Several Classes Of Fractional Differential Hemivariational Inequalities And Their Applications

Differential(hemi)variational inequality is a powerful instrument in describing problems in many fields such as fluid mechanical problems,contact friction,engineering operation research,economical dynamics,dynamic traffic networks problems,the obstacle problems for contacting bodies,etc.Differential(hemi)variational inequality has been researched by many scholars in different spaces using different methods.Considering the wide applications of fractional calculus in real world,such as the areas of enginery,economics,biology,physics,we study several classes of fractional differential hemivariational inequalities in Banach spaces by utilizing the the theory and numerical methods of fractional differential equations and hemivariational inequalities.It consists of seven chapters.In chapter 1,we review the history of differential(hemi)variational inequalities and fractional differential(hemi)variational inequalities.Moreover,we will introduce our main contribution to this dissertation.In chapter 2,a new fractional nonlinear delay evolution system driven by a hemi-variational inequality,which is called fractional delay differential hemivariational inequality,is introduced and studied.Utilizing the KKM theorem,a result concerned with upper semicontinuity and measurability of the solution set of a hemivariational inequality is established.By using a fixed point theorem for a condensing set-valued map,the nonemptiness and compactness of the set of mild solutions are also obtained for such a system under mild conditions.Finally,an example is presented to illustrate our main results.In chapter 3,a new fractional nonlinear system described by a fractional nonlinear differential equation and a quasi-hemivariational inequality,which is called fractional differential quasi-hemivariational inequality,is introduced and investigated in Banach spaces.We focus on the existence,uniqueness and stability of solutions for the new fractional nonlinear system under some mild assumptions.Moreover,we explore the well-posedness of the new fractional nonlinear system by employing the penalty method.Finally,we apply our results to a new frictional elastic contact problem with wear,in which the contact condition is governed by the generalized gradient of a nonconvex function and the wear function is described by a fractional nonlinear equation.In chapter 4,we consider a new fractional impulsive differential hemivariational inequality which captures the required characteristics of both the hemivariational inequality and the fractional impulsive differential equation within the same framework.By utilizing a surjectivity theorem and a fixed point theorem,we establish an existence and uniqueness theorem for such a problem.Moreover,we investigate the perturbation problem of the fractional impulsive differential hemivariational inequality to prove a convergence result which describes the stability of the solution in relation to perturbation data.Finally,our main results are applied to obtain some new results for a frictional contact problem with the surface traction driven by the fractional impulsive differential equation.In chapter 5,we deal with a new nonlocal fractional impulsive differential hemivariational inequality obtained by a nonlocal fractional impulsive evolution equation and a hemivariational inequality.The unique solvability of the hemivariational inequality is explored through the surjectivity theorem for set-valued maps and the existence of mild solutions for the new system is obtained by utilizing the fixed point theorem for condensing set-valued map.Moreover,the set of attractive solutions for the new system is proved to be nonempty under some suitable assumptions.Finally,our main results are applied to obtain some new results for a fractional partial differential system.In chapter 6,we introduce and study a new class of fractional differential evolutionary hemivariational inequality formulated by an evolutionary hemivariational inequality and a fractional differential equation in Banach spaces.By employing the Rothe method and the surjectivity result,we derive the existence of unique solution for such a problem under some mild conditions.Moreover,we use the fully discrete scheme to approximate the fractional differential hemivariational inequality and provide an error estimate for the approximation.Finally,the main results are applied to obtain the unique solvability as well as the numerical analysis for a viscoelastic frictional contact problem with adhesion.In chapter 7,the summary of this thesis and some future work are presented.