Existence of solutions for fractional differential equations with infinite point boundary conditions at resonance

Wei Zhang,Wenbin Liu

doi:10.1186/s13661-018-0954-6

Abstract

In this paper, by using Mawhin’s continuation theorem, we establish some sufficient conditions for the existence of at least one solution for a class of fractional infinite point boundary value problem at resonance. Moreover, an example is given to illustrate our results.

Highlights

In the past 20 years, fractional differential equations (FDEs for short) as mathematical models have been successfully used in many fields of science and engineering, which have attracted considerable attention in studying FDEs
Even though in the papers [21, 22] and [23] authors have investigated the resonance case with dimKer L = 2, all of them considered with the coupled fractional differential equations, in which the linear operator L defined as L(x, y) = (L1x, L2y) and dimKer L1 = dimKer L2 = 1
We study the one fractional differential equation resonance case with dimKer L = 2, which is obviously different from papers [21, 22] and [23]

Summary

Introduction

In the past 20 years, fractional differential equations (FDEs for short) as mathematical models have been successfully used in many fields of science and engineering (see [1,2,3,4,5,6,7]), which have attracted considerable attention in studying FDEs. In [18, 19] Zhang and Zhai et al considered the following fractional differential equation with infinite point BCs:. Motivated by the results mentioned, the purpose of this paper is to present the existence of solutions for the following infinite point BVPs by applying Mawhin’s continuous theorem:. Even though in the papers [21, 22] and [23] authors have investigated the resonance case with dimKer L = 2, all of them considered with the coupled fractional differential equations, in which the linear operator L defined as L(x, y) = (L1x, L2y) and dimKer L1 = dimKer L2 = 1. Definition 2.2 The Riemann–Liouville fractional derivative of order α > 0 for a function x : (0, +∞) → R is given by.

Main result Let

Conclusion