The Eigenvalue Problem For Nonlinear Differential Equations Involving Fractional Derivative

Differential equations with fractional order are generalization of ordinary differential equations to noninteger order. This generalization is not only mathematical curiosities but also 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. And we respectively give two important conclusions for them. Specific tasks include:First, we study eigenvalue problem of fractional equation by the mountain theorem and variational method, obtain the existence results of some new solutions and give the proof process.Secondly, we research the semi-variational inequality eigenvalue of fractional derivative by the mountain theorem and nonsmooth critical point theory, obtain the existence results of some new solutions and give the proof process.