Solutions for Impulsive Fractional Differential Equations via Variational Methods

Peiluan Li,Zheqing Li,Hui Wang

doi:10.1155/2016/2941368

Abstract

We investigate the boundary value problems of impulsive fractional order differential equations. First, we obtain the existence of at least one solution by the minimization result of Mawhin and Willem. Then by the variational methods and a very recent critical points theorem of Bonanno and Marano, the existence results of at least triple solutions are established. At last, two examples are offered to demonstrate the application of our main results.

Highlights

Fractional differential equations have been an area of great interest recently
The left and right Riemann-Liouville fractional derivatives of order α for function f denoted by tDbα f(t) function, respectively, are defined by aDtα f(t) and aDtα f (t) dn dtn αDtα−n f (t)
In order to establish a variational structure which enables us to reduce the existence of solution of problem (10) to existence of the critical point of corresponding functional, we construct the following appropriate function spaces

Summary

Introduction

Fractional differential equations have been an area of great interest recently. This is because of both the intensive development of the theory of fractional calculus itself and the applications of such constructions in various scientific fields such as physics, mechanics, chemistry, and engineering; see [1,2,3] and the references therein. Zhou [8] first considered the following fractional boundary value problems: tDαT (0Dαt u (t)) = ∇F (t, u (t)) , a.e. t ∈ [0, T] , (1) U (0) = u (T) = 0, where α ∈ (0, 1) and 0Dαt and tDTα are the left and right Riemann-Liouville fractional derivatives, respectively.

Objectives

Results

Conclusion