Existence of solutions for a class of nonlinear fractional difference equations of the Riemann–Liouville type

Abstract

Nonlinear fractional difference equations are studied deeply and extensively by many scientists by using fixed-point theorems on different types of function spaces. In this study, we combine fixed-point theory with a set of falling fractional functions in a Banach space to prove the existence and uniqueness of solutions of a class of fractional difference equations. The most important part of this article is devoted to correcting a significant mistake made in the literature in using the power rule by providing further conditions for its validity. Also, we provide specific conditions under which difference equations have attractive solutions and the solutions are also asymptotically stable. Furthermore, we construct some fractional difference examples in order to illustrate the validity of the observed results.

A nonlinear fractional partial integro‐differential equation with nonlocal initial value conditions

“…We should note that this is indeed Abel's equation of the second kind. There are some recent studies on several general classes of fractional-order kinetic equations using various approaches [6,[10][11][12]. In addition, researchers have investigated the existence and uniqueness for various types of fractional integro-differential equations, with boundary conditions and the Hilfer derivative, via different methods [13][14][15][16].…”

New Algorithms for Dealing with Fractional Initial Value Problems

“…For instance, in [6], the authors studied the numerical performances of the fractional-order mathematical model based on the Majnun and Layla romantic story. In [7], the authors combined fixed-point theory with a set of falling fractional functions in a Banach space to prove the existence and uniqueness of solutions of a class of fractional difference equations. In [8], the authors introduced the distributed-order time fractional Klein-Gordon-Zakharov system by substituting the second-order temporal derivative with a distributed-order fractional derivative.…”