# On Nonlinear Fractional Sum-Difference Equations via Fractional Sum Boundary Conditions Involving Different Orders

Authors: Saowaluk Chasreechai; Chanakarn Kiataramkul; Thanin Sitthiwirattham
Journal: Mathematical Problems in Engineering
 (2015)

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## Abstract

We study existence and uniqueness results for Caputo fractional sum-difference equations with fractional sum boundary value conditions, by using the Banach contraction principle and Schaefer’s fixed point theorem. Our problem contains different numbers of order in fractional difference and fractional sums. Finally, we present some examples to show the importance of these results.


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## Body

## 1. Introduction

In this paper we consider a Caputo fractional sum-difference equation with nonlocal fractional sum boundary value conditions of the form(1)ΔCαut=ft+α-1,ut+α-1,Ψβut+α-2,t∈N0,T≔0,1,…,T,uα-2=yu,uT+α=Δ-γgT+α+γ-3uT+α+γ-3,where 1<α≤2, 0<β≤1, 2<γ≤3, and ΔCα is the Caputo fractional difference operator of order α. For U⊆R, g∈C(Nα-2,T+α,R+∩U) and f∈C(Nα-2,T+α×U×U,U) are given functions and y:C(Nα-2,T+α,U)→U is a given functional, and for φ:Nα-2,T+α×Nα-2,T+α→[0,∞),(2)Ψβut≔Δ-βφut+β=1Γβ∑s=α-β-2t-βt-σsβ-1_φt,s+βus+β.Mathematicians have employed this fractional calculus in recent years to model and solve various applied problems. In particular, fractional calculus is a powerful tool for the processes which appear in nature, for example, biology, ecology, and other areas, and can be found in [1, 2] and the references therein. The continuous fractional calculus has received increasing attention within the last ten years or so, and the theory of fractional differential equations has been a new important mathematical branch due to its extensive applications in various fields of science, such as physics, mechanics, chemistry, and engineering. Although the discrete fractional calculus has seen slower progress, within the recent several years, a lot of papers have appeared, which has helped to build up some of the basic theory of this area; see [3–17] and references cited therein.At present, there is a development of boundary value problems for fractional difference equations which shows an operation of the investigative function. The study may also have another function which is related to the one we are interested in. These creations are incorporating with nonlocal conditions which are both extensive and more complex, for instance.Agarwal et al. [3] investigated the existence of solutions for two fractional boundary value problems:(3)Δμ-2μxt=gt+μ-1,xt+μ-1,Δxt+μ-1,t∈N0,b+2,xμ-2=0,xμ+b+1=∑k=μ-1αxk,where 1<μ≤2 and g∈C(Nμ-1,μ+b+1×R×R,R) is a given function, and (4)Δμ-3μxt=gt+μ-2,xt+μ-2,t∈N0,b+3,xμ-3=xμ+b+1=0,xα=∑k=γβxk,where 2<μ≤3, α,β,γ∈Nμ-2,μ+b, and g∈C(Nμ-2,μ+b+1×R×R,R) is a given function.Kang et al. [5] obtained sufficient conditions for the existence of positive solutions for a nonlocal boundary value problem(5)-Δμyt=λht+μ-1fyt+μ-1,t∈N0,b,yμ-2=Ψy,yμ+b=Φy,where 1<μ≤2, f∈C(0,∞,[0,∞)), h∈C(Nμ-1,μ+b-1,[0,∞)) are given functions and Ψ,Φ:Rb+3→R are given functionals.Sitthiwirattham [17] examined a Caputo fractional sum boundary value problem with a p-Laplacian of the form(6)ΔCαϕpΔCβxt=ft+α+β-1,xt+α+β-1,t∈N0,T,ΔCβxα-1=0,xα+β+T=ρΔ-γxη+γ,where 0<α, β≤1, 1<α+β≤2, 0<γ≤1, η∈Nα+β-1,α+β+T-1, ρ is a constant, f:Nα+β-2,α+β+T×R→R is a continuous function, and ϕp is the p-Laplacian operator.The plan of this paper is as follows. In Section2 we recall some definitions and basic lemmas. Also we derive a representation for the solution to (1) by converting the problem to an equivalent summation equation. In Section 3, using this representation, we prove existence and uniqueness of the solutions of boundary value problem (1) by the help of the Banach fixed point theorem and Schaefer’s fixed point theorem. Some illustrative examples are presented in Section 4.


## 2. Preliminaries

In the following, there are notations, definitions, and lemmas which are used in the main results.Definition 1.
One defines the generalized falling function bytα_≔Γ(t+1)/Γ(t+1-α), for any t and α for which the right-hand side is defined. If t+1-α is a pole of Gamma function and t+1 is not a pole, then tα_=0.Lemma 2 (see [13]).
Assume that the following factorial functions are well defined:(i)
(t-μ)tμ_=tμ+1_, where μ∈R.(ii)
Ift≤r, then tα_≤rα_ for any α>0.(iii)
tα+β_=t-βα_tβ_.Definition 3.
Forα>0 and f defined on Na≔{a,a+1,…}, the α-order fractional sum of f is defined by (7)Δ-αft≔1Γα∑s=at-αt-σsα-1_fs,where t∈Na+α and σ(s)=s+1.Definition 4.
Forα>0 and f defined on Na, the α-order Caputo fractional difference of f is defined by (8)ΔCαft≔Δ-N-αΔNft=1ΓN-α∑s=at-N-αt-σsN-α-1_ΔNfs,where t∈Na+N-α and N∈N is chosen so that 0≤N-1<α<N. If α=N, then ΔCαf(t)=ΔNf(t).Lemma 5 (see [9]).
Assume thatα>0 and 0≤N-1<α≤N. Then (9)Δ-αΔCαyt=yt+C0+C1t1_+C2t2_+⋯+CN-1tN-1_,for some Ci∈R, 0≤i≤N-1.The following lemma deals with linear variant of boundary value problem (1) and gives a representation of the solution.Lemma 6.
Let1<α≤2, 2<γ≤3, y:C(Nα-2,T+α,U)→U, and h∈C(Nα-1,T+α-1,U) be given. Then the problem(10)ΔCαut=ht+α-1,t∈N0,T,uα-2=yu,uT+α=Δ-γgT+α+γ-3uT+α+γ-3,has the unique solution(11)ut=1-t1_T+αyu+t1_T+αAu-t1_T+αΓα∑s=α-1T+α-1T+2α-1-σsα-1_hs+1Γα∑s=α-1t-1t+α-1-σsα-1_hs,where(12)Au=-yu∑s=α-2T+α-3gsT+α+γ-3+σsγ-1_1-s1_/T+α1/T+α∑s=α-2T+α-3s1_gsT+α+γ-3+σsγ-1_-Γγ-1/Γα∑s=αT+α-3∑ξ=α-1s-1gsT+α+γ-3+σsγ-1_s+α-1-σξα-1_hξ1/T+α∑s=α-2T+α-3s1_gsT+α+γ-3+σsγ-1_-Γγ+1/Γα∑s=αT+α-3∑ξ=α-1T+α-1gss1_T+α+γ-3+σsγ-1_T+2α-1-σξα-1_hξ∑s=α-2T+α-3s1_gsT+α+γ-3+σsγ-1_-Γγ.Proof.
Using Lemma5, a general solution for (10) can be written in the form(13)ut=C0+C1t1_+1Γα∑s=0t-αt-σsα-1_hs+α-1,for t∈Nα-2,α+T. Applying the first boundary condition of (10) implies(14)C0=yu. So,(15)ut=yu+C1t1_+1Γα∑s=0t-αt-σsα-1_hs+α-1.The second condition of (10) implies (16)uT+α=yu+C1T+α+1Γα∑s=0TT+α-σsα-1_hs+α-1=1Γγ∑s=α-2T+α-3T+α+γ-3-σsγ-1_gsus.A constant C1 can be obtained by solving the above equation, so (17)C1=1T+αΓγ∑s=α-2T+α-3T+α+γ-3-σsγ-1_gsus-yuT+α-1T+αΓα∑s=0TT+α-σsα-1_hs+α-1.Substituting a constant C1 into (15), we get(18)ut=1-t1_T+αyu+t1_T+α1Γγ∑s=α-2T+α-3T+α+γ-3-σsγ-1_gsus-1Γα∑s=α-1T+α-1T+2α-1-σsα-1_hs+1Γα∑s=α-1t-1t+α-1-σsα-1_hs.Let A(u)=1/Γ(γ)∑s=α-2T+α-3(T+α+γ-3-σ(s))γ-1_g(s)u(s). Then(19)Au=1Γγ∑s=α-2T+α-3gsT+α+γ-3-σsγ-1_1-s1_T+αyu+s1_T+αAu-1Γα∑ξ=α-1T+α-1T+2α-1-σξα-1_hξ+1Γα∑ξ=α-1s-1s+α-1-σξα-1_hξ,we simplify (19) becomes (12).
SubstitutingA(u) into (18), we obtain (11).


## 3. Main Results

Now we are in a position to establish the main results. First, we transform boundary value problem (1) into a fixed point problem.ForU⊆R, let (U,·) be a Banach space and let C=C(Nα-2,T+α,U) denote the Banach space of all continuous functions from Nα-2,T+α→U endowed with a topology of uniform convergence with the norm denoted by ·C. For this purpose, we consider the operator F:C→C by(20)Fut=1-t1_T+αyu+t1_T+αAu-t1_T+αΓα∑s=α-1T+α-1T+2α-1-σsα-1_fs,us,Ψβus-1+1Γα∑s=α-1t-1t+α-1-σsα-1_fs,us,Ψβus-1,where(21)Au=11/T+α∑s=α-2T+α-3s1_gsT+α+γ-3-σsγ-1_-Γγ-yu∑s=α-2T+α-3gsT+α+γ-3-σsγ-1_1-s1_T+α-1Γα∑s=αT+α-3∑ξ=α-1s-1gsT+α+γ-3-σsγ-1_s+α-1-σξα-1_fξ,uξ,Ψβuξ-1+1T+αΓα∑s=αT+α-3∑ξ=α-1T+α-1s1_gsT+α+γ-3-σsγ-1_T+2α-1-σξα-1_fξ,uξ,Ψβuξ-1.It is easy to see that problem (1) has solutions if and only if operator F has fixed points.Theorem 7.
Assume thatf:Nα-2,T+α×U×U→U is continuous and maps bounded subsets of Nα-2,T+α×U×U into relatively compact subsets of U, φ:Nα-2,T+α×Nα-2,T+α→[0,∞) is continuous with φ0=max⁡{φ(t-1,s):(t,s)∈Nα-2,T+α×Nα-2,T+α}, and y:C→U is a given functional. In addition, suppose the following:(H1)There exist constants τ1,τ2>0 such that for each t∈Nα-2,α+T and u,v∈C(22)ft,ut,Ψβut-1-ft,vt,Ψβvt-1≤τ1u-v+τ2Ψβu-Ψβv.H2 There exists a constant μ>0 such that for each u,v∈C(23)yu-yv≤μu-vC.(H3) For each t∈Nα-2,α+T(24)0<gt<K,KT+α-γ2-α-3-T+αΓγ+2ΓT>0.(H4)  Consider  Θ≔μΩ+Λτ1+τ2φ0(T+β+2)β_/Γ(β+1)<1,
where(25)Ω=2+KγT+2+3ΓT+γKT+α-γ2-α-3-T+αΓγ+2ΓT,Λ=KΓT+γ-2TΓα+1ΓT-2KT+α-γ2-α-3-T+αΓγ+2ΓTγ+1TT-1T-2T+α-2+T+αα+γ-3ΓT+α+2ΓT+α+1Γα+1ΓT+1.Then problem (1) has a unique solution on Nα-2,α+T.Proof.
We will show thatF is a contraction. For any u,v∈C and for each t∈Nα-2,α+T, we have(26)Fut-Fvt≤1-t1_T+αyu-yv+t1_T+αAu-Av+t1_T+αΓα∑s=α-1T+α-1T-1-σsα-1_fs,us,Ψβus-fs,vs,Ψβvs-1+1Γα∑s=0t-αt-σsα-1_fs,us,Ψβus-fs,vs,Ψβvs-1<μu-vC1+t1_T+α+t1_∑s=α-2T+α-3s1_gsT+α+γ-3-σsγ-1_-T+αΓγμu-vC∑s=α-2T+α-3gsT+α+γ-3-σsγ-1_1-s1_T+α+τ1+τ2φ0T+β+2β_/Γβ+1u-vCΓα∑s=αT+α-3∑ξ=α-1s-1gsT+α+γ-3-σsγ-1_s+α-1-σξα-1_+τ1+τ2φ0T+β+2β_/Γβ+1u-vCT+αΓα∑s=αT+α-3∑ξ=α-1T+α-1s1_gsT+α+γ-3-σsγ-1_T+2α-1-σξα-1_+t1_τ1+τ2φ0T+β+2β_/Γβ+1u-vCT+αΓα∑s=α-1T+α-1T+2α-1-σsα-1_+τ1+τ2φ0T+β+2β_/Γβ+1u-vCΓα∑s=α-1t-1t+α-1-σsα-1_<2μu-vC+μu-vC2KΓT+γ+3/γΓT+3KT+α-γ2-α-3/γγ+1ΓT-T+αΓγ+τ1+τ2φ0T+β+2β_Γβ+1u-vCKγT+2+3ΓT+γγγ+1ΓT+KΓT+αα+γ-3ΓT+αΓT+γ-2γγ+1Γα+1ΓT-2ΓT+1·KT+α-γ2-α-3γγ+1ΓT-T+αΓγ-1+2τ1+τ2φ0T+β+2β_/Γβ+1u-vCΓT+α+1Γα+1ΓT+1=μu-vC2+KγT+2+3ΓT+γKT+α-γ2-α-3-T+αΓγΓT+τ1+τ2φ0T+β+2β_Γβ+1u-vCKΓT+γ-2TΓα+1ΓT-2KT+α-γ2-α-3-T+αΓγ+2ΓTγ+1TT-1T-2T+α-2+T+αα+γ-3ΓT+α+2ΓT+α+1Γα+1ΓT+1=u-vCμΩ+Λτ1+τ2φ0T+β+2β_Γβ+1=u-vCΘ≤u-vC.
Consequently,F is a contraction. Therefore, by the Banach fixed point theorem, we get that F has a fixed point which is a unique solution of problem (1) on t∈Nα-2,α+T.The following result is based on Schaefer’s fixed point theorem.Theorem 8 (Arzelá-Ascoli Theorem (see [18])).
A set of function inC[a,b] with the sup norm is relatively compact if and only if it is uniformly bounded and equicontinuous on [a,b].Theorem 9 (see [18]).
If a set is closed and relatively compact then it is compact.Theorem 10 (Schaefer’s fixed point theorem (see [19])).
Assume that X is a Banach space and thatT:X→X is continuous compact mapping. Moreover assume that the set(27)⋃0≤λ≤1x∈X:x=λTx is bounded. Then T has a fixed point.Theorem 11.
Assume thatf:Nα-2,T+α×U×U→U is continuous and maps bounded subsets of Nα-2,T+α×U×U into relatively compact subsets of U and y:C→U is a given functional. In addition, suppose that (H3) holds, and suppose the following: (H5)  There exists a constant L1>0 such that for each t∈Nα-2,α+T and u∈C(28)ft,ut,Ψβut-1≤L1.(H6)  There exists a constant L2>0 such that for each u∈C(29)yu≤L2. Then problem (1) has at least one solution on Nα-2,α+T.Proof.
We will use Schaefer’s fixed point theorem to prove this result. LetF be the operator defined in (20). It is clear that F:C→C is completely continuous. So, it remains to show that the set (30)E=u∈C:u=λFuforsome0<λ<1 is bounded.
Letu∈E; then u(t)=λ(Fu)(t) for some 0<λ<1. Thus, for each t∈Nα-2,α+T, we have (31)ut=λFut<Fut≤yu·1-t1_T+α+t1_∑s=α-2T+α-3s1_gsT+α+γ-3-σsγ-1_-T+αΓγ-yu∑s=α-2T+α-3gsT+α+γ-3-σsγ-1_1-s1_T+α-1Γα∑s=αT+α-3∑ξ=α-3-1s-1gsT+α+γ-3-σsγ-1_s+α-1-σξα-1_fξ,uξ,Ψβuξ-1+1T+αΓα∑s=αT+α-3∑ξ=α-1T+α-1s1_gsT+α+γ-3-σsγ-1_T+2α-1-σξα-1_fξ,uξ,Ψβuξ-1+t1_T+αΓα∑s=α-1T+α-1T+2α-1-σsα-1_fs,us,Ψβus-1+1Γα∑s=α-1t-1t+α-1-σsα-1_fs,us,Ψβus-1<2L2+L12ΓT+α+1Γα+1ΓT+1+L2KγT+2+3ΓT+γγγ+1ΓT+L1KγT+2+3ΓT+γγγ+1ΓT+KΓT+αα+γ-3ΓT+αΓT+γ-2γγ+1Γα+1ΓT-2ΓT+1·KT+α-γ2-α-3γγ+1ΓT-T+αΓγ-1=L2Ω+L1Λ, which implies that, for each t∈Nα-2,α+T, we have (32)uC≤L2Ω+L1Λ,where Ω and Λ are defined on (25). This shows that set E is bounded. As a consequence of Schaefer’s fixed point theorem, we conclude that F has a fixed point which is a solution of problem (1).


## 4. Some Examples

In this section, in order to illustrate our results, we consider some examples.Example 1.
Consider the following fractional sum boundary value problem:(33)Δ3/2ut=e-t+1/25t+201/22·u+11+sin2u+∑s=-1t-1t-12-σs-1/2_arctancos2t-3/2πe-2s-t+7/22000πt+19/22us+12,t∈N0,4,u-12=u100e3sin2πu,u112=Δ-11/4u1781000e+200cos2178.
Hereα=3/2, T=4, β=1/2, y(u)=|u|/(100e)3sin2|πu|, γ=11/4, g(t)=1000e+200cos2t, φ(t-1,s+β)=e-2|s-t+4|/2000π, and(34)ft,ut,Ψβut-1=e-t5t+1002·u+11+sin2⁡u+arctan⁡cos2⁡t-2πt+92Δ-1/2φut+12. Let t∈N1/2,9/2; we have(35)ft,ut,Ψ1/2ut-1-ft,vt,Ψ1/2ut-1≤4404010u-v+222527Ψ1/2u-Ψ1/2v,so (H1) holds with τ1=4/404010, τ2=22/2527, and we have φ0=1/2000e8π, and(36)yu-yv=u100e3sin2⁡πu-v100e3sin2⁡πv≤1100e3u-vC,so (H2) holds with μ=1/(100e)3.
Since1000e≤g(t)≤1000e+200=K, we have (37)KT+α-γ2-α-3-T+αΓγ+2ΓT≈547.011>0;then (H3) is satisfied.
Also, we have(38)Ω≈47128.501,Λ≈14297.052.
We can show that(39)μΩ+τ1+τ2φ0T+β+2β_Γβ+1Λ+2ΓT+α+1Γα+1ΓT+1=1100e347128.501+14297.0524404010+22252713/21/2_2000e8πΓ3/2≈0.144<1.Hence, by Theorem 7, boundary value problem (33) has a unique solution.Example 2.
Consider the following fractional sum boundary value problem:(40)Δ3/2ut=t+1/21/2e-3t+1/21+t+1/21+cos2⁡u+π+∑s=-1t-1t-12-σs-1/2_t+3/2s+1/2e-s+1/22πt+3/2s+1/2-t+1/2us+12,t∈N0,3,u-12=u2+21-u2+2_π+u2,u92=Δ-8/3u11612e+sin⁡1162.
Hereα=3/2, T=3, β=1/2, y(u)=|u2+2|1-|u2+2|_/π+u2, γ=8/3 and g(t)=12e+sint2, φ(t-1,s+β)=t+1s+1/2e-s+1/22/t+1s+1/2-t, and (41)ft,ut,Ψβut-1=te-3t1+t1+cos2⁡u+π+∑s=-1t-3/2t-1-σs-1/2_φt-1,s+12us+12.
Clearly fort∈N1/2,7/2, we have (42)φ0<t+1s+1/2-t+tt+1s+1/2-t<1+1s+1/2-1≤3,ft,ut,Ψ1/2ut≤t1+t+φ09/2-1/2_Γ3/2<2.566=L1,yu=Γu2+2+1π+u2Γ2u2+2<1π=L2,12e2≤gt≤12e+12=K,1612e+12-92Γ143Γ3=55.974>0.Hence, conditions (H3), (H5), and (H6) of Theorem 11 are satisfied, and consequently boundary value problem (40) has at least one solution.



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