Oscillation of second-order functional differential equations with mixed nonlinearities and oscillatory potentials

Abstract

In this paper, we study oscillation of second-order functional differential equations with mixed nonlinearities

$$(p(t)x^{\prime }(t){)}^{\prime }+q(t)x(t-\tau )+\sum _{i=1}^n{q}_i(t)|x(t-\tau ){|}^{{\alpha }_i}\mathrm{sgn}x(t-\tau )=e(t)\text{,}$$

where $\tau ⩾0$, $p(t)\in C^1[0\text{,}\infty )$, $q(t)\text{,}{q}_i(t)\text{,}e(t)\in C[0\text{,}\infty )$, $p(t)>0$, ${\alpha }_1>\cdots >{\alpha }_m>1>{\alpha }_{m+1}>\cdots >{\alpha }_n>0(n>m⩾1)$. Without assuming that $q(t)$, $q_i(t)$ and $e(t)$ are nonnegative, the results given in [Y.G. Sun, F.W. Meng, Interval criteria for oscillation of second-order differential equations with mixed nonlinearities, Appl. Math. Comput. 198 (2008) 375–381] have been extended to the aforementioned functional differential equation.

Introduction

Consider the following second-order functional differential equation with mixed nonlinearities:

$$(p(t)x^{\prime }(t){)}^{\prime }+q(t)x(t-\tau )+\sum _{i=1}^n{q}_i(t)|x(t-\tau ){|}^{{\alpha }_i}\mathrm{sgn}x(t-\tau )=e(t)\text{,}$$

where $\tau ⩾0$, $p(t)$, $q(t)$, $q_i(t)\in C[0\text{,}\infty )$, $p(t)$ is positive and differentiable, and ${\alpha }_1>\cdots >{\alpha }_m>1>{\alpha }_{m+1}>\cdots >{\alpha }_n>0(n>m⩾1)$.

A solution $x(t)$ of Eq. (1) is said to be oscillatory if it is defined on some ray $[T\text{,}\infty )$ with $T⩾0$ and has unbounded set of zeros. Eq. (1) is said to be oscillatory if all solutions extendable throughout $[0\text{,}\infty )$ are oscillatory. When $q(t)$ and $q_i(t)$, $i=1\text{,}2\text{,}\dots \text{,}n$, are nonnegative, there are many oscillation results for Eq. (1) (see [8] and references cited therein). However, most of these results are established by assuming that there exists an oscillatory function $\varphi (t)$ such that ${\varphi }^{″}(t)=e(t)$.

For some particular cases of Eq. (1), many authors have devoted to the oscillation problem (see [1], [2], [4], [5], [6], [7], [8], [9], [10], [11], [12], [13], [14], [15], [16], [17], [18], [19], [20], [21], [22], [23], [24], [25]). Particularly, when $p(t)\equiv 1$, $q(t)\equiv 0$, and $n=1$, Eq. (1) reduces to the familiar Emden–Fowler equation

$$x^{″}(t)+q_1(t)|x(t-\tau ){|}^{{\alpha }_1}\mathrm{sgn}x(t-\tau )=e(t)\text{.}$$

When ${\alpha }_1⩾1$, the oscillation of Eq. (2) with oscillatory potentials was studied in [19]. For the particular case when $\tau =0$, the oscillation of Eq. (1) was investigated in [21], [22]. To the best of our knowledge, little has been known about oscillation of Eq. (1) when $q(t)$ and $q_i(t)$, $i=1\text{,}2\text{,}\dots \text{,}n$, are oscillatory. Here, our interest is to establish interval criteria for oscillation of Eq. (1) without assuming that $q(t)$ and $q_i(t)$ are nonnegative and there exists an oscillatory function $\varphi (t)$ such that ${\varphi }^{″}(t)=e(t)$. The arithmetic–geometric mean inequality in [3] plays a key role in the proof of the main result.