# Oscillation of Equations with Distributed Delays

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## Abstract

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.

Nonoscillation conditions are obtained in this chapter for a general equation with a distributed delay and are applied to derive nonoscillation and oscillation conditions for integrodifferential and mixed differential equations. Moreover, new nonoscillation conditions are obtained for differential equations with concentrated delays.

This chapter presents comparison theorems which allow in particular to compare oscillation properties of different classes of equations. Using these results, nonoscillation conditions for an integrodifferential equation can be deduced from known nonoscillation conditions for equations with concentrated delays. This chapter includes nonoscillation conditions for equations with positive and negative coefficients and results on the existence of slowly oscillating solutions.

## Keywords

Delay Distribution Contralateral Delay Nonoscillation Conditions Integrodifferential Equations Scalar Linear Differential Equation## References

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