MPE Mathematical Problems in Engineering 1563-5147 1024-123X Hindawi Publishing Corporation 10.1155/2015/519072 519072 Research Article On Nonlinear Fractional Sum-Difference Equations via Fractional Sum Boundary Conditions Involving Different Orders Chasreechai Saowaluk Kiataramkul Chanakarn http://orcid.org/0000-0002-8455-1402 Sitthiwirattham Thanin Ntouyas Sotiris K. Nonlinear Dynamic Analysis Research Center Department of Mathematics Faculty of Applied Science King Mongkut’s University of Technology North Bangkok Bangkok 10800 Thailand kmutnb.ac.th 2015 17112015 2015 13 07 2015 03 09 2015 06 09 2015 17112015 2015 Copyright © 2015 Saowaluk Chasreechai et al. 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 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.

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,tN0,T0,1,,T,uα-2=yu,uT+α=Δ-γgT+α+γ-3uT+α+γ-3,where 1<α2, 0<β1, 2<γ3, and ΔCα is the Caputo fractional difference operator of order α. For UR, gC(Nα-2,T+α,R+U) and fC(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)ΨβutΔ-βφut+β=1Γβs=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  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.  investigated the existence of solutions for two fractional boundary value problems:(3)Δμ-2μxt=gt+μ-1,xt+μ-1,Δxt+μ-1,tN0,b+2,xμ-2=0,xμ+b+1=k=μ-1αxk,where 1<μ2 and gC(Nμ-1,μ+b+1×R×R,R) is a given function, and (4)Δμ-3μxt=gt+μ-2,xt+μ-2,tN0,b+3,xμ-3=xμ+b+1=0,xα=k=γβxk,where 2<μ3, α,β,γNμ-2,μ+b, and gC(Nμ-2,μ+b+1×R×R,R) is a given function.

Kang et al.  obtained sufficient conditions for the existence of positive solutions for a nonlocal boundary value problem(5)-Δμyt=λht+μ-1fyt+μ-1,tN0,b,yμ-2=Ψy,yμ+b=Φy,where 1<μ2, fC(0,,[0,)), hC(Nμ-1,μ+b-1,[0,)) are given functions and Ψ,Φ:Rb+3R are given functionals.

Sitthiwirattham  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,tN0,T,ΔCβxα-1=0,xα+β+T=ρΔ-γxη+γ,where 0<α, β1, 1<α+β2, 0<γ1, ηNα+β-1,α+β+T-1, ρ is a constant, f:Nα+β-2,α+β+T×RR is a continuous function, and ϕp is the p-Laplacian operator.

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 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 by tα_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 [<xref ref-type="bibr" rid="B11">13</xref>]).

Assume that the following factorial functions are well defined:

(t-μ)tμ_=tμ+1_, where μR.

If tr, then tα_rα_ for any α>0.

tα+β_=t-βα_tβ_.

Definition 3.

For α>0 and f defined on Na{a,a+1,}, the α-order fractional sum of f is defined by (7)Δ-αft1Γαs=at-αt-σsα-1_fs,where tNa+α and σ(s)=s+1.

Definition 4.

For α>0 and f defined on Na, the α-order Caputo fractional difference of f is defined by (8)ΔCαftΔ-N-αΔNft=1ΓN-αs=at-N-αt-σsN-α-1_ΔNfs,where tNa+N-α and NN is chosen so that 0N-1<α<N. If α=N, then ΔCαf(t)=ΔNf(t).

Lemma 5 (see [<xref ref-type="bibr" rid="B7">9</xref>]).

Assume that α>0 and 0N-1<αN. Then (9)Δ-αΔCαyt=yt+C0+C1t1_+C2t2_++CN-1tN-1_,for some CiR, 0iN-1.

The following lemma deals with linear variant of boundary value problem (1) and gives a representation of the solution.

Lemma 6.

Let 1<α2, 2<γ3, y:C(Nα-2,T+α,U)U, and hC(Nα-1,T+α-1,U) be given. Then the problem(10)ΔCαut=ht+α-1,tN0,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+αΓαs=α-1T+α-1T+2α-1-σsα-1_hs+1Γαs=α-1t-1t+α-1-σsα-1_hs,where(12)Au=-yus=α-2T+α-3gsT+α+γ-3+σsγ-1_1-s1_/T+α1/T+αs=α-2T+α-3s1_gsT+α+γ-3+σsγ-1_-Γγ-1/Γαs=αT+α-3ξ=α-1s-1gsT+α+γ-3+σsγ-1_s+α-1-σξα-1_hξ1/T+αs=α-2T+α-3s1_gsT+α+γ-3+σsγ-1_-Γγ+1/Γαs=αT+α-3ξ=α-1T+α-1gss1_T+α+γ-3+σsγ-1_T+2α-1-σξα-1_hξs=α-2T+α-3s1_gsT+α+γ-3+σsγ-1_-Γγ.

Proof.

Using Lemma 5, a general solution for (10) can be written in the form(13)ut=C0+C1t1_+1Γαs=0t-αt-σsα-1_hs+α-1,for tNα-2,α+T. Applying the first boundary condition of (10) implies(14)C0=yu. So,(15)ut=yu+C1t1_+1Γαs=0t-αt-σsα-1_hs+α-1.The second condition of (10) implies (16)uT+α=yu+C1T+α+1Γαs=0TT+α-σsα-1_hs+α-1=1Γγs=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+αΓγs=α-2T+α-3T+α+γ-3-σsγ-1_gsus-yuT+α-1T+αΓαs=0TT+α-σsα-1_hs+α-1.Substituting a constant C1 into (15), we get(18)ut=1-t1_T+αyu+t1_T+α1Γγs=α-2T+α-3T+α+γ-3-σsγ-1_gsus-1Γαs=α-1T+α-1T+2α-1-σsα-1_hs+1Γαs=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=1Γγs=α-2T+α-3gsT+α+γ-3-σsγ-1_1-s1_T+αyu+s1_T+αAu-1Γαξ=α-1T+α-1T+2α-1-σξα-1_hξ+1Γαξ=α-1s-1s+α-1-σξα-1_hξ,we simplify (19) becomes (12).

Substituting A(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.

For UR, 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+αU endowed with a topology of uniform convergence with the norm denoted by ·C. For this purpose, we consider the operator F:CC by(20)Fut=1-t1_T+αyu+t1_T+αAu-t1_T+αΓαs=α-1T+α-1T+2α-1-σsα-1_fs,us,Ψβus-1+1Γαs=α-1t-1t+α-1-σsα-1_fs,us,Ψβus-1,where(21)Au=11/T+αs=α-2T+α-3s1_gsT+α+γ-3-σsγ-1_-Γγ-yus=α-2T+α-3gsT+α+γ-3-σsγ-1_1-s1_T+α-1Γαs=αT+α-3ξ=α-1s-1gsT+α+γ-3-σsγ-1_s+α-1-σξα-1_fξ,uξ,Ψβuξ-1+1T+αΓαs=αT+α-3ξ=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 that f:Nα-2,T+α×U×UU 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:CU is a given functional. In addition, suppose the following:

(H1)There exist constants τ1,τ2>0 such that for each tNα-2,α+T and u,vC(22)ft,ut,Ψβut-1-ft,vt,Ψβvt-1τ1u-v+τ2Ψβu-Ψβv.

H2 There exists a constant μ>0 such that for each u,vC(23)yu-yvμu-vC.

(H3) For each tNα-2,α+T(24)0<gt<K,KT+α-γ2-α-3-T+αΓγ+2ΓT>0.

(H4)  Consider  ΘμΩ+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 that F is a contraction. For any u,vC and for each tNα-2,α+T, we have(26)Fut-Fvt1-t1_T+αyu-yv+t1_T+αAu-Av+t1_T+αΓαs=α-1T+α-1T-1-σsα-1_fs,us,Ψβus-fs,vs,Ψβvs-1+1Γαs=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-vCs=α-2T+α-3gsT+α+γ-3-σsγ-1_1-s1_T+α+τ1+τ2φ0T+β+2β_/Γβ+1u-vCΓαs=αT+α-3ξ=α-1s-1gsT+α+γ-3-σsγ-1_s+α-1-σξα-1_+τ1+τ2φ0T+β+2β_/Γβ+1u-vCT+αΓαs=αT+α-3ξ=α-1T+α-1s1_gsT+α+γ-3-σsγ-1_T+2α-1-σξα-1_+t1_τ1+τ2φ0T+β+2β_/Γβ+1u-vCT+αΓαs=α-1T+α-1T+2α-1-σsα-1_+τ1+τ2φ0T+β+2β_/Γβ+1u-vCΓαs=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-vCΘu-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 tNα-2,α+T.

The following result is based on Schaefer’s fixed point theorem.

Theorem 8 (Arzelá-Ascoli Theorem (see [<xref ref-type="bibr" rid="B18">18</xref>])).

A set of function in C[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 [<xref ref-type="bibr" rid="B18">18</xref>]).

If a set is closed and relatively compact then it is compact.

Theorem 10 (Schaefer’s fixed point theorem (see [<xref ref-type="bibr" rid="B19">19</xref>])).

Assume that X is a Banach space and that T:XX is continuous compact mapping. Moreover assume that the set(27)0λ1xX:x=λTx is bounded. Then T has a fixed point.

Theorem 11.

Assume that f:Nα-2,T+α×U×UU is continuous and maps bounded subsets of Nα-2,T+α×U×U into relatively compact subsets of U and y:CU 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 tNα-2,α+T and uC(28)ft,ut,Ψβut-1L1.

(H6)  There exists a constant L2>0 such that for each uC(29)yuL2.

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. Let F be the operator defined in (20). It is clear that F:CC is completely continuous. So, it remains to show that the set (30)E=uC:u=λFuforsome0<λ<1 is bounded.

Let uE; then u(t)=λ(Fu)(t) for some 0<λ<1. Thus, for each tNα-2,α+T, we have (31)ut=λFut<Futyu·1-t1_T+α+t1_s=α-2T+α-3s1_gsT+α+γ-3-σsγ-1_-T+αΓγ-yus=α-2T+α-3gsT+α+γ-3-σsγ-1_1-s1_T+α-1Γαs=αT+α-3ξ=α-3-1s-1gsT+α+γ-3-σsγ-1_s+α-1-σξα-1_fξ,uξ,Ψβuξ-1+1T+αΓαs=αT+α-3ξ=α-1T+α-1s1_gsT+α+γ-3-σsγ-1_T+2α-1-σξα-1_fξ,uξ,Ψβuξ-1+t1_T+αΓαs=α-1T+α-1T+2α-1-σsα-1_fs,us,Ψβus-1+1Γαs=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 tNα-2,α+T, we have (32)uCL2Ω+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,tN0,4,u-12=u100e3sin2πu,u112=Δ-11/4u1781000e+200cos2178.

Here α=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+sin2u+arctancos2t-2πt+92Δ-1/2φut+12. Let tN1/2,9/2; we have(35)ft,ut,Ψ1/2ut-1-ft,vt,Ψ1/2ut-14404010u-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=u100e3sin2πu-v100e3sin2πv1100e3u-vC,so (H2) holds with μ=1/(100e)3.

Since 1000eg(t)1000e+200=K, we have (37)KT+α-γ2-α-3-T+αΓγ+2ΓT547.011>0;then (H3) is satisfied.

Also, we have (38)Ω47128.501,Λ14297.052.

We can show that (39)μΩ+τ1+τ2φ0T+β+2β_Γβ+1Λ+2ΓT+α+1Γα+1ΓT+1=1100e347128.501+14297.0524404010+22252713/21/2_2000e8πΓ3/20.144<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+cos2u+π+s=-1t-1t-12-σs-1/2_t+3/2s+1/2e-s+1/22πt+3/2s+1/2-t+1/2us+12,tN0,3,u-12=u2+21-u2+2_π+u2,u92=Δ-8/3u11612e+sin1162.

Here α=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+cos2u+π+s=-1t-3/2t-1-σs-1/2_φt-1,s+12us+12.

Clearly for tN1/2,7/2, we have (42)φ0<t+1s+1/2-t+tt+1s+1/2-t<1+1s+1/2-13,ft,ut,Ψ1/2utt1+t+φ09/2-1/2_Γ3/2<2.566=L1,yu=Γu2+2+1π+u2Γ2u2+2<1π=L2,12e2gt12e+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.

Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

Acknowledgments

The authors would like to thank the editor and the referees for their useful comments. This research was funded by King Mongkut’s University of Technology North Bangkok (Contract no. KMUTNB-GOV-58-50).

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