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Advanced functional evolution equations and inclusions. (English) Zbl 1326.34012
Developments in Mathematics 39. Cham: Springer (ISBN 978-3-319-17767-0/hbk; 978-3-319-17768-7/ebook). xxi, 408 p. (2015).
In this monograph, the authors present their results on various partial functional and neutral functional differential equations and inclusions in infinite dimensional spaces obtained in the last years. The book is organized in 13 chapters, an extensive bibliography and an index.
Chapter 1 contains the notations, definitions and auxiliary results including some theorems from the semigroups theory, fixed point theorems and the nonlinear alternatives of Leray-Schauder type in Banach spaces and Fréchet spaces which are used throughout the book. Chapter 2 deals with the existence and uniqueness of mild solutions for some first order classes of partial functional, neutral functional, integro-differential and neutral integro-differential equations on the positive line $$\mathbb{R}_+$$ with local or nonlocal conditions, where the past interval is bounded and the delay is finite. Chapter 3 gives sufficient conditions for the existence and uniqueness of mild solutions on the positive half line $$\mathbb{R}_+$$ for two classes of first order partial functional and neutral functional differential equations with infinite delay. The controllability for four partial functional equations and neutral functional equations is also investigated. Chapter 4 is focused on the existence of mild solutions for three perturbed partial functional and neutral functional evolution equations on the semi-infinite interval $$\mathbb{R}_+$$ with finite or infinite delay. In Chapter 5, the authors present sufficient conditions for the existence of mild solutions on the semi-infinite interval $$\mathbb{R}_+$$ for two classes of first order partial functional and neutral functional differential evolution inclusions with finite delay. Chapter 6 is concerned with the existence of mild solutions of two classes of partial functional and neutral functional evolution inclusions with infinite delay on the semi-infinite interval $$\mathbb{R}_+$$. Chapter 7 provides existence results for the mild solutions and extremal mild solutions of some first order classes of impulsive semi-linear functional differential inclusions with local or nonlocal conditions and finite delay in a separable Banach space. The controllability of a semi-linear functional differential inclusion with impulses and finite delay is also studied. In Chapter 8, the authors establish sufficient conditions for the existence of integral solutions and extremal integral solutions for two non-densely defined impulsive functional differential inclusions in separable Banach spaces with local or nonlocal conditions. The controllability of these inclusions with multi-valued jumps, and the topological structure of the solutions set are also investigated. Chapter 9 is devoted to the existence of mild solutions, integral solutions and extremal solutions for some classes of impulsive semi-linear functional equations and neutral functional equations with finite, infinite or state-dependent delay, and local or nonlocal conditions, where the linear operators from the equations are non-densely defined. Chapter 10 gives sufficient conditions for the existence of mild solutions and extremal mild solutions for two densely defined or non-densely defined impulsive functional differential inclusions and neutral functional differential inclusions in separable Banach spaces with infinite delay. The controllability of such inclusions is also studied. In Chapter 11, the authors present existence results for the mild solutions and integral solutions for some functional differential inclusions with infinite delay or state-dependent delay, and with multi-jumps in a Banach space. The controllability of an impulsive differential evolution inclusion with infinite delay is also investigated. Chapter 12 is focused on the global existence of mild solutions for four classes of functional differential equations and inclusions with finite or state-dependent delay. Finally, Chapter 13 is concerned with the global existence of mild solutions for two classes of second order semi-linear functional equations with finite or state-dependent delay.
The presented theoretical results are illustrated by some examples. This book will be useful to researchers and graduate students interested in functional evolution equations and inclusions.

##### MSC:
 34-02 Research exposition (monographs, survey articles) pertaining to ordinary differential equations 34K05 General theory of functional-differential equations 34K09 Functional-differential inclusions 34K30 Functional-differential equations in abstract spaces 34K40 Neutral functional-differential equations 47N20 Applications of operator theory to differential and integral equations 34K35 Control problems for functional-differential equations 34K45 Functional-differential equations with impulses
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