Oscillation of higher order nonlinear dynamic equations on time scales

Abstract

Some new criteria for the oscillation of n th order nonlinear dynamic equations of the form

$$x^{{\mathsf{\Delta }}^n}\left(t\right)+q\left(t\right){\left(x^{\sigma }\left(\xi \left(t\right)\right)\right)}^{\lambda }=0$$

are established in delay ξ(t) ≤ t and non-delay ξ(t) = t cases, where n ≥ 2 is a positive integer, λ is the ratio of positive odd integers. Many of the results are new for the corresponding higher order difference equations and differential equations are as special cases.

Mathematics Subject Classification (2011): 34C10; 34C15.

1. Introduction

Consider the n th order nonlinear delay dynamic equation

$$x^{{\mathsf{\Delta }}^n}\left(t\right)+q\left(t\right){\left(x^{\sigma }\left(\xi \left(t\right)\right)\right)}^{\lambda }=0$$

(1.1)

on an arbitrary time-scale $\mathbb{T}\subseteq ℝ$ with sup $\mathbb{T}=\infty$ and $0\in \mathbb{T}$, where n ≥ 2 is a positive integer, λ is the ratio of positive odd integers, $q:\mathbb{T}\to {ℝ}^{+}=\left(0,\infty \right)$ and $\xi :\mathbb{T}\to \mathbb{T}$ are real-valued rd-continuous functions, ξ(t) ≤ t, ξ^Δ(t) ≥ 0, and lim_t→∞ξ(t) = ∞. Throughout the article by t ≥ s for t, $s\in \mathbb{T}$ we shall mean $t\in \left[s,\infty \right)\cap \mathbb{T}:={\left[s,\infty \right)}_{\mathbb{T}}$. For the forward jump operator σ, we use the usual notation x^σ = x ○ σ.

We recall that a solution x of Equation (1.1) is said to be nonoscillatory if there exists a $t_0\in \mathbb{T}$ such that x(t)x(σ(t)) > 0 for all t ≥ t₀; otherwise, it is said to be oscillatory. Equation (1.1) is said to be oscillatory if all its solutions are oscillatory.

Recently, there has been an increasing interest in studying the oscillatory behavior of first-and second-order dynamic equations on time-scales, see [1–7]. However, there are very few results regarding the oscillation of higher order equations. Therefore, the purpose of this article is to obtain new criteria for the oscillation of Equation (1.1). This topic is fairly new for dynamic equations on time scales. For a general background on time scale calculus, we may refer to [8, 9].

The article is organized as follows: In Section 2, some preliminary lemmas and notations are given, while Section 3 is devoted to the study of Equation (1.1) via comparison with a set of second-order dynamic equations whose oscillatory character is known and have been investigated extensively in the literature. In Section 4, we establish new oscillation criteria for Equation (1.1) when ξ(t) = t for linear, sublinear, and superlinear cases. Further results are presented in Section 5 when there is a special restriction on the function q. We should note that many of our results of this article are new for the corresponding higher order nonlinear differential and difference equations. In fact, the obtained results extend, unify and correlate many of the existing results in the literature.

2. Preliminaries

We shall employ the following lemmas. The first lemma is the well-known Kiguradze's lemma.

Lemma 2.1. Let $x\in C_{rd}^m\left(\left[t_0,\infty \right),{ℝ}^{+}\right)$. If $x^{{\mathsf{\Delta }}^m}\left(t\right)$ is of constant sign on ${\left[t_0,\infty \right)}_{\mathbb{T}}$ and not identically zero on ${\left[t_1,\infty \right)}_{\mathbb{T}}$ for any t₁ ≥ t₀, then there exist a t _x ≥ t₀ and an integer ℓ, 0 ≤ ℓ ≤ m with m + ℓ even for $x^{{\mathsf{\Delta }}^m}\left(t\right)\ge 0$, or m + ℓ odd for $x^{{\mathsf{\Delta }}^m}\left(t\right)\le 0$ such that

$$\ell >0implies{x}^{{\mathsf{\Delta }}^k}\left(t\right)>0fort\ge t_x,k\in \left\{1,2,\dots ,\ell -1\right\}$$

(2.1)

and

$$\ell \le m-1implies{\left(-1\right)}^{\ell +k}{x}^{{\mathsf{\Delta }}^k}\left(t\right)>0fort\ge t_x,k\in \left\{\ell ,\ell +1,\dots ,m-1\right\}.$$

(2.2)

Lemma 2.2. If the inequality

$$x^{\mathsf{\Delta }\mathsf{\Delta }}+Q\left(t\right)x^{\lambda }\le 0,$$

(2.3)

where Q is a positive real-valued, rd-continuous function on , has an eventually positive solution, then the equation

$$x^{\mathsf{\Delta }\mathsf{\Delta }}+Q\left(t\right)x^{\lambda }=0$$

(2.4)

also has an eventually positive solution.

Proof. Let x(t) be an eventually positive solution of inequality (2.3). It is easy to see that x^Δ(t) > 0 eventually. Let t₀ be sufficiently large so that x(t) > 0 and y(t) =: x^Δ(t) > 0 for $t\in {\left[t_0,\infty \right)}_{\mathbb{T}}$. Then in view of

$$x\left(t\right)=x\left(t_0\right)+\underset{t_0}{\overset{t}{\int }}y\left(s\right)\mathsf{\Delta }s,$$

(2.3) becomes

$$y^{\mathsf{\Delta }}\left(t\right)+Q\left(t\right){\left(x\left(t_0\right)+\underset{t_0}{\overset{t}{\int }}y\left(s\right)\mathsf{\Delta }s\right)}^{\lambda }\le 0,t\in {\left[t_0,\infty \right)}_{\mathbb{T}}.$$

(2.5)

Integrating (2.5) from t to u ≥ t ≥ t₀ and letting u → ∞, we have

$$y\left(t\right)\ge F\left(t,y\left(t\right)\right),t\in {\left[t_0,\infty \right)}_{\mathbb{T}},$$

where

$$F\left(t,y\right):=\underset{t}{\overset{\infty }{\int }}Q\left(v\right){\left(x\left(t_0\right)+\underset{t_0}{\overset{v}{\int }}y\left(s\right)\mathsf{\Delta }s\right)}^{\lambda }\mathsf{\Delta }v.$$

Next, we define a sequence of successive approximations {z _j (t)} as follows:

$$z_0\left(t\right)=y\left(t\right)$$

$$z_{j+1}\left(t\right)=F\left(t,z_j\left(t\right)\right),j=0,1,2,\dots .$$

It is easy to show that

$$0<z_j\left(t\right)\le y\left(t\right)\mathsf{\text{and }}{z}_{j+1}\left(t\right)\le z_j\left(t\right),j=0,1,2,\dots .$$

Thus the sequence {z _j (t)} is nonincreasing and bounded for each t ≥ t₀. This means we may define z(t) = lim_j→∞z _j (t) ≥ 0. Since 0 ≤ z(t) ≤ z _j (t) ≤ y(t) for all j ≥ 0, we find that

$$\underset{t_0}{\overset{t}{\int }}{z}_j\left(s\right)\mathsf{\Delta }s\le \underset{t_0}{\overset{t}{\int }}y\left(s\right)\mathsf{\Delta }s.$$

By the Lebesgue dominated convergence theorem on time scales, one can easily obtain

$$z\left(t\right)=F\left(t,z\left(t\right)\right).$$

Therefore,

$$z^{\mathsf{\Delta }}\left(t\right)=-Q\left(t\right)m^{\lambda }\left(t\right),$$

(2.6)

where

$$m\left(t\right)=x\left(t_0\right)+\underset{t_0}{\overset{t}{\int }}z\left(s\right)\mathsf{\Delta }s.$$

Then, m(t) > 0 and m^Δ(t) = z(t). Equation (2.6) then gives

$$m^{\mathsf{\Delta }\mathsf{\Delta }}\left(t\right)+Q\left(t\right)m^{\lambda }\left(t\right)=0.$$

Hence, Equation (2.4) has a positive solution m(t). This completes the proof. □

Lemma 2.3 ([4]). Suppose |x|^Δ is of one sign on ${\left[t_0,\infty \right)}_{\mathbb{T}}$ and α > 0, α ≠ 1. Then

$$\frac{|x{|}^{\mathsf{\Delta }}}{{\left(\left|x^{\sigma }\right|\right)}^{\alpha }}\le \frac{{\left(|x{|}^{1-\alpha }\right)}^{\mathsf{\Delta }}}{\left(1-\alpha \right)}\le \frac{|x{|}^{\mathsf{\Delta }}}{\left(|x{|}^{\alpha }\right)},t\ge t_0.$$

(2.7)

It will be convenient to employ the Taylor monomials (see [[8], Sect. 1.6]) $h_n,g_n:{\mathbb{T}}^2\to ℝ$, $n\in {\mathbb{N}}_0$, which are defined recursively as follows:

$$h_0\left(t,s\right)=g_0\left(t,s\right)=1,$$

$${h_n}_{+\mathsf{\text{1}}}\left(t,s\right)=\underset{s}{\overset{t}{\int }}{h}_n\left(\tau ,s\right)\mathsf{\Delta }\tau ,{g_n}_{+\mathsf{\text{1}}}\left(t,s\right)=\underset{s}{\overset{t}{\int }}{g}_n\left(\sigma \left(\tau \right),s\right)\mathsf{\Delta }\tau ,t,s\in \mathbb{T},n\in {\mathbb{N}}_0.$$

It is clear that h₁(t, s) = g₁(t, s) = t - s for any time-scales, but simple formulas in general do not hold for n ≥ 2. It is also known that

$$h_n\left(t,s\right)={\left(-1\right)}^n{g}_n\left(s,t\right).$$

3. Comparison criteria for delay dynamic equations

In this section, we shall consider the equation

$$x^{{\mathsf{\Delta }}^n}\left(t\right)+q\left(t\right)x^{\lambda }\left(\xi \left(t\right)\right)=0.$$

(3.1)

For $t_0\in \mathbb{T}$ and ℓ ∈ {1, 2, ..., n - 1}, we define

$$q_{\ell }\left(t,t_0\right)=\underset{t}{\overset{\infty }{\int }}{\tau }^{-\lambda }{Q}_{\ell }\left(\tau ,t,t_0\right)\mathsf{\Delta }\tau ,t\in {\left[t_0,\infty \right)}_{\mathbb{T}},$$

where

$$Q_{\ell }\left(\tau ,t,t_0\right)=g_{n-\ell -2}\left(\sigma \left(\tau \right),t\right)R_{\ell }^{\lambda }\left(\tau ,t_0\right)q\left(\tau \right),\tau \ge t.$$

with

$$R_{\ell }\left(\tau ,t_0\right)=\left\{\begin{array}{cc}\underset{t_0}{\overset{\xi \left(\tau \right)}{\int }}s{h}_{\ell -2}\left(\xi \left(\tau \right),\sigma \left(s\right)\mathsf{\Delta }s,\right.\hfill & \ell \ge 2\hfill \\ \xi \left(\tau \right),\hfill & \ell =1.\hfill \end{array}\right.$$

Theorem 3.1. Let $t_0\in \mathbb{T}$. Suppose that for every ℓ ∈ {1, 2, ..., n - 1},

$$\underset{\infty }{\overset{}{\int }}{Q}_{\ell }\left(\tau ,t_0,t_0\right)\mathsf{\Delta }\tau =\infty .$$

(3.2)

Then, Equation (3.1) is oscillatory if

(i) for n even, the equation

$$y^{\mathsf{\Delta }\mathsf{\Delta }}+q_{\ell }\left(t,t_0\right)y^{\lambda }=0,$$

(3.3)

for all ℓ ∈ {1, 3, ..., n - 1} is oscillatory;

(ii) for n odd, the Equation (3.3) for all ℓ ∈ {2, 4, ..., n - 1} is oscillatory, and

$$\underset{t\to \infty }{\mathsf{\text{lim}}\mathsf{\text{sup}}}\underset{\xi \left(t\right)}{\overset{t}{\int }}{h}_{n-1}^{\lambda }\left(\xi \left(s\right),\xi \left(t\right)\right)q\left(s\right)\mathsf{\Delta }s>\left\{\begin{array}{l}0when0<\lambda <1\\ 1when\lambda =1.\end{array}\right.$$

(3.4)

Proof. Let x(t) be a nonoscillatory solution of Equation (3.1). Without loss of generality, we may assume that x(t) > 0 and x(ξ(t)) > 0 for t ≥ t₀, since otherwise the substitution w = -x transforms Equation (3.1) into an equation of the same form subject to the assumptions of the theorem.

By Lemma 2.1, there exist a t₁ ≥ t₀ and an integer ℓ ∈ {0, 1, ..., n} with n + ℓ odd such that (2.1) and (2.2) hold for all t ≥ t₁. We see that

$$x^{{\mathsf{\Delta }}^{\ell -1}}\left(t\right)>0,x^{{\mathsf{\Delta }}^{\ell }}\left(t\right)>0,x^{{\mathsf{\Delta }}^{\ell +1}}\left(t\right)<0\mathsf{\text{for }}t\ge t_1,$$

and by Taylor's formula

$$\begin{array}{cc}\hfill x\left(t\right)& =\sum _{k=0}^{l-2}{x}^{{\mathsf{\Delta }}^k}\left(t_1\right)h_k\left(t,t_1\right)+\underset{t_1}{\overset{t}{\int }}{h}_{\ell -2}\left(t,\sigma \left(\tau \right)\right)x^{{\mathsf{\Delta }}^{\ell -1}}\left(\tau \right)\mathsf{\Delta }\tau \hfill \\ \ge \underset{t_1}{\overset{t}{\int }}{h}_{\ell -2}\left(t,\sigma \left(\tau \right)\right)x^{{\mathsf{\Delta }}^{\ell -1}}\left(\tau \right)\mathsf{\Delta }\tau \mathsf{\text{for }}\ell >1.\hfill \end{array}$$

(3.5)

We claim that

$$\frac{x^{{\mathsf{\Delta }}^{\ell -1}\left(t\right)}}{t}\mathsf{\text{ is strictly decreasing for }}t\ge t_1\mathsf{\text{ and }}\ell >0.$$

(3.6)

To prove it, set $X\left(t\right)=x^{{\mathsf{\Delta }}^{\ell -1}}\left(t\right)-t{x}^{{\mathsf{\Delta }}^{\ell }}\left(t\right)$. Because

$${\left(\frac{x^{{\mathsf{\Delta }}^{\ell -1}}}{t}\right)}^{\mathsf{\Delta }}=\frac{t{x}^{{\mathsf{\Delta }}^{\ell }}-x^{{\mathsf{\Delta }}^{\ell -1}}}{t\sigma \left(t\right)}=-\frac{X\left(t\right)}{t\sigma \left(t\right)},$$

it suffices to show that X(t) is strictly positive. Suppose on the contrary that X(t) < 0. Then $x^{{\mathsf{\Delta }}^{\ell -1}}/t$ is strictly increasing and hence

$$x^{{\mathsf{\Delta }}^{\ell -1}}\left(t\right)\ge ct\mathsf{\text{for }}t\ge t_1,$$

(3.7)

where $c=x^{{\mathsf{\Delta }}^{\ell -1}}\left(t_1\right)/t_1>0$. Using (3.7) in (3.5), we have

$$x\left(\xi \left(t\right)\right)\ge c\underset{t_1}{\overset{\xi \left(t\right)}{\int }}\tau h_{\ell -2}\left(\xi \left(t\right),\sigma \left(\tau \right)\right)\mathsf{\Delta }\tau .$$

(3.8)

Let ℓ = 1, then (3.7) gives x(ξ(t)) ≥ cξ(t) for t ≥ t₁ by increasing the size of t₁ if necessary. Thus, we obtain

$$x\left(\xi \left(t\right)\right)\ge c{R}_{\ell }\left(t,t_1\right)\mathsf{\text{for }}t\ge t_1\mathsf{\text{ and }}\ell >0.$$

(3.9)

On the other hand, by Taylor's formula we may write that

$$\begin{array}{cc}\hfill x^{{\mathsf{\Delta }}^{l+1}}\left(t\right)& =\sum _{k=0}^{n-\ell -2}{x}^{{\mathsf{\Delta }}^{\ell +k+1}}\left(s\right)h_k\left(t,s\right)+\underset{t}{\overset{s}{\int }}{h}_{n-\ell -2}\left(t,\sigma \left(\tau \right)\right)\left(-x^{{\mathsf{\Delta }}^n}\left(\tau \right)\right)\mathsf{\Delta }\tau \hfill \\ =\sum _{k=0}^{n-\ell -2}{x}^{{\mathsf{\Delta }}^{\ell +k+1}}\left(s\right){\left(-1\right)}^k{g}_k\left(s,t\right)+\underset{t}{\overset{s}{\int }}{\left(-1\right)}^{n-\ell -2}{g}_{n-\ell -2}\left(\sigma \left(\tau \right),t\right)\left(-x^{{\mathsf{\Delta }}^n}\left(\tau \right)\right)\mathsf{\Delta }\tau \hfill \\ \le -\underset{t}{\overset{\infty }{\int }}{g}_{n-\ell -2}\left(\sigma \left(\tau \right),t\right)q\left(\tau \right)x^{\lambda }\left(\xi \left(\tau \right)\right)\mathsf{\Delta }\tau .\hfill \end{array}$$

(3.10)

From (3.9) and (3.10), we have

$$-x^{{\mathsf{\Delta }}^{\ell +1}}\left(t_1\right)\ge c^{\lambda }\underset{t_1}{\overset{\infty }{\int }}{g}_{n-\ell -2}\left(\sigma \left(\tau \right),t_1\right)q\left(\tau \right)R_{\ell }^{\lambda }\left(\tau ,t_1\right)\mathsf{\Delta }\tau ,$$

(3.11)

which contradicts (3.2), and hence completes the proof of the claim.

Now in view of (3.6) it follows from (3.5) that

$$x\left(t\right)\ge \frac{x^{{\mathsf{\Delta }}^{\ell -1}}\left(t\right)}{t}\underset{t_1}{\overset{t}{\int }}\tau h_{\ell -2}\left(t,\sigma \left(\tau \right)\right)\mathsf{\Delta }\tau ,t\ge t_1.$$

(3.12)

Replacing t by ξ(t) in (3.12) and using (3.6), we have

$$x\left(\xi \left(t\right)\right)\ge x^{{\mathsf{\Delta }}^{\ell -1}}\left(t\right)\underset{t_1}{\overset{\xi \left(t\right)}{\int }}\frac{\tau }{t}{h}_{\ell -2}\left(\xi \left(t\right),\sigma \left(\tau \right)\right)\mathsf{\Delta }\tau ,\ell >1$$

(3.13)

for all t ≥ t₂ for some t₂ ≥ t₁.

If ℓ = 1, then we may write that

$$x\left(\xi \left(t\right)\right)=\frac{x^{{\mathsf{\Delta }}^{\ell -1}}\left(\xi \left(t\right)\right)}{\xi \left(t\right)}\xi \left(t\right)\ge \frac{x^{{\mathsf{\Delta }}^{\ell -1}}\left(t\right)}{t}\xi \left(t\right),t\ge t_2.$$

(3.14)

Thus, from (3.13) and (3.14) for all t ≥ t₂,

$$x\left(\xi \left(t\right)\right)\ge \frac{x^{{\mathsf{\Delta }}^{\ell -1}}\left(t\right)}{t}{R}_{\ell }\left(t,t_1\right),\ell >0.$$

(3.15)

Substituting (3.15) into (3.10) gives

$$-x^{{\mathsf{\Delta }}^{\ell +1}}\left(t\right)\ge {\left(x^{{\mathsf{\Delta }}^{\ell -1}}\left(t\right)\right)}^{\lambda }\underset{t}{\overset{\infty }{\int }}{\tau }^{-\lambda }{g}_{n-\ell -2}\left(\sigma \left(\tau \right),t\right)R_{\ell }^{\lambda }\left(\tau ,t_1\right)q\left(\tau \right)\mathsf{\Delta }\tau ,t\ge t_2.$$

(3.16)

Set $w\left(t\right)=x^{{\mathsf{\Delta }}^{\ell -1}}\left(t\right)$ in (3.16), then w(t) > 0 satisfies

$$w^{\mathsf{\Delta }\mathsf{\Delta }}+q_{\ell }\left(t,t_1\right)w^{\lambda }\le 0,t\ge t_2.$$

By Lemma 2.2, the equation

$$w^{\mathsf{\Delta }\mathsf{\Delta }}+q_{\ell }\left(t,t_1\right)w^{\lambda }=0$$

has a nonoscillatory solution. But this is impossible by the hypothesis.

Finally, we let ℓ = 0. This is the case, when n is odd. By applying Taylor's formula and using (2.2) with ℓ = 0, we can easily find

$$x\left(u\right)\ge h_{n-1}\left(u,v\right)x^{{\mathsf{\Delta }}^{n-1}}\left(v\right)$$

(3.17)

for v ≥ u ≥ t₁, which implies that

$$x\left(\xi \left(s\right)\right)\ge h_{n-1}\left(\xi \left(s\right),\xi \left(t\right)\right)x^{{\mathsf{\Delta }}^{n-1}}\left(\xi \left(t\right)\right),t>s\ge t_{\mathsf{\text{3}}}.$$

(3.18)

for some t₃ ≥ t₁. Integrating equation (3.1) from ξ(t) ≥ t₃ to t ≥ t, we get

$$x^{{\mathsf{\Delta }}^{n-1}}\left(\xi \left(t\right)\right)\ge \underset{\xi \left(t\right)}{\overset{t}{\int }}q\left(s\right)x^{\lambda }\left(\xi \left(s\right)\right)\mathsf{\Delta }s.$$

(3.19)

Using (3.18) in (3.19), we have

$$x^{{\mathsf{\Delta }}^{n-1}}\left(\xi \left(t\right)\right)\ge {\left(x^{{\mathsf{\Delta }}^{n-1}}\left(\xi \left(t\right)\right)\right)}^{\lambda }\underset{\xi \left(t\right)}{\overset{t}{\int }}{h}_{n-1}^{\lambda }\left(\xi \left(s\right),\xi \left(t\right)\right)q\left(s\right)\mathsf{\Delta }s$$

or

$${\left(x^{{\mathsf{\Delta }}^{n-1}}\left(\xi \left(t\right)\right)\right)}^{1-\lambda }\ge \underset{\xi \left(t\right)}{\overset{t}{\int }}{h}_{n-1}^{\lambda }\left(\xi \left(s\right),\xi \left(t\right)\right)q\left(s\right)\mathsf{\Delta }s$$

Taking the lim sup as t → ∞, we obtain a contradiction to condition (3.4). □

The following immediate result can be extracted from Theorem 3.1.

Corollary 3.1. Let n be an odd and condition (3.4) hold. Then every bounded solution of Equation (3.1) is oscillatory.

Next, we claim that inequality (3.15) can be replaced by

$$x\left(\xi \left(t\right)\right)\ge \frac{1}{t}{h}_{\ell }\left(\xi \left(t\right),t_1\right)x^{{\mathsf{\Delta }}^{\ell -1}}\left(t\right).$$

(3.20)

To prove this, we write that

$$x^{{\mathsf{\Delta }}^{\ell -2}}\left(t\right)\ge \underset{t_1}{\overset{t}{\int }}{x}^{{\mathsf{\Delta }}^{\ell -1}}\left(s\right)\mathsf{\Delta }s=\underset{t_1}{\overset{t}{\int }}s\left(\frac{x^{{\mathsf{\Delta }}^{\ell -1}}\left(s\right)}{s}\right)\mathsf{\Delta }s$$

and hence by (3.6) we find

$$x^{{\mathsf{\Delta }}^{\ell -2}}\left(t\right)\ge h_2\left(t,t_1\right)\left(\frac{x^{{\mathsf{\Delta }}^{\ell -1}}\left(t\right)}{t}\right).$$

Integrating this inequality (ℓ - 2)-times from t₁ to t ≥ t₁ and using (3.6), we obtain

$$x\left(t\right)\ge h_{\ell }\left(t,t_1\right)\left(\frac{x^{{\mathsf{\Delta }}^{\ell -1}}\left(t\right)}{t}\right).$$

Thus, there exists a t₂ ≥ t₁ such that

$$x\left(\xi \left(t\right)\right)\ge h_{\ell }\left(\xi \left(t\right),t_1\right)\frac{x^{{\mathsf{\Delta }}^{\ell -1}}\left(\xi \left(t\right)\right.}{\left.\xi \left(t\right)\right)}\ge \frac{1}{t}{h}_{\ell }\left(\xi \left(t\right),t_1\right)x^{{\mathsf{\Delta }}^{\ell -1}}\left(t\right),t\ge t_2.$$

This completes the proof of our claim.

Set

$$Q_{\ell }^{*}\left(\tau ,t,t_0\right)=g_{n-\ell -2}\left(\sigma \left(\tau \right),t\right)h_l^{\lambda }\left(\xi \left(\tau \right),t_0\right)q\left(\tau \right),\tau \ge t$$

and

$$q_{\ell }^{*}\left(t,t_0\right)=\underset{t}{\overset{\infty }{\int }}{\tau }^{-\lambda }{Q}_{\ell }^{*}\left(\tau ,t,t_0\right)\mathsf{\Delta }\tau ,t\ge t_0.$$

In view of Theorem 3.1 and inequality (3.20) we may state the following theorem.

Theorem 3.2. In Theorem 3.1, let q _ℓ and Q _ℓ be replaced by $q_{\ell }^{*}$ and $Q_{\ell }^{*}$, respectively. Then the conclusions of Theorem 3.1 hold.

Let $\mathbb{T}=ℝ$, i.e., the continuous case. Here Equation (3.1) becomes

$$x^{\left(n\right)}\left(t\right)+q\left(t\right)x^{\lambda }\left(\xi \left(t\right)\right)=0$$

(3.21)

and the functions $q_{\ell }^{*}$ and $Q_{\ell }^{*}$ take the form

$$q_{\ell }^c\left(t,t_0\right)=\underset{t}{\overset{\infty }{\int }}{\tau }^{-\lambda }{Q}_{\ell }^c\left(\tau ,t,t_0\right)d\tau$$

and

$$Q_{\ell }^c\left(\tau ,t,t_0\right)=\frac{{\left(\xi \left(\tau \right)-t_0\right)}^{\lambda \ell }}{{\left(\ell !\right)}^{\lambda }}\frac{{\left(\tau -t\right)}^{n-\ell -2}}{\left(n-\ell -2\right)!}q\left(\tau \right).$$

From Theorem 3.2 we have the following theorem.

Theorem 3.3. Let $t_0\in \mathbb{T}$. Suppose that for ℓ ∈ {1, 2, ..., n - 1},

$$\underset{\infty }{\overset{}{\int }}{Q}_{\ell }^c\left(\tau ,t_0,t_0\right)d\tau =\infty .$$

(3.22)

Then, Equation (3.21) is oscillatory if

(i) for n even, the equation

$$\mathrm{y"}+q_{\ell }^c\left(t,t_0\right)y=0,$$

(3.23)

for all ℓ ∈ {1, 3, ..., n - 1} is oscillatory;

(ii) for n odd, the Equation (3.23) for all ℓ ∈ {2, 4, ..., n - 1} is oscillatory and

$$\underset{t\to \infty }{\mathsf{\text{lim}}\mathsf{\text{sup}}}\underset{\xi \left(t\right)}{\overset{t}{\int }}{\left(\frac{{\left(\xi \left(s\right)-\xi \left(t\right)\right)}^{n-1}}{\left(n-1\right)!}\right)}^{\lambda }q\left(s\right)\mathsf{\Delta }s>\left\{\begin{array}{l}0when0<\lambda <1\\ 1when\lambda =1.\end{array}\right.$$

(3.24)

Next, we let $\mathbb{T}=\mathbb{Z}$, i.e., the discrete case. Then, Equation (3.1) reads as

$${\mathsf{\Delta }}^{n}x\left(m\right)+q\left(m\right)x^{\lambda }\left(\xi \left(m\right)\right.=0$$

(3.25)

and the functions $q_{\ell }^{*}$ and $Q_{\ell }^{*}$ become

$$q_{\ell }^d\left(m,m_0\right)=\sum _{j=m}^{\infty }{j}^{-\lambda }{Q}_{\ell }^d\left(j,m,m_0\right)$$

and

$$Q_{\ell }^d\left(j,m,m_0\right)=\frac{{\left[{\left(\xi \left(j\right)-m_0\right)}^{\left(\ell \right)}\right]}^{\lambda }}{{\left(\ell !\right)}^{\lambda }}\frac{{\left(j-m+n-\ell -2\right)}^{\left(n-\ell -2\right)}}{\left(n-\ell -2\right)!}q\left(j\right),$$

where t^(m)= t(t - 1)(t - 2) ... (t - m + 1) is the usual factorial function.

Theorem 3.4. Let $m_0\in \mathbb{Z}$. Suppose that for ℓ ∈ {1, 2, ..., n - 1}

$$\sum _{j=m_0}^{\infty }{Q}_{\ell }^d\left(j,m_0,m_0\right)=\infty .$$

(3.26)

Then, Equation (3.25) is oscillatory if

(i) for n even, the second-order difference equation

$${\mathsf{\Delta }}^{2}y\left(m\right)+q_{\ell }^d\left(m,m_0\right)y^{\lambda }\left(m\right)=0,$$

(3.27)

for all ℓ ∈ {1, 3, ..., n - 1} is oscillatory;

(ii) for n odd, the Equation (3.27) for all ℓ ∈ {2, 4, ..., n - 1} is oscillatory and

$$\underset{m\to \infty }{\mathsf{\text{lim}}\mathsf{\text{sup}}}\sum _{j=\xi \left(m\right)}^m{\left(\frac{{\left(\xi \left(j\right)-\xi \left(m\right)\right)}^{\left(n-1\right)}}{\left(n-1\right)!}\right)}^{\lambda }q\left(j\right)>\left\{\begin{array}{l}0when0<\lambda <1\\ 1when\lambda =1.\end{array}\right.$$

(3.28)

Remark 1. The oscillation of Equation (3.1) is obtained via a comparison with a set of second-order dynamic equations whose oscillatory behavior has been studied extensively in the literature. In fact, there are many sufficient conditions for the oscillation of Equation (3.3) which can be employed rather easily.

4. Even order dynamic equations without delay

In this section, we present new oscillation criteria for (3.1) when n is even. That is, we consider

$$x^{{\mathsf{\Delta }}^{2n}}+q\left(t\right){\left(x^{\sigma }\right)}^{\lambda }=0.$$

(4.1)

For $t\in \mathbb{T}$, we define

$${\widehat{Q}}_{\ell }\left(t\right)=\underset{t}{\overset{\infty }{\int }}\underset{s_{2n-\ell -1}}{\overset{\infty }{\int }}\dots \underset{s_1}{\overset{\infty }{\int }}q\left(s\right)\mathsf{\Delta }s\mathsf{\Delta }{s}_1\dots \mathsf{\Delta }{s}_{2n-\ell -1},\ell \in \left\{1,3,\dots ,2n-1\right\}.$$

(4.2)

Theorem 4.1. Let λ > 1 and $t_0\in \mathbb{T}$. If for every integer ℓ ∈ {1, 3, ..., 2n - 1},

$$\underset{t_0}{\overset{\infty }{\int }}{h}_{\ell -1}\left(s,t_0\right){\widehat{Q}}_{\ell }\left(s\right)\mathsf{\Delta }s=\infty ,$$

(4.3)

then Equation (4.1) is oscillatory.

Proof. Let x(t) be a nonoscillatory solution of Equation (4.1), say, x(t) > 0 for t ≥ t₀. From Equation (4.1), we see that $x^{{\mathsf{\Delta }}^{2n}}\left(t\right)\le 0$ for t ≥ t₀, where $x^{{\mathsf{\Delta }}^{2n}}\left(t\right)$ is not identically zero for all large t. Using Lemma 2.1 there exist a t₁ ≥ t₀ and an integer ℓ ∈ {1, 3, ..., 2n - 1} such that (2.1) and (2.2) hold for all t ≥ t₁. From (2.1), we see that $x^{{\mathsf{\Delta }}^{\ell }}\left(t\right)>0$ and decreasing on ${\left[t_1,\infty \right)}_{\mathbb{T}}$. Now,

$$x^{{\mathsf{\Delta }}^{\ell -1}}\left(s\right)-x^{{\mathsf{\Delta }}^{\ell -1}}\left(t_1\right)=\underset{t_1}{\overset{s}{\int }}{x}^{{\mathsf{\Delta }}^{\ell }}\left(\tau \right)\mathsf{\Delta }\tau \ge h_1\left(s,t_1\right)x^{{\mathsf{\Delta }}^{\ell }}\left(s\right),$$

or

$$x^{{\mathsf{\Delta }}^{\ell -1}}\left(s\right)\ge h_{\mathsf{\text{1}}}\left(s,t_{\mathsf{\text{1}}}\right)x^{{\mathsf{\Delta }}^{\ell }}\left(s\right),s\ge t_{\mathsf{\text{1}}}.$$

(4.4)

Integrating (4.4) (ℓ - 2)-times from t₁ to s ≥ t₁, we have

$$x^{\mathsf{\Delta }}\left(s\right)\ge h_{\ell -1}\left(s,t_{\mathsf{\text{1}}}\right)x^{{\mathsf{\Delta }}^{\ell }}\left(s\right),s\ge t_{\mathsf{\text{1}}}.$$

(4.5)

Next, we integrate Equation (4.1) from s₁ ≥ t₁ to v ≥ s₁ and let v → ∞ to get

$$x^{{\mathsf{\Delta }}^{2n-1}}\left(s_1\right)\ge \underset{s_1}{\overset{\infty }{\int }}q\left(\tau \right)x^{\lambda }\left(\sigma \left(\tau \right)\right)\mathsf{\Delta }\tau \ge \left(\underset{s_1}{\overset{\infty }{\int }}q\left(\tau \right)\mathsf{\Delta }\tau \right)x^{\lambda }\left(\sigma \left(s_1\right)\right).$$

Integrating this inequality from s₂ ≥ t₁ to v ≥ s₂ and then letting v → ∞ and using (2.2), we get

$$-x^{{\mathsf{\Delta }}^{2n-2}}\left(s_2\right)\ge \left(\underset{s_2}{\overset{\infty }{\int }}\underset{s_1}{\overset{\infty }{\int }}q\left(\tau \right)\mathsf{\Delta }\tau \mathsf{\Delta }{s}_1\right)x^{\lambda }\left(\sigma \left(s_2\right)\right).$$

Continuing this process, one can easily find

$$x^{{\mathsf{\Delta }}^{\ell }}\left(s\right)\ge \left(\underset{s}{\overset{\infty }{\int }}\underset{s_{2n-\ell -1}}{\overset{\infty }{\int }}\cdots \underset{s_1}{\overset{\infty }{\int }}q\left(\tau \right)\mathsf{\Delta }\tau \mathsf{\Delta }{s}_1\dots \mathsf{\Delta }{s}_{2n-\ell -1}\right)x^{\lambda }\left(\sigma \left(s\right)\right),$$

or

$$x^{{\mathsf{\Delta }}^{\ell }}\left(s\right)\ge {\widehat{Q}}_{\ell }\left(s\right)x^{\lambda }\left(\sigma \left(s\right)\right),s\ge t_1.$$

(4.6)

From (4.5) and (4.6), we find

$$x^{-\lambda }\left(\sigma \left(s\right)\right)x^{\mathsf{\Delta }}\left(s\right)\ge h_{\ell -1}\left(s,t_{\mathsf{\text{1}}}\right){\widehat{Q}}_{\ell }\left(s\right),s\ge t_{\mathsf{\text{1}}},$$

and hence

$$\underset{t_1}{\overset{t}{\int }}{x}^{-\lambda }\left(\sigma \left(s\right)\right)x^{\mathsf{\Delta }}\left(s\right)\mathsf{\Delta }s\ge \underset{t_1}{\overset{t}{\int }}{h}_{\ell -1}\left(s,t_1\right){\widehat{Q}}_{\ell }\left(s\right)\mathsf{\Delta }s.$$

By employing the first inequality in Lemma 2.3, we get

$$\underset{t_1}{\overset{t}{\int }}\frac{{\left(x^{1-\lambda }\left(s\right)\right)}^{\mathsf{\Delta }}}{1-\lambda }\mathsf{\Delta }s\ge \underset{t_1}{\overset{t}{\int }}{h}_{\ell -1}\left(s,t_1\right){\widehat{Q}}_{\ell }\left(s\right)\mathsf{\Delta }s,$$

and so

$$\underset{t_1}{\overset{\infty }{\int }}{h}_{\ell -1}\left(s,t_1\right){\widehat{Q}}_{\ell }\left(s\right)\mathsf{\Delta }s\le \frac{x^{1-\lambda }\left(t_1\right)}{\lambda -1}<\infty .$$

But this contradicts condition (4.3). The proof is complete. □

Theorem 4.2. Let λ > 1 and $t_0\in \mathbb{T}$. If for every integer ℓ ∈ {1, 3, ..., 2n - 1},

$$\underset{t_0}{\overset{\infty }{\int }}{h}_{\ell -1}\left(s,t_0\right)\left(\underset{s}{\overset{\infty }{\int }}{g}_{2n-\ell -1}\left(\sigma \left(\tau \right),s\right)q\left(\tau \right)\mathsf{\Delta }\tau \right)\mathsf{\Delta }s=\infty ,$$

(4.7)

then Equation (4.1) is oscillatory.

Proof. Let x(t) be a nonoscillatory solution of Equation (1.1), say, x(t) > 0 for t ≥ t₀. By Taylor's formula, we see that

$$x^{{\mathsf{\Delta }}^{\ell }}\left(s\right)\ge -\underset{s}{\overset{\infty }{\int }}{g}_{2n-\ell -1}\left(\sigma \left(\tau \right),s\right)x^{{\mathsf{\Delta }}^{2n}}\left(\tau \right)\mathsf{\Delta }\tau ,s\ge t_1.$$

(4.8)

Using Equation (4.1) in (4.8), we get

$$\begin{array}{cc}\hfill x^{{\mathsf{\Delta }}^{\ell }}\left(s\right)& \ge \underset{s}{\overset{\infty }{\int }}{g}_{2n-\ell -1}\left(\sigma \left(\tau \right),s\right)q\left(\tau \right)x^{\lambda }\left(\sigma \left(\tau \right)\right)\mathsf{\Delta }\tau \hfill \\ \ge \left(\underset{s}{\overset{\infty }{\int }}{g}_{2n-\ell -1}\left(\sigma \left(\tau \right),s\right)q\left(\tau \right)\mathsf{\Delta }\tau \right)x^{\lambda }\left(\sigma \left(s\right)\right),s\ge t_1.\hfill \end{array}$$

(4.9)

Combining (4.8) with (4.9), we find

$$x^{\mathsf{\Delta }}\left(s\right)\ge h_{\ell -1}\left(s,t_1\right)\left(\underset{s}{\overset{\infty }{\int }}{g}_{2n-\ell -1}\left(\sigma \left(\tau \right),s\right)q\left(\tau \right)\mathsf{\Delta }\tau \right)x^{\lambda }\left(\sigma \left(s\right)\right),s\ge t_1.$$

Dividing both sides by x^λ(σ(s)) and integrating from t₁ to t ≥ t₁, we have

$$\underset{t_1}{\overset{t}{\int }}{x}^{-\lambda }\left(\sigma \left(s\right)\right)x^{\mathsf{\Delta }}\left(s\right)\mathsf{\Delta }s\ge \underset{t_1}{\overset{t}{\int }}{h}_{\ell -1}\left(s,t_1\right)\underset{s}{\overset{\infty }{\int }}{g}_{2n-\ell -1}\left(\sigma \left(\tau \right),s\right)q\left(\tau \right)\mathsf{\Delta }\tau \mathsf{\Delta }s.$$

The rest of the proof is similar to that of Theorem 4.1 and hence it is omitted. This completes the proof. □

Next, we apply Theorems 4.1 and 4.2 to obtain oscillation criteria for Equation (4.1) when λ ≤ 1.

Theorem 4.3. Let λ ≤ 1 and $t_0\in \mathbb{T}$. Assume that there exists a positive constant α such that α + λ > 1. If for every ℓ ∈ {1, 3, ..., 2n - 1}, condition (4.3) or (4.7) holds with q(t) replaced by $cq\left(t\right)h_{\ell }^{-\alpha }\left(t,0\right)$, where c is any positive constant, then Equation (4.1) is oscillatory.

Proof. Let x(t) be a nonoscillatory solution of Equation (4.1) and assume that there exists a t₀ > 0 such that x(t) > 0 for t ≥ t₀ and (2.1) and (2.2) hold for t ≥ t₀. From (2.1) and the decreasing nature of $x^{{\mathsf{\Delta }}^{\ell }}\left(t\right)$, there exists a constant c₁ > 0 such that $x^{{\mathsf{\Delta }}^{\ell }}\left(t\right)\le c_1$ for t ≥ t₀. Integrating this inequality ℓ - times from t₀ to t, we have

$$x\left(t\right)\le c{h}_{\ell }\left(t,0\right),t\ge t_0,$$

(4.10)

where c is a positive constant. Now, from Equation (4.1), we have

$$\begin{array}{cc}\hfill 0& =x^{{\mathsf{\Delta }}^{\mathsf{\text{2}}n}}\left(t\right)+q\left(t\right)x^{-\alpha }\left(\sigma \left(t\right)\right)x^{\lambda +\alpha }\left(\sigma \left(t\right)\right)\hfill \\ \ge x^{{\mathsf{\Delta }}^{\mathsf{\text{2}}n}}\left(t\right)+c^{-\alpha }q\left(t\right)h_{\ell }^{-\alpha }\left(\sigma \left(t\right),0\right)x^{\lambda +\alpha }\left(\sigma \left(t\right)\right),t\ge t_0.\hfill \end{array}$$

(4.11)

By applying Theorems 4.1 and 4.2 with inequality (3.20), we arrive at the desired conclusion. This completes the proof. □

Theorem 4.4. Let λ < 1 and $t_0\in \mathbb{T}$. If for every ℓ ∈ {1, 3, ..., 2n - 1},

$$\underset{t_0}{\overset{\infty }{\int }}q\left(t\right){\left(\underset{t_0}{\overset{t}{\int }}{h}_{\ell -1}\left(t,\sigma \left(u\right)\right)g_{2n-\ell -1}\left(t,u\right)\mathsf{\Delta }u\right)}^{\lambda }\mathsf{\Delta }t=\infty ,$$

(4.12)

then Equation (4.1) is oscillatory.

Proof. Let x(t) be a nonoscillatory solution of Equation (4.1), say, x(t) > 0 for t ≥ t₀. As in the proof of Theorem 4.1, we see that (2.1) and (2.2) hold for t ≥ t₁ ≥ t₀. It is easy to see that

$$x\left(t\right)\ge \underset{t_1}{\overset{t}{\int }}{h}_{\ell -1}\left(t,\sigma \left(u\right)\right)x^{{\mathsf{\Delta }}^{\ell }}\left(u\right)\mathsf{\Delta }u$$

and

$$x^{{\mathsf{\Delta }}^{\ell }}\left(u\right)\ge g_{\mathsf{\text{2}}n-\ell -1}\left(t,u\right)x^{{\mathsf{\Delta }}^{\mathsf{\text{2}}n-1}}\left(t\right),t\ge u\ge t_{\mathsf{\text{1}}}.$$

Therefore,

$$x\left(t\right)\ge \left(\underset{t_1}{\overset{t}{\int }}{h}_{\ell -1}\left(t,\sigma \left(u\right)\right)g_{2n-\ell -1}\left(t,u\right)\mathsf{\Delta }u\right)x^{{\mathsf{\Delta }}^{2n-1}}\left(t\right)\mathsf{\text{for }}t\ge t_1.$$

Using this inequality in Equation (4.1), we get

$$\begin{array}{cc}\hfill -{\left(x^{{\mathsf{\Delta }}^{\mathsf{\text{2}}n-1}}\left(t\right)\right)}^{\mathsf{\Delta }}& =q\left(t\right)x^{\lambda }\left(\sigma \left(t\right)\right)\ge q\left(t\right)x^{\lambda }\left(t\right)\hfill \\ \ge q\left(t\right){\left(\underset{t_1}{\overset{t}{\int }}{h}_{\ell -1}\left(t,\sigma \left(u\right)\right)g_{2n-\ell -1}\left(t,u\right)\mathsf{\Delta }u\right)}^{\lambda }{\left(x^{{\mathsf{\Delta }}^{2n-1}}\left(t\right)\right)}^{\lambda },t\ge t_1.\hfill \end{array}$$

Set $w\left(t\right)=x^{{\mathsf{\Delta }}^{2n-1}}\left(t\right)$, then

$$-w^{\lambda }\left(t\right)w^{\mathsf{\Delta }}\left(t\right)\ge q\left(t\right){\left(\underset{t_1}{\overset{t}{\int }}{h}_{\ell -1}\left(t,\sigma \left(u\right)\right)g_{2n-\ell -1}\left(t,u\right)\mathsf{\Delta }u\right)}^{\lambda },t\ge t_1.$$

Finally, in view of a chain rule, we integrate the last inequality from t₁ to t to get

$$\infty >\frac{w^{1-\lambda }\left(t_1\right)}{1-\lambda }\ge \underset{t_1}{\overset{t}{\int }}q\left(s\right){\left(\underset{t_1}{\overset{s}{\int }}{h}_{\ell -1}\left(s,\sigma \left(u\right)\right)g_{2n-\ell -1}\left(s,u\right)\mathsf{\Delta }u\right)}^{\lambda }\mathsf{\Delta }s,$$

a contradiction with condition (4.12). □

As an example, we shall reformulate some of the above results for the case $\mathbb{T}=\mathbb{Z}$, i.e., the discrete case. The Equation (4.1) takes the form

$${\mathsf{\Delta }}^{2n}x\left(m\right)+q\left(t\right)x^{\lambda }\left(m+1\right)=0$$

(4.13)

and establish new criteria for the oscillation of Equation (4.13).

We let

$${\widehat{Q}}_{\ell }^d\left(m\right)=\sum _{s_{2n-\ell -1}=t}^{\infty }\cdots \sum _{s_1=s_2}^{\infty }\sum _{u=s_1}^{\infty }q\left(u\right),\ell \in \left\{\mathsf{\text{1}},\mathsf{\text{3}},\dots ,\mathsf{\text{2}}n--\mathsf{\text{1}}\right\},m\ge m_0.$$

Theorem 4.5. Let λ > 1 and $m_0\in \mathbb{Z}$. If for every ℓ ∈ {1, 3, ..., 2n - 1},

$$\sum _{s=m_0}^{\infty }{s}^{\left(\ell -1\right)}{\widehat{Q}}_{\ell }^d\left(s\right)=\infty ,$$

(4.14)

then Equation (4.13) is oscillatory.

Theorem 4.6. Let λ > 1 and $m_0\in \mathbb{Z}$. If for every ℓ ∈ {1, 3, ..., 2n - 1},

$$\sum _{s=m_0}^{\infty }{s}^{\left(\ell -1\right)}\sum _{\tau =s}^{\infty }{\left(\tau -s+1\right)}^{\left(2n-\ell -1\right)}q\left(\tau \right)=\infty ,$$

(4.15)

then Equation (4.13) is oscillatory.

Theorem 4.7. Let λ < 1 and $m_0\in \mathbb{Z}$. If

$$\sum _{s=m_0}^{\infty }{\left(s^{\lambda }\right)}^{\left(2n-1\right)}q\left(s\right)=\infty ,$$

(4.16)

then Equation (4.13) is oscillatory.

Theorem 4.8. Let λ ≤ 1 and $m_0\in \mathbb{Z}$. Assume that there exists a positive constant α such that α + λ > 1. If for every ℓ ∈ {1, 3, ..., 2n - 1} condition (4.14) or (4.15) holds with q(t) be replaced by c q(t)(t)^(ℓ)/ℓ!)^-α, where c is any positive constant, then Equation (4.13) is oscillatory.

Remark 2. For Equation (4.1) of odd order, one may obtain results for the oscillatory and asymptotic behavior, while for complete oscillation, we may consider Equation (1.1) and employ the technique given in Theorem 3.1. The details are left to the reader.

5. Further oscillation criteria

In this section, we consider

$$x^{{\mathsf{\Delta }}^n}+q\left(t\right){\left(x^{\sigma }\right)}^{\lambda }=0,$$

(5.1)

subject to the condition

$$\underset{t_0}{\overset{\infty }{\int }}\underset{v}{\overset{\infty }{\int }}\underset{u}{\overset{\infty }{\int }}q\left(s\right)\mathsf{\Delta }s\mathsf{\Delta }u\mathsf{\Delta }v=\infty .$$

(5.2)

Note that if x(t), t ≥ t₀ is a positive solution of Equation (5.1), then by Lemma 2.1, Equations (2.1), and (2.2) hold for t ≥ t₁. Here, we claim that ℓ = n - 1. Otherwise, we find $x^{{\mathsf{\Delta }}^{n-1}}\left(t\right)>0$, $x^{{\mathsf{\Delta }}^{n-2}}\left(t\right)<0$ and $x^{{\mathsf{\Delta }}^{n-3}}\left(t\right)>0$ on ${\left[t_1,\infty \right)}_{\mathbb{T}}$. Integrating Equation (5.1) from t ≥ t₁ to u ≥ t and letting u → ∞, we have

$$x^{{\mathsf{\Delta }}^{n-1}}\left(t\right)\ge \underset{t}{\overset{\infty }{\int }}q\left(s\right)x^{\lambda }\left(\sigma \left(s\right)\right)\mathsf{\Delta }s.$$

(5.3)

Since x is increasing on ${\left[t_1,\infty \right)}_{\mathbb{T}}$, there exists a constant c > 0 such that

$$x\left(t\right)\ge c,t\in {\left[t_1,\infty \right)}_{\mathbb{T}}.$$

(5.4)

Using (5.4) in (5.3), we get

$$-x^{{\mathsf{\Delta }}^{n-1}}\left(t\right)\le -c^{\lambda }\underset{t}{\overset{\infty }{\int }}q\left(s\right)\mathsf{\Delta }s.$$

Integrating this inequality twice, once from v ≥ t to w ≥ v and letting w → ∞ and then from t₁ to t ≥ t₁, we have

$$x^{{\mathsf{\Delta }}^{n-3}}\left(t\right)\le -c^{\lambda }\underset{t_1}{\overset{t}{\int }}\underset{v}{\overset{\infty }{\int }}\underset{s}{\overset{\infty }{\int }}q\left(u\right)\mathsf{\Delta }u\mathsf{\Delta }s\mathsf{\Delta }v\to \infty \mathsf{\text{as }}t\to \infty ,$$

which contradicts (5.2). Thus, we must have ℓ = n - 1, i.e.,

Thus, we have

$$x^{{\mathsf{\Delta }}^{n-2}}\left(t\right)=x^{{\mathsf{\Delta }}^{n-2}}\left(t_1\right)+\underset{t_1}{\overset{t}{\int }}{x}^{{\mathsf{\Delta }}^{n-1}}\left(s\right)\mathsf{\Delta }s\ge h_1\left(t,t_1\right)x^{{\mathsf{\Delta }}^{n-1}}\left(t\right),t\ge t_1.$$

Integrating this inequality (n - 2)-times from t₁ to t, we obtain

$$x\left(t\right)\ge {h_n}_{-\mathsf{\text{1}}}\left(t,t_{\mathsf{\text{1}}}\right)x^{{\mathsf{\Delta }}^{n-1}}\left(t\right),t\ge t_{\mathsf{\text{1}}}.$$

(5.5)

Now, by making use of earlier results in [6], we obtain the following interesting theorems.

Theorem 5.1. Let condition (5.2) hold. If there exists a positive nondecreasing, differentiable function $\eta \in C_{rd}\left(\mathbb{T},{ℝ}^{+}\right)$ such that for any t₁ ≥ t₀,

$$\underset{t\to \infty }{\mathsf{\text{lim}}\mathsf{\text{sup}}}\underset{t_1}{\overset{t}{\int }}\left[\eta \left(s\right)q\left(s\right)-{\eta }^{\mathsf{\Delta }}\left(s\right)\frac{A\left(s,t_0\right)}{h_{n-1}\left(s,t_0\right)}\right]\mathsf{\Delta }s=\infty ,$$

(5.6)

where

$$A\left(t,t_0\right)=\left\{\begin{array}{cc}{c}_1,c_{1}isanypositiveconstant,\hfill & when\lambda >1\hfill \\ 1,\hfill & when\lambda =1\hfill \\ c_2{h}_{n-1}^{1-\lambda }\left(t,t_0\right),c_{2}isanypositiveconstant,\hfill & when\lambda <1,\hfill \end{array}\right.$$

then Equation (5.1) is oscillatory.

Proof. Let x(t) be a nonoscillatory solution of Equation (5.1), say, x(t) > 0 for t ≥ t₁ ≥ t₀.

Define

$$w\left(t\right)=\eta \left(t\right)\frac{x^{{\mathsf{\Delta }}^{n-1}}\left(t\right)}{x^{\lambda }\left(t\right)},t\ge t_1.$$

(5.7)

It is easy to see that for t ≥ t₁,

$$\begin{array}{cc}\hfill w^{\mathsf{\Delta }}& ={\left(\frac{\eta }{x^{\lambda }}\right)}^{\mathsf{\Delta }}{\left(x^{{\mathsf{\Delta }}^{n-1}}\right)}^{\sigma }+\left(\frac{\eta }{x^{\lambda }}\right)\left(x^{{\mathsf{\Delta }}^n}\right)\hfill \\ =-\eta q{\left(\frac{x^{\sigma }}{x}\right)}^{\lambda }+{\left(x^{{\mathsf{\Delta }}^{n-1}}\right)}^{\sigma }\left[\frac{{\eta }^{\mathsf{\Delta }}{x}^{\lambda }-\eta \left(x^{\lambda }\right)\mathsf{\Delta }}{x^{\lambda }{\left(x^{\sigma }\right)}^{\lambda }}\right].\hfill \end{array}$$

(5.8)

By [[8], Theorem 1.90],

$${\left(x^{\lambda }\right)}^{\mathsf{\Delta }}=\lambda x^{\mathsf{\Delta }}\underset{0}{\overset{1}{\int }}{\left[x+\mu h{x}^{\mathsf{\Delta }}\right]}^{\lambda -1}dh>0.$$

(5.9)

Using (5.9) in (5.8) we have

$$w^{\mathsf{\Delta }}\left(t\right)\le -\eta \left(t\right)q\left(t\right)+{\eta }^{\mathsf{\Delta }}\left(t\right)\frac{{\left(x^{{\mathsf{\Delta }}^{n-1}}\left(t\right)\right)}^{\sigma }}{{\left(x^{\sigma }\left(t\right)\right)}^{\lambda }}\le -\eta \left(t\right)q\left(t\right)+{\eta }^{\mathsf{\Delta }}\left(t\right)\frac{x^{{\mathsf{\Delta }}^{n-1}}\left(t\right)}{x^{\lambda }\left(t\right)},t\ge t_1,$$

and hence in view of (5.5), we find

$$w^{\mathsf{\Delta }}\left(t\right)\le -\eta \left(t\right)q\left(t\right)+\frac{{\eta }^{\mathsf{\Delta }}\left(t\right)}{h_{n-1}\left(t,t_1\right)}{x}^{1-\lambda }\left(t\right),t>t_1.$$

(5.10)

Let λ > 1. Since there exist c > 0 and t₂ ≥ t₁ such that x(t) ≥ c for all t ≥ t₂, we have x^1-λ(t) ≤ c^1-λ: = c₁ for all t ≥ t₂. If λ = 1, then x^1-λ(t) = 1 for all t ≥ t₁. If λ < 1, then there exist b > 0 and t₃ ≥ t₁ such that $x^{{\mathsf{\Delta }}^{n-1}}\left(t\right)\le b$ for all t ≥ t₃, and hence $x^{1-\lambda }\left(t\right)\le c_2{h}_{n-1}^{1-\lambda }\left(t,t_1\right)$ for all t ≥ t₃, where c₂ : = b^1-λ. Combining all these we see that

$$x^{1-\lambda }\left(t\right)\le A\left(t,t_1\right),t\ge t_4$$

(5.11)

for some t₄ ≥ max{t₂, t₃}. From (5.10) and (5.11),

$$w^{\mathsf{\Delta }}\left(t\right)\le -\eta \left(t\right)q\left(t\right)+{\eta }^{\mathsf{\Delta }}\left(t\right)\frac{A\left(t,t_1\right)}{h_{n-1}\left(t,t_1\right)},t\ge t_4.$$

Integrating this inequality from t₄ to t, we find

$$\underset{t_4}{\overset{t}{\int }}\left[\eta \left(s\right)q\left(s\right)-{\eta }^{\mathsf{\Delta }}\left(s\right)\frac{A\left(s,t_1\right)}{h_{n-1}\left(s,t_1\right)}\right]\mathsf{\Delta }s\le w\left(t_4\right).$$

Taking limit superior as t → ∞, we obtain a contradiction to condition (5.6). This completes the proof. □

In the following result, we employ the lemma below, see [10].

Lemma 5.1. If X and Y are nonnegative and α > 1, then

$$X^{\alpha }-\alpha X{Y}^{\alpha -1}+\left(\alpha -1\right)Y^{\alpha }\ge 0,$$

(5.12)

where equality holds if and only if X = Y.

Theorem 5.2. Let condition (5.2) hold. If there exists a positive, nondecreasing, differentiable function $\eta \in C_{rd}\left(\mathbb{T},{ℝ}^{+}\right)$ such that for any t₁ ≥ t₀,

$$\underset{t\to \infty }{\mathsf{\text{lim}}\mathsf{\text{sup}}}\underset{t_1}{\overset{t}{\int }}\left[\eta \left(s\right)q\left(s\right)-\frac{{\left({\eta }^{\mathsf{\Delta }}\left(s\right)\right)}^{\lambda +1}}{{\left(\lambda +1\right)}^{\lambda +1}{\left(h_{n-2}\left(s,t_0\right)\eta \left(s\right)B\left(s,t_0\right)\right)}^{\lambda }}\right]\mathsf{\Delta }s=\infty ,$$

(5.13)

where

$$B\left(t,t_0\right)=\left\{\begin{array}{cc}{c}_1,c_{1}isanypositiveconstant,\hfill & when\lambda >1\hfill \\ 1,\hfill & when\lambda =1\hfill \\ c_2{\left(h_{n-1}^{\sigma }\left(t,t_0\right)\right)}^{\lambda +1},c_{2}isanypositiveconstant,\hfill & when\lambda <1,\hfill \end{array}\right.$$

(5.14)

then Equation (5.1) is oscillatory.

Proof. Let x(t) be a nonoscillatory solution of Equation (5.1), say, x(t) > 0 for t ≥ t₀. Let w be as in (5.7). Then (5.8) and (5.9) hold. We also have

$$w^{\mathsf{\Delta }}\le -\eta q+\frac{{\eta }^{\mathsf{\Delta }}}{\eta }{w}^{\sigma }-\lambda \eta \left(\frac{x^{\mathsf{\Delta }}}{x}\right){\left(w/\eta \right)}^{\sigma }.$$

(5.15)

Using the fact that ℓ = n - 1 and

$$\frac{x^{{\mathsf{\Delta }}^{n-1}}}{x}\ge {\left({\left(\frac{w}{\eta }\right)}^{\sigma }\right)}^{1/\lambda }{\left(x^{\sigma }\right)}^{\lambda -1}$$

in (5.15), we obtain

$$w^{\mathsf{\Delta }}\le -\eta q+\frac{{\eta }^{\mathsf{\Delta }}}{\eta }{w}^{\sigma }-\lambda \eta h_{n-2}{\left({\left(\frac{w}{\eta }\right)}^{\sigma }\right)}^{1+1/\lambda }{\left(x^{\sigma }\right)}^{\lambda -1}.$$

(5.16)

If λ > 1, then from x^σ(t) ≥ x^σ(t₁) for t ≥ t₁, we have (x^σ(t))^λ-1≥ c₁ = (x^σ(t₁))^λ-1. In case λ = 1, (x^σ(t))^λ-1= 1 for all t ≥ t₁. Finally, let λ < 1. We see that there exist t₂ ≥ t₁ and b > 0 such that $x^{{\mathsf{\Delta }}^{n-1}}\left(t\right)\le b$ for all t ≥ t₂. It follows that x(t) ≤ bh_n-1(t, t₁) for all t ≥ t₂, and hence ${\left(x^{\sigma }\left(t\right)\right)}^{\lambda -1}\ge b^{\lambda -1}{\left(h_{n-1}^{\sigma }\left(t,t_1\right)\right)}^{\lambda -1}$ for all t ≥ t₂, where c₂ = b^λ-1. Putting all these together, we have

$${\left(x^{\sigma }\left(t\right)\right)}^{\lambda -1}\ge B\left(t,t_1\right),t\ge t_2.$$

(5.17)

In view of (5.17) and (5.16), we find

$$w^{\mathsf{\Delta }}\left(t\right)\le -\eta \left(t\right)q\left(t\right)+\frac{{\eta }^{\mathsf{\Delta }}\left(t\right)}{\eta \left(t\right)}{w}^{\sigma }\left(t\right)-\lambda \eta \left(t\right)h_{n-2}\left(t,t_1\right)B\left(t,t_1\right){\left({\left(\frac{w\left(t\right)}{\eta \left(t\right)}\right)}^{\sigma }\right)}^{1+1/\lambda },t\ge t_2.$$

(5.18)

Now, setting

$$X={\left(\lambda \eta h_{n-2}B\right)}^{\lambda /\left(\lambda +1\right)}{\left(\frac{w}{\eta }\right)}^{\sigma }\mathsf{\text{and }}Y={\left(\frac{\lambda }{\lambda +1}\right)}^{\lambda }{\left({\eta }^{\mathsf{\Delta }}\right)}^{\lambda }{\left({\left(\frac{1}{\lambda h_{n-2}B}\right)}^{\lambda /\left(\lambda +1\right)}\right)}^{\lambda }$$

and α = (λ + 1)/λ > 1 in Lemma 5.1, we have

$$\lambda \eta h_{n-2}B{\left({\left(\frac{w}{\eta }\right)}^{\sigma }\right)}^{1+1/\lambda }-{\eta }^{\mathsf{\Delta }}{\left(\frac{w}{\eta }\right)}^{\sigma }+\frac{{\left({\eta }^{\mathsf{\Delta }}\right)}^{\lambda +1}}{{\left(\lambda +1\right)}^{\lambda +1}{\left(\eta h_{n-1}B\right)}^{\lambda }}\ge 0.$$

Therefore, from (5.18)

$$w^{\mathsf{\Delta }}\le -\eta q+\frac{1}{{\left(\lambda +1\right)}^{\lambda +1}}-\frac{{\left({\eta }^{\mathsf{\Delta }}\right)}^{\lambda +1}}{{\left(\eta h_{n-1}B\right)}^{\lambda }},t\ge t_2.$$

Integrating this inequality from t₂ to t results in

$$\underset{t_2}{\overset{t}{\int }}\left[\eta \left(s\right)q\left(s\right)-\frac{1}{{\left(\lambda +1\right)}^{\lambda +1}}\frac{{\left({\eta }^{\mathsf{\Delta }}\left(s\right)\right)}^{\lambda +1}}{{\left(\eta \left(s\right)h_{n-2}\left(s,t_1\right)B\left(s,t_1\right)\right)}^{\lambda }}\right]\mathsf{\Delta }s\le w\left(t_2\right),$$

which contradicts (5.15). This completes the proof. □

Finally, we present the following result.

Theorem 5.3. Let condition (5.2) hold. If there exists a positive, nondecreasing differentiable function η such that for any t₁ ≥ t₀,

$$\underset{t\to \infty }{\mathsf{\text{lim}}\mathsf{\text{sup}}}\underset{t_1}{\overset{t}{\int }}\left[\eta \left(s\right)q\left(s\right)-\frac{{\left({\eta }^{\mathsf{\Delta }}\left(s\right)\right)}^{\lambda }}{4\lambda \eta \left(s\right)B\left(s,t_0\right)h_{n-2}\left(s,t_0\right){\left(h_{n-1}^{\sigma }\left(s,t_0\right)\right)}^{\lambda }}\right]\mathsf{\Delta }s=\infty ,$$

(5.19)

where B(t, t₀) is as in (5.14), then Equation (5.1) is oscillatory.

Proof. Let x(t) be a nonoscillatory solution of Equation (5.1), say, x(t) > 0 for t ≥ t₀. Proceeding as in the proof of Theorem 5.2, we obtain

$$\begin{array}{cc}\hfill w^{\mathsf{\Delta }}& \le -\eta q+{\eta }^{\mathsf{\Delta }}{\left(\frac{w}{\eta }\right)}^{\sigma }-\lambda \eta h_{n-2}B{\left({\left(\frac{w}{\eta }\right)}^{\sigma }\right)}^{1+1/\lambda }\hfill \\ =-\eta q+{\eta }^{\mathsf{\Delta }}{\left(\frac{w}{\eta }\right)}^{\sigma }-\lambda \eta h_{n-2}B\frac{{\left(w^{\sigma }\right)}^{1/\lambda -1}}{{\left({\eta }^{\sigma }\right)}^{1/\lambda +1}}{\left(w^{\sigma }\right)}^2,\hfill \end{array}$$

where B = B(t, t₁) and h_n-2= h_n-2(t, t₁). Since

$$w^{1/\lambda -1}\left(t\right)={\lambda }^{1/\lambda -1}{\left(\frac{x^{{\mathsf{\Delta }}^{n-1}}\left(t\right)}{x\left(t\right)}\right)}^{1-\lambda }\ge {\eta }^{1/\lambda -1}\left(t\right)h_{n-1}^{\lambda -1}\left(t,t_1\right),$$

it follows that

$$\begin{array}{cc}\hfill w^{\mathsf{\Delta }}& \le -\eta q+{\eta }^{\mathsf{\Delta }}{\left(\frac{w}{\eta }\right)}^{\sigma }-\lambda \eta B{h}_{n-1}{\left(h_{n-1}^{\sigma }\right)}^{\lambda -1}{\left(\frac{w^{\sigma }}{{\eta }^{\sigma }}\right)}^2\hfill \\ =-\eta q-{\left[{\left(\lambda \eta B{h}_{n-2}{\left(h_{n-1}^{\sigma }\right)}^{\lambda -1}\right)}^{1/2}{\left(\frac{w}{\eta }\right)}^{\sigma }-\frac{{\eta }^{\mathsf{\Delta }}}{2{\left(\lambda \eta B{h}_{n-2}{\left(h_{n-1}^{\sigma }\right)}^{\lambda -1}\right)}^{1/2}}\right]}^2\hfill \\ +\frac{{\left({\eta }^{\mathsf{\Delta }}\right)}^2}{4\lambda \eta B{h}_{n-2}{\left(h_{n-1}^{\sigma }\right)}^{\lambda -1}}\hfill \\ \le -\eta q+\frac{{\left({\eta }^{\mathsf{\Delta }}\right)}^2}{4\lambda \eta B{h}_{n-2}{\left(h_{n-1}^{\sigma }\right)}^{\lambda -1}},t\ge t_2.\hfill \end{array}$$

Integrating this inequality from t₂ to t, we have

$$\underset{t_2}{\overset{t}{\int }}\left[\eta \left(s\right)q\left(s\right)-\frac{{\left({\eta }^{\mathsf{\Delta }}\left(s\right)\right)}^2}{4\lambda \eta \left(s\right)B\left(s,t_1\right)h_{n-2}\left(s,t_1\right){\left(h_{n-1}^{\sigma }\left(s,t_1\right)\right)}^{\lambda -1}}\right]\mathsf{\Delta }s\le w\left(t_2\right),$$

which contradicts (5.19). This completes the proof. □

Remark 3. We note that the oscillation criteria given in this article are new for the corresponding difference equations and some of these results are new for the corresponding differential and/or delay differential equations. The results can be extended easily to equations of the form

$$x^{{\mathsf{\Delta }}^n}\left(t\right)+f\left(t,x\left(\xi \left(t\right)\right)\right.=0,$$

when $f:\mathbb{T}\times ℝ\to ℝ$ is continuous and f is strongly superlinear or f is strongly sublinear, see [4].

As examples, we have reformulated some of the obtained results for the time-scales $\mathbb{T}=ℝ$ (i.e., the continuous case) and $\mathbb{T}=\mathbb{Z}$ (i.e., the discrete case). One may obtain more results by employing other types of time scales such as $\mathbb{T}=h\mathbb{Z}$ with h > 0, $\mathbb{T}=q^{{\mathbb{N}}_0}$ with q > 1, and $\mathbb{T}={\mathbb{N}}_0^2$, see [8]. The details are left to the reader.