Continuity of solutions to discrete fractional initial value problems

Abstract

a b s t r a c tIn this paper, we consider a fractional initial value problem (IVP) in the case where the order ν of the fractional difference satisfies 0 < ν ≤ 1. We show that solutions of this IVP satisfy a continuity condition both with respect to the order of the difference, ν, and with respect to the initial conditions, and we deduce several important corollaries from this theorem. Thus, we address a complication that arises in the fractional case but not in the classical (integer-order) case.

Multiplicity and uniqueness for a class of discrete fractional boundary value problems

“…Then a number of papers appeared investigating the discrete fractional boundary value problems, such as [1], [6], [12], [16], [17], [18], [19], [20], [23], [21], [24], [15], [26], [31], [33], [5], [14], [22].…”

Existence of a positive solution to a first-order -Laplacian BVP on a time scale

“…There has recently been a considerable amount of research conducted on both discrete and continuous fractional boundary value problems and, more generally, fractional difference and differential equations and certain of their properties; see, for example, [31][32][33][34][35][36][37][38][39][40][41][42][43][44][45][46] and the references therein. In many cases, fractional differential equations provide for more realistic models of physical phenomena.…”

Analysis of discrete fractional operators

“…In this time period, the theory of discrete fractional calculus has been developed in many directions parallel to the theory of fractional calculus such as fractional difference equations, discrete Mittag-Leffler functions, inequalities with discrete fractional operators, see [1][2][3][4][5][6][7][8][9][10][11][12][13][14][15][16][17][18] and the references therein.…”