Existence And Properties Of Solutions For Functional Differential Equations In Banach Spaces

This work is devoted to study the abstract semi-linear and nonlinear functionaldi?erential equations in Banach spaces, followed by that of the ergodic theoryfor semigroups of nonlinear non-Lipschitzian operators。There are five chapters in this dissertation. In Chapter 1 a class of semi-linear functional di?erential equations with nonlocal conditions is discussed. Byutilizing the theory of the measure of noncompactness in the space of continuousfunctions and Schauder's fixed point theorem, existence of mild solutions andcompactness of solution set are obtained without the assumption of the com-pactness or equicontinuity of the associated semigroups.Chapter 2 is concerned with neutral functional di?erential and integro-di?erential equations. By employing the theory of analytic semigroups, Haus-dor?'s measure of noncompactness and Darbo-Sadovskii's fixed point theorem,existence of mild solutions to neutral functional di?erential and integro-di?erentialequations with nonlocal conditions. The models discussed in this chapter enableus to handle simultaneously several classes of equations, such as neutral di?er-ential equations with delay and nonlocal neutral di?erential equations.Viability for a class of semilinear di?erential equations with infinite delay inBanach spaces is studied in Chapter 3. Based on the Scorda Dragoni's propertyfor Caratheodory type functions and Lebesgue type derivative, it is verified thatthe necessary and su?cient condition for a tube in a Banach space to be viablefor a class of semilinear di?erential equations with infinite delay is the tangencycondition.Chapter 4 is devoted to establish the existence of unique strong solution of aclass of nonlinear evolution equations. A straightforward approximation schemeis available to show that the solution of such equation is the uniform limit of thecontinuously di?erentiable solutions of approximation equations involving theYosida approximants of m?accretive operators. The final chapter is concerned with the ergodic theorem for commutativesemigroups of non-Lipschitzian mappings. Strong ergodic theorem for semigroupof asymptotically non-expansive type in the intermediate sense mapping is es-tablished in a uniformly convex Banach space, which extends and unifies manypreviously known results.