Functional Analysis and Differential Equations in Abstract Spaces provides an elementary treatment of this very classical topic-but presented in a rather unique way. The author offers the functional analysis interconnected with specialized sections on differential equations, thus creating a self-contained text that includes most of the necessary functional analysis background, often with quite complete proofs.Beginning with some basic functional analysis-Hilbert and Banach spaces and their linear operators-Dr. Zaidman then presents some results about the abstract Cauchy problem, in implicit or explicit form, and related semigroups of operators, weak and ultraweak solutions, the uniqueness of the Cauchy problem, the uniqueness of bounded ultraweak solutions, and the well-posed ultraweak Cauchy problem. He goes on to present some results on almost-periodic solutions and an asymptotic result for a differential inequality in ultraweak form.D esigned to inspire interest in this elegant and rapidly growing field of mathematics, this volume presents the material at a relatively elementary level-requiring a minimum of knowledge and ability in the field-yet with depth sufficient for understanding various special topics in operator differential equations.
Many of the research results appear for the first time in book form and some for the first time anywhere. Researchers in the theories of differential equations in abstract spaces, semigroups of operators, and evolution equations, along with researchers in mathematical physics and quantum mechanics will find this work both enlightening and accessible.
IntroductionHilbert Space, Definitions, First ResultsBanach Spaces and Linear OperatorsDuality, the Hahn-Banach Extension TheoremSome Singular (Implicit) Abstract Differential EquationsSemigroups of Class C0 and the Abstract Cauchy ProblemCompact Linear OperatorsContinuous Symmetric Operators in Hilbert SpacesSemidynamical Systems and C0-SemigroupsUniqueness of Weak Solutions for a Second-Order Abstract Cauchy Problem in Hilbert SpaceUniqueness of Ultraweak Solutions for a First Order Abstract Cauchy Problem in Banach SpaceUniqueness of Bounded Ultraweak Solutions in Hilbert SpaceAlmost-Periodic Solutions IThe Well-Posed Ultraweak Cauchy Problem and Related Semigroups of OperatorsAsymptotic Result for Ultraweak Differential Inequalities in Hilbert SpaceAlmost-Periodic Solutions IIVariational Form of the Equation Au = a in Hilbert SpaceSolution of the Weak Neumann Problem in an Arbitrary Open Set of RnReferencesSubject IndexIndex of Symbols
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