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The theory of linear Volterra integro-differential 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. A major part of the research is devoted to the study of various types of abstract (multi-term) fractional differential equations with Caputo fractional derivatives, primarily from their invaluable importance in modeling of various phenomena appearing in physics, chemistry, engineering, biology and many other sciences. The book also contributes to the theories of abstract first and second order differential equations, as well as to the theories of higher order abstract differential equations and incomplete abstract Cauchy problems, which can be viewed as parts of the theory of abstract Volterra integro-differential equations only in its broad sense. The operators examined in our analyses need not be densely defined and may have empty resolvent set.

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. Most of the subject matter is intended to be accessible to readers whose backgrounds include functions of one complex variable, integration theory and the basic theory of locally convex spaces. An important feature of this book as compared to other monographs and papers on abstract Volterra integro-differential equations is, undoubtedly, 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 a plenty of examples and open problems. The numbering of theorems, propositions, lemmas, corollaries, and definitions are by chapter and section. The bibliography is provided alphabetically by author name and a reference to an item is of the form,

The book does not claim to be exhaustive. Degenerate Volterra equations, the solvability and asymptotic behaviour of Volterra equations on the line, almost periodic and positive solutions of Volterra equations, semilinear and quasilinear problems, as some of many topics are not covered in the book. The author’s justification for this is that it is not feasible to encompass all aspects of the theory of abstract Volterra equations in a single monograph.

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 Nn□□1

Topological dynamics of certain classes of abstract time-fractional PDEs with unilateral backward shifts

Index

The theory of linear Volterra integro-differential 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. A major part of the research is devoted to the study of various types of abstract (multi-term) fractional differential equations with Caputo fractional derivatives, primarily from their invaluable importance in modeling of various phenomena appearing in physics, chemistry, engineering, biology and many other sciences. The book also contributes to the theories of abstract first and second order differential equations, as well as to the theories of higher order abstract differential equations and incomplete abstract Cauchy problems, which can be viewed as parts of the theory of abstract Volterra integro-differential equations only in its broad sense. The operators examined in our analyses need not be densely defined and may have empty resolvent set.

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. Most of the subject matter is intended to be accessible to readers whose backgrounds include functions of one complex variable, integration theory and the basic theory of locally convex spaces. An important feature of this book as compared to other monographs and papers on abstract Volterra integro-differential equations is, undoubtedly, 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 a plenty of examples and open problems. The numbering of theorems, propositions, lemmas, corollaries, and definitions are by chapter and section. The bibliography is provided alphabetically by author name and a reference to an item is of the form,

The book does not claim to be exhaustive. Degenerate Volterra equations, the solvability and asymptotic behaviour of Volterra equations on the line, almost periodic and positive solutions of Volterra equations, semilinear and quasilinear problems, as some of many topics are not covered in the book. The author’s justification for this is that it is not feasible to encompass all aspects of the theory of abstract Volterra equations in a single monograph.

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 Nn□□1

Topological dynamics of certain classes of abstract time-fractional PDEs with unilateral backward shifts

Index

The theory of linear Volterra integro-differential 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. A major part of the research is devoted to the study of various types of abstract (multi-term) fractional differential equations with Caputo fractional derivatives, primarily from their invaluable importance in modeling of various phenomena appearing in physics, chemistry, engineering, biology and many other sciences. The book also contributes to the theories of abstract first and second order differential equations, as well as to the theories of higher order abstract differential equations and incomplete abstract Cauchy problems, which can be viewed as parts of the theory of abstract Volterra integro-differential equations only in its broad sense. The operators examined in our analyses need not be densely defined and may have empty resolvent set.

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. Most of the subject matter is intended to be accessible to readers whose backgrounds include functions of one complex variable, integration theory and the basic theory of locally convex spaces. An important feature of this book as compared to other monographs and papers on abstract Volterra integro-differential equations is, undoubtedly, 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 a plenty of examples and open problems. The numbering of theorems, propositions, lemmas, corollaries, and definitions are by chapter and section. The bibliography is provided alphabetically by author name and a reference to an item is of the form,

The book does not claim to be exhaustive. Degenerate Volterra equations, the solvability and asymptotic behaviour of Volterra equations on the line, almost periodic and positive solutions of Volterra equations, semilinear and quasilinear problems, as some of many topics are not covered in the book. The author’s justification for this is that it is not feasible to encompass all aspects of the theory of abstract Volterra equations in a single monograph.

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 Nn□□1

Topological dynamics of certain classes of abstract time-fractional PDEs with unilateral backward shifts

Index

PREFACE

NOTATION

INTRODUCTION

PRELIMINARIES

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

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)

(a; k)-Regularized C-resolvent families and abstract multi-term fractional 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

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 -times C-regularized resolvent families

Hypercyclic and topologically mixing properties of solutions of (2) with Aj = cjI; cj 2 C; j 2 Nn□□1

Index

PREFACE

NOTATION

INTRODUCTION

PRELIMINARIES

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

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)

(a; k)-Regularized C-resolvent families and abstract multi-term fractional 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

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 -times C-regularized resolvent families

Hypercyclic and topologically mixing properties of solutions of (2) with Aj = cjI; cj 2 C; j 2 Nn□□1

Index

PREFACE

NOTATION

INTRODUCTION

PRELIMINARIES

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

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)

(a; k)-Regularized C-resolvent families and abstract multi-term fractional 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

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 -times C-regularized resolvent families

Hypercyclic and topologically mixing properties of solutions of (2) with Aj = cjI; cj 2 C; j 2 Nn□□1

Index