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.

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.

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.

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 R^d 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 = 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(O) 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(O) 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(O) 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(O) 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 G_E Proof of Theorem 8.2.1 A reaction-diffusion system in L¹(O) A reaction-diffusion system in L²(O) 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 approxi…