Infinitely many solutions for nonlinear fractional boundary value problems via variational methods

Guoqing Chai

doi:10.1186/s13662-016-0917-0

Abstract

In this paper, the author considers the following nonlinear fractional boundary value problem: $$\left \{ \textstyle\begin{array}{l} \frac{d}{dt} ( {\frac{1}{2}{ }_{0}D_{t}^{-\beta} ({u}'(t))+\frac{1}{2}{ }_{t}D_{T}^{-\beta} ({u}'(t))} )+\nabla F(t,u(t))=0, \quad\mbox{a.e. } t\in[0,T], u(0)=u(T)=0, \end{array}\displaystyle \right . $$ where ${ }_{0}D_{t}^{-\beta} $ and ${ }_{t}D_{T}^{-\beta} $ are the left and right Riemann-Liouville fractional integrals of order $0\le\beta<1$ , respectively, $\nabla F(t,x)$ is the gradient of F at x. By applying the variant fountain theorems, the author obtains the existence of infinitely many small or high energy solutions to the above boundary value problem.

Highlights

Fractional calculus has applications in many areas, including fluid flow, electrical networks, probability and statistics, chemical physics and signal processing and so on; see [ – ] and the references therein
There have been many papers dealing with the existence of solutions of nonlinear initial value problems of fractional equations by applying nonlinear analysis such as fixed point theorems, lower and upper solution method, monotone iterative method, coincidence degree theory
Up to now, there are few results on the solutions to fractional boundary value problems that are established by the variational methods; see for example, [ – ]

Summary

Introduction

Fractional calculus has applications in many areas, including fluid flow, electrical networks, probability and statistics, chemical physics and signal processing and so on; see [ – ] and the references therein. Jiao and Zhou [ ] were first to show that the critical point theory is an effective approach to track the existence of solutions to the following fractional boundary value problem (BVP for short): D–t β In [ ], by applying critical point theorems, Li, Sun and Zhang studied the existence of solutions to the following problem: D–t β λu(t). By the critical point theory, Jiao and Zhou in [ ] established the existence of solutions for the following fractional boundary value problem: tDαT ( Dαt u(t)) = ∇F(t, u(t)), a.e. t ∈ [ , ], u( ) = u(T) =. Different from the work mentioned above, in this paper, the author attempts to apply the variant fountain theorems to study the existence of infinitely many small or high energy solutions to BVP In Section , the author establishes the existence of infinitely many small or high energy solutions for BVP ( . ) and gives two examples to show the effectiveness of the results obtained

Preliminaries

Conclusion