# On Evolution Equations in Banach Spaces and Commuting Semigroups

Abstract (Summary)

This dissertation is concerned with several questions about the qualitative behavior of mild solutions of differential equations with multi-dimensional time on Banach spaces. For the differential equations [See PDF for equations.] where A and B are linear (in general, unbounded) operators defined on a Banach space E, we give a definition of the mild solution of (*). In order for Eq.(*) to have a (mild) solution, we introduce the condition (Ds- A)f2 = (Dt- B)f1 (where Dsand Dt are the partial differential operators with respect to s and t, respectively) which is understood in a generalized (mild) sense. We extend the notion of admissible subspaces, for closed translation-invariant subspaces M of BUC(R^2,E ) with respect to Eq.(*), and we give characterization of the admissibility in terms of solvability of the operator equations AX - XDMR = C and AX - XDMt = CT. As a tool to investigate almost periodic functions of several variables which is important for the study of asymptotic behavior of Eq.(*), we give answer to a question raised by Basit in 1971, which is an extension of the classical Bohl-Bohr theorem (to almost periodic functions with two or more variables). We also extend a theorem due to Loomis (who obtained it for functions with values in a finite dimensional space) to functions with values in Banach space E, under conditions introduced by Kadets. As an application, we obtain a result on the almost periodicity of a double integral of an almost periodic function defined on the plane. Applying to Eq.(*), we show that the boundedness implies the almost periodicity for solutions of (*) in the finite dimensional case. For the infinite dimensional case, and under the conditions that A and B are bounded linear operators, we reduce the question of the almost periodicity (resp, almost automorphicity) of the differential equations (*) for given almost periodic (resp, almost automorphic) functions f1 and f2 to the question of the almost periodicity (resp, almost automorphicity) of the homogeneous system (i.e., f1 = f2 = 0). Finally, in chapter 6, we introduce the notion of C-admissible subspaces and obtain various conditions of C-admissibilities, generalizing well known results of Schuler-Vu and others.
Bibliographical Information:

Advisor:

School:Ohio University

School Location:USA - Ohio

Source Type:Master's Thesis

Keywords:differential equations with multi time c admissibility of subspaces analytic semigroup integral almost periodic functions solutions

ISBN:

Date of Publication:01/01/2005