Oscillation criteria for second-order nonlinear differential equations with damping

Abstract

For a second-order nonlinear differential equation with a damping term, we obtain new oscillation criteria without an assumption that has been required for related results reported in the literature over the last two decades. Two approaches refining the standard integral averaging technique suggested in the paper can be used to derive a variety of simpler oscillation theorems for different classes of nonlinear differential equations, and both sets of oscillation criteria established in the paper are of independent interest. Several examples are provided to illustrate the relevance of the new theorems.

Introduction

In this paper, we are concerned with the problem of oscillation of a nonlinear second-order differential equation with damping

$${\left(r\left(t\right)x^{\prime }\left(t\right)\right)}^{\prime }+p\left(t\right)x^{\prime }\left(t\right)+q\left(t\right)f\left(x\left(t\right)\right)=0\text{,}$$

where $t\ge t_0>0,r\left(t\right)\in C^1([t_0,\infty );(0,\infty )),p\left(t\right),q\left(t\right)\in C([t_0,\infty );\mathbb{R}),f\left(x\right)\in C(\mathbb{R};\mathbb{R})$ and $xf\left(x\right)>0$ for $x\ne 0$. Throughout the paper, we assume that solutions of (1) exist for any $t\ge t_0$. A solution $x\left(t\right)$ of Eq. (1) is called oscillatory if it has arbitrarily large zeros; otherwise we call it non-oscillatory. We call differential equation (1) oscillatory if all its solutions oscillate.

The theory of oscillation is an important branch of the qualitative theory of differential equations. Its foundations were laid down by the well-known results regarding zeros of solutions of self-adjoint second-order differential equations published in 1836 by Sturm [1]. Since then, oscillatory properties of solutions to different classes of linear and nonlinear ordinary, functional, partial, discrete, impulsive differential equations have attracted the attention of many researchers. A raised interest to this topic has been reflected, for instance, in monographs on oscillation by Agarwal et al. [2], [3], Erbe et al. [4], Győri and Ladas [5], Kreith [6], Ladde et al. [7], Swanson [8]; chapters in monographs on differential equations by Bellman [9], Coppel [10], Kiguradze and Chanturiya [11]; survey papers by Wong [12], [13], [14], etc. In the last three decades, oscillation of differential equations with damping has become an important area of research due to the fact that such equations arise in many real life problems. We refer the reader to the research papers [15], [16], [17], [18], [19], [20], [21], [22], [23], [24], [25], [26], [27], [28], [29], [30], [31], [32], [33], [34], [35], [36], [37], [38], [39], [40], [41], [42], [43], [44], [45], [46], [47], [48] and the references cited therein.

Oscillation of nonlinear differential equations with a linear damping term of the form (1) has been addressed in the papers by Grace and Lalli [25], Elabbassy et al. [18], Kirane and Rogovchenko [30], Li and Agarwal [32], Li et al. [33], Rogovchenko [40], Sun [41], Tiryaki and Zafer [42], Yang [45], to mention a few, whereas oscillation criteria for the equation

$${\left(r\left(t\right)\psi \left(x\left(t\right)\right)x^{\prime }\left(t\right)\right)}^{\prime }+p\left(t\right)x^{\prime }\left(t\right)+q\left(t\right)f\left(x\left(t\right)\right)=0$$

were suggested, for instance, in the papers by Grace [19], [21], Grace and Lalli [22], [23], Kirane and Rogovchenko [29], Manojlović [34] and others. Several authors were concerned with equations with nonlinear damping terms; see, for example, Baker [15], Bobisud [16], Grace et al. [26], Elabbasy et al. [18], Rogovchenko and Rogovchenko [37], [38], Tiryaki and Zafer [43].

To establish oscillation criteria, one usually uses either an integral averaging technique involving integrals and weighted integrals of coefficients of a given differential equation (see, for instance, [18], [19], [20], [21], [22], [23], [24], [25], [26], [27], [29], [30], [31], [33], [34], [37], [38], [39], [40], [42], [44], [45], [46], [47]), or interval oscillation methods which use the information on weighted averages of coefficients over subintervals of $\mathbb{R}$ rather than on the entire real line (see [32], [41], [43], [48]). Other efficient approaches include comparison theorems and linearization techniques; see, for example, the monographs by Agarwal et al. [2], [3] and the references quoted there.

In 1986, Yan [45] established an important extension of the celebrated Kamenev oscillation criterion [28] for the equation with linear damping term

$${\left[r\left(t\right)x^{\prime }\left(t\right)\right]}^{\prime }+p\left(t\right)x^{\prime }\left(t\right)+q\left(t\right)x\left(t\right)=0\text{.}$$

Theorem 1 [45, Theorem 2]

Suppose that there exist a positive continuously differentiable function $h\left(t\right)$ on $[t_0,\infty )$ and $\alpha \in (1,\infty )$ such that

$$\left(C_1\right)\underset{t\to \infty }{lim\; sup}{t}^{-\alpha }{\int }_{t_0}^t{(t-\tau )}^{\alpha }h\left(\tau \right)q\left(\tau \right)d\tau <\infty \text{,}$$

there exists a continuous function $\varphi \left(t\right)$ on $[t_0,\infty )$ such that

$$\left(C_2\right)\underset{t\to \infty }{lim\; inf}{t}^{-\alpha }{\int }_s^t[{(t-\tau )}^{\alpha }h\left(\tau \right)q\left(\tau \right)-\frac{1}{4}{(t-\tau )}^{\alpha -2}\frac{r\left(\tau \right)}{h\left(\tau \right)}{[(t-\tau )\frac{h\left(\tau \right)p\left(\tau \right)}{r\left(\tau \right)}+\alpha h\left(\tau \right)-(t-\tau )h^{\prime }\left(\tau \right)]}^2]d\tau \ge \varphi \left(s\right)\text{,}$$

for all $t>t_0$ , and

$$\left(C_3\right)\underset{t\to \infty }{lim}{\int }_{t_0}^t\frac{{\varphi }_{+}^2\left(\tau \right)}{h\left(\tau \right)r\left(\tau \right)}d\tau =\infty \text{,}$$

where ${\varphi }_{+}\left(t\right)=max(\varphi \left(t\right),0)$ . Then Eq.(3)is oscillatory.

Using general weight functions from the class $\mathcal{W}$, introduced in the paper by Philos [36] instead of Kamenev’s weight function ${(t-\tau )}^{\alpha }$ exploited by Yan, Grace [21] extended Theorem 1 to Eqs. (1), (2). Let $D=\left\{(t,s)\right|-\infty <s\le t<+\infty \}$. We say that a continuous function $H(t,s),H:D\to [0,+\infty )$, belongs to the class $\mathcal{W}$ if:

• $H(t,t)=0$ and $H(t,s)>0$ for $-\infty <s<t<+\infty$;

• $H$ has a continuous partial derivative $\partial H/\partial s$ satisfying, for some $h\in L_{\mathrm{loc}}(D,\mathbb{R})$, the condition $\partial H/\partial s=-h(t,s){\left(H(t,s)\right)}^{1/2}$.

Theorem 2 [21, Theorem 6]

Suppose that $f^{\prime }\left(x\right)\ge K>0$ for $x\ne 0$ , and let $H\in \mathcal{W}$ satisfy $$

$$\left(C_4\right)0<\underset{s\ge t_0}{inf}\left[\underset{t\to \infty }{lim\; inf}\frac{H(t,s)}{H(t,t_0)}\right]\le \infty \text{.}$$

Assume further that there exist functions $v\in C^1([t_0,\infty );(0,\infty ))$ and $\varphi \in C([t_0,\infty );\mathbb{R})$ such that

$$\left(C_5\right)\underset{t\to \infty }{lim\; sup}\frac{1}{H(t,t_0)}{\int }_{t_0}^{t}r\left(s\right)v\left(s\right){[h(t,s)-\gamma \left(s\right){\left(H(t,s)\right)}^{1/2}]}^{2}ds<\infty \text{,}$$

for all $t>t_0$,

$$\left(C_6\right)\underset{t\to \infty }{lim\; sup}\frac{1}{H(t,T)}{\int }_T^t[H(t,s)v\left(s\right)q\left(s\right)-\frac{r\left(s\right)v\left(s\right)}{4K}{(h(t,s)-\gamma \left(s\right){\left(H(t,s)\right)}^{1/2})}^2]ds\ge \varphi \left(T\right)\text{,}$$

for every $T\ge t_0$ , and

$$\left(C_7\right)\underset{t\to \infty }{lim}{\int }_{t_0}^t\frac{{\varphi }_{+}^2\left(\tau \right)}{v\left(\tau \right)r\left(\tau \right)}d\tau =\infty \text{,}$$

where $\gamma \left(t\right)=(r\left(t\right)v^{\prime }\left(t\right)-p\left(t\right)v\left(t\right))/\left(r\left(t\right)v\left(t\right)\right)$ and ${\varphi }_{+}\left(t\right)$ is as inTheorem 1. Then Eq.(1)is oscillatory.

Theorem 2 is a powerful general test for oscillation which has been further refined in the recent paper by the first author [40]; cf. also [42].

Theorem 3 [40, Theorem 2]

Let the functions $f$ and $H$ be as inTheorem 2, and suppose that (C₄)  holds. Assume further that there exist functions $g\in C^1([t_0,\infty );\mathbb{R})$ and $\varphi \in C([t_0,\infty );\mathbb{R})$ such that

$$\left(C_8\right)\underset{t\to \infty }{lim\; sup}\frac{1}{H(t,t_0)}{\int }_{t_0}^{t}v\left(s\right)r\left(s\right){(h(t,s)+\frac{p\left(s\right)}{r\left(s\right)}{\left(H(t,s)\right)}^{1/2})}^{2}ds<\infty \text{,}$$

for all $t>t_0$,

$$\left(C_9\right)\underset{t\to \infty }{lim\; sup}\frac{1}{H(t,T)}{\int }_T^t[H(t,s)\psi \left(s\right)-\frac{v\left(s\right)r\left(s\right)}{4K}{(h(t,s)+\frac{p\left(s\right)}{r\left(s\right)}{\left(H(t,s)\right)}^{1/2})}^2]ds\ge \varphi \left(T\right)\text{,}$$

for any $T\ge t_0$ , where ${\varphi }_{+}\left(t\right)$ is as inTheorem 1, $v\left(t\right)=exp(-2{\int }^{t}g\left(u\right)du)$ and $\psi \left(t\right)=K^{-1}v\left(t\right)(Kq\left(t\right)-p\left(t\right)g\left(t\right)-{\left[r\left(t\right)g\left(t\right)\right]}^{\prime }+r\left(t\right)g^2\left(t\right))$ . If

$$\left(C_{10}\right)\underset{t\to \infty }{lim\; sup}{\int }_{t_0}^t\frac{{\varphi }_{+}^2\left(\tau \right)}{v\left(\tau \right)r\left(\tau \right)}d\tau =\infty \text{,}$$

Eq.(1)is oscillatory.

As demonstrated in [40], conditions in Theorem 3 are less restrictive than those in Theorem 2 and oscillation theorems in the cited paper are more flexible compared to those due to Grace [21] and Grace and Lalli [24] because no specific assumptions on the damping coefficient $p\left(t\right)$ are imposed. Theorem 3 also proves to be more efficient compared to the result due to Li et al. [33, Theorem 2.2] obtained by using a similar technique.

The purpose of this paper is to further improve Theorem 1, Theorem 2, Theorem 3, as well as other related results regarding oscillation of Eq. (1) by refining the standard integral averaging technique. We suggest two different approaches which allow one to remove condition (C₈) in Theorem 3 and, respectively, conditions (C₅) in Theorem 2 and (C₁) in Theorem 1. A modified integral averaging technique enables one to simplify essentially the proofs of oscillation criteria. A number of carefully selected examples are inserted in the text to illustrate the main results.