Application of Fixed-Point Index Theory for a Nonlinear Fractional Boundary Value Problem with an Advanced Argument

Li Wu,Chuanzhi Bai,Bo Yang

doi:10.1155/2021/6781541

Abstract

In this paper, we investigate the existence of positive solutions of a class of fractional three-point boundary value problem with an advanced argument by using fixed-point index theory. Our results improve and extend some known results in the literature. Two examples are given to demonstrate the effectiveness of our results.

Highlights

In [3], due to the well-known Guo–Krasnoselskii fixedpoint theorem [20], Wang et al proved the existence of a positive solution to the following three-point BVPs of nonlinear fractional-order differential equation with an advanced argument:
E paper is organized in this order
A function u is a solution of the following boundary value problem: CDαu(t) + g(t) 0, 0 < t < 1, (8)

Summary

Introduction

In [3], due to the well-known Guo–Krasnoselskii fixedpoint theorem [20], Wang et al proved the existence of a positive solution to the following three-point BVPs of nonlinear fractional-order differential equation with an advanced argument:. In [2], Ma considered the following three-point boundary value problem: E existence of at least one positive solution for BVP (1) is obtained by virtue of the Krasnosel’skii fixed-point theorem. ⎩ u(0) u′′(0) 0, (2) u(1) βu(η), where 2 < α ≤ 3, η ∈ (0, 1), 0 < β < (1/η), CDα is the Caputo fractional derivative, and f: [0, ∞) ⟶ [0, ∞) is a continuous function.

Results

Conclusion