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

Abstract

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.

The relationship between sequential fractional differences and convexity

“…In the case where 0 < ν < 1 a recent paper by Atici and Uyanik [11] provides some additional significant contributions regarding the connection between the fractional difference and the monotonicity of functions. More generally, there has been a growing and broadening interest in the discrete fractional calculus over the past 10 years or so, beginning with the initial investigations of Atici and Eloe [5,6,7,8,9], and continuing in a variety of directions such as operational properties of fractional differences [1,2,3,4,20,31,42], Laplace transforms [14], fractional boundary value problems [15,21,22,24,26,28,38,39,40], extensions to other time scales such as q Z [14,18,19,23], asymptotic behavior of solutions to fractional initial value problems [35,36,37], chaotic dynamics of fractional-order dynamical systems [41], and applications to modeling in the biological sciences [10]; one may also consult the book by Goodrich and Peterson [30] for a broad overview of these and other related topics.…”

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

“…Theorem 3.3 Suppose 1 < p < 2, t.Á C ˛ ˇC 3/ ¤ . C 1/.Á ˛ ˇC 3/ and the following condition holds: .H 3 / There exists a nonnegative function g 2 C OE˛Cˇ 2,˛CˇC T N˛Cˇ 2 and m :D 1 7) and there exists a constant k with 0…”

“…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.…”

On Separate Fractional Sum-Difference Equations with n-Point Fractional Sum-Difference Boundary Conditions via Arbitrary Different Fractional Orders

“…Weidong [12] examined the sequential fractional boundary value problem with a p-Laplacian Recently, Sitthiwirattham [19,20] investigated three-point fractional sum boundary value problems for sequential fractional difference equations of the forms…”