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

Nontrivial solutions for boundary value problems of a fourth order difference equation with sign-changing nonlinearity

“…In this paper, we established the existence of nontrivial solutions for the boundary value problems of the fourth order difference equation (1.1) with sign-changing nonlinearity using the topological degree theory. Under some conditions concerning the first eigenvalue corresponding to the relevant linear problem, the results here improve and generalize those obtained in [1][2][3][4][5][6][7][8][9][10][11].…”

“…, T} and R = (-∞, +∞) (it is assumed to be continuous from the topological space T T 2 ×R into the topological space R, the topology on T T 2 being the discrete topology). Difference equations with discrete boundary value conditions have been widely studied in the literature; see, for example, [1][2][3][4][5][6][7][8][9][10][11] and the references therein. However, as mentioned in [6], very few results are available with sign-changing nonlinearities; see [6][7][8][9][10][11].…”

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.…”

Positive solutions for a class of fractional difference systems with coupled boundary conditions

“…They used the Guo-Krasnosel'skii fixed point theorem to obtain the existence of positive solutions for (1.3), and they also presented intervals for parameters λ and μ for the positive solutions. However, as is mentioned by Christopher S. Goodrich in [28], there has been little work done in fractional difference equations, we only refer to [29][30][31][32][33][34][35][36][37][38][39][40][41][42][43]. For example, in [29] the authors studied discrete fractional calculus and offered some important properties of the fractional sum and the fractional difference operators.…”