Oscillations of higher order nonlinear functional differential equations with impulses

doi:10.1016/j.amc.2007.01.029

Applied Mathematics and Computation

Volume 190, Issue 1, 1 July 2007, Pages 370-381

https://doi.org/10.1016/j.amc.2007.01.029 Get rights and content

Abstract

The present paper is devoted to the investigation of the oscillation of a kind of very extensively higher order nonlinear functional differential equations with impulses, some interesting results are obtained. The results extend and improve the earlier publications. Two examples are given to illustrate the theory.

Introduction

The oscillations of classical second order nonlinear (linear) ODE (FDE) has been extensively studied in the literature, e.g., Refs. [1], [2], [3], [4], [5], [6], [7], [8], [9], [10].

However, impulsive effect likewise exists in a wide variety of evolutionary processes in which states are changed abruptly at certain moments of time, involving such field as medicine and biology, economics, mechanics, electronics, and telecommunications, etc. Many interesting results on impulsive effect have been gained in [11], [12], [13], [14], [15], [16], [17], [18], [19], [20], [21], [22], [23]. Those papers have only first or second order ODE (FDE) with impulses. But papers devoted to the study of the oscillations of differential equation of higher order with impulses are quite rare.

Papers devoted to the study of the oscillations of higher order nonlinear ODE with impulses has been done in [24]. But it is not considered with delay effect and the main difficulty for higher order FDE comes from every order differential coefficient relation. Therefore, it is necessary to consider both impulsive effect and delay effect on the oscillations of differential equation.

In the paper, we mainly study a kind of very extensively higher order nonlinear FDE with impulses under the conditions (A), (B), (C). We can always find some suitable impulse functions such that all the solutions of the equation can become oscillatory under the impulse control. The results extend and improve the earlier publications. Two example are given to illustrate the theory.

Section snippets

Main results

We consider the following systems: $\{\begin{matrix} [r (t) | x^{(2 n - 1)} (t) |^{α - 1} x^{(2 n - 1)} (t)]^{'} + f (t, x (t), x (t - τ)) = 0, t ⩾ t_{0}, t \neq t_{k}, \\ x^{(i)} (t_{k}^{+}) = g_{k (i)} (x^{(i)} (t_{k})), i = 0, 1, \dots, 2 n - 1, k = 1, 2 \dots, \\ x^{(i)} (t_{0}^{+}) = x_{0}^{(i)}, \\ x (t) = ϕ (t), t_{0} - τ ⩽ t ⩽ t_{0}, \end{matrix}$ where $\begin{matrix} x^{(i)} (t) = \lim_{h \to 0^{-}} \frac{x^{(i - 1)} (t + h) - x^{(i - 1)} (t)}{h}, \\ x^{(i)} (t^{+}) = \lim_{h \to 0^{+}} \frac{x^{(i - 1)} (t + h) - x^{(i - 1)} (t)}{h} . \end{matrix}$ $ϕ : [t_{0} - τ, t_{0}] \to R$ has at most finite discontinuous points of the first kind and is left continuous at these points. $α > 0, τ > 0, 0 < t_{0} < t_{1} < t_{2} < \dots < t_{k} < \dots, k = 1, 2, \dots, \lim_{k \to \infty} t_{k} = + \infty, x^{(0)} (t) = x (t)$ , n is a natural number. Here, we always assume that the following

Example

Example 1

Consider $\{\begin{matrix} x^{(2 n)} (t) + \frac{1}{4 t} x^{3} (t - \frac{1}{2}) = 0, t ⩾ \frac{1}{2}, t \neq k, k = 1, 2, \dots \\ x (k^{+}) = x (k), x^{(i)} (k^{+}) = \frac{k}{k + 1} x^{(i)} (k), i = 1, \dots, 2 n - 1 . \\ x (\frac{1}{2}) = x_{0}, x^{(i)} (\frac{1}{2}) = x_{0}^{(i)}, \\ x (t) = ϕ (t), t \in [\frac{1}{2}, 1] \end{matrix}$ where $a_{k}^{(0)} = b_{k}^{(0)} = 1, a_{k}^{(i)} = b_{k}^{(i)} = \frac{k}{k + 1}, i = 1, 2, \dots, 2 n - 1, p (t) = \frac{1}{4 t}, t_{k} = k, φ (x) = x^{3}, r (t) = 1, α = 1, t_{0} = \frac{1}{2}$ . It is obvious that the condition (A), (B) are satisfied, for condition (C) when $i > 1, a_{k}^{(i)} = b_{k}^{(i - 1)} = \frac{k}{k + 1}$ $(t_{1} - t_{0}) + (t_{2} - t_{1}) + (t_{3} - t_{2}) + \dots + (t_{m + 1} - t_{m}) + \dots = \frac{1}{2} + 1 + \dots + 1 + \dots = + \infty .$ When $i = 1, a_{k}^{(1)} = \frac{k}{k + 1}, b_{k}^{(0)} = 1$ $(t_{1} - t_{0}) + \frac{1}{2} (t_{2} - t_{1}) + \frac{1}{3} (t_{3} - t_{2}) + \dots + \frac{1}{m + 1} (t_{m + 1} - t_{m}) + \dots = \frac{1}{2} + \frac{1}{2} + \frac{1}{3} + \dots + \frac{1}{m + 1} + \dots = + \infty .$ From the above,the condition

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^☆: Foundation item: This work was partially supported by Natural Science Foundation of Guangdong (011471) and Natural Science Foundation of Guangdong Higher Education (0120), Science and Technology Plan Project of Guangzhou (2006J1-C0341).

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Oscillations of higher order nonlinear functional differential equations with impulses☆

Abstract

Introduction

Section snippets

Main results

Example

J. Math. Annl Appl.

J. Math. Anal. Appl.

J. Math. Anal. Appl.

J. Math. Anal. Appl.

J. Math. Anal. Appl.

On second-order nonlinear oscillations

Pacific J. Math.

Oscillation solutions of certain nonlinear differential equations of the second order

Mat. Fyz. Casopis Sloven. Akad. Vied.

Oscillation of solutions of second order nonlinear differential equations

Pacific J. Math.

On second order nonlinear oscillation

Funkcial. Ekavac.

Oscillation of solutions of second order nonlinear equations with signvariable coefficients

Differential’nye Uravneniya

A criterion for the oscillation of solutions of second order ordinary differential equations

Mat. Zametki

Some specifically nonlinear oscillation theorem

Mat. Zametki

A note on second order nonlinear oscillation

Math. Japan.