Almost Automorphic and Almost Periodic Functions in Abstract Spaces

by N'Guerekata, Gaston M.

• ISBN13: 9780306466861
• ISBN10: 0306466864
• Format: Hardcover
• Copyright: 2001-10-01
• Publisher: Plenum Pub Corp

Summary

Almost Automorphic and Almost Periodic Functions in Abstract Spaces introduces and develops the theory of almost automorphic vector-valued functions in Bochner's sense and the study of almost periodic functions in a locally convex space in a homogenous and unified manner. It also applies the results obtained to study almost automorphic solutions of abstract differential equations, expanding the core topics with a plethora of groundbreaking new results and applications. For the sake of clarity, and to spare the reader unnecessary technical hurdles, the concepts are studied using classical methods of functional analysis.

Table of Contents

Introduction and Preliminaries	1	(10)
Introduction	1	(1)
Banach spaces	1	(3)
Linear operators	4	(1)
Functions with values in a Banach space	5	(2)
Semigroups of linear operators	7	(1)
Topological vector spaces	8	(3)
Almost Automorphic Functions with Values in a Banach Space	11	(40)
Definition and elementary properties	11	(9)
The function f(t, x(t))	20	(3)
Weakly almost automorphic functions	23	(3)
Differentiation and integration	26	(11)
Asymptotically almost automorphic functions	37	(2)
Differentiation and integration of asymptotically almost automorphic functions	39	(3)
Dynamical systems and C0-semigroups	42	(9)
Almost Periodic Functions with Values in a Linear Topological Space	51	(18)
Almost periodic functions	51	(6)
Weakly almost periodic functions	57	(12)
The Equation x'(t) = Ax(t) + f(t)	69	(20)
The case A = λ ∈ C	69	(2)
The case A is a bounded linear operator	71	(4)
The case A is the infinitesimal generator of a C0-group of operators	75	(9)
The equation x' = (A + B)x + f	84	(5)
The Equation x' = f(t, x)	89	(4)
Existence of almost automorphic solutions	89	(1)
The linear case	90	(3)
A Case of One-to-One Correspondence between Almost Automorphic and Asymptotically Almost Automorphic Solutions	93	(6)
The correspondence	94	(5)
Almost Periodic Solutions of the Equation x' = Ax + f in Locally Convex Spaces	99	(14)
The exponential of a bounded linear operator	99	(10)
The equation x' = Ax + f	109	(4)
Almost Periodic Solutions of Differential Equations in Normed Spaces	113	(18)
The homogeneous equations x' = Ax and x'' = Ax	113	(7)
The non-homogeneous equation x' = Ax + f	120	(4)
The nonlinear equation x'(t) = Ax(t) + f(t, x(t))	124	(7)
References	131	(6)
Index	137