Infinitely Many Solutions Of Fractional Boundary Value Problems

Differential equations with fractional order are generalization of ordinary differential equations to noninteger order. This generalization is not only mathematical curiosities but has extensive applications in many areas of science and engineering such as in viscoelasticity, electrical circuits, and neuron modeling. To some extent, fractional order differential equations in order to describe phenomenons which can not be defined by integer order differential equations.Fractional calculus and fractional differential equations can find many applications in various fields of physical science such as viscoelasticity, diffusion, control, relaxation processes, and modeling phenomena in engineering. Recently, many results were obtained dealing with the existence and multiplicity of solutions of nonlinear fractional (including Leray-Schauder nonlinear alternative), topological degree theory(including coincidence degree theory),and comparison method (including upper and lower solutions methods and monotone iterative method). However, it seems that the popular methods mentioned above are not appropriate for discussing the problem, and the equivalent integral equation is not easy to be obtained. Therefore, we need other more effective ways to study these issues.In this paper,we research two kinds of fractional order differential equation,respectively is the smooth fractional order differential equation and the nonsmooth fractional order differential equation. The variational method and the critical point theory were applied in the process of the research of the equations, and proposes the important theory respectively. The aim of the present paper is to establish the existence of infinitely many distinct positive solutions for the equations under suitable oscillatory assumptions on the potential F at zero or at infinity. Indeed, our main results give sufficient conditions on the oscillatory terms such that problem has infinitely many positive solutions. As a byproduct, these solutions can be constructed in such a way that their norms in Ea tend to zero (to infinity, respectively) whenever the nonlinearity oscillates at zero (at infinity,respectively).The content of this paper can be divided into four parts. Firstly, the related knowledge of fractional differential equations and the research status at home and abroad are introduced.Secondly, The references therein of fractional differential equations, variational method and the critical point, generalized gradients variational method and the critical point. Thirdly,establish the existence of infinitely many distinct positive solutions for the smooth fractional differential equation under suitable oscillatory assumptions on the potential F at zero or at infinity. Lastly, establish the existence of infinitely many distinct positive solutions for the nonsmooth fractional differential equation under suitable oscillatory assumptions on the potential F at zero or at infinity.