非线性分数阶半变分不等式的变分法研究

In this item, we shall study the existence and multiplicity of solutions for nonlinear fractional hemivariational inequalities through using the nonsmooth variational methods. It arises form its applications as a model for physical phenomena exhibiting anomalous diffusion. First, we shall study the variational structure for the problems, because of the reasion of discontinuity,the variational functionals are only locally Lipschitz, hence, classical critical point theory and minimax method can not applied, and hence we shall study this item from the following three hands:.(1)The existence and multiplicity of periodic solutions will be studied by nonsmooth critical point theory for locally Lipschitz functionals;.(2)The existence and multiplicity of subharmonic solutions will be studied by nonsmooth critical point theory for locally Lipschitz functionals;.(3)The existence and multiplicity of homoclinic solutions will be studied by nonsmooth critical point theory for locally Lipschitz functionals.

本项目拟采用非光滑变分方法研究非线性分数阶半变分不等式解的存在性和多重性。这类问题源于描述反常扩散的物理模型。本项目首先将研究其变分结构，由于不连续性的原因，此类问题的变分泛函仅仅是局部Lipschitz的，传统的临界点理论和极小极大方法不适用于其研究。因此，本项目将从下面三方面进行研究：.(1)采用局部Lipschitz泛函的非光滑临界点理论研究其周期解的存在性和多重性；.(2)采用局部Lipschitz泛函的非光滑临界点理论研究其次调和解的存在性和多重性；.(3)采用局部Lipschitz泛函的非光滑临界点理论研究其同宿解的存在性和多重性。

该项目主要研究了下述问题：(1) 我们建立了两类分数阶半变分不等式的非光滑变分结构，应用非光滑山路定理、对称山路定理，Clark定理研究了其解的存在性和无穷多解的存在性；(2) 在非线性项不满足对称性的条件下，我们应用非光滑临界点理论研究了其无穷多解的存在性；(3) 应用临界点理论研究了分数阶Hamiltonian系统无穷多个同宿解的存在性。