On nonlocal fractional sum-difference boundary value problems for Caputo fractional functional difference equations with delay

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1 Kaewwisetkul and Sitthiwirattham Advances in Difference Equations 207) 207:29 DOI 0.86/s R E S E A R C H Open Access On nonlocal fractional sum-difference boundary value problems for Caputo fractional functional difference equations with delay Bunjong Kaewwisetkul and Thanin Sitthiwirattham 2,3* * Correspondence: thanin.s@sci.kmutnb.ac.th 2 Department of Mathematics, Faculty of Applied Science, King Mongkut s University of Technology North Bangkok, Bangkok, Thailand 3 Mathematics Department, Faculty of Science and Technology, Suan Dusit University, Bangkok, Thailand Full list of author information is available at the end of the article Abstract In this article, we study the existence and uniqueness results for a nonlocal fractional sum-difference boundary value problem for a Caputo fractional functional difference equation with delay, by using the Schauder fixed point theorem and the Banach contraction principle. Finally, we present some examples to display the importance of these results. MSC: 39A05; 39A2 Keywords: Caputo fractional functional difference equations; boundary value problem; delay; existence Introduction In this paper, we consider a nonlocal fractional sum boundary value problem for a Caputo fractional functional difference equation with delay of the form α C ut)=f[ t + α,u t+α, β C ut + α β)], γ Cuα γ )=0, ut + α)=ρ ω uη + ω), t N 0,T := {0,,...,T},.) Ɣω)α 3)[α 2)2 γ )+2]+T+α)[T α+5)2 γ ) 2]) and u α 3 = ψ, where ρ η, α s=α 3 η+ω σ s))ω α 3)[α 2)2 γ )+2]+η+ω)[η+ω 2α+5)2 γ ) 2]) 2, 3), β, γ, ω 0, ), η N α,t+α are given constants, F CN α 3,T+α C r R, R) and ψ is an element of the space C + r α 3):={ ψ C r : ψα 3)=0, β C ψs β +)=0,s N α r 3,α 3}. For r N 0,T+3 we denote C r is the Banach space of all continuous functions ψ : N α r 3,α 3 R endowed with the norm ψ Cr = max ψs). s N α r 3,α 3 The Authors) 207. This article is distributed under the terms of the Creative Commons Attribution 4.0 International License which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original authors) and the source, provide a link to the Creative Commons license, and indicate if changes were made.

2 Kaewwisetkuland SitthiwiratthamAdvances in Difference Equations 207) 207:29 Page 2 of 4 If u : N α r 3,α 3 R, then for any t N α 3,T+α we denote by u t the element of C r defined by u t θ)=ut + θ) forθ N r,0. Fractional difference calculus or discrete fractional calculus is a very new field for mathematicians. Basic definitions and properties of fractional difference calculus can be found in the book []. Some real-world phenomena are being studied with the assistance of fractional difference operators, one may refer to [2, 3] and the references therein. Good papers related to discrete fractional boundary value problems can be found in [4 2]andthereferences cited therein. At present, the development of boundary value problems for fractional difference equations which show an operation of the investigative function. The study may also have other functions related to the ones we are interested in. These creations are incorporating with nonlocal conditions which are both extensive and more complex. For example, Goodrich [6] considered the discrete fractional boundary value problem ν yt)=λf t + ν,yt + ν )), t N,b+, yν 2)= N i= F i yt i )), yν + b +)= M i= F2 i yt2 i )),.2) where b > 0 is an integer, and λ > 0 is a parameter, ν, 2] is a real number, f : N ν,ν+b R R is a continuous function, Fi, F2 i : R [0, ] are continuous functions for each i and satisfy some growth conditions to be specified later, and {ti }N i=, {t2 i }M i= N ν,ν+b.the existence of positive solutions are obtained by Krasnosel skii fixed point theorem. Reunsumrit et al. [8] obtained sufficient conditions for the existence of positive solutions for the three-point fractional sum boundary value problem for Caputo fractional difference equations via an argument with a shift α C ut)+at + α )f u θt + α ) )) =0, t N 0,T, uα 3)= 2 uα 3)=0,.3) ut + α)=λ β uη + β), where 2 < α 3, 0 < β, η N α,t+α, α C is the Caputo fractional difference operator of order α,andf :[0, ) [0, ) is a continuous function. The existence of at least one positive solution is proved by using Krasnoselskii s fixed point theorem. Recently, Sitthiwirattham [6] investigated three-point fractional sum boundary value problems for sequential fractional difference equations of the forms α C [φ p β Cx)]t)=f t + α + β,xt + α + β )), β C xα )=0, xα + β +.4) T)=ρ γ xη + γ ), where t N 0,T,0<α, β, < α + β 2, 0 < γ, η N α+β,α+β+t, ρ is a constant, f : N α+β 2,α+β+T R R is a continuous function, φ p is the p-laplacian operator. Existence

3 Kaewwisetkuland SitthiwiratthamAdvances in Difference Equations 207) 207:29 Page 3 of 4 and uniqueness of solutions are obtained by the Banach fixed point theorem and Schaefer s fixed point theorem. The results mentioned above are the motivation for this research. The plan of this paper is as follows. In Section 2 we recall some definitions and basic lemmas. Also, we derive a representation for the solution of.) by converting the problem to an equivalent summation equation. In Section 3, we prove existence and uniqueness results of the problem.) by using the Schauder fixed point theorem and the Banach contraction principle. Some illustrative examples are presented in Section 4. 2 Preliminaries In the following, there are notations, definitions, and lemma which are used in the main results. We briefly recall the necessary concepts from the discrete fractional calculus; see [] for further information. Definition 2. We define the generalized falling function by t α := Ɣt+), for any t and α Ɣt+ α) for which the right-hand side is defined. If t + α is a pole of the Gamma function and t +isnotapole,thent α =0. Lemma 2. []) Assume the following factorial functions are well defined. If t r, then t α r α for any α >0. Definition 2.2 For α >0andf defined on N a := {a, a+,...},theαth-order fractional sum of f is defined by α f t)= α a f t):= where t N a+α and σ s)=s +. t α ) α f t σ s) s), s=a Definition 2.3 For α >0andf defined on N a,theαth-order Caputo fractional difference of f is defined by α C f t):= N α) a N f t)= ƔN α) t N α) s=a t σ s) ) N α N f s), where t N a+n α and N N is chosen so that 0 N <α < N.Ifα = N,then α C f t)= N f t). Lemma 2.2 [9]) Assume that α >0and f defined on N a. Then α a+n α α C yt)=yt)+c 0 + C t a) + C 2 t a) C N t a) N, for some C i R,0 i N and 0 N <α N. To define the solution of the boundary value problem.) we need the following lemma that deals with a linear variant of the boundary value problem.)andgivesarepresentation of the solution.

4 Kaewwisetkuland SitthiwiratthamAdvances in Difference Equations 207) 207:29 Page 4 of 4 Ɣω)α 3)[α 2)2 γ )+2]+T+α)[T α+5)2 γ ) 2]) Lemma 2.3 Let ρ η, α 2, 3), s=α 3 η+ω σ s))ω α 3)[α 2)2 γ )+2]+η+ω)[η+ω 2α+5)2 γ ) 2]) γ, ω 0, ) and f CN α,t+α, R) be given. Then the problem α C ut)=f t + α ), t N 0,T, 2.) uα 3)=0, γ Cuα γ )=0, 2.2) ut + α)=ρ ω uη + ω), η N α,t+α, has the unique solution ut)= [ ) α 3) α 2)2 γ )+2 +2t 3 α)2 γ ) ) + t 2] [ T ) α f ρ T + α σ s) s + α ) Ɣω) ] η α η α ) ω ) α f η + ω α σ ξ) ξ + α σ s) s + α ) + ξ=s t α ) α f t σ s) s + α ), 2.3) for t N α 3,T+α, where =3 α) [ α 2)2 γ )+2 ] +T + α) [ 2 T α + 5)2 γ ) ] + ρ Ɣω) η ) ω [ ] η + ω σ s) α 3) α 2)2 γ )+2 +η + ω) s=α 3 [ η + ω 2α + 5)2 γ ) 2 ]). 2.4) Proof Using the fractional sum of order α 2, 3) for 2.) and from Lemma 2.2,weobtain ut)=c + C 2 t + C 3 t 2 + t α ) α f t σ s) s + α ), 2.5) for t N α 3,T+α. By substituting t = α 3into2.5) and employing the first condition of 2.2), we obtain C + C 2 α 3) + C 3 α 3) 2 =0. 2.6) Using the Caputo fractional difference of order 0 < γ <for2.5), we obtain γ C ut)= Ɣ γ ) + for t N α γ 2,T+α γ +. ) γ [ t σ s) C2 +2C 3 s ] t+γ s=α 3 Ɣ γ )Ɣα ) t+γ s=α s α+ ) γ ) α 2f t σ s) s σ ξ) ξ + α ), 2.7) ξ=0

5 Kaewwisetkuland SitthiwiratthamAdvances in Difference Equations 207) 207:29 Page 5 of 4 By substituting t = α γ into2.7) and employing the second condition of 2.2)implies 2 γ )C 2 +2 [ +α 3)2 γ ) ] C 3 =0. 2.8) Finally, taking the fractional sum of order 0 < ω <for2.5), we obtain ω ut)= Ɣω) + t ω ) ω [ t σ s) C + C 2 s + C 3 s 2] s=α 3 t ω Ɣω) = Ɣω) + s α ) ω ) α f t σ s) s σ ξ) ξ + α ) s=α ξ=0 t ω ) ω [ t σ s) C + C 2 s + C 3 s 2] s=α 3 Ɣω) t α ω ) ω t α σ ξ) t α ω ξ=s ξ + α σ s) ) α f s + α ), 2.9) for t N α γ +ω 2,T+α γ +ω+. By substituting t = T + α, η + ω into 2.5),2.9), respectively, and employing the last condition of 2.2)implies [ ] ρ η ) ω η + ω σ s) C Ɣω) s=α 3 [ ] + T + α) ρ η ) ω η η + ω σ s) + ω) C 2 Ɣω) s=α 3 [ ] + T + α) 2 ρ η ) ω η η + ω σ s) + ω) 2 C 3 Ɣω) = s=α 3 T ) α f t + α σ s) s + α ) η α ρ s α ) ω + η + ω α σ ξ) Ɣω) s=α ξ=s ξ + α σ s) ) α f s + α ). 2.0) The constants C, C 2 and C 3 can be obtained by solving the system of equations 2.6), 2.8) and 2.0), C = α 3)[ α 2)2 γ )+2 ] Q[f ], C 2 = 2 [ 3 α)2 γ ) ] Q[f ], C 3 = Q[f ],

6 Kaewwisetkuland SitthiwiratthamAdvances in Difference Equations 207) 207:29 Page 6 of 4 where is defined by 2.4) and the functional Q[f ]isdefinedas Q[f ]= T T + α σ s) ) α f s + α ) ρ Ɣω) η α η α ) ω ) α f η + ω α σ ξ) ξ + α σ s) s + α ). ξ=s Substituting the constants C, C 2 and C 3 into 2.5), we obtain 2.3). Corollary 2. Problem 2.)-2.2) has a unique solution of the form T ut)= Gt, s)hs + α ), 2.) where Gt, s)= g t, s), s N 0,t α N 0,η α, g 2 t, s), s N t α+,η α, g 3 t, s), s N η α+,t α, g 4 t, s), s N t α+,t N η α+,t, 2.2) with g i t, s), i 4, as [ ) ) g t, s)=: α 3) α 2)2 γ ) α)2 γ ) t + t 2] [ ] T η α ) α ρ ) ω ) α + α σ s) η + ω α σ ξ) ξ + α σ s) Ɣω) ξ=s + t σ s) ) α, [ ) ) g 2 t, s)=: α 3) α 2)2 γ ) α)2 γ ) t + t 2] [ ] T η α ) α ρ ) ω ) α + α σ s) η + ω α σ ξ) ξ + α σ s), Ɣω) g 3 t, s)=: g 4 t, s)=: [ ) ) α 3) α 2)2 γ ) α)2 γ ) t + t 2] T + α σ s) ) α ) α, + t σ s) [ ) ) α 3) α 2)2 γ ) α)2 γ ) t + t 2] T + α σ s) ) α. ξ=s Lemma 2.4 [22]) AboundedsetinR n is relatively compact; a closed bounded set in R n is compact. Lemma 2.5 [22]) If a set is closed and relatively compact, then it is compact.

7 Kaewwisetkuland SitthiwiratthamAdvances in Difference Equations 207) 207:29 Page 7 of 4 Lemma 2.6 Schauder fixed point theorem [22]) Assume that K is a convex compact set in a Banach space X and that T : K K is a continuous mapping. Then T has a fixed point. 3 Main results In this section, we wish to establish the existence results for the problem.). To accomplish this, we define the Banach space X = { u : u CN α r 3,T+α, R), β C u CN α β r 2,T+α β+, R), 0 < β < } with the norm defined by u X = u + β C u, 3.) where u = max t Nα r 3,T+α ut) and β C u = max t N α r 3,T+α ν Cut β +). For u α 3 = ψ, in view of the definitions of u t and ψ,weobtain u α 3 = u α 3 θ)=uθ + α 3)=ψθ + α 3) forθ N r,0. 3.2) Thus, we have ut)=ψt) fort N α r 3,α ) Since F CN α 3,T+α C r R, R), set F[t, u t, β Cut β +)]:=f t) in Lemma 2.3. We see by Lemma 2.3 that a function u is a solution of boundary value problem.) ifand only if it satisfies T ut)= Gt, s)f[s + α,u s+α, β Cus + α β)], t N α 3,T+α, ψt), t N α r 3,α ) Define an operator T : X X as follows: T T u)t)= Gt, s)f[s + α,u s+α, β Cus + α β)], t N α 3,T+α, ψt), t N α r 3,α 3, 3.5) and = = max t N α r 3,α 3 max t N α r 3,α 3 { T Gt, s)φs + α ) }, 3.6) { T t Gt β +,s)φs + α ) }, 3.7) ϒ = T + α) α Ɣ2 β) Ɣα +) ρt + α + ω) ƔT)Ɣα + ω +) { T + α)ɣ2 β) [ 3 γ )T + α +2 γ ) ] +2T + α)+2 [ 3 α)2 γ ) ]} + T + α)α+ {+ } αt + α β +) β. 3.8) T +)Ɣ2 β)

8 Kaewwisetkuland SitthiwiratthamAdvances in Difference Equations 207) 207:29 Page 8 of 4 Theorem 3. Assume the following properties: A) There exists a nonnegative function φ CN α 3,T+α ) such that F[t, x, y] φt)+λ x τ + λ 2 y τ 2, for each x C r, y R where λ, λ 2 are negative constants and 0<τ, τ 2 <; or A2) there exists a nonnegative function φ CN α 3,T+α ) such that F[t, x, y] φt)+λ x τ + λ 2 y τ 2, for each x C r, y R where λ, λ 2 are negative constants and τ, τ 2 >. Then boundary value problem.) has at least one solution. Proof We shall use the Schauder fixed point theorem to prove that the operator T defined by 3.5) has a fixed point. We divide the proof into three steps. Step I. Verify T maps bounded sets into bounded sets. Suppose A) holds, choose { ) T + α β +) β L max 3 +,3λ ϒ) τ,3λ 2 ϒ) τ 2 }, 3.9) Ɣ2 β) and define the P = {u X : u L, L >0}. For any u P,weobtain T u)t) T = Gt, s)f [ s + α,u s+α, β C us + α β)] T Gt, s)φs + α ) + λ u s+α τ + λ 2 β C us + α β) τ ) 2 { α 3) α 2)2 γ )+2 ) +2 3 α)2 γ ) ) t + t 2 T ) α ρ T + α σ s) Ɣω) } η α η α ) ω ) α η + ω α σ ξ) ξ + α σ s) + t α ) α t σ s) ξ=s + λ u s+α τ + λ 2 β C us + α β) τ ){ 2 T + α) α Ɣα +) ρt + α + ω) ƔT)Ɣα + ω +) [ T + α) [ 3 γ )T + α +2 γ ) ]] } T + α)α +. Ɣα +) Next, we consider t T u)t) T t Gt, s) F [ s + α,u s+α, β C us + α β)]

9 Kaewwisetkuland SitthiwiratthamAdvances in Difference Equations 207) 207:29 Page 9 of 4 T t Gt, s)φs + α ) + λ u s+α τ + λ 2 β C us + α β) τ ) 2 { ) 2 3 α)2 γ ) +2t T ) α T + α σ s) } η α η α ρ ) ω ) α η + ω α σ ξ) ξ + α σ s) + tα Ɣω) ξ=s + λ u s+α τ + λ 2 β C us + α β) τ ) 2 { 2 T + α) α Ɣα +) ρt + α + ω) ƔT)Ɣα + ω +) [ T + α +3 α)2 γ ) ] } T + α)α + and β C T u) t β +) Ɣ β) t ) β t β + σ s) T u)s) { t β +) β + λ u s+α τ + λ 2 β C us + α [ β) τ 2) T + α) α Ɣ2 β) Ɣ2 β) + T + α) α Ɣα +) ρt + α + ω) ƔT)Ɣα + ω +) 2T + α)+2 [ 3 α)2 γ ) ] ]} { T + α β +) β + λ u s+α τ + λ 2 β C us + α [ β) τ 2) T + α) α Ɣ2 β) Ɣ2 β) + 2 T + α) α Ɣα +) ρt + α + ω) ] ƔT)Ɣα + ω +) [ ]} T + α +3 α)2 γ ). Hence, we obtain T ut) X + T + α β +) β Ɣ2 β) + λ u s+α τ + λ 2 β C us + α β) τ 2 ) ϒ L 3 + λ u s+α τ + λ 2 β C us + α β) τ 2 ) ϒ L 3 + L 3 + L = L, 3.0) 3 which implies that T : P P. For the second cases, if A2) holds, choose { ) ) ) } T + α β +) β τ τ 2 L max 3 +,,, 3.) Ɣ2 β) 3λ ϒ 3λ 2 ϒ

10 Kaewwisetkuland SitthiwiratthamAdvances in Difference Equations 207) 207:29 Page 0 of 4 and by the same argument as above, we obtain T ut) X + T + α β +) β + λ u s+α τ + λ 2 β C Ɣ2 β) us + α β) τ ) 2 ϒ L 3 + L 3 + L = L, 3.2) 3 which implies that T : P P. Step II. The continuity of the operator T follows from the continuity of F and G. Step III. By Lemma 2.4 and Lemma 2.5, P is compact. Hence, by the Schauder fixed point theorem, we can conclude that problem.)hasat least one solution. The proof is completed. The second result is the existence and uniqueness of a solution to problem.), by using the Banach contraction principle. Theorem 3.2 Assume the following properties: A3) There exists a constant κ >0such that F[t, u, u 2 ] F[t, v, v 2 ] κ u v + u 2 v 2 ), for each u, v C r, u 2, v 2 R. A4) κ + 2 )<, where T + α)α = Ɣα +) + T + α) α Ɣα +) ρt + α + ω) ƔT)Ɣα + ω +) T + α) [ 3 γ )T + α +2 γ ) ], 3.3) T + α β +) β 2 = T + α) α Ɣ2 β) Ɣα +) ρt + α + ω) ƔT)Ɣα + ω +) { 2 [ ] T + α) α } T + α +3 α)2 γ ) ) Then the problem.) has a unique solution. Proof Consider the operator T : X X defined by 3.5). Clearly, the fixed point of the operator T is the solution of boundary value problem.). We will use the Banach contraction principle to prove that T has a fixed point. We first show that T is a contraction. For each t N α 3,T+α,wehave T u)t) T v)t) = T Gt, s) F [ s + α,u s+α, β Cus + α β)] F [ s + α,v s+α, β C vs + α β)] κ u v X { α 3) α 2)2 γ )+2 ) +2 3 α)2 γ ) ) t + t 2

11 Kaewwisetkuland SitthiwiratthamAdvances in Difference Equations 207) 207:29 Page of 4 T T + α σ s) ) α ρ Ɣω) } η α η α ) ω ) α η + ω α σ ξ) ξ + α σ s) + t α ) α t σ s) ξ=s { T + α) α κ u v X Ɣα +) + T + α) α Ɣα +) ρt + α + ω) ƔT)Ɣα + ω +) T + α) [ 3 γ )T + α +2 γ ) ]} = κ u v X. Next, we consider β C T u) t β +) β C T v) t β +) t ) β [ ] t β + σ s) T u)s) T v)s) Ɣ β) Ɣ β) t t β + σ s) ) β[ T ξ=0 s Gs, ξ) F [ ξ + α,u ξ+α, β C uξ + α β +)] F [ ξ + α,v ξ+α, ] β C vξ + α β +)] κ u v X Ɣ β) t s Gs, ξ) ] ) T β[ t β + σ s) ξ=0 T + α β +) β κ u v X T + α) α Ɣ2 β) Ɣα +) ρt + α + ω) ƔT)Ɣα + ω +) { 2 [ ] T + α) α } T + α +3 α)2 γ ) + = κ u v X 2. Obviously, for each t N α r 3,α 3,wehave T u)t) T v)t) =0. Therefore, we obtain T u)t) T v)t) X κ x y X + 2 ). By A4) implies T is a contraction. Hence, by the Banach contraction principle, we see that T has a fixed point which is a unique solution of the problem.). 4 Examples In this section, to illustrate our results, we consider some examples.

12 Kaewwisetkuland SitthiwiratthamAdvances in Difference Equations 207) 207:29 Page 2 of 4 Example 4. Consider the following fractional difference boundary value problem: 2 5 C ut)= t + 3 ) 2 + e t+ 3 2 ) 2 u ) 4 = 3 2 C u 3 4 t + 5 u 2 )3 t+ 3 τ + e 2t+ 5 2 ) C u t + ) τ 2, t N 0,5, 6 ) ) 5 =0, u = ) u ) Set α = 5 2, β = 2 3, γ = 3 4, ω = 4 5, η = 7 2, T =5, ρ = = Ɣω)α 3)[α 2)2 γ )+2]+T + α)[t α + 5)2 γ ) 2]) η s=α 3 η + ω σ s))ω α 3)[α 2)2 γ )+2]+η + ω)[η + ω 2α + 5)2 γ ) 2]), and F[t, u t, β C u t β+]=t 2 + e t t+) 3 u t τ + e 2t+) 2 3 C u t+ 3 τ 2. For t N 2, 5 2,wehave F [ ) t, u t, β C u ] t β e u t τ + e 2 3 C u t + 3) τ 2, so φt) 95 4, λ = 8 e, λ 2 = e.for0<τ, τ 2 <, A) is satisfied and for τ, τ 2 >,A2)is satisfied. Therefore, by Theorem 3., boundary value problem 4.) has at least one solution. Example 4.2 Consider the following fractional difference boundary value problem: 2 5 u t C ut)= 2 C ut + 6 ), t N 0,5, t )2 [ + u t C ut + 6 ) ] u ) ) ) 4 = C u =0, u = ) u ) Set α = 5 2, β = 2 3, γ = 3 4, ω = 4 5, η = 7 2, T =5, ρ = = Ɣω)α 3)[α 2)2 γ )+2]+T + α)[t α + 5)2 γ ) 2]) η s=α 3 η + ω σ s))ω α 3)[α 2)2 γ )+2]+η + ω)[η + ω 2α + 5)2 γ ) 2]), and F[t, u t, β C u t β+]= For t N 2, 5 2,wehave 2 u t + 3 C ut+ 3 ) t+200) 2 2. [+ u t + 3 C ut+ 3 ) ] F [ t, u t, β C u] F [ [ t, v t, β C v] 4 u t v t + 59, C u t + ) C v t + )], 3

13 Kaewwisetkuland SitthiwiratthamAdvances in Difference Equations 207) 207:29 Page 3 of 4 and for t N 2, 5 2,wehave F [ t, u t, β C u] F [ [ t, v t, β C v] 4 u t v t + 48, C u t + ) C v t + )], 3 so A3) holds with κ = 4. Also, we can show that 59, , 6.027, 2,202.36, and κ + 2 ) <. Therefore A4) holds, by Theorem 3.2, boundary value problem 4.2) hasauniquesolution. Acknowledgements This research was funded by King Mongkut s University of Technology North Bangkok. Contract no. KMUTNB-GEN Competing interests The authors declare that they have no competing interests. Authors contributions The authors declare that they carried out all the work in this manuscript and read and approved the final manuscript. Author details Department of Mathematics, Faculty of Liberal Arts, Rajamangala University of Technology Rattanakosin, Nakhon Pathom, Thailand. 2 Department of Mathematics, Faculty of Applied Science, King Mongkut s University of Technology North Bangkok, Bangkok, Thailand. 3 Mathematics Department, Faculty of Science and Technology, Suan Dusit University, Bangkok, Thailand. Publisher s Note Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations. Received: 6 April 207 Accepted: 8 July 207 References. Goodrich, CS, Peterson, AC: Discrete Fractional Calculus. Springer, New York 205) 2. Wu, GC, Baleanu, D: Discrete fractional logistic map and its chaos. Nonlinear Dyn. 75, ) 3. Wu, GC, Baleanu, D: Chaos synchronization of the discrete fractional logistic map. Signal Process. 02, ) 4. Agarwal, RP, Baleanu, D, Rezapour, S, Salehi, S: The existence of solutions for some fractional finite difference equations via sum boundary conditions. Adv. Differ. Equ. 204, ) 5. Goodrich, CS: On a discrete fractional three-point boundary value problem. J. Differ. Equ. Appl. 8, ) 6. Goodrich, CS: On semipositone discrete fractional boundary value problems with non-local boundary conditions. J. Differ. Equ. Appl. 9, ) 7. Lv, W: Existence of solutions for discrete fractional boundary value problems with a p-laplacian operator. Adv. Differ. Equ. 202, ) 8. Ferreira, R: Existence and uniqueness of solution to some discrete fractional boundary value problems of order less than one. J. Differ. Equ. Appl. 9, ) 9. Abdeljawad, T: On Riemann and Caputo fractional differences. Comput. Math. Appl. 623), ) 0. Atici, FM, Eloe, PW: Two-point boundary value problems for finite fractional difference equations. J. Differ. Equ. Appl. 7, ). Atici, FM, Eloe, PW: A transform method in discrete fractional calculus. Int. J. Differ. Equ. 22), ) 2. Anastassiou, GA: Nabla discrete fractional calculus and nabla inequalities. Math. Comput. Model. 5, ) 3. Sitthiwirattham, T, Tariboon, J, Ntouyas, SK: Existence results for fractional difference equations with three-point fractional sum boundary conditions. Discrete Dyn. Nat. Soc. 203, Article ID ) 4. Sitthiwirattham, T, Tariboon, J, Ntouyas, SK: Boundary value problems for fractional difference equations with three-point fractional sum boundary conditions. Adv. Differ. Equ. 203, ) 5. Sitthiwirattham, T: Existence and uniqueness of solutions of sequential nonlinear fractional difference equations with three-point fractional sum boundary conditions. Math. Methods Appl. Sci. 38, ) 6. Sitthiwirattham, T: Boundary value problem for p-laplacian Caputo fractional difference equations with fractional sum boundary conditions. Math. Methods Appl. Sci. 396), )

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