Burlica / Necula / Vrabie Delay Differential Evolutions Subjected to Nonlocal Initial Conditions

Erscheinungsjahr 2016
ISBN: 978-1-4987-4646-5
Verlag: CRC Press
Format: PDF
Kopierschutz: Adobe DRM (»Systemvoraussetzungen)

E-Book, Englisch, 388 Seiten

Reihe: Monographs and Research Notes in Mathematics

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.

Autoren/Hrsg.

Burlica, Monica-Dana

Necula, Mihai

Vrabie, Ioan I.

Ro¿u, Daniela

Fachgebiete

• Mathematik | Informatik Mathematik Mathematische Analysis Differentialrechnungen und -gleichungen
• Mathematik | Informatik Mathematik Numerik und Wissenschaftliches Rechnen Angewandte Mathematik, Mathematische Modelle

Inhaltsverzeichnis

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
Furth

Über Autor(innen)

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.