Existence And Properties Of Solutions For Differential Inclusions In Banach Spaces

Differential inclusions also called multivalued functional differential equa-tions is an important branch in the theory of nonlinear analysis, the theory ofwhich has developed fast because of its extensive practical applications in manyfields such as engineering, economics, optimal control and optimization theory.There are many results about the theory of differential inclusions, such as theexistence of solutions, continuation of solutions, dependence on initial condi-tions and parameters. Since a differential inclusion usually has many solutionsstarting at a given point, new issues appear, such as investigation of topologicalproperties of the set of solutions, selection of solutions with given properties, etc.However, many of them are obtained under the assumption that the semigroupis compact or equicontinuous. Naturally, one may ask that whether there are thesimilar results without the assumption on the compactness or equicontinuity ofthe semigroup. In this work, we mainly deal with these problems. The resultsobtained are presented as follows.In Chapter 1, we study the existence of solutions to the following semilinearevolution differential inclusionin a real Banach space X. Here {A(t)}t∈[0,T] is a family of linear operators andF is a multifunction.The main tools in the approach followed in this Chapter are measure ofnoncompactness, the theory of semilinear differential equations and multivaluedanalysis. A brief reminder of these is provided in Section 1.1. In Section 1.2, wegive our main results, i.e. existence of mild solutions and continuation of solutions(see Theorem 1.10, Theorem 1.11, Corollary 1.14). In our proof, we first definea new regular measure of noncompactness. Then, we make use of the measureof noncompactness of the section to calculate the modulus of equicontinuity. Therefore, we do not need the equicontinuity of the evolution system {U(t,s) :0≤s≤t≤T} in our proof and also obtain the above main results, whichextend those in [50, 65, 78], where they need the compactness of the evolutionsystem or the semigroup.Chapter 2 deals with the existence results of integral solutions and the con-tinuous dependence of the solution set on the initial function of the followingnonlinear differential inclusion with infinite delayin a real Banach space X. Here we assume that A is an m-accretive operatorsuch that ffA generates an equicontinuous semigroup, F is a multifunction, B isa phase space and Xff is uniformly convex. We make us of the measure of non-compactness and the multivalued fixed point theory to get the existence resultsof the above problem without the compactness assumption on the semigroup,which extend the results of [39, 44, 48] to the case of infinite delay and extendthe results of [35] to the fully nonlinear case.In Section 2.1, we recall some basic definitions and a multivalued fixed pointtheory. In Section 2.2, we define the integral solution operator and give somebasic properties of which. And we give the existence of local integral solutions forthe problem we considered in Section 2.3 (see Theorem 2.12). In Section 2.4, weprove the existence of global integral solution and obtain the compactness of thesolution set (see Theorem 2.14). Finally, we obtain the continuous dependence ofthe solution set P(ff) on the initial function ff in Section 2.5 (see Theorem 2.18).Chapter 3 is devoted to the study of the nonlocal initial value problemwhere A is the infinitesimal generator of a strongly continuous semigroup ofbounded linear operators T(t) in Banach space X, f : [0,b]×X→X andg : C(0,b;X)→X. Our basic tools are the methods and results for semilinear differential equa-tions in Banach spaces, the properties of noncompact measures and fixed pointtechniques. We try to make use of the properties of noncompact measures inproof. It seems that we first make use of the measure of noncompactness to dealwith the nonlocal problem, which not only makes us to remove the compactnessof the semigroup, even the equicontinuity of the semigroup, but also enables usto avoid the diffculties associated with unbounded operators when t = 0. InSection 3.1, we recall some definitions and facts about the measure of noncom-pactness and semilinear differential equations. Our main results will be givenin Section 3.2 (see Theorem 3.7, Theorem 3.10, Theorem 3.12, Theorem 3.15 andTheorem 3.16), which extend and improve many results about this problem.In Chapter 4, we discuss the weak and strong convergence of the integralsolutions of the following nonlinear evolution inclusionwhere {A(t) : t≥0} is a family of m-accretive operators in a real Banach spaceX, x0∈D(A(0)) and F is a strongly measurable multivalued mapping.In Section 4.1, we give some facts about the integral solutions of the non-autonomous system in Banach spaces. Since the integral solutions we consideredhere are not the almost-orbits of the semigroup, the methods in [33, 68, 69, 66, 72,73, 81] can not be applied to this case. We make full use of properties of integralsolutions to obtain the ergodic theorem and the suffcient and necessary con-ditions to the weak convergence in Section 4.2 (see Theorem 4.5, Theorem 4.7).The suffcient condition to the strong convergence of integral solutions is obtainedin Section 4.3 (see Theorem 4.10).Chapter 5 is concerned with the topological structure of the solution set tothe constrained nonlinear differential inclusionin a real Banach space X, where A : D(A) ff X→2X is an m-accretive operatorwith ffA generating a semigroup of nonexpansive mappings S(t) : D(A)→D(A) for t≥0, D ff D(A) is a nonempty subset and F : [0,T]×D→2X is amultivalued function.In this Chapter, under the tangency condition and the appropriate assump-tions on D, we make use of the methods in [6] to obtain that the solution setof the constrained nonlinear differential inclusion is an Rδ-set in Section 5.3 (seeTheorem 5.8). It seems that this is the first result about the constrained nonlin-ear differential inclusions. In Section 5.4, we discuss the semilinear case. And inview of our proof, we can remove many key conditions of Theorem 16 in [6] (seeRemark 5.20). Finally, as an application, we obtain the periodic solutions of thedifferential inclusions (see Theorem 5.25 and Theorem 5.26).