Stability of vector differential delay equations [electronic resource] / Michael I. Gilʹ

Differential equations with delay naturally arise in various applications, such as control systems, viscoelasticity, mechanics, nuclear reactors, distributed networks, heat flows, neural networks, combustion, interaction of species, microbiology, learning models, epidemiology, physiology, and many o...

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Bibliographic Details
Online Access:	Full Text (via Springer)
Main Author:	Gilʹ, M. I. (Mikhail Iosifovich)
Format:	Electronic eBook
Language:	English
Published:	Basel : London : Birkhäuser ; Springer [distributor], 2013.
Series:	Frontiers in mathematics.
Subjects:	Delay differential equations.

Table of Contents:

Preliminaries
Some Results of the Matrix Theory
General Linear Systems
Time-invariant Linear Systems with Delay
Properties of Characteristic Values
Equations Close to Autonomous and Ordinary Differential Ones
Periodic Systems
Linear Equations with Oscillating Coefficients
Linear Equations with Slowly Varying Coefficients
Nonlinear Vector Equations
Scalar Nonlinear Equations
Forced Oscillations in Vector Semi-Linear Equations
Steady States of Differential Delay Equations
Multiplicative Representations of Solutions.
Machine generated contents note: 1. Preliminaries
1.1. Banach and Hilbert spaces
1.2. Examples of normed spaces
1.3. Linear operators
1.4. Ordered spaces and Banach lattices
1.5. abstract Gronwall lemma
1.6. Integral inequalities
1.7. Generalized norms
1.8. Causal mappings
1.9. Compact operators in a Hilbert space
1.10. Regularized determinants
1.11. Perturbations of determinants
1.12. Matrix functions of bounded variations
1.13. Comments
2. Some Results of the Matrix Theory
2.1. Notations
2.2. Representations of matrix functions
2.3. Norm estimates for resolvents
2.4. Spectrum perturbations
2.5. Norm estimates for matrix functions
2.6. Absolute values of entries of matrix functions
2.7. Diagonalizable matrices
2.8. Matrix exponential
2.9. Matrices with non-negative off-diagonals
2.10. Comments
3. General Linear Systems
3.1. Description of the problem
3.2. Existence of solutions
3.3. Fundamental solutions
3.4. generalized Bohl-Perron principle
3.5. Lp-version of the Bohl-Perron principle
3.6. Equations with infinite delays
3.7. Proof of Theorem 3.6.1
3.8. Equations with continuous infinite delay
3.9. Comments
4. Time-invariant Linear Systems with Delay
4.1. Statement of the problem
4.2. Application of the Laplace transform
4.3. Norms of characteristic matrix functions
4.4. Norms of fundamental solutions of time-invariant systems
4.5. Systems with scalar delay-distributions
4.6. Scalar first-order autonomous equations
4.7. Systems with one distributed delay
4.8. Estimates via determinants
4.9. Stability of diagonally dominant systems
4.10. Comments
5. Properties of Characteristic Values
5.1. Sums of moduli of characteristic values
5.2. Identities for characteristic values
5.3. Multiplicative representations of characteristic functions
5.4. Perturbations of characteristic values
5.5. Perturbations of characteristic determinants
5.6. Approximations by polynomial pencils
5.7. Convex functions of characteristic values
5.8. Comments
6. Equations Close to Autonomous and Ordinary Differential Ones
6.1. Equations "close" to ordinary differential ones
6.2. Equations with small delays
6.3. Nonautomomous systems "close" to autonomous ones
6.4. Equations with constant coefficients and variable delays
6.5. Proof of Theorem 6.4.1
6.6. fundamental solution of equation (4.1)
6.7. Proof of Theorem 6.6.1
6.8. Comments
7. Periodic Systems
7.1. Preliminary results
7.2. main result
7.3. Norm estimates for block matrices
7.4. Equations with one distributed delay
7.5. Applications of regularized determinants
7.6. Comments
8. Linear Equations with Oscillating Coefficients
8.1. Vector equations with oscillating coefficients
8.2. Proof of Theorem 8.1.1
8.3. Scalar equations with several delays
8.4. Proof of Theorem 8.3.1
8.5. Comments
9. Linear Equations with Slowly Varying Coefficients
9.1. "freezing" method
9.2. Proof of Theorem 9.1.1
9.3. Perturbations of certain ordinary differential equations
9.4. Proof of Theorems 9.3.1
9.5. Comments
10. Nonlinear Vector Equations
10.1. Definitions and preliminaries
10.2. Stability of quasilinear equations
10.3. Absolute Lp-stability
10.4. Mappings defined on Ω (g) [∩] L2
10.5. Exponential stability
10.6. Nonlinear equations "close" to ordinary differential ones
10.7. Applications of the generalized norm
10.8. Systems with positive fundamental solutions
10.9. Nicholson-type system
10.10. Input-to-state stability of general systems
10.11. Input-to-state stability of systems with one delay in linear parts
10.12. Comments
11. Scalar Nonlinear Equations
11.1. Preliminary results
11.2. Absolute stability
11.3. Aizerman-Myshkis problem
11.4. Proofs of Lemmas 11.3.2 and 11.3.4
11.5. First-order nonlinear non-autonomous equations
11.6. Comparison of Green's functions to second-order equations
11.7. Comments
12. Forced Oscillations in Vector Semi-Linear Equations
12.1. Introduction and statement of the main result
12.2. Proof of Theorem 12.1.1
12.3. Applications of matrix functions
12.4. Comments
13. Steady States of Differential Delay Equations
13.1. Systems of semilinear equations
13.2. Essentially nonlinear systems
13.3. Nontrivial steady states
13.4. Positive steady states
13.5. Systems with differentiable entries
13.6. Comments
14. Multiplicative Representations of Solutions
14.1. Preliminary results
14.2. Volterra equations
14.3. Differential delay equations
14.4. Comments
Appendix A General Form of Causal Operators
Appendix B Infinite Block Matrices
B.1. Definitions
B.2. Properties of π-Volterra operators
B.3. Resolvents of π-triangular operators
B.4. Perturbations of block triangular matrices
B.5. Block matrices close to triangular ones
B.6. Diagonally dominant block matrices
B.7. Examples.

Stability of vector differential delay equations [electronic resource] / Michael I. Gilʹ

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