Multiple Solutions for a Nonlinear Fractional Boundary Value Problem via Critical Point Theory

Yang Wang,Yujun Cui,Yansheng Liu

doi:10.1155/2017/8548975

Abstract

This paper is concerned with the existence of multiple solutions for the following nonlinear fractional boundary value problem: DT-αaxD0+αux=fx,ux, x∈0,T, u0=uT=0, where α∈1/2,1, ax∈L∞0,T with a0=ess infx∈0,Tax>0, DT-α and D0+α stand for the left and right Riemann-Liouville fractional derivatives of order α, respectively, and f:0,T×R→R is continuous. The existence of infinitely many nontrivial high or small energy solutions is obtained by using variant fountain theorems.

Highlights

Consider the following nonlinear fractional boundary value problem (BVP for short): DTα− (a (x) D0α+ u (x)) = f (x, u (x)), x ∈ [0, T], (1)u (0) = u (T) = 0, where α ∈ (1/2, 1], a(x) ess inf x∈[0,T]a(x) > 0, DTα− and ∈ D0α+L∞[0, T] with a0 stand for the left= and right Riemann-Liouville fractional derivatives of order α, respectively, and f : [0, T] × R → R is continuous.Fractional calculus has numerous applications in various fields, including signal processing and control, fractal theory, neural network model, mechanics and engineering, and chemical physics
The existence of solutions for nonlinear fractional differential equations was obtained by using nonlinear analysis, including some fixed theorems, coincidence degree theory, the monotone iterative method, and the method of upper-lower solutions
It is worth mentioning that the methods above cannot be used to deal with BVP (1) because it is very difficult to find the equivalent integral equation

Summary

Introduction

Variational methods have proved to be a very effective approach in dealing with the existence and multiple solutions for fractional boundary value problem including both left and right fractional derivatives; see [6,7,8,9,10,11,12,13,14,15,16]. Inspired by the above references, we apply variant fountain theorems to study the existence of infinitely many small or high energy solutions for BVP (1).

Preliminaries

Main Results

Examples