Abstract Volterra Integro-Differential Equations

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