Existence And Asymptotic Properties Of Solutions For Differential Inclusions With Multivalued Perturbations In Banach Spaces

The theory of differential inclusions in Banach spaces is a very active branchin that of nonlinear analysis. Since the 1970s, many famous mathematicians (such asV.Barbu, J.P.Aubin, T.Kato, N.H.Pavel, etc.) in US, Romania and Japan and so onhave started to be engaged in a series of studies(see [2, 9, 13, 71]) in this subject. Inrecent dozens of years, the research work in the field is extremely crucial to nonlinearproblems and control theory in modern physics and engineering technology. Volterraequations(see [2]), partial differential equations(see [9, 13, 71]) and many problems incontrol theory and optimization theory can be rewritten in the abstract form as differ-ential inclusions. Therefore, the study of the existence of solutions (including strongsolutions, weak solutions, mild solutions and integral solutions) and asymptotic prop-erties for differential inclusions is very important. In this thesis, we mainly discuss theexistence and properties of solutions for nonlocal differential inclusions with multival-ued perturbtation in Banach spaces . It consists of four chapters.In Chapter 1, we discuss the existence of mild solutions for the following nonlocalinitial valued problem :where F is an upper semicontinuous multifunction, g : C([0, T]; E)→E is a givenEffvalued function and the linear operator (usually unbounded) A : D(A) (?) E→E isthe densely defined generator of a compact semigroup S (t) for t > 0 in a real Banachspace E.we prove a new result on the existence of mild solutions for the nonlocal multi-valued differential inclusion in a Banach space by using the technique of multi-valuedfixed point theorem and compactness methods (see Theorem 1.2). In our proof, we tryto construct a new and special set-valued mapping. Secondly, by careful analysis, themultivalued mapping is an upper semicontinuous and compact operator with closedand convex values on a given disc. Finally, we make full use of the properties of the set-valued mapping to manage to construct a relatively compact solution sequence inC([0, T]; E). Thus, we can obtain the main result. Moreover, under the asymptoticconditions and strong boundedness conditions on F and g, we also obtain the sameresults, respectively (see Corollary 1.6 and Remark 1.7). Our results extend the mainones in [5, 64] to the nonlocal multi-valued case. Since the Lipschitz type conditionon perturbation F is not required, our results are new even in the case that the pertur-bation F is single-valued. Finally, Using the established results, we take two examplesin order to point out the effectiveness of the abstract results proved in the former.Chapter 2 is devoted to continuing to study the class of nonlocal multivaluedproblem when F is an upper-Carathe′odory multifunction , g : C([0, T]; E)→E is agiven operator and A : D(A) ff E→E is the densely defined infinitesimal generatorof a strongly continuous semigroup of bounded linear operators {S(t) : t∈[0, T]} ingeneral Banach spaces.In this chapter, we deal with the existence of mild solutions for the semilineardifferential inclusions in general Banach spaces by using multi-valued fixed point the-orem on upper semicontinuity, the measure of noncompactness and some known re-sults about the theory of semilinear differential inclusions and multivalued analysis(see Lemma 2.9 and Theorem 2.7). Throughout this chapter, the new inequality inLemma 2.9 is very key to the proof of Theorem 2.7. In Theorem 2.7, we haven't anyhypothesis for the Banach space E. Moreover, we don't also assume the semigroupS(t) is compact. Therefore our results extend those in [22, 28, 30, 88, 89, 91] to thecase that the nonlocal multi-valued differential inclusions.The purpose of Chapter 3 is to present the existence of mild solutions for the fol-lowing nonlocal evolution differential inclusion with the upper semicontinuous non-linearity:in a real Banach space. Here the family of linear unbounded operators {A(t)}t∈[0,d]generates a strongly continuous evolution system U(t, s) and F is a multifunction.In this chapter, we first prove the existence result of the above evolution inclusionunder the assumption the nonlocal condition g is completely continuous (see Theorem 3.5). In Theorem 3.5, we only suppose that the linear part of the inclusion generates astrongly continuous evolution system U(t, s), which is not assumed to be equicontin-uous or compact. In proof, we introduce a new measure of noncompactness, which isvery crucial to construct a nonempty, compact and convex subset in C([0, d]; X). Thisallows us to improve considerably the results in the second chapter by replacing theequicontinuity of C0ffsemigroup by a strictly weaker condition. Subsequently, we dealwith the same problem concerning the case that g is Lipschitz continuous (see Theo-rem 3.11). It is worth mentioning that we try to make full use of the estimations to theHausdorff measure of noncompactness and the properties of the superposition operatorin our proof of Theorem 3.11. Thus, we may derive the existence of mild solutionswithout the assumption of separability on Banach space X and that of compactnesson the associated evolution system U(t, s). Therefore our results improve and extendsome known results in this field (see, for example, [7, 22, 28, 30, 41, 47, 88, 91]) andthe references therein). Finally, in the last section we discuss an example of semilinearpartial differential equations.Chapter 4 is concerned with this existence and asymptotic properties of integralsolutions for the following nonlinear nonlocal initial value problem:in a real Banach space when A : D(A) ff X→X is a nonlinear mffdissipative operatorwhich generates a contraction semigroup S (t) and F is a weakly upper semicontinuousmultifunction with respect to its second variable in a real Banach space with uniformlyconvex dual X~*.In this chapter, we prove the existence of integral solutions of the above Cauchyproblem . Moreover, we discuss the asymptotic properties of integral solutions. InSection 4.1, we first recall some facts about geometric properties of Banach spaces.In the sequel, we introduce some basic definitions and give the existence and unique-ness result and Be′nilan inequalities about the integral solutions of the nonautonomousdissipative system in Banach spaces. In Section 4.2, we discuss the case that g iscompletely continuous and S (t) is equicontinuous (see Theorem 4.15). The existenceresult for the above problem is stated when g is Lipschitz and F is Lipschitz-type map- ping about the Hausdorff metric in Section 4.3 (see Theorem 4.17). Finally, we firstdiscuss the asymptotic properties of almost nonexpansive curves. In the sequel, we tryto look for the relations between almost nonexpansive curves and integral solutions ofgiven dissipative system. Then, by the relations between almost nonexpansive curvesand integral solutions, we study the asymptotic behavior of integral solutions in Sec-tion 4.4 (see Theorem 4.23 and Theorem 4.26). These results extend and improvesome known results in [22, 58, 79, 80, 86, 87, 91, 92].