Delay Differential Evolutions Subjected to Nonlocal Initial Conditions: 1st Edition (Hardback) book cover

Delay Differential Evolutions Subjected to Nonlocal Initial Conditions

1st Edition

By Monica-Dana Burlică, Mihai Necula, Daniela Roșu, Ioan I. Vrabie

Chapman and Hall/CRC

362 pages

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pub: 2016-06-20
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Description

Filling a gap in the literature, Delay Differential Evolutions Subjected to Nonlocal Initial Conditions reveals important results on ordinary differential equations (ODEs) and partial differential equations (PDEs). It presents very recent results relating to the existence, boundedness, regularity, and asymptotic behavior of global solutions for differential equations and inclusions, with or without delay, subjected to nonlocal implicit initial conditions.

After preliminaries on nonlinear evolution equations governed by dissipative operators, the book gives a thorough study of the existence, uniqueness, and asymptotic behavior of global bounded solutions for differential equations with delay and local initial conditions. It then focuses on two important nonlocal cases: autonomous and quasi-autonomous. The authors next discuss sufficient conditions for the existence of almost periodic solutions, describe evolution systems with delay and nonlocal initial conditions, examine delay evolution inclusions, and extend some results to the multivalued case of reaction-diffusion systems. The book concludes with results on viability for nonlocal evolution inclusions.

Reviews

"This book will be useful to researchers and graduate students interested in delay evolution equations and inclusions subjected to nonlocal initial conditions." - Sotiris K. Ntouyas (Ioannina)

Table of Contents

Preliminaries

Topologies on Banach spaces

A Lebesgue-type integral for vector-valued functions

The superposition operator

Compactness theorems

Multifunctions

C0-semigroups

Mild solutions

Evolutions governed by m-dissipative operators

Examples of m-dissipative operators

Strong solutions

Nonautonomous evolution equations

Delay evolution equations

Integral inequalities

Brezis–Browder Ordering Principle

Bibliographical notes and comments

Local Initial Conditions

An existence result for ODEs with delay

An application to abstract hyperbolic problems

Local existence: The case f Lipschitz

Local existence: The case f continuous

Local existence: The case f compact

Global existence

Examples

Global existence of bounded C0-solutions

Three more examples

Bibliographical notes and comments

Nonlocal Initial Conditions: The Autonomous Case

The problem to be studied

The case f and g Lipschitz

Proofs of the main theorems

The transport equation in Rd

The damped wave equation with nonlocal initial conditions

The case f Lipschitz and g continuous

Parabolic problems governed by the p-Laplacian

Bibliographical notes and comments

Nonlocal Initial Conditions: The Quasi-Autonomous Case

The quasi-autonomous case with f and g Lipschitz

Proofs of Theorems 4.1.1, 4.1.2

Nonlinear diffusion with nonlocal initial conditions

Continuity with respect to the data

The case f continuous and g Lipschitz

An example involving the p-Laplacian

The case f Lipschitz and g continuous

The case A linear, f compact, and g nonexpansive

The case f Lipschitz and compact, g continuous

The damped wave equation revisited

Further investigations in the case ℓ = ω

The nonlinear diffusion equation revisited

Bibliographical notes and comments

Almost Periodic Solutions

Almost periodic functions

The main results

Auxiliary lemmas

Proof of Theorem 5.2.1

The w-limit set

The transport equation in one dimension

An application to the damped wave equation

Bibliographical notes and comments

Evolution Systems with Nonlocal Initial Conditions

Single-valued perturbed systems

The main result

The idea of the proof

An auxiliary lemma

Proof of Theorem 6.2.1

Application to a reaction-diffusion system in L2(Ω)

Nonlocal initial conditions with linear growth

The idea of the proof

Auxiliary results

Proof of Theorem 6.7.1

A nonlinear reaction-diffusion system in L1(Ω)

Bibliographical notes and comments

Delay Evolution Inclusions

The problem to be studied

The main results and the idea of the proof

Proof of Theorem 7.2.1

A nonlinear parabolic differential inclusion

The nonlinear diffusion in L1(Ω)

The case when F has affine growth

Proof of Theorem 7.6.1

A differential inclusion governed by the p-Laplacian

A nonlinear diffusion inclusion in L1(Ω)

Bibliographical notes and comments

Multivalued Reaction-Diffusion Systems

The problem to be studied

The main result

Idea of the proof of Theorem 8.2.1

A first auxiliary lemma

The operator ΓE

Proof of Theorem 8.2.1

A reaction-diffusion system in L1(Ω)

A reaction-diffusion system in L2(Ω)

Bibliographical notes and comments

Viability for Nonlocal Evolution Inclusions

The problem to be studied

Necessary conditions for viability

Sufficient conditions for viability

A sufficient condition for null controllability

The case of nonlocal initial conditions

An approximate equation

Proof of Theorem 9.5.1

A comparison result for the nonlinear diffusion

Bibliographical notes and comments

Bibliography

Index

About the Authors

Monica-Dana Burlica is an associate professor in the Department of Mathematics and Informatics at the “G. Asachi” Technical University of Iasi. She received her doctorate in mathematics from the University “Al. I. Cuza” of Iasi. Her research interests include differential inclusions, reaction-diffusion systems, viability theory, and nonlocal delay evolution equations.

Mihai Necula is an associate professor in the Faculty of Mathematics at the University "Al. I. Cuza” of Iasi. He received his doctorate in mathematics from the University “Al. I. Cuza” of Iasi. His research interests include differential inclusions, viability theory, and nonlocal delay evolution equations.

Daniela Rosu is an associate professor in the Department of Mathematics and Informatics at the “G. Asachi” Technical University of Iasi. She received her doctorate in mathematics from the University "Al. I. Cuza” of Iasi. Her research interests include evolution equations, viability theory, and nonlocal delay evolution equations.

Ioan I. Vrabie is a full professor in the Faculty of Mathematics at the University "Al. I. Cuza” of Iasi and a part-time senior researcher at the "O. Mayer" Mathematical Institute of the Romanian Academy. He received his doctorate in mathematics from the University “Al. I. Cuza” of Iasi. He has been a recipient of several honors, including The First Prize of the Balkan Mathematical Union and the “G. Titeica” Prize of the Romanian Academy. His research interests include evolution equations, viability theory, and nonlocal delay evolution equations.

About the Series

Chapman & Hall/CRC Monographs and Research Notes in Mathematics

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Subject Categories

BISAC Subject Codes/Headings:
MAT003000
MATHEMATICS / Applied
MAT007000
MATHEMATICS / Differential Equations