Abstract Volterra Integro-Differential Equations.pdf
The theory of linear Volterra Integro-differental equations has been developing rapidly in the last three decades. This book provides an easy-to-read, concise introduction to the theory of ill-posed abstract Volterra Integro-differential equations. It is accessible to readers whose backgrounds include functions of one complex variable, integration theory and the basic theory of locally convex spaces. Divided into three chapters, the book is a logical continuation of some previously published monographs in the field of ill-posed abstract Cauchy problems. It is not written as a traditional text, but rather as a guidebook suitable as an introduction for advanced graduate students in mathematics or engineering science, researchers in abstract partial differential equations, and experts from other areas. An important feature of this book as compared to other monographs and papers on abstract Volterra integro-differential equations is the consideration of solutions, and their hypercyclic properties, in locally convex spaces. Each chapter is further divided in sections and subsections and, with the exception of the introductory one, contains plenty of examples and open problems. The numbering of theorems, propositions, lemmas, corollaries, definitions, etc., are by chapter and section.
PREFACE NOTATION INTRODUCTION PRELIMINARIES Vector-valued functions, closed operators and integration in sequentially complete locally convex spaces Laplace transform in sequentially complete locally convex spaces Operators of fractional differentiation, Mittag-Leffler and Wright functions (a; k)-REGULARIZED C-RESOLVENT FAMILIES IN LOCALLY CONVEX SPACES Definition and main structural properties of (a; k)-regularized C-resolvent families Wellposedness of related abstract Cauchy problems Convoluted C-semigroups and convoluted C-cosine functions in locally convex spaces Differential and analytical properties of (a; k)-regularized C-resolvent families Systems of abstract time-fractional equations q-Exponentially equicontinuous (a; k)-regularized C-resolvent families Abstract differential operators generating fractional resolvent families Perturbation theory for abstract Volterra equations Bounded perturbation theorems Unbounded perturbation theorems Time-dependent perturbations of abstract Volterra equations Approximation and convergence of (a; k)-regularized C-resolvent families (a; k)-Regularized (C1;C2)-existence and uniqueness families Complex powers of (a; b;C)-nonnegative operators and fractional resolvent families generated by them Complex powers of a C-sectorial operator A satisfying 0 2 (C(A)) The Balakrishnan operators Complex powers of almost C-nonnegative operators The case m = 1 Semigroups generated by fractional powers of almost C-sectorial operators Fractional powers of (a; b;C)-nonnegative operators and semigroups generated by them The existence and growth of mild solutions of operators generating fractionally integrated C-semigroups and cosine functions in locally convex spaces Representation of powers Abstract multi-term fractional differential equations k-Regularized (C1;C2)-existence and uniqueness propagation families for (2) k-Regularized (C1;C2)-existence and uniqueness families for (2) Approximation and convergence of k-regularized C-resolvent propagation families; further results, examples and applications (a; k)-Regularized C-resolvent families and abstract multi-term fractional differential equations HYPERCYCLIC AND TOPOLOGICALLY MIXING PROPERTIES OF CERTAIN CLASSES OF VOLTERRA INTEGRO-DIFFERENTIAL EQUATIONS Hypercyclic and topologically mixing properties of abstract rst order equations Disjoint hypercyclic semigroups Hypercyclic and topologically mixing properties of abstract second order equations C-Distribution cosine functions, almost C-distribution cosine functions and integrated C-cosine functions Hypercyclicity and chaos for C-distribution cosine functions and integrated C-cosine functions Hypercyclic and chaotic properties of cosine functions Disjoint hypercyclicity of C-distribution cosine functions Hypercyclic and topologically mixing properties of abstract multi-term fractional differential equations Hypercyclic and topologically mixing properties of -times C-regularized resolvent families Hypercyclic and topologically mixing properties of solutions of (2) with Aj = cjI; cj 2 C; j 2 Nn1 Topological dynamics of certain classes of abstract time-fractional PDEs with unilateral backward shifts Index