A NEW CLASS OF FOUR-POINT FRACTIONAL SUM BOUNDARY VALUE PROBLEMS FOR NONLINEAR SEQUENTIAL FRACTIONAL DIFFERENCE EQUATIONS INVOLVING SHIFT OPERATORS

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1 Kragujevac Journal of Mathematics Volume 23) 208), Pages A NEW CLASS OF FOUR-POINT FRACTIONAL SUM BOUNDARY VALUE PROBLEMS FOR NONLINEAR SEQUENTIAL FRACTIONAL DIFFERENCE EQUATIONS INVOLVING SHIFT OPERATORS J. REUNSUMRIT AND T. SITTHIWIRATTHAM 2 Abstract. In this article, we study the existence result for a Riemann-Liouville fractional difference equation with four-point fractional sum boundary value conditions, by using the Sadovskii s fixed point theorem. Our problem contains the shift operators on fractional difference operators that are different orders. Finally, we present an example to show the importance of these result.. Introduction In this paper we consider a Riemann-Liouville fractional difference equation with nonlocal four-point fractional sum boundary value conditions of the form [ α 0 Eγ β α+γut) ) + E β γ.) α+β φt)vt))] =ft + α + β + γ, ut + α + β + γ ), vt + α + β + γ )), uα + β + γ 2) = ρu), uξ) = [ ν α+β+γ+ν3 vs + ν) us + ν) ] T +α+β+γ, where t N 0,T := {0,,..., T }, 0 < α + β < α + γ < α + β + γ 2, 0 < ν, ξ, η N α+β+γ,t +α+β+γ, ξ < η, functions f C N α+β+γ2,t +α+β+γ R R +, R), v, φ C N α+β+γ2,t +α+β+γ, R + ), a functional ρ : C N α+β+γ2,t +α+β+γ, R) R are given, and the shift operator E τ ut) := ut τ). Key words and phrases. Existence, fractional difference equations, fractional sum, boundary value problem. 200 Mathematics Subject Classification. Primary: 39A05. Secondary: 39A2. Received: December 22, 206. Accepted: April,

2 372 J. REUNSUMRIT AND T. SITTHIWIRATTHAM Fractional difference equations have been interested many mathematicians since they can use for describing many problems in the real-world phenomena such as physics, mechanics, chemistry, control systems, electrical networks, and flow in porous media. In recent years, mathematicians have used this fractional calculus to model and solve various related problems. In particular, fractional calculus is a powerful tool for the processes which appears in nature, e.g., biology, ecology and other areas, can be found in [29, 30], and the references therein. Some good papers dealing with boundary value problems for fractional difference equations, which have helped to build up some of the basic theory of this field, see the textbooks [, 6] and the papers [5]-[8] and references cited therein. For example, Goodrich [] considered the discrete fractional boundary value problem.2) µ µ 2 µ 3 yt) = ft + µ + µ 2 + µ 3, yt + µ + µ 2 + µ 3 )), y0) = 0 = yb + 2), where t N 2µ µ 2 µ 3,b+2µ µ 2 µ 3, 0 < µ, µ 2, µ 3 <, < µ 2 + µ 3 < 2, < µ + µ 2 + µ 3 < 2 and f : N 0 R [0, + ) is a continuous function. Existence of positive solutions are obtained by the the Krasnosel skii fixed point theorem. Dong et al. [] investigated the existence of solutions to the following fractional boundary value problem.3) t ν) ν xt) = fxt + ν )), t N0,T, T ν t xν 2) = [ t ν νxt) ] t=t = 0, where 0 < ν, t ν ν are T ν t are respectively, the left fractional difference and the right fractional difference operators and f : R R is continuous. Sitthiwirattham [23, 2] examined two fractional sum boundary value problems for fractional difference equations of the forms.) and.5) α C[φ p β Cx)]t) = ft + α + β, xt + α + β )), β Cxα ) = 0, xα + β + T ) = ρ γ xη + γ), α α β α+β + λe β)xt) = ft + α + β, xt + α + β )), xα + β 2) = 0, xα + β + T ) = ρ γ α+βxη + γ), where t N 0,T, 0 < α, β, < α + β 2, 0 < γ, η N α+β,α+β+t, ρ, f : N α+β2,α+β+t R R is a given continuous function, E β xt) = xt + β ) and φ p is the p-laplacian operator. The results mentioned above are the motivation for this research. The plan of this paper is as follows. In the next section, we recall some definitions and basic lemmas. In Section 3, we prove the existence of solutions to the boundary value problem.) by the help of the Sadovskii s fixed point theorem. An illustrative example is presented in the last section.

3 A NEW CLASS OF FOUR-POINT FRACTIONAL SUM BOUNDARY VALUE PROBLEMS Preliminaries In the following, there are notations, definitions, and lemmas which are used in the main results. Definition 2.. The generalized falling function is defined by t α := Γt+), for any Γt+α) t and α for which the right-hand side is defined. If t + α is a pole of the Gamma function and t + is not a pole, then t α = 0. Lemma 2.. [7] Assume the following factorial functions are well defined. i) t µ) t µ = t µ+, where µ R. ii) If t r, then t α r α for any α > 0. iii) t α+β = t β) α t β. Definition 2.2. The Gauss hypergeometric function 2 F a, b; c; x) is a function which can be defined in the form of a hypergeometric series c k+ c k = P k) Qk) k + a)k + b) = x, c 0 =. k + c) The resulting Gauss hypergeometric function is written by a) n b) n 2F a, b; c; x) = xn c) n n!, where z) n = Γz+n) Γz) n=0 is the Pochhammer symbol. Theorem 2.. [9] The Gauss hypergeometric theorem) Let 2F a, b; c; ) be the Gauss hypergeometric function with x =. Then where c a b > 0, a, b, c R. 2F a, b; c; ) = Γc) Γc a b) Γc a) Γc b), Definition 2.3. For α > 0 and f defined on N a fractional sum of f is defined by where t N a+α and σs) = s +. α a+αft) := tα t σs)) α fs), Γα) s=a := {a, a +,...}, the α-order Definition 2.. For α > 0 and f defined on N a, the α-order Riemann-Liouville fractional difference of f is defined by α a+nαft) := N Nα) a+nα ft) = t+α t σs)) α fs), Γα) where t N a+nα and N N is chosen so that 0 N < α N. s=a

4 37 J. REUNSUMRIT AND T. SITTHIWIRATTHAM Lemma 2.2. [7] Let 0 N < α N and y defined on N αn. Then α αn α 0 yt) = yt) + C t α + C 2 t α2 + + C N t αn, for some C i R, with i N. The following lemma deals with a linear variant of the boundary value problem.) and gives a representation of the solution. Lemma 2.3. Let 0 < α + β < α + γ <, < α + β + γ 2, 0 < ν, ξ < η, ξ, η N α+β+γ,t +α+β+γ, h C N α+β+γ,t +α+β+γ, R), v, φ C N α+β+γ2,t +α+β+γ, R + ) and ρ : C N α+β+γ2,t +α+β+γ, R) R be given. Then the problem [ α 0 Eγ β α+γut) ) + E β γ α+β φt)vt))] = ht + α + β + γ ), uα + β + γ 2) = ρu), 2.) uξ) = [ ν α+β+γ+ν3 vs + ν) us + ν) ] T +α+β+γ, has the unique solution ρu) t β ut) = α + β + γ 2) β + 2.2) where 2.3) and 2.) + ξβ tβ s=α+γt σs)) β s γ) α ξβ s=α+γξ σs)) β s γ) α ρu) ξ β α + β + γ 2) β + Au) + Γβ)Γγ) p=α+γ s=α+β+γ Γβ)Γα) Γβ)Γγ) Γβ)Γα) ξβ ξ σp)) β p + β γ σs)) γ φs)vs) p+β p=α+γ s=α+β+γ tβ p=α+β s=α+β+γ tβ p+β p=α+γ s=α+β+γ Au) = Γν) ΘT, ξ, η) T +α+β+γ ξ σp)) β p + α + β σs)) α hs) t σp)) β p + β γ σs)) γ φs)vs) t σp)) β p + α + β σs)) α hs), T + α + β + γ + ν σs)) ν vs) {Υ ρ s, ξ) + ψs, ξ)ψ v,φ ξ) ψs, ξ)i h ξ) + J h s) Φ v,φ s)}, sβ ψs, ξ) = p=α+γs σp)) β p γ) α ξβ p=α+γξ σp)) β p γ), α

5 A NEW CLASS OF FOUR-POINT FRACTIONAL SUM BOUNDARY VALUE PROBLEMS ) 2.6) 2.7) 2.8) 2.9) 2.0) 2.) P v [ψs, ξ)] = T +α+β+γ T + α + β + γ + ν σs)) ν vs)ψs, ξ), Γν) ΘT, ξ, η) = P v [ψs, ξ)], Υ ρ s, ξ) = [ s β ψs, ξ) ξ β] ρu, v) α + β + γ 2) β, Ψ v,φ ξ) = Γβ)Γγ) ξβ p=α+γ s=α+β+γ p + β γ σs)) γ φs)vs), I h ξ) = Γβ)Γα) ξβ p+β p=α+γ ω=α+β+γ ξ σp)) β ξ σp)) β p + α + β σω)) α hω), J h s) = Γβ)Γα) sβ p+β p=α+γ ω=α+β+γ s σp)) β p + α + β σω)) α hω), Φ v,φ s) = Γβ)Γγ) sβ p=α+β ω=α+β+γ p + β γ σω)) γ φω)vω). s σp)) β Proof. Using Lemma 2.2 and the fractional sum of order 0 < α < for 2.), we obtain E γ β α+γut) ) + E β γ α+β φt)vt)) =C t α + tα t σs)) α hs + α + β + γ ), Γα) s=0 for t N α,t +α. From E τ ut) = ut τ), we have 2.2) β α+γ ut) + γ α+β φt + β γ)vt + β γ) =C t γ) α + tγα t γ σs)) α hs + α + β + γ ), Γα) s=0 for t N α+γ,t +α+γ. Once again, using Lemma 2.2 and the fractional sum of order 0 < β < α for 2.2), we obtain ut) =C 2 t β + C Γβ) tβ s=α+γ t σs)) β s γ) α

6 376 J. REUNSUMRIT AND T. SITTHIWIRATTHAM 2.3) + Γβ)Γγ) Γβ)Γα) tβ p=α+γ s=α+β+γ tβ pβ p=α+γ s=α+β+γ for t N α+β+γ2,t +α+β+γ and < α + β + γ 2. Applying the first boundary condition of 2.) implies 2.) C 2 = The second condition of 2.) implies ρu) ξ β α + β + γ 2) β + C Γβ) + Γβ)Γγ) Γβ)Γα) ξβ t σp)) β p + β γ σs)) γ φs)vs) t σp)) β p + α + β σs)) α hs), ρu) α + β + γ 2) β. ξβ s=α+γ p=α+γ s=α+β+γ ξβ p+β p=α+γ s=α+β+γ ξ σs)) β s γ) α ξ σp)) β p + β γ σs)) γ φs)vs) ξ σp)) β p + α + β σs)) α hs) = T +α+β+γ T + α + β + γ + ν σs)) ν vs) us). Γν) Γβ) The constant C can be obtained by solving the above equation, so ρu) ξ β C = ξβ s=α+γξ σs)) β s γ) α α + β + γ 2) β 2.5) + T +α+β+γ Γν) ξβ p=α+γ s=α+β+γ Γβ)Γα) ξβ T + α + β + γ + ν σs)) ν vs) us) + Γβ)Γγ) ξ σp)) β p + β γ σs)) γ φs)vs) p+β p=α+γ s=α+β+γ Substituting the constants C, C 2 into 2.), we obtain ut) ρu) t β tβ = α + β + γ 2) β + s=α+γt σs)) β s γ) α ξβ s=α+γξ σs)) β s γ) α ξ σp)) β p + α + β σs)) α hs).

7 A NEW CLASS OF FOUR-POINT FRACTIONAL SUM BOUNDARY VALUE PROBLEMS 377 ρu) ξ β α + β + γ 2) β + T +α+β+γ T + α + β + γ + ν σs)) ν vs)us) Γν) ) Γβ)Γγ) Γβ)Γα) Γβ)Γγ) Γβ)Γα) ξβ p=α+γ s=α+β+γ ξβ p+β p=α+γ s=α+β+γ tβ p=α+β s=α+β+γ tβ p+β p=α+γ s=α+β+γ ξ σp)) β p + β γ σs)) γ φs)vs) ξ σp)) β p + α + β σs)) α hs) t σp)) β p + β γ σs)) γ φs)vs) t σp)) β p + α + β σs)) α hs). Let then we have Au) = T +α+β+γ T + α + β + γ + ν σs)) ν vs) us), Γν) Au) = T +α+β+γ T + α + β + γ + ν σs)) ν vs) Γν) 2.7) Υ ρs, ξ) + ψs, ξ)au) + ψs, ξ)ψ v,φ ξ) ψs, ξ)i h ξ) + J h s) Φ v,φ s), where ψs, ξ), Υ ρ s, ξ), Ψ v,φ ξ), I h ξ), J h s) and Φ v,φ s) are defined as 2.), 2.7)- 2.), respectively. We simplify 2.7) into 2.3). Finally, substituting Au) into 2.6), we obtain 2.2). This completes the proof. In the following, we shall give some definitions and lemma, which are associated with the Sadovskii fixed point theorem as follow. Definition 2.5. Let M be a bounded set in metric space X; d), then Kuratowski measure of noncompactness, αm) is defined as inf{ɛ : M covered by a finitely many sets such that the diameter of each set ɛ}. Definition 2.6. Let Φ : DΦ) X X be a bounded and continuous operator on Banach space X. Then Φ is called a condensing map if αφb)) < αb) for all bounded sets B DΦ), where α denotes the Kuratowski measure of noncompactness.

8 378 J. REUNSUMRIT AND T. SITTHIWIRATTHAM Lemma 2.. [3] The map K + C is a k-set contraction with 0 k <, and thus also condensing, if i) K, C : D X X are operators on the Banach space X; ii) K is k-contractive, i.e., Kx Ky k x y for all x, y D and fixed k [0, ); iii) C is compact. Lemma 2.5. [7] Arzelá-Ascoli theorem) A set of function in C[a, b] with the sup norm, is relatively compact if and only it is uniformly bounded and equicontinuous on [a, b]. Lemma 2.6. [7] If a set is closed and relatively compact then it is compact. Theorem 2.2. [22] Sadovskii s fixed point theorem) Let B be a convex, bounded and closed subset of a Banach space X and Φ : B B be a condensing map. Then Φ has a fixed point. 3. The existence of solutions to the four-point fractional sum boundary value problem.) Now, we wish to establish the existence result for the problem.). To accomplish this, we denote that C = CN α+β+γ2,t +α+β+γ, R) is a Banach space of all functions u with the norm defined by u C = u v, where u = max { ut) : ut) }, t N α+β+γ2,t +α+β+γ v = max{ vt) : v CN α+β+γ2,t +α+β+γ, R + ) is a given function}. Also define an operator F : C C by 3.) and ρu) t β F u)t) = α + β + γ 2) β + ξβ Fu)t) =F u)t) + F 2 u)t), tβ s=α+γt σs)) β s γ) α ξβ s=α+γξ σs)) β s γ) α ρu) ξ β α + β + γ 2) β + Âu) + Γβ)Γγ) p=α+γ s=α+β+γ Γβ)Γα) ξβ ξ σp)) β p + β γ σs)) γ φs)vs) p+β p=α+γ s=α+β+γ ξ σp)) β p + α + β σs)) α

9 3.2) 3.3) A NEW CLASS OF FOUR-POINT FRACTIONAL SUM BOUNDARY VALUE PROBLEMS 379 F 2 u)t) = where 3.) 3.5) 3.6) fs, us), vs)) =Υ ρ t, ξ) + ψt, ξ)âu) + ψt, ξ)ψ vξ) ψt, ξ)îfξ), Γβ)Γγ) φs)vs) + tβ p=α+β s=α+β+γ Γβ)Γα) tβ p=α+γ s=α+β+γ t σp)) β p + β γ σs)) γ p+β p + α + β σs)) α fs, us), vs)) =Ĵft) Φ v t), Âu) = Γν)ΘT, ξ, η) { vs) Î f ξ) = Γβ)Γα) T +α+β+γ t σp)) β T + α + β + γ + ν σs)) ν Υ ρ s, ξ) + ψs, ξ)ψ v,φ ξ) ψs, ξ)îfξ) + Ĵfs) Φ v,φ s) ξβ p+β p=α+γ s=α+β+γ fs, us), vs))), Ĵ f s) = Γβ)Γα) tβ p=α+γ s=α+β+γ fs, us), vs)), p+β ξ σp)) β p + α + β σs)) α t σp)) β p + α + β σs)) α with ψs, ξ), P v T, η), Qξ), ΘT, ξ, η), Υ ρ s, ξ), Ψ v,φ s, ξ) and Φ v,φ s, ξ) are defined as 2.)-2.7) and 2.). The problem.) has solutions if and only if the operator F has fixed points. In the following theorem, we shall give the existence result for the problem.), by the help of Sadovskii s fixed point theorem. Theorem 3.. Assume that functions f C N α+β+γ2,t +α+β+γ R R +, R), v, φ C N α+β+γ2,t +α+β+γ, R + ) and functional ρ : C N α+β+γ2,t +α+β+γ, R) R are given. Also, suppose f, ρ and φ satisfying the following conditions. H ) There exists a constant L > 0 such that ft, u, v) ft, u 2, v) L u u 2 v, for each t N α+β+γ2,t +α+β+γ and u, u 2 R. H 2 ) There exists a constant λ > 0 such that ρu ) ρu 2 ) λ u u 2 v, },

10 380 J. REUNSUMRIT AND T. SITTHIWIRATTHAM for each u, u 2 C. H 3 ) 0 < n φt) N for each t N α+β+γ2,t +α+β+γ. Then the problem.) has at least one solution on N α+β+γ2,t +α+β+γ, provided that where 3.7) 3.8) 3.9) χ := v { λω + LΩ 2 + NΩ 3 } <, Ω = Υ ) P + Θ, Θ Ω 2 = ψ max Θ Ω 3 = ψ max Θ Ĵ + Î) P + Θ ), Φ + Ψ ) P + Θ ), with ψ max, P, Θ, Υ, Ψ, Î, Ĵ and Φ are defined as 3.0), 3.2)-3.8), respectively. Proof. Let B R = {u C : u C R} be a closed bounded and convex subset of C, where R will be fixed later. We define a map F : B R C as Fu)t) = F u)t) + F 2 u)t), where F and F 2 are defined by 3.2) and 3.3) respectively. Notice that the problem.) is equivalent to a fixed point problem Fu) = u. Step. FB R ) B R. Set max ft, 0, 0) = K, sup ρu) = M and choose a constant R t N α+β+γ2,t +α+β+γ u C satisfied MΩ + K Ω 2 + R Ĵ) [ λω v + L Ω 2 + Ĵ) + N Ω 3 + Φ )]. We first consider that the values of ψs, ξ), P v, Q, Θs, T, ξ, η), Υ ρ s, ξ), Ψ ρ,φ ξ), Î f s, ξ), Ĵfs) and Φ v,φ s), as follow s α γ ) ψs, ξ) = β 2F α, α + β + γ s; α + γ s + ; ) ξ α γ ) β 2F α, α + β + γ ξ; α + γ ξ + ; ), 3.0) 3.) 3.2) ψ max =ψα + β + γ, ξ) β ) β 2F α, 0; β; ) = ξ α γ ) β 2F α, α + β + γ ξ; α + γ ξ + ; ), ψ min =ψα + β + γ, ξ) = 0, P v v T +α+β+γ T + α + β + γ + ν σs)) ν Γν) v ΓT + α + β + γ + ν η + ) = =: v P, Γν + )ΓT + α + β + γ η + )

11 A NEW CLASS OF FOUR-POINT FRACTIONAL SUM BOUNDARY VALUE PROBLEMS 38 ΘT, ξ, η) ϕmin v P ) = = 3.3) = : Θ, ) Υ ρ s, ξ) ψξβ s β ρu) ρ0) + ρ0) α + β + γ 2) β ψ max ξ β s β λ v u + M) α + β + γ 2) β ψ max ξ β T + α + β + γ) β λ u C + M) α + β + γ 2) β 3.) = : λ u C + M) Υ, 3.5) 3.6) 3.7) Ψ v,φ ξ) v φ ξ α γ)β 2F γ, α + β + γ ξ ; α + γ ξ; ) Γβ) v u Nξ α γ)β 2F γ, α + β + γ ξ ; α + γ ξ; ) Γβ) = u CNξ α γ) β 2F γ, α + β + γ ξ ; α + γ ξ; ) Γβ) = : u C NΨ, Î f ξ) Γβ)Γα) ξβ p+β p=α+γ ω=α+β+γ ξ σp)) β p + α + β σω)) α fω, uω), vω)) fω, 0, 0) + fω, 0, 0) ) L v u + K) ξ α γ ) β Γβ) 2 F α +, α + β + γ ξ; α + β ξ + ; ) = : L u C + K) Î, Ĵ f s) Γβ)Γα) sβ p+β p=α+γ ω=α+β+γ s σp)) β p + α + β σω)) α fω, uω), vω)) fω, 0, 0) + fω, 0, 0) ) L v u + K) s α γ ) β Γβ) 2F α +, α + β + γ s; α + β s + ; ) L u C + K) β ) β 2F α +, ; γ; ) Γβ) = : L u C + K) Ĵ,

12 382 J. REUNSUMRIT AND T. SITTHIWIRATTHAM 3.8) Φ v,φ s) Next, we consider 3.9) v φ s α γ)β 2F γ, α + β + γ s ; α + γ s; ) Γβ) v N β )β 2F γ, 0; β; ) Γβ) = u CN β ) β 2F γ, 0; β; ) Γβ) = : u C NΦ. Âu) P v Θ Υρ s, ξ) + ψs, ξ)ψ v,φ ξ) ψs, ξ)îfξ) + Ĵfs) Φ v,φ s) ). Now, we will show that FB R ) B R. For each t N α+β+γ2,t +α+β+γ and u B R, we have F u)t) = Υρ t, ξ) + ψs, ξ)âu) + ψs, ξ)ψ v,φξ) ψs, ξ)îfs, ξ) [Ĵ Υ ρt, ξ) + ψs, ξ)p v Θ + ψs, ξ)îfs, ξ) ] P v ψs, ξ)îfs, ξ) Θ [ ψs, ξ)ψv,φ ξ) Φ ] P v + ψs, ξ)ψ v,φ ξ) Θ λ u C + M) Υ [ ] P + Θ + L u C + K) ϕ [Ĵ max + Θ Θ Î) P + Θ )] ϕ [ ) )] max + N u C Φ + Ψ P + Θ Θ 3.20) = λ u C + M) Ω + L u C + K) Ω 2 + N u C Ω 3, and 3.2) Consequently, F 2 u)t) Ĵft) = Φ v,φ t) v Fu)t) = v { F u)t) + F2 u)t) } 3.22) L u C + K) Ĵ + N u CΦ. = v { λ u C + M) Ω + L u C + K) [ Ω 2 + Ĵ] + N u C [Ω 3 + Φ] } R. Therefore Fu)t) C R, it follows that FB R ) B R. Step 2. F is continuous and χ-contractive.

13 A NEW CLASS OF FOUR-POINT FRACTIONAL SUM BOUNDARY VALUE PROBLEMS 383 Let ɛ > 0 be given. Since ρ, f and v are continuous, so ρ, f and v are uniformly continuous on B R. Therefore, there exists δ = min { } δ, δ 2, δ 3 > 0 such that ρu ) ρu 2 ) ɛ <, whenever u u 2 v < δ, 3 v λω ft, u, v) ft, u 2, v) ɛ <, whenever max { u u 2 v } < δ 2, 3 v LΩ 2 C ɛ u u 2 <, whenever u u 2 v < δ 3. 3 v NΩ 3 Thus, for all t N α+β+γ2,t +α+β+γ and for all u, u 2 B R, we obtain F u )t) F u 2 )t) C = v F u )t) F u 2 )t) v {λω ρu ) ρu 2 ) + LΩ2 ft, u, v) ft, u 2, v) } + u NΩ3 u C 2 < ɛ 3 + ɛ 3 + ɛ 3 = ɛ. This means that F is continuous on B R. Next, we show that F is χ-contractive. For any u, u 2 B R and for each t N α+β+γ2,t +α+β+γ, we have F u )t) F u 2 )t) C = v F u )t) F u 2 )t) v { } λω u u C 2 + LΩ u 2 u C 2 + NΩ u 3 u C 2 C = χ u u 2. By the given assumption: χ <, it follows that F is χ-contractive. Step 3. F 2 is compact. In Step. it has been shown that F 2 is uniformly bounded. Now we show that F 2 maps bounded sets into equicontinuous sets of C. For any ɛ > 0, there exists δ = min { δ, δ } 2 > 0 such that Ĵft 2 ) Ĵft ) ɛ 2 v f, whenever t 2 t < δ, Φ v,φ t 2 ) Φ v,φ t ) ɛ 2 v 2 N, whenever t 2 t < δ 2. Hence, for any t, t 2 N α+β+γ2,t +α+β+γ any u, v B R, we have F 2 ut 2 ) F 2 ut ) C = v F 2 ut 2 ) F 2 ut ) { v f Ĵft 2 ) Ĵft ) } + v N Φv,φ t 2 ) Φ v,φ t ) ɛ 2 + ɛ 2 = ɛ.

14 38 J. REUNSUMRIT AND T. SITTHIWIRATTHAM Thus F 2 )B R ) is an equicontinuous set. Therefore it follows by Lemma 2.6 and the Arzelá-Ascoli theorem that F 2 is compact on B R. Step. F is is condensing. Since F is continuous, χ-contraction and F 2 is compact, therefore, by Lemma 2., F : B R B R with F = F + F 2 is a condensing map on B R. Consequently, by Theorem 3., the map F has a fixed point which, implies that the problem.) has a solution.. Example In this section, in order to illustrate our result, we consider an example. Consider the following fractional sum boundary value problem.) 0 [E 3 u 3 ) = u ) 3 = ) ) ] 2 3 ut) + E φt)vt) = et C i ut i ), t i = i 3 i=0, [ 2 7 v 2 s + ) u s + )] 25, 2 2 s= 2 ) vt + ) ) 2 t + 0 u u +, where t N 0,5, vt) = e sinπt), φt) = et are given functions, and C 00π i are given positive constants with 7 i=0 C i < e. 20 Here α =, T = 5, β = 2, γ =, ν =, ξ = 3, η = 2, ft, ut), vt)) = e t u and ρu) = 7 t+25) 2 u + i=0 C i ut i ). Let t N 3, 25 and u, u 2 C, then e vt) e and ft, u t), vt)) ft, u 2 t), vt)) 6 So, H ) holds with L = e / Also, we get 7 7 ρu ) ρu 2 ) = C i u t i ) C i u 2 t i ) i=0 i=0 7 C i u u 2 < i=0 6 e3/ 020 u u 2. e 20 u u 2 = 20 v u u 2. So, H 2 ) holds with λ = Since φt) = N, then H 3 ) is satisfied. We can show that ) /3 3 2F ψ max =, 0; ; ) 3 /3 =.383, 5 3) 2F, 2; 5; ) 3

15 A NEW CLASS OF FOUR-POINT FRACTIONAL SUM BOUNDARY VALUE PROBLEMS 385 ψ min =0, P = Γ 5) 2 Γ 3 =.5, )Γ2) 2 Θ =,.383 ) /3 ) / Υ = ) /3 = , 3 /3 8 ) 3) 2 Ψ = Γ 2) 2F 3, 3; 8 3 ; =.728, 3 /3 5 ) 3) 5 Î = Γ 2) 2F, 2; 5 3 ; = 2.795, 3 Ĵ = Φ = ) /3 Γ 2 3 ) 2F ) /3 Γ 2 3 ) 2F and Ω = , Ω 2 = and Ω 3 = Therefore, we have ) 5, 5; 3 3 ; = , ) 2, 6; ; = 2.082, χ = v { λω + LΩ 2 + NΩ 3 } = 0.296e = <. Hence, by Theorem 3., the boundary value problem.) has at least one solution on N α+β+γ2,t +α+β+γ. Acknowledgements. This research was funded by King Mongkut s University of Technology North Bangkok. Contract no. KMUTNB-GEN References [] T. Abdeljawad, Dual identities in fractional difference calculus within Riemann, Adv. Difference Equ ), Artical ID 36, 6 pages. [2] T. Abdeljawad, On Riemann and Caputo fractional differences, Comput. Math. Appl. 623) 20), [3] T. Abdeljawad and D. Baleanu. Fractional differences and integration by parts, J. Comput. Anal. Appl. 33) 20), [] R. P. Agarwal, Difference Equations and Inequalities: Theory, Methods and Applications, CRC Press, New York, [5] R. P. Agarwal, D. Baleanu, S. Rezapour and S. Salehi, The existence of solutions for some fractional finite difference equations via sum boundary conditions, Adv. Difference Equ ), Artical ID 282, 6 pages.

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17 A NEW CLASS OF FOUR-POINT FRACTIONAL SUM BOUNDARY VALUE PROBLEMS 387 [28] Lv. Weidong, Existence of solutions for discrete fractional boundary value problems witha p- Laplacian operator, Adv. Difference Equ ), Artical ID63, 0 pages. [29] G. C. Wu and D. Baleanu, Chaos synchronization of the discrete fractional logistic map, Signal Process ), [30] G. C. Wu and D. Baleanu, Discrete fractional logistic map and its chaos, Nonlinear Dyn ), [3] E. Zeidler, Nonlinear Functional Analysis and its Application, Fixed Point Theorems, Springer- Verlag, New York, 986. Department of Mathematics, Faculty of Applied Science, King Mongkut s University of Technology North Bangkok, Bangkok, Thailand address: jiraporn.r@sci.kmutnb.ac.th 2 Mathematics Department, Faculty of Science and Technology, Suan Dusit University, Bangkok, Thailand address: thanin_sit@dusit.ac.th