We study oscillatory properties of solutions of the Emden-Fowler type differential equation $$u^{\left(n\right)}\left(t\right)+p\left(t\right){\left|u\left(\sigma \left(t\right)\right)\right|}^{\lambda }signu\left(\sigma \left(t\right)\right)=0,$$ where $0<\lambda <1$, $p\in L_{\mathrm{loc}}({ℝ}_{+};ℝ)$, $\sigma \in C({ℝ}_{+};{ℝ}_{+})$ and $\sigma \left(t\right)\ge t$ for $t\in {ℝ}_{+}$. Sufficient (necessary and sufficient) conditions of new type for oscillation of solutions of the above equation are established. Some results given in this paper generalize the results obtained in the paper by Kiguradze and Stavroulakis (1998).

For the equation $$y^{\left(n\right)}+{\left|y\right|}^k\mathrm{sgn}y=0,k>1,n=3,4,$$ existence of oscillatory solutions $$y={(x^{*}-x)}^{-\alpha }h(log(x^{*}-x)),\alpha =\frac{n}{k-1},x<x^{*},$$ is proved, where $x^{*}$ is an arbitrary point and $h$ is a periodic non-constant function on $ℝ$. The result on existence of such solutions with a positive periodic non-constant function $h$ on $ℝ$ is formulated for the equation $$y^{\left(n\right)}={\left|y\right|}^k\mathrm{sgn}y,k>1,n=12,13,14.$$

The author considers the quasilinear differential equations $$\begin{array}{c}{\left(r\left(t\right)\varphi \left(x^{\text{'}}\right)\right)}^{\text{'}}+q\left(t\right)f\left(x\right)=0,t\ge a\\ \multicolumn{2}{c}{\text{and}}\\ {\left(r\left(t\right)\varphi \left(x^{\text{'}}\right)\right)}^{\text{'}}+F(t,x)=\pm g\left(t\right),t\ge a.\end{array}$$ By means of topological tools there are established conditions ensuring the existence of nonnegative asymptotic decaying solutions of these equations.

The paper describes asymptotic properties of a strongly nonlinear system $\dot{x}=f(t,x)$, $(t,x)\in ℝ\times {ℝ}^n$. The existence of an $\lfloor n/2\rfloor$ parametric family of solutions tending to zero is proved. Conditions posed on the system try to be independent of its linear approximation.

The asymptotic properties of solutions of the equation $u^{\text{'}\text{'}\text{'}}\left(t\right)=p_1\left(t\right)u\left({\tau }_1\left(t\right)\right)+p_2\left(t\right)u^{\text{'}}\left({\tau }_2\left(t\right)\right)$, are investigated where $p_i:[a,+\infty [\to R(i=1,2)$ are locally summable functions, ${\tau }_i:[a,+\infty [\to R(i=1,2)$ measurable ones and ${\tau }_i\left(t\right)\ge t(i=1,2)$. In particular, it is proved that if $p_1\left(t\right)\le 0$, $p_2^2\left(t\right)\le \alpha \left(t\right)\left|p_1\left(t\right)\right|$, $${\int }_a^{+\infty }{[{\tau }_1\left(t\right)-t]}^2{p}_1\left(t\right)dt<+\infty \text{and}{\int }_a^{+\infty }\alpha \left(t\right)dt<+\infty ,$$ then each solution with the first derivative vanishing at infinity is of the Kneser type and a set of all such solutions forms a one-dimensional linear space.

The (modified) two-parametric Mittag-Leffler function plays an essential role in solving the so-called fractional differential equations. Its asymptotics is known (at least for a subset of its domain and special choices of the parameters). The aim of the paper is to introduce a discrete analogue of this function as a solution of a certain two-term linear fractional difference equation (involving both the Riemann-Liouville as well as the Caputo fractional $h$-difference operators) and describe its...

We define a non-smooth guiding function for a functional differential inclusion and apply it to the study the asymptotic behavior of its solutions.