Oscillation criteria for second-order nonlinear dynamic equations

Abstract

This paper concerns the oscillation of solutions to the second-order dynamic equation

$${(r(t)x^{\mathrm{\Delta }}(t))}^{\mathrm{\Delta }}+p(t)x^{\mathrm{\Delta }}(t)+q(t)f(x^{\sigma }(t))=0,$$

on a time scale $\mathbb{T}$ which is unbounded above. No sign conditions are imposed on $r(t)$, $p(t)$, and $q(t)$. The function $f\in C(\mathbb{R},\mathbb{R})$ is assumed to satisfy $xf(x)>0$ and $f^{\mathrm{\prime }}(x)>0$ for $x\ne 0$. In addition, there is no need to assume certain restrictive conditions and also the both cases

$${\int }_{t_0}^{\mathrm{\infty }}\frac{\mathrm{\Delta }t}{r(t)}=\mathrm{\infty }\text{and}{\int }_{t_0}^{\mathrm{\infty }}\frac{\mathrm{\Delta }t}{r(t)}<\mathrm{\infty }$$

are considered. Our results will improve and extend results in (Baoguo et al. in Can. Math. Bull. 54:580-592, 2011; Bohner et al. in J. Math. Anal. Appl. 301:491-507, 2005; Hassan et al. in Comput. Math. Anal. 59:550-558, 2010; Hassan et al. in J. Differ. Equ. Appl. 17:505-523, 2011) and many known results on nonlinear oscillation. These results have significant importance to the study of oscillation criteria on discrete time scales such as $\mathbb{T}=\mathbb{Z}$, $\mathbb{T}=h\mathbb{Z}$, $h>0$, or $\mathbb{T}=\{t:t=q^k,k\in {\mathbb{N}}_0,q>1\}$ and the space of harmonic numbers $\mathbb{T}=H_n$. Some examples illustrating the importance of our results are also included.

MSC:34K11, 39A10, 39A99.

1 Introduction

The theory of time scales, which has recently received a lot of attention, was introduced by Stefan Hilger in his PhD dissertation written under the direction of Bernd Aulbach (see [1]). Since then a rapidly expanding body of literature has sought to unify, extend, and generalize ideas from discrete calculus, quantum calculus, and continuous calculus to arbitrary time scale calculus. Recall that a time scale $\mathbb{T}$ is a nonempty closed subset of the reals, and the cases when this time scale is the reals or the integers represent the classical theories of differential and of difference equations. Not only does the new theory of the so-called ‘dynamic equations’ unify the theories of differential equations and difference equations, but it also extends these classical cases to cases ‘in between’, e.g., to the so-called q-difference equations when $\mathbb{T}=q^{{\mathbb{N}}_0}$, and can be applied to different types of time scales like $\mathbb{T}=h\mathbb{Z}$, $\mathbb{T}={\mathbb{N}}_0^2$, and the space of harmonic numbers $\mathbb{T}=\{{\mathbb{H}}_n\}$. In this work, knowledge and understanding of time scales and time scale notation is assumed. For an excellent introduction to the calculus on time scales, see Bohner and Peterson [2, 3]. We recall the following concepts related to the notion of time scales. A time scale $\mathbb{T}$ is an arbitrary nonempty closed subset of the real numbers ℝ. For $t\in \mathbb{T}$, we define the forward and backward jump operators $\sigma :\mathbb{T}\to \mathbb{T}$ and $\rho :\mathbb{T}\to \mathbb{T}$ by

$$\sigma (t)=inf\{s\in \mathbb{T}:s>t\},\rho (t)=sup\{s\in \mathbb{T}:s<t\},$$

where $inf\mathrm{\varnothing }:=sup\mathbb{T}$ and $sup\mathrm{\varnothing }=inf\mathbb{T}$, where ∅ denotes the empty set. A point $t\in \mathbb{T}$, $t>inf\mathbb{T}$, is said to be left-dense if $\rho (t)=t$, right-dense if $t<sup\mathbb{T}$ and $\sigma (t)=t$, left-scattered if $\rho (t)<t$, and right-scattered if $\sigma (t)>t$. A function $g:\mathbb{T}\to \mathbb{R}$ is said to be right-dense continuous (rd-continuous) provided g is continuous at right-dense points and at left-dense points in $\mathbb{T}$, left-hand limits exist and are finite. The set of all such rd-continuous functions is denoted by $C_{\mathrm{rd}}(\mathbb{T})$.

We are concerned with the oscillatory behavior of the following second-order dynamic equation

$${(r(t)x^{\mathrm{\Delta }}(t))}^{\mathrm{\Delta }}+p(t)x^{\mathrm{\Delta }}(t)+q(t)f(x^{\sigma }(t))=0,$$

(1.1)

on a time scale $\mathbb{T}$ which is unbounded above, where r, p, and q are real-valued, right-dense continuous functions on $\mathbb{T}$. The function $f\in C(\mathbb{R},\mathbb{R})$ is assumed to satisfy $xf(x)>0$ and $f^{\mathrm{\prime }}(x)>0$ for $x\ne 0$.

By a solution of (1.1), we mean a nontrivial real-valued function $x\in C_{\mathrm{rd}}^1[T_x,\mathrm{\infty })$, $T_x\ge t_0$ which has the property that $r{x}^{\mathrm{\Delta }}\in C_{\mathrm{rd}}^1[T_x,\mathrm{\infty })$ and satisfies equation (1.1) on $[T_x,\mathrm{\infty })$, where $C_{\mathrm{rd}}$ is the space of rd-continuous functions. The solutions vanishing in some neighborhood of infinity will be excluded from our consideration. A solution x of (1.1) is said to be oscillatory if it is neither eventually positive nor eventually negative. Otherwise it is said to be nonoscillatory. There has been a great deal of research into obtaining criteria for oscillation of all solutions of dynamic equations on time scales. It is usually assumed that r, p, q are nonnegative functions. We refer the reader to the papers [4–11, 22–33] and the references cited therein. On the other hand, very little is known about equations when no explicit sign assumptions are made with respect to the coefficient functions p, q, and r. In the papers [12–14], it is shown that one may relate oscillation and boundedness of solutions of the nonlinear equation (1.1) to a related linear equation, which, in the case $r=1$ and $p=0$, reduces to

$$x^{\mathrm{\Delta }\mathrm{\Delta }}(t)+\lambda q(t)x^{\sigma }(t)=0,$$

where $\lambda >0$, for which many oscillation criteria are known. However, it was assumed that the nonlinearity has the property

$$f^{\mathrm{\prime }}(x)\ge \frac{f(x)}{x},\text{for }x\ne 0.$$

Bohner, Erbe, and Peterson [15] studied the second-order nonlinear dynamic equation

$$x^{\mathrm{\Delta }\mathrm{\Delta }}(t)+p(t)x^{\mathrm{\Delta }\sigma }(t)+q(t)f(x^{\sigma }(t))=0,$$

(1.2)

where p is a positively regressive function and $q_1$ satisfies condition (A), that is,

$$\underset{t\to \mathrm{\infty }}{lim\hspace{0.17em}inf}{\int }_T^t{q}_1(s)\mathrm{\Delta }s\ge 0\text{and}\not\equiv 0$$

for all large T and

$${\int }_{t_0}^{\mathrm{\infty }}\frac{\mathrm{\Delta }t}{r_1(t)}={\int }_{t_0}^{\mathrm{\infty }}{q}_1(t)\mathrm{\Delta }t=\mathrm{\infty },$$

where $q_1(t):=q(t)e_p(t,t_0)$ and $r_1(t):=e_p(t,t_0)$. Oscillation criterion for equation (1.2) is shown in [15] when $f^{\mathrm{\prime }}$ satisfied condition (B), if for each $k>0$ there exists $m>0$ such that $f^{\mathrm{\prime }}(x)\ge m$ provided $x\ge k$. We say $\mathbb{T}$ satisfies condition (C) if there is an $M>0$ such that $\chi (t)\le M\mu (t)$, $t\in \mathbb{T}$, where χ is the characteristic function of the set $\hat{\mathbb{T}}=\{t\in \mathbb{T}:\mu (t)>0\}$. We note that if $\mathbb{T}$ satisfies condition (C), then the subset $\check{\mathbb{T}}$ of $\mathbb{T}$ defined by

$$\check{\mathbb{T}}=\{t\in \mathbb{T}:t>0\text{ is right-scattered or left-scattered}\}$$

is necessarily countable and $\hat{\mathbb{T}}\subset \check{\mathbb{T}}$. Then we can rewrite $\check{\mathbb{T}}$ by

$$\check{\mathbb{T}}=\{t_i\in \mathbb{T}:0<t_1<t_2<\cdots <t_n<\cdots \},$$

and so

$$\mathbb{T}=\check{\mathbb{T}}\cup [\bigcup _{n\in A}(t_{n-1},t_n)],$$

where A is the set of all integers for which the real open interval $(t_{n-1},t_n)$ is contained in $\mathbb{T}$. To be precise, we have

$$A:=\{n\in \mathbb{N}:(t_{n-1},t_n)\subset \mathbb{T}\}.$$

Baoguo, Erbe, and Peterson established in [16] some oscillation criteria of Kiguradze-type in particular, for the second-order superlinear dynamic equation

$$x^{\mathrm{\Delta }\mathrm{\Delta }}(t)+q(t)f(x^{\sigma }(t))=0,$$

where $\mathbb{T}$ satisfies condition (C) and $f(x)$ satisfies the superlinearity condition

$$0<{\int }_{ϵ}^{\mathrm{\infty }}\frac{dx}{f(x)},{\int }_{-\mathrm{\infty }}^{-ϵ}\frac{dx}{f(x)}<\mathrm{\infty },\text{for all }ϵ>0.$$

(1.3)

Hassan, Erbe, and Peterson [17] improved these results and generalized these to the superlinear dynamic equation (1.1), where the coefficient functions r, p, q are allowed to change sign for large t, and $\mathbb{T}$ satisfies condition (C). Also, Hassan, Erbe, and Peterson [18] applied these results to the sublinear dynamic equation (1.1), where the functions r, p, q are also allowed to change sign for large t, and $\mathbb{T}$ satisfies condition (C), and $f(x)$ satisfies the sublinearity condition

$$0<{\int }_0^{ϵ}\frac{dx}{f(x)};{\int }_{-ϵ}^0\frac{dx}{f(x)}<\mathrm{\infty },\text{for all }ϵ>0.$$

(1.4)

A number of sufficient conditions for oscillation were obtained in [15, 16] for the case when

$${\int }_{t_0}^{\mathrm{\infty }}\frac{\mathrm{\Delta }t}{r(t)}=\mathrm{\infty },$$

and in [17, 18] for the case when

$${\int }_{t_0}^{\mathrm{\infty }}\frac{\mathrm{\Delta }t}{\varphi (t)r(t)}=\mathrm{\infty },$$

where ϕ is a rd-continuous function. Therefore, it will be of great interest to establish oscillation criteria for (1.1) for both of the cases

$${\int }_{t_0}^{\mathrm{\infty }}\frac{\mathrm{\Delta }t}{\varphi (t)r(t)}=\mathrm{\infty }$$

(1.5)

and

$${\int }_{t_0}^{\mathrm{\infty }}\frac{\mathrm{\Delta }t}{\varphi (t)r(t)}<\mathrm{\infty },$$

(1.6)

where the function ϕ will be defined in the next section. We will still assume that the functions $r(t)$, $p(t)$, and $q(t)$ change sign for arbitrarily large values of t, and conditions (A), (B), and (C) are not needed. Hence, our results will improve and extend results in [15–18] and many known results on nonlinear oscillation. In addition, linear, sublinear, and superlinear results will be presented.

These results have significant importance to the study of oscillation criteria on discrete time scales such as $\mathbb{T}=\mathbb{Z}$, $\mathbb{T}=h\mathbb{Z}$, $h>0$, or $\mathbb{T}=\{t:t=q^k,k\in {\mathbb{N}}_0,q>1\}$ and the space of harmonic numbers $\mathbb{T}=H_n$. In particular, we give examples where the coefficient function r changes sign for large t as well as p and q and without conditions (1.3) and (1.4).

2 Main results

Before stating our main results, we begin with the following lemma (Second Mean Value Theorem) which will play an important role in the proof of our main results.

Lemma 2.1 ([[3], Theorem 5.45])

Let h be a bounded function that is integrable on ${[a,b]}_{\mathbb{T}}$. Let $m_H$ and $M_H$ be the infimum and supremum of the function $H(t):={\int }_a^{t}h(s)\mathrm{\Delta }s$ on ${[a,b]}_{\mathbb{T}}$ respectively. Suppose that g is a nonnegative and nonincreasing function on ${[a,b]}_{\mathbb{T}}$. Then there is some number Λ with $m_H\le \mathrm{\Lambda }\le M_H$ such that

$${\int }_a^{b}h(t)g(t)\mathrm{\Delta }t=g(a)\mathrm{\Lambda }.$$

Our first result is stated in terms of an auxiliary function $\varphi =\varphi (t)$.

Theorem 2.1 Assume that f satisfies (1.3). If there exists a $C_{\mathrm{rd}}^1$ function ϕ such that

$$\varphi (t)r(t)>0,P(t)\ge 0,P^{\mathrm{\Delta }}(t)\le 0,\mathit{\text{for }}t\in {[t_0,\mathrm{\infty })}_{\mathbb{T}},$$

(2.1)

where $P(t):={\varphi }^{\mathrm{\Delta }}(t)r(t)-{\varphi }^{\sigma }(t)p(t)$ and

$${\int }_{t_0}^{\mathrm{\infty }}\frac{\mathrm{\Delta }t}{\varphi (t)r(t)}={\int }_{t_0}^{\mathrm{\infty }}{\varphi }^{\sigma }(t)q(t)\mathrm{\Delta }t=\mathrm{\infty },$$

(2.2)

then every solution of equation (1.1) is oscillatory.

Proof Assume (1.1) has a nonoscillatory solution x on ${[t_0,\mathrm{\infty })}_{\mathbb{T}}$. Then, without loss of generality, there is a solution x of (1.1) and a $T\in {[t_0,\mathrm{\infty })}_{\mathbb{T}}$ such that $x(t)>0$ on ${[T,\mathrm{\infty })}_{\mathbb{T}}$. Define

$$w(t):=\frac{\varphi (t)r(t)x^{\mathrm{\Delta }}(t)}{f(x(t))}.$$

Then from the product and quotient rules and the Pötzsche chain rule [[2], Theorem 1.90], we get

$$\begin{array}{rcl}{w}^{\mathrm{\Delta }}(t)& =& {\left[\frac{\varphi (t)}{f(x(t))}\right]}^{\sigma }{(r(t)x^{\mathrm{\Delta }}(t))}^{\mathrm{\Delta }}+{\left[\frac{\varphi (t)}{f(x(t))}\right]}^{\mathrm{\Delta }}r(t)x^{\mathrm{\Delta }}(t)\\ =& {\left[\frac{\varphi (t)}{f(x(t))}\right]}^{\sigma }{(r(t)x^{\mathrm{\Delta }}(t))}^{\mathrm{\Delta }}\\ +[\frac{{\varphi }^{\mathrm{\Delta }}(t)}{f(x^{\sigma }(t))}-\frac{\varphi (t){\int }_0^1{f}^{\mathrm{\prime }}(x_h(t))dh{x}^{\mathrm{\Delta }}(t)}{f(x(t))f(x^{\sigma }(t))}]r(t)x^{\mathrm{\Delta }}(t),\end{array}$$

where $x_h(t):=(1-h)x(t)+h{x}^{\sigma }(t)>0$ for $0\le h\le 1$, $t\in {[T,\mathrm{\infty })}_{\mathbb{T}}$. Therefore, from (1.1), we have

$$w^{\mathrm{\Delta }}(t)=-{\varphi }^{\sigma }(t)q(t)+\frac{P(t)x^{\mathrm{\Delta }}(t)}{f(x^{\sigma }(t))}-\frac{\varphi (t)r(t){\int }_0^1{f}^{\mathrm{\prime }}(x_h(t))dh{(x^{\mathrm{\Delta }}(t))}^2}{f(x(t))f(x^{\sigma }(t))}.$$

(2.3)

Since $vf(v)>0$, $f^{\mathrm{\prime }}(v)>0$, for all $v\ne 0$ and $\varphi (t)r(t)>0$, for all $t\ge T$, we get

$$w^{\mathrm{\Delta }}(t)\le -{\varphi }^{\sigma }(t)q(t)+\frac{P(t)x^{\mathrm{\Delta }}(t)}{f(x^{\sigma }(t))},\text{for }t\in {[T,\mathrm{\infty })}_{\mathbb{T}}.$$

Integrating the above inequality from T to t (≥T), we obtain

$$w(t)\le w(T)-{\int }_T^t{\varphi }^{\sigma }(s)q(s)\mathrm{\Delta }s+{\int }_T^t\frac{P(s)x^{\mathrm{\Delta }}(s)}{f(x^{\sigma }(s))}\mathrm{\Delta }s.$$

(2.4)

We claim that${\int }_T^t\frac{P(s)x^{\mathrm{\Delta }}(s)}{f(x^{\sigma }(s))}\mathrm{\Delta }s$ is bounded above for all $t\ge T$. Since $P(t)\ge 0$ and $P^{\mathrm{\Delta }}(t)\le 0$, we have from Lemma 2.1 that for each $t\in {[T,\mathrm{\infty })}_{\mathbb{T}}$,

$${\int }_T^t\frac{P(s)x^{\mathrm{\Delta }}(s)}{f(x^{\sigma }(s))}\mathrm{\Delta }s=P(T)\mathrm{\Lambda }(t),$$

(2.5)

where $m\le \mathrm{\Lambda }(t)\le M$, and where m and M denote the infimum and supremum, respectively, of the function ${\int }_T^s\frac{x^{\mathrm{\Delta }}(\tau )}{f(x^{\sigma }(\tau ))}\mathrm{\Delta }\tau$ for $s\in {[T,t)}_{\mathbb{T}}$. Define

$$F(x(s)):={\int }_{x(T)}^{x(s)}\frac{d\tau }{f(\tau )},$$

and so

$${\left(F(x(s))\right)}^{\mathrm{\Delta }}={\int }_0^1{F}^{\mathrm{\prime }}(x_h(s))dh{x}^{\mathrm{\Delta }}(s)={\int }_0^1\frac{1}{f(x_h(s))}dh{x}^{\mathrm{\Delta }}(s).$$

(2.6)

For a fixed point $s\in {[T,\mathrm{\infty })}_{\mathbb{T}}$, we have

$$x_h(s)=(1-h)x(s)+h{x}^{\sigma }(s)\{\begin{array}{cc}\ge x^{\sigma }(s),\hfill & \text{if }{x}^{\mathrm{\Delta }}(s)\le 0,\hfill \\ \le x^{\sigma }(s),\hfill & \text{if }{x}^{\mathrm{\Delta }}(s)\ge 0,\hfill \end{array}$$

and so

$$\frac{x^{\mathrm{\Delta }}(s)}{f(x_h(s))}\ge \{\begin{array}{cc}\frac{x^{\mathrm{\Delta }}(s)}{f(x^{\sigma }(s))},\hfill & \text{if }{x}^{\mathrm{\Delta }}(s)\le 0,\hfill \\ \frac{x^{\mathrm{\Delta }}(s)}{f(x^{\sigma }(s))},\hfill & \text{if }{x}^{\mathrm{\Delta }}(s)\ge 0.\hfill \end{array}$$

Then

$$\frac{x^{\mathrm{\Delta }}(s)}{f(x_h(s))}\ge \frac{x^{\mathrm{\Delta }}(s)}{f(x^{\sigma }(s))},\text{for }s\in {[T,\mathrm{\infty })}_{\mathbb{T}}.$$

And so, from (2.6), we have

$${\left(F(x(s))\right)}^{\mathrm{\Delta }}\ge \frac{x^{\mathrm{\Delta }}(s)}{f(x^{\sigma }(s))},\text{for }s\in {[T,\mathrm{\infty })}_{\mathbb{T}}.$$

Also,

$$F(x(s))={\int }_{x(T)}^{x(s)}\frac{d\tau }{f(\tau )}<\{\begin{array}{cc}0,\hfill & \text{if }x(s)\le x(T),\hfill \\ {\int }_{x(T)}^{\mathrm{\infty }}\frac{d\tau }{f(\tau )},\hfill & \text{if }x(s)>x(T).\hfill \end{array}$$

Then, from (1.3), we get

$$F(x(s))={\int }_{x(T)}^{x(s)}\frac{d\tau }{f(\tau )}\le L_1,\text{for }s\in {[T,\mathrm{\infty })}_{\mathbb{T}}.$$

Hence, it follows that

$${\int }_T^t\frac{x^{\mathrm{\Delta }}(s)}{f(x^{\sigma }(s))}\mathrm{\Delta }s\le F(x(t))-F(x(T))\le L_1,$$

and from (2.5), for all $t\in {[T,\mathrm{\infty })}_{\mathbb{T}}$, we have

$${\int }_T^t\frac{P(s)x^{\mathrm{\Delta }}(s)}{f(x^{\sigma }(s))}\mathrm{\Delta }s=P(T)\mathrm{\Lambda }(t)\le P(T)L_1.$$

(2.7)

From (2.4) and (2.7), we get for $t\in {[T,\mathrm{\infty })}_{\mathbb{T}}$,

$$w^{\mathrm{\Delta }}(t)\le w^{\mathrm{\Delta }}(T)+P(T)L_1-{\int }_T^t{\varphi }^{\sigma }(s)q(s)\mathrm{\Delta }s.$$

In view of condition (2.2), it follows from the last inequality that there exists a sufficiently large $T_1\ge T$ such that

$$x^{\mathrm{\Delta }}(t)<0,\text{for }t\in {[T_1,\mathrm{\infty })}_{\mathbb{T}}.$$

Also, from (2.2), there exists $T_2\ge T_1$ such that

$${\int }_{T_2}^t{\varphi }^{\sigma }(s)q(s)\mathrm{\Delta }s\ge 0,\text{for all }t\ge T_2.$$

(2.8)

Indeed, (1.1) yields on integration

(2.9)

Now by the integration by parts, we have

(2.10)

and by the integration by parts and the Pötzsche chain rule and then by (2.8), we get

$$\begin{array}{rcl}{\int }_{T_2}^t{\varphi }^{\sigma }(s)q(s)f(x^{\sigma }(s))\mathrm{\Delta }s& =& f(x(t)){\int }_{T_2}^t{\varphi }^{\sigma }(\tau )q(\tau )\mathrm{\Delta }\tau \\ -{\int }_{T_2}^t\left[{\int }_0^1{f}^{\mathrm{\prime }}(x_h(s))dh\right]x^{\mathrm{\Delta }}(s){\int }_{T_2}^s{\varphi }^{\sigma }(\tau )q(\tau )\mathrm{\Delta }\tau \mathrm{\Delta }s\\ \ge & f(x(t)){\int }_{T_2}^t{\varphi }^{\sigma }(s)q(s)\mathrm{\Delta }s\ge 0,\text{for }t\in {[T_2,\mathrm{\infty })}_{\mathbb{T}},\end{array}$$

(2.11)

since $vf(v)>0$ and $f^{\mathrm{\prime }}(v)>0$ for all $v\ne 0$ and $x^{\mathrm{\Delta }}(t)<0$ for $t\ge T_2$. Using (2.9) and (2.10) in (2.11), we get

$$\varphi (t)r(t)x^{\mathrm{\Delta }}(t)\le \varphi (T_2)r(T_2)x^{\mathrm{\Delta }}(T_2),$$

and so

$$x(t)-x(T_2)\le \varphi (T_2)r(T_2)x^{\mathrm{\Delta }}(T_2){\int }_{T_2}^t\frac{\mathrm{\Delta }s}{\varphi (s)r(s)}.$$

Since $\varphi (T_2)r(T_2)x^{\mathrm{\Delta }}(T_2)<0$, we conclude from (2.2) that ${lim}_{t\to \mathrm{\infty }}x(t)=-\mathrm{\infty }$, which is a contradiction. This completes the proof. □

In the case $\mathbb{T}=\mathbb{R}$, $r(t)=1$, $p(t)=0$, and $f(x)=x^{\alpha }$, $\alpha >1$, Theorem 2.1 is due to Kiguradze [19].

If $\mathbb{T}=\mathbb{Z}$, $r(t)=1$, $p(t)=0$, $q(t)\ge 0$, $f(x)=x^{\alpha }$, $\alpha >1$, and $\varphi (t)=t-1$, then Theorem 2.1 includes Theorem 4.1 in Hooker and Patula [[20], Theorem 4.1] and Mingarelli [21].

Suppose that there exists a $C_{\mathrm{rd}}^1$ function ϕ such that, for $t\in {[t_0,\mathrm{\infty })}_{\mathbb{T}}$,

$$\varphi (t)r(t)>0,{\varphi }^{\mathrm{\Delta }}(t)r(t)={\varphi }^{\sigma }(t)p(t).$$

(2.12)

Then, in this case, we have $P(t)\equiv 0$, and so we do not need to assume the superlinearity conditions (1.3), and so the result applies to linear, sublinear, and superlinear case.

Corollary 2.1 If there exists a $C_{\mathrm{rd}}^1$ function ϕ such that (2.2) and (2.12) hold, then every solution of equation (1.1) is oscillatory.

If we do not assume superlinearity condition (1.3) and condition (2.12), then we can conclude that all bounded solutions are oscillatory.

Corollary 2.2 If there exists a $C_{\mathrm{rd}}^1$ function ϕ such that (2.1) and (2.2) hold, then every bounded solution of equation (1.1) is oscillatory.

In the following, we assume that there exists a $C_{\mathrm{rd}}^1$ function ϕ such that

$${\int }_{t_0}^{\mathrm{\infty }}\frac{\mathrm{\Delta }t}{\varphi (t)r(t)}<\mathrm{\infty },$$

(2.13)

holds and establish some sufficient conditions for equation (1.1).

Theorem 2.2 Assume that f satisfies (1.3). If there exists a $C_{\mathrm{rd}}^1$ function ϕ such that (2.1) and (2.13) hold and if

$${\int }_{t_0}^{\mathrm{\infty }}{\varphi }^{\sigma }(t)q(t)\mathrm{\Delta }t=\mathrm{\infty },$$

(2.14)

and

$${\int }_{t_0}^{\mathrm{\infty }}\frac{1}{\varphi (t)r(t)}[{\int }_{t_0}^t{\varphi }^{\sigma }(s)q(s)\mathrm{\Delta }s]\mathrm{\Delta }t=\mathrm{\infty },$$

(2.15)

then every solution of equation (1.1) is either oscillatory or tends to zero.

Proof Assume (1.1) has a nonoscillatory solution x on ${[t_0,\mathrm{\infty })}_{\mathbb{T}}$. Then, without loss of generality, there is a solution x of (1.1) and a $T\in {[t_0,\mathrm{\infty })}_{\mathbb{T}}$ such that $x(t)>0$ on ${[T,\mathrm{\infty })}_{\mathbb{T}}$. As in the proof of Theorem 2.1, by using (2.14), we have that there exists a sufficiently large $T_1\ge T$ such that

$$x^{\mathrm{\Delta }}(t)<0,\text{for }t\in {[T_1,\mathrm{\infty })}_{\mathbb{T}},$$

and so

$$w^{\mathrm{\Delta }}(t)\le -{\varphi }^{\sigma }(t)q(t)+\frac{P(t)x^{\mathrm{\Delta }}(t)}{f(x^{\sigma }(t))}\le -{\varphi }^{\sigma }(t)q(t),$$

which yields

$$\begin{array}{rcl}{\int }_{T_1}^t{\varphi }^{\sigma }(s)q(s)\mathrm{\Delta }s& \le & \varphi (T_1)\frac{r(T_1)x^{\mathrm{\Delta }}(T_1)}{f(x(T_1))}-\varphi (t)\frac{r(t)x^{\mathrm{\Delta }}(t)}{f(x(t))}\\ \le & -\varphi (t)r(t)\frac{x^{\mathrm{\Delta }}(t)}{f(x(t))},\text{for }t\in {[T_1,\mathrm{\infty })}_{\mathbb{T}}.\end{array}$$

That is,

$$\frac{1}{\varphi (t)r(t)}{\int }_{T_1}^t{\varphi }^{\sigma }(s)q(s)\mathrm{\Delta }s\le -\frac{x^{\mathrm{\Delta }}(t)}{f(x(t))},$$

so integration from $T_1$ to t ($t\ge T_1$) yields

$${\int }_{T_1}^t\frac{1}{\varphi (s)r(s)}[{\int }_{T_1}^s{\varphi }^{\sigma }(u)q(u)\mathrm{\Delta }u]\mathrm{\Delta }s\le -{\int }_{T_1}^t\frac{x^{\mathrm{\Delta }}(s)}{f(x(s))}\mathrm{\Delta }s.$$

(2.16)

Define

$$F(x(t)):={\int }_{x(T_1)}^{x(t)}\frac{ds}{f(s)}.$$

Then

$$\begin{array}{rcl}{\left(F(x(t))\right)}^{\mathrm{\Delta }}& =& {\int }_0^1{F}^{\mathrm{\prime }}(x_h(t))dh{x}^{\mathrm{\Delta }}(t)={\int }_0^1\frac{1}{f(x_h(t))}dh{x}^{\mathrm{\Delta }}(t)\\ \le & {\int }_0^1\frac{1}{f(x(t))}dh{x}^{\mathrm{\Delta }}(t)=\frac{x^{\mathrm{\Delta }}(t)}{f(x(t))}.\end{array}$$

(2.17)

Since $x^{\mathrm{\Delta }}(t)<0$, we have ${lim}_{t\to \mathrm{\infty }}x(t)=L\ge 0$. If we assume $L>0$, then

$$-F(x(t))={\int }_{x(t)}^{x(T_1)}\frac{ds}{f(s)}\le {\int }_L^{x(T_1)}\frac{ds}{f(s)}<\mathrm{\infty },$$

so

$${\int }_{T_1}^t\frac{1}{\varphi (s)r(s)}[{\int }_{T_1}^s{\varphi }^{\sigma }(u)q(u)\mathrm{\Delta }u]\mathrm{\Delta }s\le -{\int }_{T_1}^t{\left(F(x(s))\right)}^{\mathrm{\Delta }}\mathrm{\Delta }s=-F(x(t))<\mathrm{\infty },$$

and let $t\to \mathrm{\infty }$ to get a contradiction to (2.15). This completes the proof. □

Corollary 2.3 If there exists a $C_{\mathrm{rd}}^1$ function ϕ such that (2.12), (2.13), (2.14), and (2.15) hold, then every solution of equation (1.1) is either oscillatory or tends to zero.

Corollary 2.4 If there exists a $C_{\mathrm{rd}}^1$ function ϕ such that (2.1), (2.13), (2.14), and (2.15) hold, then every bounded solution of equation (1.1) is either oscillatory or tends to zero.

Next we present oscillation criteria for equation (1.1) where f satisfies sublinearity condition (1.4).

Theorem 2.3 Assume that f satisfies (1.4). If there exists a $C_{\mathrm{rd}}^1$ function ϕ such that (2.12), (2.13), (2.14), and (2.15), then every solution of equation (1.1) is oscillatory.

Proof Assume (1.1) has a nonoscillatory solution x on ${[t_0,\mathrm{\infty })}_{\mathbb{T}}$. Then, without loss of generality, there is a solution x of (1.1) and a $T\in {[t_0,\mathrm{\infty })}_{\mathbb{T}}$ such that $x(t)>0$ on ${[T,\mathrm{\infty })}_{\mathbb{T}}$. As in the proof of Theorems 2.1 and 2.2, we have that there exists a sufficiently large $T_1\ge T$ such that

$$x^{\mathrm{\Delta }}(t)<0,\text{for }t\in {[T_1,\mathrm{\infty })}_{\mathbb{T}},$$

and, from (2.16) and (2.17), we get

$${\int }_{T_1}^t\frac{1}{\varphi (s)r(s)}[{\int }_{T_1}^s{\varphi }^{\sigma }(u)q(u)\mathrm{\Delta }u]\mathrm{\Delta }s\le -{\int }_{T_1}^t{\left(F(x(s))\right)}^{\mathrm{\Delta }}\mathrm{\Delta }s=-F(x(t)).$$

Since $x^{\mathrm{\Delta }}(t)<0$, then ${lim}_{t\to \mathrm{\infty }}x(t)\ge 0$ and so

$$-F(x(t))={\int }_{x(t)}^{x(T_1)}\frac{ds}{f(s)}\le {\int }_0^{x(T_1)}\frac{ds}{f(s)}\stackrel{(\text{1.4})}{<}\mathrm{\infty }.$$

Then

$${\int }_{T_1}^t\frac{1}{\varphi (s)r(s)}[{\int }_{T_1}^s{\varphi }^{\sigma }(u)q(u)\mathrm{\Delta }u]\mathrm{\Delta }s<\mathrm{\infty },$$

which is a contradiction to (2.15). This completes the proof. □

Theorem 2.4 Assume that f satisfies (1.4). If there exists a $C_{\mathrm{rd}}^1$ function ϕ such that (2.1), (2.13), (2.14), and (2.15) hold, then every bounded solution of equation (1.1) is oscillatory.

The results are very general. With appropriate choices of $\varphi (t)$, we can obtain several sufficient conditions for the oscillation of equation (1.1).

3 Examples

In this section, we give two examples to illustrate our main results. We note that in the first example, all of the coefficient functions $r(t)$, $p(t)$, $q(t)$ change sign for arbitrarily large values of t, and the function f may be linear, superlinear, or sublinear, since we do not assume nonlinearity conditions (1.3) and (1.4).

Example 3.1 Let $t_0>0$ and $\mathbb{T}$ be a discrete time scale satisfying $\mu >0$ on $\mathbb{T}$. Consider the second-order nonlinear dynamic equation with damping

$${(r(t_n)x^{\mathrm{\Delta }}(t_n))}^{\mathrm{\Delta }}+p(t_n)x^{\mathrm{\Delta }}(t_n)+q(t_n)f(x^{\sigma }(t_n))=0,$$

(3.1)

where f satisfies $f^{\mathrm{\prime }}(x)>0$ and $xf(x)>0$ for $x\ne 0$. Define, for $t_n\in \mathbb{T}$, $n\in \mathbb{N}$,

$$\varphi (t_n):={(-1)}^n{t}_n^{{\beta }_1},r(t_n):=\frac{{(-1)}^n}{t_n^{{\beta }_2}},p(t_n):=\frac{{(-1)}^n}{t_n^{{\beta }_2}\mu (t_n)}(1+{\left(\frac{t_n}{t_{n+1}}\right)}^{{\beta }_1}),$$

and

$$q(t_n):=\frac{1}{t_{n+1}^{{\beta }_1}}(\frac{1}{t_n{t}_{n+1}}-\frac{{(-1)}^n}{t_n^{{\beta }_3}}),$$

where ${\beta }_i\in \mathbb{R}$, $i=1,2,3$ such that ${\beta }_1-{\beta }_2\le 1$ and ${\beta }_3\le 1$. Therefore,

$$\varphi (t_n)r(t_n)=t_n^{{\beta }_1-{\beta }_2}>0,{\varphi }^{\mathrm{\Delta }}(t_n)r(t_n)={\varphi }^{\sigma }(t_n)p(t_n),$$

so (2.12) is satisfied. Also,

$${\int }_{t_0}^{\mathrm{\infty }}\frac{\mathrm{\Delta }{t}_n}{\varphi (t_n)r(t_n)}={\int }_{t_0}^{\mathrm{\infty }}\frac{\mathrm{\Delta }{t}_n}{t_n^{{\beta }_1-{\beta }_2}}=\mathrm{\infty },$$

and

$${\int }_{t_0}^{\mathrm{\infty }}{\varphi }^{\sigma }(t_n)q(t_n)\mathrm{\Delta }{t}_n={\int }_{t_0}^{\mathrm{\infty }}(\frac{1}{t_n^{{\beta }_3}}-\frac{{(-1)}^n}{t_n{t}_{n+1}})\mathrm{\Delta }{t}_n=\mathrm{\infty },$$

by Example 5.60 in [3]. Then, by Theorem 2.1, every solution of equation (3.1) is oscillatory.

Example 3.2 Let $\mathbb{T}=\mathbb{N}$ and consider the difference equation

$$\mathrm{\Delta }(r(n)\mathrm{\Delta }x(n))+p(n)\mathrm{\Delta }x(n)+q(n)f(x(n+1))=0,n\ge n_0>1,$$

(3.2)

where f satisfies $f^{\mathrm{\prime }}(x)>0$ and $xf(x)>0$ for $x\ne 0$ and nonlinearity condition (1.3). Define

$$\varphi (n):={(n-1)}^{\beta },r(n):=n^{\delta -\beta },p(n):=0,$$

and

$$q(n):=\frac{a}{n^{\alpha +\gamma }}+\frac{b{(-1)}^n}{n^{\gamma }},$$

where $\alpha ,\delta \in [0,1]$ and $a,b,\gamma ,\beta \in \mathbb{R}$. It is easy to see that (2.1) and ${\int }_{t_0}^{\mathrm{\infty }}\frac{\mathrm{\Delta }t}{\varphi (t)r(t)}=\mathrm{\infty }$ hold. Note that

$${\int }_{t_0}^{\mathrm{\infty }}{\sigma }^{\beta }(t)q(t)\mathrm{\Delta }t=\sum _{k=n_0}^{\mathrm{\infty }}(\frac{a}{k^{\alpha +\gamma -\beta }}+\frac{b{(-1)}^k}{k^{\gamma -\beta }}).$$

If $\beta =\gamma$, we get

$${\int }_{t_0}^{\mathrm{\infty }}{\sigma }^{\beta }(t)q(t)\mathrm{\Delta }t=\mathrm{\infty }.$$

Then, by Theorem 2.1, every solution of equation (3.2) is oscillatory.