Delay Differential Evolutions Subjected to Nonlocal Initial Conditions  book cover
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Delay Differential Evolutions Subjected to Nonlocal Initial Conditions




ISBN 9781498746441
Published June 20, 2016 by Chapman and Hall/CRC
362 Pages

 
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Book 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.

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

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Author(s)

Biography

Monica-Dana Burlică is an associate professor in the Department of Mathematics and Informatics at the “G. Asachi” Technical University of Iaşi. She received her doctorate in mathematics from the University “Al. I. Cuza” of Iaşi. 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 Iaşi. He received his doctorate in mathematics from the University “Al. I. Cuza” of Iaşi. His research interests include differential inclusions, viability theory, and nonlocal delay evolution equations.

Daniela Roşu is an associate professor in the Department of Mathematics and Informatics at the “G. Asachi” Technical University of Iaşi. She received her doctorate in mathematics from the University "Al. I. Cuza” of Iaşi. 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 Iaşi 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 Iaşi. He has been a recipient of several honors, including The First Prize of the Balkan Mathematical Union and the “G. Ţiţeica” Prize of the Romanian Academy. His research interests include evolution equations, viability theory, and nonlocal delay evolution equations.

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)