Asymptotic properties of solutions such as stability/ instability,oscillation/ nonoscillation, existence of solutions with specific asymptotics, maximum principles present a classical part in the theory of higher order functional differential equations. The use of these equations in applications is one of the main reasons for the developments in this field. The control in the mechanical processes leads to mathematical models with second order delay differential equations. Stability and stabilization of second order delay equations are one of the main goals of this book. The book is based on the authors’ results in the last decade.
- Stability, oscillatory and asymptotic properties of solutions are studied in correlation with each other.
- The first systematic description of stability methods based on the Bohl-Perron theorem.
- Simple and explicit exponential stability tests.
In this book, various types of functional differential equations are considered: second and higher orders delay differential equations with measurable coefficients and delays, integro-differential equations, neutral equations, and operator equations. Oscillation/nonoscillation, existence of unbounded solutions, instability, special asymptotic behavior, positivity, exponential stability and stabilization of functional differential equations are studied. New methods for the study of exponential stability are proposed. Noted among them inlcude the W-transform (right regularization), a priory estimation of solutions, maximum principles, differential and integral inequalities, matrix inequality method, and reduction to a system of equations.
The book can be used by applied mathematicians and as a basis for a course on stability of functional differential equations for graduate students.
Table of Contents
Introduction to Stability Methods.
Stability: A priori Estimation Method.
Stability: Reduction to a System of Equations.
Stability: W-transform Method I.
Stability: W-transform Method II.
Exponential Stability for Equations with Positive and Negative Coeffcients.
Connection Between Nonoscillation and Stability.
Stabilization for Second Order Delay Models, Simple Delay Control.
Stabilization by Delay Distributed Feedback Control.
Wronskian of Neutral FDE and Sturm Separation Theorem.
Vallee-Poussin Theorem for Delay and Neutral DE.
Sturm Theorems and Distance between Adjacent Zeros.
Unbounded Solutions and Instability of Second Order DDE.
Upper and Lower Estimates of Distances Between Zeros and Floquet Theory for Second Order DDE. Distribution of Zeros and Unboundedness of Solutions to Partial DDE.
Second Order Equations: Oscillation and Boundary Value Problems.
Stability of Third Order DDE.
Operator Differential Equations.
Properties A and B of Equations with a Linear Minorant.
On Kneser-Type Solutions.
Monotonically Increasing Solutions.
Specific Properties of FDE.
A Useful Theorems from Analysis.
B Functional-differential Equations.
Leonid Berezansky studied mathematics at Novosibirsk State University, USSR. In 1980, he received his Ph.D. degree in Mathematics, at the Institute of Mathematics and Mechanics of the Academy of Sciences of Kazakhstan, Alma-Ata, and, in 1991, a Doctor of Science degree at the Institute of Mathematics Academy of Science of USSR, Sverdlovsk with the thesis, Existence of Solutions and Stability of Linear Functional Di erential Equations. In 1991- 1992 he was a full-professor at the Novosibirsk Water Transport Institute, USSR. In 1993 he got a professor tenure (Kameya Programm) at Ben-Gurion University of the Negev, now retired. He is the author of one monograph and more then 150 papers on mathematics and is an associate editor in several scientific journals.
Alexander Domoshnitsky studied mathematics at Perm State University, USSR. He received his Ph.D. degree in Mathematics in 1984 at Tbilisi State University, USSR (now Georgia). In 1984-1990, he held positions of Senior Lecturer and Dozent (Associate Professor) of Perm Polytechnical Institute, Perm, USSR. In 1998-2009, he was Senior Lecturer and Associate Professor of the Department of Mathematics and Computer Sciences, Ariel University Center of Samaria, Ariel, Israel. Since 2009 he is a Professor of the Department of Mathematics, Ariel University, Ariel, Israel. He is the author of one monograph and more that 120 papers on mathematics. He is associate editor in several scientific journals.
Roman Koplatadze studied at Tbilisi State University, USSR (now Georgia). He received his Ph.D. degree on Mathematics in 1974 in Tbilisi State University, Georgia, and Doctor of Sciences in 1995 A. Razmadze Mathematical Institute of Georgian Academy of Sciences. In 1970 - 1976 he held the position of junior researcher, in the Department of Ordinary Differential Equations of I. Vekua Institute of Applied Mathematics of Tbilisi State University; 1976-1983 Senior Researcher of the same department; 1983-1990 Leading Researcher of the same department; 1991-1995 Head of the same department; 1996-2006 Leading Researcher of A. Razmadze Mathematical Institute of Georgian Academy of Sciences, Professor; since 2006-present I. Vekua Institute of Applied Mathematics of Iv, Javakhishvili Tbilisi State University, Leading of the same department and Professor at the Department of Mathematics of Iv, Javakhishvili Tbilisi State University. He is the author of two monographs and more that 130 papers on mathematics. He is deputy editor of the journal Seminar of Ilia Vekua Institute of Applied Mathematics.