Monotone iterative method for nonlinear fractional p‐Laplacian differential equation in terms of ψ‐Caputo fractional derivative equipped with a new class of nonlinear boundary conditions

Abstract

The aim of this paper is to deal with the existence of extremal solutions for a novel class of nonlinear fractional p‐Laplacian differential equation in terms of ψ‐Caputo fractional derivative equipped with a new class of nonlinear boundary conditions. Initially, we focus on the linear problem and we give an explicit form of the solutions, from which we derive new comparison results which is crucial for the proof of the main theorem of this paper. Secondly, with the help of the monotone iterative method along with its accompanying upper and lower solutions, the existence of extremal solutions for the aforementioned problem is investigated. Then, based on the proposed method, we can construct two types of successive approximations which converge uniformly monotonically to minimal and maximal solutions of the given problem. Finally, we illustrate our results with an example.