Multiple Positive Solutions for Nonlinear Fractional Differential Equations with Integral Boundary Value Conditions and a Parameter

Juan Mao,Daliang Zhao

doi:10.1155/2019/2787569

Abstract

By using the theory of fixed-point index on cone for differentiable operators, spectral radii of some related linear integral operators, and properties of Green’s function, the existence of multiple positive solutions to a nonlinear fractional differential system with integral boundary value conditions and a parameter is established. At last, some examples are also provided to illustrate the validity of our main results.

Highlights

In recent years, the interest in the study of fractional differential equations has been growing rapidly since it has many applications in biology, mechanics, electrochemistry, and dynamical processes in self-similar structures, etc.; see [1,2,3], for instance
Fractional differential equations serve as an excellent tool for the description of hereditary properties of various materials and processes
In [15], Yang studied a class of fractional derivatives of constant and variable orders for the first time

Summary

Introduction

The interest in the study of fractional differential equations has been growing rapidly since it has many applications in biology, mechanics, electrochemistry, and dynamical processes in self-similar structures, etc.; see [1,2,3], for instance. By deriving properties of the Green’s function and by means of Guo-Krasnoselskii’s fixedpoint theorem, the author established some existence results of at least one positive solution provided that f(t, x) satisfies some growth conditions. Motivated by all the above-mentioned works, we aim to establish some existence criteria of multiple positive solutions for the following nonlinear fractional BVP with integral boundary conditions and a parameter: Dα0+ u (t) + f (t, u (t)) = 0, t ∈ (0, 1) , u(j) (0) = 0, 0 ≤ j ≤ 3, j ≠ 1, (2). By using the differential operator method, spectrum theory, and the properties of Green’s function, we firstly investigate some existence results of multiple positive solutions for the considered fractional system.

Preliminaries and Some Lemmas

Main Results

Examples