On Evolution Equations in Banach Spaces and Commuting Semigroups

by Alsulami, Saud M.

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.