Delay Differential Evolutions Subjected to Nonlocal Initial Conditions

Content: 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 = omega 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(OMEGA)  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(OMEGA)  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(OMEGA)  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(OMEGA)  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 GAMMAE Proof of Theorem 8.2.1  A reaction-diffusion system in L1(OMEGA)  A reaction-diffusion system in L2(OMEGA) 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