Leonid Berezansky

Nonoscillation Theory of Functional Differential Equations with Applications

1. Introduction to Oscillation Theory.- 2. Scalar Delay Differential Equations on Semiaxes.- 3. Scalar Delay Differential Equations on Semiaxis with Positive and Negative Coefficients.- 4. Oscillation of Equations with a Distributed Delay.- 5. Scalar Advanced and Mixed Differential Equations on Semiaxes.- 6. Neutral Differential Equations.- 7. Second Order Delay Differential Equations.- 8. Second Order Delay Differential Equations with Damping Terms.- 9. Vector Delay Differential Equations.- 10. Linearized Methods for Nonlinear Equations with a Distributed Delay.- 11. Nonlinear Models - Modifications of Delay Logistic Equations.- 12. First Order Linear Delay Impulsive Differential Equation.- 13. Second Order Linear Delay Impulsive Differential Equations.- 14. Linearized Oscillation Theory for Nonlinear Delay Impulsive Equations.- 15. Maximum Principles and Nonoscillation Intervals for First Order Volterra Functional Differential Equations.- 16. Systems of Functional Differential Equations on Finite Intervals.- 17. Nonoscillation Interval for n-th Order Functional Differential Equations.- Appendix A.- Appendix B.

On Impulsive Beverton-Holt Difference Equations and their Applications

The asymptotic properties of the impulsive Beverton-Holt difference equation where p is a fixed positive integer, are considered. The results are applied to an impulsive logistic equation with non-constant coefficients In particular, sufficient extinction and non-extinction conditions are obtained for both equations.

Sufficient conditions for the global stability of nonautonomous higher order difference equations

We present some explicit sufficient conditions for the global stability of the zero solution in nonautonomous higher order difference equations. The linear case is discussed in detail. We illustrate our main results with some examples. In particular, the stability properties of the equilibrium in a nonlinear model in macroeconomics is addressed.

On exponential stability of a linear delay differential equation with an oscillating coefficient

Abstract New explicit exponential stability conditions are obtained for the nonautonomous linear equation x ( t ) + a ( t ) x ( h ( t ) ) = 0 , where h ( t ) ≤ t and a ( t ) is an oscillating function. We apply the comparison method based on the Bohl–Perron type theorem. Coefficients and delays are not assumed to be continuous. Some real-world applications and several examples are also discussed.

On the Critical Case in Oscillation for Differential Equations with a Single Delay and with Several Delays

New nonoscillation and oscillation criteria are derived for scalar delay differential equations 𝑥(𝑡)

On oscillation properties of delay differential equations with positive and negative coefficients

Abstract For a scalar delay differential equation x (t)+a(t)x h(t) −b(t)x g(t) =0, a(t)⩾0, b(t)⩾0, h(t)⩽t, g(t)⩽t, a connection between the following properties is established: nonoscillation of the differential equation and the corresponding differential inequalities, positiveness of the fundamental function and existence of a nonnegative solution for a certain explicitly constructed nonlinear integral inequality. A comparison theorem and explicit nonoscillation and oscillation results are presented.

Oscillation of Equations with Distributed Delays

Chapter 4 deals with nonoscillation properties of scalar linear differential equations with a distributed delay. It is usually believed that equations with a distributed delay, which involve differential equations with several variable delays, integrodifferential equations and mixed equations with concentrated delays and integral terms, provide a more realistic description for models of population dynamics and mathematical biology in general.

A Final Result on the Oscillation of Solutions of the Linear Discrete Delayed Equation Δ()=−()(−) with a Positive Coefficient

A linear ( 𝑘 + 1 ) th-order discrete delayed equation Δ 𝑥 ( 𝑛 ) = − 𝑝 ( 𝑛 ) 𝑥 ( 𝑛 − 𝑘 ) where 𝑝 ( 𝑛 ) a positive sequence is considered for 𝑛 → ∞ . This equation is known to have a positive solution if the sequence 𝑝 ( 𝑛 ) satisfies an inequality. Our aim is to show that, in the case of the opposite inequality for 𝑝 ( 𝑛 ) , all solutions of the equation considered are oscillating for 𝑛 → ∞ .

On oscillation of a logistic equation with several delays

Abstract For a scalar delay logistic equation y (t)=y(t) ∑ k=1 m r k (t) 1− y(h k (t)) K , h k (t)⩽t, a connection between oscillating properties of this equation, the corresponding differential inequalities and the linear equation x (t)+ ∑ k=1 m r k (t)x(h k (t))=0, is established. Explicit nonoscillation and oscillation conditions are presented.

New Stability Conditions for Linear Differential Equations with Several Delays

New explicit conditions of asymptotic and exponential stability are obtained for the scalar nonautonomous linear delay differential equation ∑𝑥(𝑡)