Method of Upper and Lower Solutions for Nonlinear Caputo Fractional Difference Equations and Its Applications

Abstract

In this paper, we extend the applications of the method of upper and lower solutions for a class of nonlinear nabla fractional difference equations involving Caputo derivative. We obtain the existence of coupled minimal and maximal solutions which constructed by two monotone sequences. In order to illustrate our main results, we present two numerical examples in the end.

Initial Value Problem For Nonlinear Fractional Differential Equations With ψ-Caputo Derivative Via Monotone Iterative Technique

“…At the present day, different kinds of fixed point theorems are widely used as fundamental tools in order to prove the existence and uniqueness of solutions for various classes of nonlinear fractional differential equations for details, we refer the reader to a series of papers [24][25][26][27][28][29][30] and the references therein, but here we focus on those using the monotone iterative technique, coupled with the method of upper and lower solutions. This method is a very useful tool for proving the existence and approximation of solutions to many applied problems of nonlinear differential equations and integral equations (see [31][32][33][34][35][36][37][38][39][40][41][42]). However, as far as we know, there is no work yet reported on the existence of extremal solutions for the Cauchy problem with ψ-Caputo fractional derivative.…”

Monotone Iterative Method for ψ-Caputo Fractional Differential Equation with Nonlinear Boundary Conditions

“…To the best of our observations, the monotone iterative technique combined with the method of upper and lower solutions has not been used to study the existence of solutions for ψ-Caputo fractional differential equation with nonlinear boundary conditions. For more expository details on monotone iterative method, the readers can consult some interesting research works [31][32][33][34][35].…”

Existence, uniqueness, approximation of solutions and Ealpha-Ulam stability results for a class of nonlinear fractional differential equations involving psi-Caputo derivative with initial conditions

“…In recent years, there are several contributions focusing on fractional differential equations, mainly on the existence, uniqueness, and stability of solutions. For more details, the readers are referred to the previous studies [4,5,6,7,9,15,17,18,19,20,21,22,23,24,35,36,41,43,49,50,51,52,53,54,55] and the references therein. The main techniques used in these studies are fixed-point techniques, Leray-Schauder theory, coincidence theory, or monotone iterative technique combined with the method of upper and lower solutions.…”