Boundary value problems for fractional difference equations with three-point fractional sum boundary conditions

Discrete Dynamics in Nature and Society

Tariboon²,

Ntouyas³

2013

Self Cite

We consider a discrete fractional boundary value problem of the formΔαu(t)=f(t+α-1,u(t+α-1)), t∈[0,T]ℕ0:={0,1,…,T}, u(α-2)=0, u(α+T)=Δ-βu(η+β),where1<α≤2,β>0,η∈[α-2,α+T-1]ℕα-2:={α-2,α-1,…,α+T-1}, andf:[α-1,α,…,α+T-1]ℕα-1×ℝ→ℝis a continuous function. The existence of at least one solution is proved by using Krasnoselskii's fixed point theorem and Leray-Schauder's nonlinear alternative. Some illustrative examples are also presented.

Section: Introductionsupporting

confidence: 86%

Section: Introductionmentioning

confidence: 99%

Existence Results for Fractional Difference Equations with Three-Point Fractional Sum Boundary Conditions

Discrete Dynamics in Nature and Society

Tariboon²,

Ntouyas³

2013

Self Cite

“…Some real-world phenomena are being studied with the help of discrete fractional operators. A good account of papers dealing with discrete fractional boundary value problems can be found in [1][2][3][4][5][6][7][8][9][10][11][12][13] and references cited therein.…”

Section: Introductionmentioning

confidence: 99%

Boundary value problem forp−Laplacian Caputo fractional difference equations with fractional sum boundary conditions

2015

Math. Meth. Appl. Sci.

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In this paper, we consider a discrete fractional boundary value problem of the form 8 < :where 0 <˛,ˇÄ 1, 1 <˛CˇÄ 2, 0 < Ä 1, Á 2 OE˛Cˇ 1, T C˛Cˇ 1 N˛Cˇ 1 , is a constant, C and Č denote the Caputo fractional differences of order˛andˇ, respectively, f : OE˛Cˇ 2, T C˛Cˇ N˛Cˇ 2 R ! R is a continuous function, and p is the p-Laplacian operator. The existence of at least one solution is proved by using Banach fixed point theorem and Schaefer's fixed point theorem. Some illustrative examples are also presented.

“…Sitthiwirattham et al . in – considered a discrete fractional boundary value problem of the form:

({, \begin{matrix} Δ^{α} x (t) = f (t + α - 1, x (t + α - 1)), t \in N_{0, T}, \\ x (α - 2) = 0, x (α + T) = Δ^{- β} x (η + β), \end{matrix})

where 1 < α ≤2,

f : ({double-struckN}_{α - 1, α + T - 1} \times double-struckR) \to double-struckR

is a continuous function. Existence and uniqueness of solutions are obtained by the contraction mapping theorem, the nonlinear contraction theorem, Schaefer's fixed point theorem, Krasnoselskii's fixed point theorem, and Leray–SchauderŠs nonlinear alternative.…”

Section: Introductionmentioning

confidence: 99%

“…Sitthiwirattham et al in [4]- [5] considered a discrete fractional boundary value problem of the form:…”

Section: Introductionmentioning

confidence: 99%

Existence and uniqueness of solutions of sequential nonlinear fractional difference equations with three‐point fractional sum boundary conditions

Math Methods in App Sciences

2014

Self Cite

In this paper, we consider a discrete fractional boundary value problem of the form:where 0 <˛,ˇÄ 1, 1 <˛CˇÄ 2, and are constants, > 0, Á 2 N˛Cˇ 1,˛CˇCT 1 :D f˛Cˇ 2,˛Cˇ 1, : : : ,˛Č CT 2,˛CˇCT 1g, f : N˛Cˇ 1,˛CˇCT 1 R ! R is a continuous function, and Eˇx.t/ D x.t Cˇ 1/. The existence and uniqueness of solutions are proved by using Banach's fixed point theorem. An illustrative example is also presented.