Consider the second order superlinear dynamic equation

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

where $p\,\in \,C(\mathbb{T},\,\mathbb{R})$, $\mathbb{T}$ is a time scale, $f\,:\,\mathbb{R}\,\to \,\mathbb{R}$ is continuously differentiable and satisfies ${{f}^{'}}(x)>0$, and $x\,f\,(x)\,>\,0$ for $x\,\ne \,0$. Furthermore, $f(x)$ also satisfies a superlinear condition, which includes the nonlinear function $f(x)\,=\,{{x}^{\alpha }}$ with $\alpha \,>\,1$, commonly known as the Emden–Fowler case. Here the coefficient function $p(t)$ is allowed to be negative for arbitrarily large values of $t$. In addition to extending the result of Kiguradze for $\left( * \right)$ in the real case $\mathbb{T}\,=\,\mathbb{R}$, we obtain analogues in the difference equation and $q$-difference equation cases.

In the classical theorems about lower and upper vector functions for systems of linear differential equations very heavy restrictions on the signs of coefficients are assumed. These restrictions in many cases become necessary if we wish to compare all the components of a solution vector. The formulas of the integral representation of the general solution explain that these theorems claim actually the positivity of all elements of the Green's matrix. In this paper we define a principle of partial monotonicity (comparison of only several components of the solution vector), which assumes only the positivity of elements in a corresponding row of the Green's matrix. The main theorem of the paper claims the equivalence of positivity of all elements in the nth row of the Green's matrices of the initial and two other problems, non-oscillation of the nth component of the solution vector and a corresponding assertion about differential inequality of the de La Vallee Poussin type. Necessary and sufficient conditions of the partial monotonicity are obtained. It is demonstrated that our sufficient tests of positivity of the elements in the nth row of the Cauchy matrix are exact in corresponding cases. The main idea in our approach is a construction of an equation for the nth component of the solution vector. In this sense we can say that an analog of the classical Gauss method for solving systems of functional differential equations is proposed in the paper.

In this work, oscillatory and asymptotic behaviours of all solutions of higher-order neutral differential equations are compared with first-order delay differential equations, depending on two different ranges of the coefficient associated with the neutral part. Some simple examples are given to compare our results with the existing results in the literature and to illustrate the significance and applicability of our new results. Our results generalise, improve and correct some of the existing results in the literature.

By means of generalized Riccati transformation techniques and generalized exponential functions, some oscillation criteria are given for the nonlinear dynamic equation \[ (p(t)x^{\Delta} (t))^{\Delta}+q(t)(f\circ x^{\sigma})=0 \] on time scales. The results are also applied to linear and nonlinear dynamic equations with damping, and some sufficient conditions are obtained for the oscillation of all solutions.