0 (13) Obviously, () is a solution of (9)ifitisafixedpointofthe ∈[−1,+−1] . operator . N−1 Theorem 9. In view of (−1)+(+)=0and Δ( − 1) + Δ( + Assume that. ( ) >0 |(, ) − −1)=0,wehave 1 There exists a constant such that (,)|≤ |−| ∈[−1,+] for each N−1 and all 1 , . ∈ ∑(+−) −1 (−1) (+) −1 −2 − (+2) −1 =0, 0 1 Then, the BVP (9) has a unique solution on provided that Γ () =0 −1 1 (−2) 3Γ (++1) + Γ (+) ∑(+−−2) (+−1) −21 =0. < + ( −Γ()). Γ (−1) =0 2Γ (+1) Γ (+1) 2Γ (−1) Γ (+1) (14) (17) Then, Proof. Let , ;thenforeach ∈ ∈[−1,+]N ,we −1 1 (−1) have 0 = ∑(+−−1) (+−1) 2Γ () =0 −() −1 ( )( ) ( ) +2−1 (−2) − ∑(+−−2) (+−1) , (15) 2Γ (−1) 1 − =0 ≤ ∑(−−1)(−1) Γ () =0 1 −1 = ∑(+−−2)(−2) (+−1) . 1 × (+−1,(+−1)) 2Γ (−1) =0 − ( + −1, (+−1)) Substituting the values of 0 and 1 into (12), we obtain (11). Conversely, if () is a solution of (11), by a direct 1 computation, it follows that the solution given by (11) satisfies + ∑(+−−1)(−1) (10). The proof is completed. 2Γ () =0 4 Discrete Dynamics in Nature and Society × (+−1,(+−1)) unique fixed point which is a unique solution of the BVP9 ( ). This completes the proof. − ( + −1, (+−1)) Theorem 10. Assume that. −1 |+2−1−| (−2) (2) :[−1,+]N + ∑(+−−2) There exists a bounded function −1 → |(, )| ≤ ()|| ∈ 2Γ (−1) =0 such that for all . Then, the BVP (9) has at least a solution on provided that × (+−1,(+−1)) − ( + −1, (+−1)) 3Γ (++1) + Γ (+) ∗ < + ( −Γ()), − 2Γ (+1) Γ (+1) 2Γ (−1) Γ (+1) ≤ ∑(−−1)(−1) (21) Γ () =0 × (+−1) −(+−1) ∗ where = max{() : ∈ [ − 1,N +] }. −1 + ∑(+−−1)(−1) 2Γ () =0 Proof. Let >0; define the set × (+−1) −(+−1) −1 |+2−1−| ={() | [−1,+]N → , ‖‖ ≤}. (22) + ∑(+−−2)(−2) −1 2Γ (−1) =0 × (+−1) −(+−1) To prove this theorem, we only need to show that maps − in . − (−1) − ≤ ∑(−−1) + For (),wehave ∈ Γ () =0 2Γ () (−1) × ∑(+−−1) |()()| =0 − −1 () (−1) |+2−1−| ⋅− (−2) ≤ ∑(−−1) | (+−1)| + ∑(+−−2) Γ () =0 2Γ (−1) =0 () (−1) − Γ (+1) + ∑(+−−1) | (+−1)| ≤ ⋅ 2Γ () 2Γ () Γ (−+1) =0 − Γ (++1) (+2−1−) () + ⋅ + 2Γ () Γ (+1) 2Γ (−1) −1 (+) − 1 Γ (+) (−2) + ⋅ ( −Γ()) × ∑(+−−2) | (+−1)| 2Γ (−1) Γ (+1) =0 3Γ (++1) − ≤[ () ‖‖ (−1) 2Γ (+1) Γ (+1) ≤ ∑(−−1) Γ () =0 + Γ (+) + ( −Γ()) ]− . () ‖‖ 2Γ (−1) Γ (+1) + ∑(+−−1)(−1) 2Γ () (18) =0 According to (17), we obtain (+2−1−) () ‖‖ −1 + ∑(+−−2)(−2) 2Γ −1 ()() −()() < − . (19) ( ) =0 Then, () ‖‖ Γ (+1) () ‖‖ Γ (++1) ≤ ⋅ + ⋅ − ≤ − (20) 2Γ () Γ (−+1) 2Γ () Γ (+1) ()(+) ‖‖ 1 Γ (+) which implies that is a contraction. Therefore, the Banach + ⋅ ( −Γ()) fixed-point theoremLemma ( 7)guaranteesthat has a 2Γ (−1) Γ (+1) Discrete Dynamics in Nature and Society 5 3Γ (++1) ≤[ (i) |(+−1)|≤,∈[0,]N , 2Γ (+1) Γ (+1) 0 (ii) Δ () = ( + − 1, ( + − 1)) + (+−1),∈ + Γ (+) + ( −Γ())]() ‖‖ [0, ]N , 2Γ (−1) Γ (+1) 0 3Γ (++1) similar remark for inequality (25). ≤[ 2Γ (+1) Γ (+1) Theorem 15. Assume that (1) holds. Let ∈be a solution + Γ (+) of inequality (24) and let ∈be a solution of the following + ( −Γ())]∗. 2Γ (−1) Γ (+1) boundary value problem: (23) Δ ()=(+−1,(+−1)) ,∈[0, ] , 1<<2, N0 From (21), we have |()()|;then, < ‖‖ ≤ which (−1) =(−1) ,(+) =(+) . implies that maps in . 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 (+1) Γ (+1) < . Similar to the definitions of the Ulam stability for fractional 2Γ (++1) (29) differential equation14 [ ],weintroducefourtypesoftheUlam stability definitions for fractional difference equation.
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