The Solvability And Properties Of Solution Sets For Nonlinear Nonlocal Evolution Inclusions

Nonlinear evolution inclusions,a class of time-varying nonlinear multivalued partial d-ifferential equations,is widely used in many fields such as biology,physics and engineering The study of its solvability and properties of solution sets is very important to the devel-opment of these fields.Nonlinear nonlocal evolution equations,such as those involving an integral term of Volterra type,is an important kind of such evolution inclusions.The nonlocal term reflects the time memory effect and physical feedback propertiesIn this paper,we study the solvability and several properties of solution sets for two classes of nonlinear nonlocal evolution inclusions,i.e.,nonlinear evolution inclusions of Volterra type and nonautonomous evolution inclusions of Volterra type.We then give exam-ples to verify the validity of the abstract results.The main results of this paper are divided into four parts:In Part 1,we study the global solvability for nonlinear evolution inclusions of Volterra type subjected to a nonlocal initial condition:1).The global existence and boundedness results of C0-solutions for the evolution inclusions subjected to the local initial condition are obtained;2).We then prove the global existence and boundedness of C0-solutions for the original problem are obtained by applying a fixed point theorem for upper semicontinuous multifunctions with contractible values.Here,the nonlinear semigroup generated by the principal operator is only assume to be equicontinuous rather than compact.To overcome the lack of compactness,we need to impose a noncompactness measure condition on the nonlinearity.In addition,we assume that the nonlinear semigroup is exponentially stable,which leads to that the C0-solutions obtained have an upper bound independent of time In Part 2,we study the global solvability for nonautonomous evolution inclusions of Volterra type subjected to a nonlocal initial condition:The global existence and bounded-ness results of integral solutions for the evolution inclusions subjected to local and nonlocal initial conditions are successively obtained,by using the corresponding topological fixed point theorems.Here,the evolution process generated by the principal operator is exponen-tially stable,but not compact.In order to overcome the lack of compactness of the evolution process,we impose a noncompactness measure condition on the nonlinearityIn Part 3,we establish the nonemptiness and compactness of the C~0-solution set for the nonlinear evolution inclusions of Volterra type subjected to a nonlocal initial condition.This is done by means of the topological structure of C~0-solution sets of a modified version of the evolution inclusion subjected to local initial conditions and several geometric properties of the corresponding solution map:1).It is proved that the C~0-solution set is nonempty,compact and R_?;2).Moreover,the corresponding solution map is an R_?-map with nonempty and compact values.In this part,the nonlinear semigroup generated by the principal operator does not assume exponential stability condition.Although the solution obtained has no upper bound independent of time,it is sufficient to further investigate the topology of the solution set.In addition,we need to impose an invariant condition on the nonlinearity when studying the original problemIn the last part,we study the topological structure of integral solution sets for the nonau-tonomous evolution inclusions of Volterra type and several geometric properties of corre-sponding solution map:1).The integral solution set is proved to be nonempty and compact;2).Moreover,it is an R_?-set;3).It is proved that the corresponding solution map is an R_?-map with nonempty and compact values.Here,the evolution process generated by the principal operator does not need any compactness and exponential stability assumptions.