Oscillation, Nonoscillation, Stability and Asymptotic Properties for Second and Higher Order Functional Differential Equations

Cover
Half Title
Title Page
Copyright Page
Contents
Authors
Preface
1. Introduction to Stability Methods
	1.1 Introduction
	1.2 Preliminaries
	1.3 A priori estimation method
		1.3.1 Delay-independent conditions
		1.3.2 Delay-dependent conditions
	1.4 Reduction to a system of differential equations
	1.5 W-transform method
		1.5.1 Delay-independent conditions
		1.5.2 Delay-dependent conditions
	1.6 Remarks and exercises
		1.6.1 Possible topics for a course of stability FDE
		1.6.2 Exercises
2. Stability: A priori Estimation Method
	2.1 Introduction
	2.2 Preliminaries
	2.3 Estimation of solutions
		2.3.1 Estimates of x
		2.3.2 Estimates of x
		2.3.3 Estimate of  x
	2.4 Exponential stability conditions
	2.5 Some generalizations
		2.5.1 Equations with several delays
		2.5.2 Equation with integral terms
		2.5.3 Equation with distributed delays
	2.6 Equations with perturbations by a damping term
		2.6.1 Estimation of solutions
		2.6.2 Exponential stability conditions
	2.7 Neutral differential equations
		2.7.1 Introduction and preliminaries
		2.7.2 Explicit stability conditions
	2.8 Remarks and open problems
3. Stability: Reduction to a System of Equations
	3.1 Introduction
	3.2 Application of M-matrix
		3.2.1 Introduction
		3.2.2 Equations without delay in damping terms
		3.2.3 Equations with delay in damping terms
	3.3 1+1/e stability conditions
		3.3.1 Introduction
		3.3.2 Main results
	3.4 Nonlinear equations
	3.5 Sunflower model and its modifications
	3.6 Remarks and open problems
4. Stability: W-transform Method I
	4.1 Introduction and preliminaries
	4.2 Main results
		4.2.1 Equations without delays in the damping terms
		4.2.2 Equations with delays in the damping terms
	4.3 Remarks and some topics for future research
5. Stability: W-transform Method II
	5.1 Introduction
	5.2 Formulations of main results
	5.3 Values of integrals of the modulus of Cauchy functions for auxiliary equations
	5.4 Proofs of main theorems
	5.5 Comments and open problems
6. Exponential Stability for Equations with Positive and Negative Coefficients
	6.1 Introduction
	6.2 Positivity of the Cauchy functions and stability
		6.2.1 Tests of positivity
		6.2.2 Auxiliary results
		6.2.3 Main results
	6.3 Application of W-method
		6.3.1 Main results
		6.3.2 Proofs of main theorems
	6.4 Transformations to equations with a damping term
		6.4.1 Delay differential equations
		6.4.2 Integro-differential equations and equations with distributed delays
		6.4.3 Equation with a damping term
	6.5 Remarks and open problems
7. Connection Between Nonoscillation and Stability
	7.1 Introduction
	7.2 Preliminaries
	7.3 Nonoscillation criteria
	7.4 Exponential stability of delay differential equations
	7.5 Exponential stability of integro-differential equations and equations with distributed delays
	7.6 A priori esimation method
		7.6.1 Introduction
		7.6.2 Estimates of x, x, x
		7.6.3 Exponential stability conditions
	7.7 Conclusions and open problems
8. Stabilization for Second Order Delay Models, Simple Delay Control
	8.1 Introduction
	8.2 Preliminaries
	8.3 Damping control
	8.4 Classical proportional control
	8.5 Summary
9. Stabilization by Delay Distributed Feedback Control
	9.1 Introduction
	9.2 Impossibility of stabilization by the control (9.3) in the case of K1 (t,s) = β1e−α1 (t−s) and m = 1
	9.3 About stability of model differential equations
	9.4 Cauchy function of the equation (9.15)
	9.5 Stabilization by the control in the form (9.3) in the case of controls with bounded memory
	9.6 Stabilization by the control in the form (9.3) in the case of controls with delays in upper limits
	9.7 Stability of integro-differential equations with variable coefficients
	9.8 Remarks
10. Wronskian of Neutral FDE and Sturm Separation Theorem
	10.1 Homogeneous functional differential equation
	10.2 Wronskian of the fundamental system for neutral functional differential equation
	10.3 Nonvanishing Wronskian through small delays and small differences between delays for neutral delay equations
	10.4 Sturm separation theorems for delay neutral equations through small delays and small difference between delays
11. Vallee-Poussin Theorem for Delay and Neutral DE
	11.1 Introduction
	11.2 Theorem about six equivalences
	11.3 Remarks
12. Sturm Theorems and Distance Between Adjacent Zeros
	12.1 Introduction
	12.2 Sturm separation theorem for binomial delay differential equation with nondecreasing deviation
	12.3 Distance between zeros of solutions and Sturm separation theorem on this basis
	12.4 Nondecreasing Wronskian
	12.5 Distance between zeros of solutions and Sturm theorem for neutral equations
	12.6 Sturm separation theorem through difference between delays
		12.6.1 Introduction
		12.6.2 Main results
		12.6.3 Proofs
	12.7 Sturm separation theorem for integro-differential equation x′′ (t) +Δ∫h (t) Δ (t) K (t,s) x (s) ds=0
	12.8 A possibility to preserve oscillation properties of binomial equation for second order equation x′′(t) + (Qx)(t) = 0 with general operator Q
	12.9 Sturm separation theorem for neutral equation with wise constant deviation of argument
	12.10 Sturm theorem for integro-differential equation x ′ ′ (t) + Δ∫0h (t) p (t) q (s) x (g (s)) ds=0
	12.11 Remarks
13. Unbounded Solutions and Instability of Second Order DDE
	13.1 Introduction
	13.2 Preliminaries
	13.3 Main results
	13.4 Growth of Wronskian and existence of unbounded solutions
	13.5 Estimates of Wronskian
	13.6 Proofs and corollaries
	13.7 Some other instability results
		13.7.1 Asymptotically small coefficients
		13.7.2 Application of positivity of the fundamental solution
		13.7.3 Equation with a negative damping term
		13.7.4 Reducing to a system of two first order equations
	13.8 Remarks
14. Upper and Lower Estimates of Distances Between Zeros and Floquet Theory for Second Order DDE
	14.1 Introduction
	14.2 Periodic problem
	14.3 Upper estimates of distance between two adjacent zeros
	14.4 Unboundedness of all solutions on the basis of Floquet theory and distances between zeros
	14.5 Remarks
15. Distribution of Zeros and Unboundedness of Solutions to Partial DDE
	15.1 Introduction
	15.2 Zeros and unboundedness of solutions
	15.3 Proofs
	15.4 Remarks
16. Second Order Equations: Oscillation and Boundary Value Problems
	16.1 Introduction
	16.2 Oscillation of second order linear delay differential equation
		16.2.1 Introduction
		16.2.2 Preliminary lemmas
		16.2.3 Oscillations caused by the delay
		16.2.4 General oscillation criteria
		16.2.5 Oscillations due to the second order nature of the equation (16.1)
	16.3 Second order homogeneous nonstability type differential equations
		16.3.1 On a singular boundary value problem
		16.3.2 Existence of bounded solutions
	16.4 Comments
17. Stability of Third Order DDE
	17.1 Introduction
	17.2 Preliminaries
	17.3 Cauchy function of an autonomous third order ordinary differential equation
	17.4 Stability of third order delay equations
	17.5 Proofs
	17.6 Conclusions, discussion and some topics for future research
18. Operator Differential Equations
	18.1 Some auxiliary statements
		18.1.1 Preliminary definitions
		18.1.2 On some classes of nonoscillatory functions
		18.1.3 On some classes of mappings from C (R+;R) into Lloc (R+;R)
	18.2 Comparison theorems
		18.2.1 Minorant case
		18.2.2 Superposition case
	18.3 Sufficient conditions
		18.3.1 Ineffective sufficient conditions
		18.3.2 Effective sufficient conditions
	18.4 Necessary and sufficient conditions
		18.4.1 Effective conditions
19. of Equations with a Linear Minorant
	19.1 Linear differential inequalities with a deviating argument
		19.1.1 Auxiliary lemmas
		19.1.2 On solutions of differential inequalities
	19.2 Linear differential inequalities with property A (B)
		19.2.1 Equations with property A
		19.2.2 Equations with property B
	19.3 Equations with a linear minorant having properties A and B
		19.3.1 Some auxiliary lemmas
		19.3.2 Functional differential equations with a linear minorant having properties A and B
		19.3.3 Sufficient conditions for the existence of a nonoscillatory solution
20. On Kneser-Type Solutions
	20.1 Some auxiliary statements
		20.1.1 On nonincreasing solutions
	20.2 On the existence of Kneser-type solutions
		20.2.1 Functional differential equations with linear minorant
		20.2.2 Linear inequalities with deviated arguments
		20.2.3 Nonlinear equations
21. Monotonically Increasing Solutions
	21.1 Auxiliary statements
		21.1.1 Some auxiliary lemmas
	21.2 On monotonically increasing solutions
		21.2.1 Equation with a linear minorant
		21.2.2 Differential inequalities with deviating arguments
		21.2.3 Nonlinear equations
22. Specific Properties of FDE
	22.1 Equations with property A
		22.1.1 Nonlinear equations
		22.1.2 Equations with a linear minorant
	22.2 Equations with property B
		22.2.1 Nonlinear equations
		22.2.2 Equations with a linear minorant
	22.3 Oscillatory equations
		22.3.1 Equations with a linear minorant
		22.3.2 Equations of the Emden-Fowler type
	22.4 Existence of an oscillatory solution
		22.4.1 Existence of a proper solution
		22.4.2 Existence of a monotonically increasing solution
		22.4.3 Existence of a proper oscillatory solution
Appendix A: Useful Theorems from Analysis
	A.1 Vector spaces
	A.2 Functional spaces
	A.3 Linear operators in functional spaces
	A.4 Nonlinear operators
	A.5 Gronwall-Bellman and Coppel inequalities
Appendix B: Functional-differential Equations
	B.1 Linear functional differential equations
		B.1.1 Differential equations with several concentrated delays
		B.1.2 Integro-differential equations with delays
		B.1.3 Equations with a distributed delay
		B.1.4 Second order scalar delay differential equations
	B.2 Nonlinear delay differential equations
	B.3 Stability theorems
	B.4 Nonoscillation results
Bibliography
Index