Oscillation of second-order nonlinear dynamic equations with positive and negative coefficients

on an arbitrary time scale $\mathbb{T}$ with $sup\mathbb{T}=\mathrm{\infty }$, subject to the following conditions:

(C₁) $t_0\in \mathbb{T}$ and ${[t_0,\mathrm{\infty })}_{\mathbb{T}}:=\{t\in \mathbb{T}:t\ge t_0\}$ is a time scale interval in $\mathbb{T}$;

(C₂) $r\in C_{\mathrm{rd}}({[t_0,\mathrm{\infty })}_{\mathbb{T}},(0,\mathrm{\infty }))$ and ${\int }_{t_0}^{\mathrm{\infty }}\frac{1}{r(t)}\mathrm{\Delta }t=\mathrm{\infty }$;

(C₃) $p,q\in C_{\mathrm{rd}}({[t_0,\mathrm{\infty })}_{\mathbb{T}},[0,\mathrm{\infty }))$;

(C₄) $\xi ,\delta \in C_{\mathrm{rd}}(\mathbb{T},\mathbb{T})$, ${lim}_{t\to \mathrm{\infty }}\xi (t)={lim}_{t\to \mathrm{\infty }}\delta (t)=\mathrm{\infty }$, δ has the inverse function ${\delta }^{-1}\in C_{\mathrm{rd}}(\mathbb{T},\mathbb{T})$, $v:={\delta }^{-1}\circ \xi \in C_{\mathrm{rd}}(\mathbb{T},\mathbb{T})$, ${\xi }^{\mathrm{\Delta }},v^{\mathrm{\Delta }}\in C_{\mathrm{rd}}({[t_0,\mathrm{\infty })}_{\mathbb{T}},(0,\mathrm{\infty }))$, $\xi (t),v(t)\le t$ for $t\in {[t_0,\mathrm{\infty })}_{\mathbb{T}}$, $\xi ({[t_0,\mathrm{\infty })}_{\mathbb{T}})={[\xi (t_0),\mathrm{\infty })}_{\mathbb{T}}$ and $v({[t_0,\mathrm{\infty })}_{\mathbb{T}})={[v(t_0),\mathrm{\infty })}_{\mathbb{T}}$, where $\xi ({[t_0,\mathrm{\infty })}_{\mathbb{T}}):=\{\xi (t):t\in {[t_0,\mathrm{\infty })}_{\mathbb{T}}\}$ and $v({[t_0,\mathrm{\infty })}_{\mathbb{T}}):=\{v(t):t\in {[t_0,\mathrm{\infty })}_{\mathbb{T}}\}$;

(C₅) $f,h\in C(\mathbb{R},\mathbb{R})$, there exist positive constants $L_1$, $L_2$ and M such that $f(u)/u\ge L_1$, $0<h(u)/u\le L_2$ and $|h(u)|\le M$ for $u\ne 0$, and $L_{1}p(t)-L_{2}q(v(t))v^{\mathrm{\Delta }}(t)>0$ for $t\in {[t_0,\mathrm{\infty })}_{\mathbb{T}}$;

(C₆) ${\int }_t^{\mathrm{\infty }}[\frac{1}{r(s)}{\int }_{v(s)}^{s}q(u)\mathrm{\Delta }u]\mathrm{\Delta }s<\mathrm{\infty }$ for every sufficiently large $t\in \mathbb{T}$.

Recall that a solution of (1) is a nontrivial real function x such that $x\in C_{\mathrm{rd}}^1({[t_x,\mathrm{\infty })}_{\mathbb{T}},\mathbb{R})$ and $r{x}^{\mathrm{\Delta }}\in C_{\mathrm{rd}}^1({[t_x,\mathrm{\infty })}_{\mathbb{T}},\mathbb{R})$ for a certain $t_x\ge t_0$ and satisfying (1) for $t\ge t_x$. Our attention is restricted to those solutions of (1) which exist on ${[t_x,\mathrm{\infty })}_{\mathbb{T}}$ and satisfy $sup\{|x(t)|:t>t_{\ast }\}>0$ for any $t_{\ast }\ge t_x$. A solution x of (1) is said to be oscillatory if it is neither eventually positive nor eventually negative; otherwise, it is nonoscillatory. Equation (1) is said to be oscillatory if all its solutions are oscillatory.

For convenience of the readers and completeness of the paper, we recall the following basic concepts and results for the calculus on time scales. More details can be found in [1, 2].

A time scale $\mathbb{T}$ is an arbitrary nonempty closed subset of the real numbers ℝ. We assume throughout that $\mathbb{T}$ has the topology that it inherits from the standard topology on the real numbers ℝ. Some examples of time scales are as follows: the real numbers ℝ, the integers ℤ, the positive integers ℕ, the nonnegative integers ${\mathbb{N}}_0$, $[0,1]\cup [2,3]$, $[0,1]\cup \mathbb{N}$, $h\mathbb{Z}:=\{hk:k\in \mathbb{Z},h>0\}$ and $\overline{q^{\mathbb{Z}}}:=\{q^k:k\in \mathbb{Z},q>1\}\cup \{0\}$. But the rational numbers ℚ, the complex numbers ℂ and the open interval $(0,1)$ are not time scales. Many other interesting time scales exist, and they give rise to plenty of applications (see [1]).

i.e., $f^{\sigma }:=f\circ \sigma$.

Open time scale intervals and half-open time scale intervals etc. are defined accordingly.

In this case, we say that $f^{\mathrm{\Delta }}(t)$ is the (delta) derivative of f at t and that f is (delta) differentiable at t.

where $g^{\sigma }=g\circ \sigma$ and $g{g}^{\sigma }\ne 0$.

The calculus on time scales was introduced by Hilger [3] with the motivation of providing a unified approach to continuous and discrete calculus. The theory of dynamic equations on time scales not only unifies the theories of differential equations and difference equations, but it is also able to extend these classical cases to cases ‘in between’, e.g., to the so-called q-difference equations. Dynamic equations on time scales have an enormous potential for modeling a variety of applications; see, for example, the monograph by Bohner and Peterson [1]. For advances in dynamic equations on time scales, one can see the book by Bohner and Peterson [2].

where σ is the forward jump operator on $\mathbb{T}$, $x^{\sigma }:=x\circ \sigma$, $r(t)\ne 0$, r and p are rd-continuous functions and ${lim}_{s\to t^{-}}r(s)\ne 0$ at all left-dense and right-scattered points. They established a necessary and sufficient condition for the oscillation of (10) by using the so-called trigonometric transformation.

Medico and Kong [5, 6] also investigated the oscillation of (10). They supposed that $r,p\in C_{\mathrm{rd}}(\mathbb{T},\mathbb{R})$ with $r(t)>0$. Medico and Kong [5] gave some Kamenev-type and interval criteria for the oscillation of (10). Their results covered those for differential equations and offered new oscillation criteria for difference equations. Medico and Kong [6] extended the work in [5] by modifying the class of kernel functions and deriving new criteria of Sun type (see [7]).

on time scales in terms of the coefficients and the graininess function, where r and p are positive real-valued rd-continuous functions, ${\int }_{t_0}^{\mathrm{\infty }}\frac{1}{r(t)}\mathrm{\Delta }t=\mathrm{\infty }$ or ${\int }_{t_0}^{\mathrm{\infty }}\frac{1}{r(t)}\mathrm{\Delta }t<\mathrm{\infty }$, and $f:\mathbb{R}\to \mathbb{R}$ such that $xf(x)>0$ and $f(x)/x\ge K>0$ for $x\ne 0$.

Erbe and Peterson [9] also discussed the oscillation of (11), where $r(t)>0$. When no explicit sign assumptions are made with respect to the coefficient p, they established some sufficient conditions for the oscillation of (11) when ${liminf}_{t\to \mathrm{\infty }}{\int }_T^{t}p(t)\mathrm{\Delta }t>0$ for sufficiently large T.

on a time scale $\mathbb{T}$, where ${\xi }_0\in \mathbb{R}$, $t-{\xi }_0\in \mathbb{T}$, $p:\mathbb{T}\to [0,\mathrm{\infty })$ is a real-valued rd-continuous function, $f:\mathbb{R}\to \mathbb{R}$ is continuous, $f(u)$ is nondecreasing, $f(-u)=-f(u)$ for $u\in \mathbb{R}$, and $uf(u)>0$ for $u\ne 0$. They established the equivalence of the oscillation of (12) and (13), from which they obtained some oscillation criteria and a comparison theorem for (12).

on a time scale interval ${[t_0,\mathrm{\infty })}_{\mathbb{T}}$, where $p\in C_{\mathrm{rd}}(\mathbb{T},\mathbb{R})$ is a positive function, $\xi \in C_{\mathrm{rd}}(\mathbb{T},\mathbb{T})$ is an increasing function such that $\xi (t)<t$ and ${lim}_{t\to \mathrm{\infty }}\xi (t)=\mathrm{\infty }$, and $f\in C(\mathbb{R},\mathbb{R})$ satisfies $f(x)/x\ge L$ for a certain positive constant L and for all $x\ne 0$.

where r and p are real rd-continuous positive functions defined on $\mathbb{T}$, the so-called delay function ξ satisfies $\xi :{[t_0,\mathrm{\infty })}_{\mathbb{T}}\to \mathbb{T}$ is rd-continuous, $\xi (t)\le t$ for $t\in {[t_0,\mathrm{\infty })}_{\mathbb{T}}$, ${lim}_{t\to \mathrm{\infty }}\xi (t)=\mathrm{\infty }$, and $f:\mathbb{R}\to \mathbb{R}$ is a continuous function satisfying $uf(u)>0$ for all $u\ne 0$ and $|f(u)|\ge K|u|$. The authors obtained some new oscillation criteria which improved the results established by Zhang and Zhu [10] and Şahiner [11].

Jia et al. [13] also dealt with the oscillation of (13), where $p\in C_{\mathrm{rd}}(\mathbb{T},\mathbb{R})$, $\mathbb{T}$ is a time scale, and $f:\mathbb{R}\to \mathbb{R}$ is continuously differentiable and satisfies $f^{\prime }(x)>0$ and $xf(x)>0$ for $x\ne 0$. Jia et al. [13] obtained several Kiguradze-type oscillation theorems for (13).

where $B\in C_{\mathrm{rd}}({[t_0,\mathrm{\infty })}_{\mathbb{T}},\mathbb{R})$ is allowed to oscillate.

where $sup\mathbb{T}=\mathrm{\infty }$, $A\in C_{\mathrm{rd}}({[t_0,\mathrm{\infty })}_{\mathbb{T}},\mathbb{R})$, $B,C\in C_{\mathrm{rd}}({[t_0,\mathrm{\infty })}_{\mathbb{T}},{(0,\mathrm{\infty })}_{\mathbb{R}})$, $F\in C_{\mathrm{rd}}(\mathbb{R},\mathbb{R})$, and $\alpha ,\beta ,\gamma \in C_{\mathrm{rd}}({[t_0,\mathrm{\infty })}_{\mathbb{T}},\mathbb{T})$ are strictly increasing and unbounded functions. The authors weakened the assumptions on the coefficients that are assumed to hold in the literature and improved some known results by providing necessary and sufficient conditions for the solutions of the equation to oscillate or to converge to zero. The coefficient associated with the neutral part was considered in three distinct ranges, in one of which the coefficient is allowed to oscillate.

to oscillate or to tend to zero as t tends to infinity, where $n\ge 2$ is an integer, $q>0$ and $u\ge 0$. Both bounded and unbounded solutions were considered in this paper.

For some recent other results on the oscillation, nonoscillation and asymptotic behavior of solutions of different types of dynamic equations, we refer the reader to the papers [17–44] and the references cited therein.

The results in [4–6, 8–16] are very valuable. But these results also have some disadvantages. For example, the oscillation criteria of Došlý and Hilger [4] are unsatisfactory since additional assumptions have to be imposed on the unknown solutions.

The results of Zhang and Zhu [10] are valid only when the graininess function $\mu (t)$ is bounded, which is a restrictive condition (e.g., the results cannot be applied to $\mathbb{T}=q^{{\mathbb{N}}_0}:=\{q^k:k\in {\mathbb{N}}_0,q>1\}$, where $\mu (t)=(q-1)t$ is unbounded; see [12]).

is oscillatory if $\beta >\frac{1}{4}$, but (17) does not give this result.

The restriction $f^{\prime }(x)>0$ for $x\ne 0$ is required in [13]. This condition does not hold and cannot be applied in the case when $f(x)=x(\frac{1}{9}+\frac{1}{1+x^2})$, since $f^{\prime }(x)=\frac{(x^2-2)(x^2-5)}{9{(1+x^2)}^2}$ changes sign four times.

The results in [14, 15] cannot be applied to higher-order delay dynamic equations with positive and negative coefficients. The results in [16] fail to be applied to general time scales.

Besides the above-mentioned disadvantages, it is clear that (10)-(15) are some special cases of (1), and that the results in [4–6, 8–13] cannot be applied to general cases of (1). Therefore, it is of great interest to investigate the oscillation of (1). To the best of our knowledge, nothing is known regarding the oscillatory behavior of (1) on time scales up to now. Following the trend shown in [4–6, 8–16], in this paper we deal with the oscillation of (1). We obtain some oscillation criteria for (1) by developing a generalized Riccati substitution technique. Our results are essentially new and extend and improve some results in [4–6, 8–13]. We also illustrate our main results with several examples.

In what follows, for convenience, when we write a functional inequality or equality without specifying its domain of validity, we assume that it holds for all sufficiently large t.

Lemma 2.1 (Substitution [[1], Theorem 1.98])

where ${\eta }^{-1}$ is the inverse function of η and ${\phantom{}}^{\tilde{\mathrm{\Delta }}}$ denotes the derivative on $\tilde{\mathbb{T}}$.

Lemma 2.2 (Existence of antiderivatives [[1], Theorem 1.74])

is an antiderivative of f.

Lemma 2.3 (Chain rule [[1], Theorem 1.93])

where ${\phantom{}}^{\tilde{\mathrm{\Delta }}}$ denotes the derivative on $\tilde{\mathbb{T}}$.

where $a_{+}^{\mathrm{\Delta }}(s):=max\{a^{\mathrm{\Delta }}(s),0\}$. Then every solution of (1) is oscillatory.

which contradicts (18). Thus, the proof is complete. □

where $\epsilon >0$ is an arbitrary constant and $a_{+}^{\mathrm{\Delta }}(s):=max\{a^{\mathrm{\Delta }}(s),0\}$. Then all the solutions of (1) are oscillatory.

Thus, we find ${lim\hspace{0.17em}sup}_{t\to \mathrm{\infty }}{\int }_{t_5}^t[a(s)Q(s)-\epsilon a_{+}^{\mathrm{\Delta }}(s)]\mathrm{\Delta }s\le w(t_5)<\mathrm{\infty }$, which contradicts (40). Hence, the proof is complete. □

for every sufficiently large T, where $a_{+}^{\mathrm{\Delta }}(s):=max\{a^{\mathrm{\Delta }}(s),0\}$ and $g_{+}(t,s):=max\{g(t,s),0\}$. Then all the solutions of (1) are oscillatory.