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. Each chapter is further divided into sections and subsections, and contains plenty of examples and open problems.
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. Each chapter is further divided into sections and subsections, and contains plenty of examples and open problems.
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 conv
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 conv
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