Iterative Analysis of the Unique Positive Solution for a Class of Singular Nonlinear Boundary Value Problems Involving Two Types of Fractional Derivatives with p-Laplacian Operator

Abstract

This article is concerned with a class of singular nonlinear fractional boundary value problems with p-Laplacian operator, which contains Riemann–Liouville fractional derivative and Caputo fractional derivative. The boundary conditions are made up of two kinds of Riemann–Stieltjes integral boundary conditions and nonlocal infinite-point boundary conditions, and different fractional orders are involved in the boundary conditions and the nonlinear term, respectively. Based on the method of reducing the order of fractional derivative, some properties of the corresponding Green’s function, and the fixed point theorem of mixed monotone operator, an interesting result on the iterative sequence of the uniqueness of positive solutions is obtained under the assumption that the nonlinear term may be singular in regard to both the time variable and space variables. And we obtain the dependence of solution upon parameter. Furthermore, two numerical examples are presented to illustrate the application of our main results.