Mathematics

On oscillation of solutions to quasi-linear Emden–Fowler type higher-order differential equations

I. V. Astashova

Lomonosov Moscow State University, 1, Leninskie Gory, Moscow, 119991, Russian Federation

Abstract: Existence and behavior of oscillatory solutions to nonlinear equations with regular and singular power nonlinearity are investigated. In particular, the existence of oscillatory solutions is proved for the equation
\begin{gather*} y^{(n)}+P(x,y,y',\ldots,y^{(n-1)})|y|^k\ {\rm sign}\,y=0,\\ n\ge 2,\,\,\,k\in \mathbb {R},\,\,\, k>1,\,\,\, P\neq0,\,\,\,\,P\in C(\mathbb{R}^{n+1}). \end{gather*}
A criterion is formulated for oscillation of all solutions to the quasilinear even-order differential equation
\begin{gather*} y^{(n)}+\sum_{i=0}^{n-1}a_{j}(x)\;y^{(i)}+p(x)\;|y|^{k} {\rm sign} y=0,\\ p\in C(\mathbb{R}),\,\,a_j\in C(\mathbb{R}),\,\,\,j=0,\dots,{n-1},\,\,\, k>1,\,\, n=2m,\,\, m\in\mathbb{N}, \end{gather*}
which generalizes the well-known Atkinson's and Kiguradze's criteria.
The existence of quasi-periodic solutions is proved both for regular ($k>1$) and singular ($0<k<1$) nonlinear equations
$$ y^{(n)}+p_0\,|y|^{k} {\rm sign} y=0, \quad n>2,\quad k\in \mathbb {R},\quad k>0,\,\,\,k\neq1, \quad p_0\in \mathbb {R}, $$
with $(-1)^{n}p_0>0.$ A result on the existence of periodic oscillatory solutions is formulated for this equation with $n=4,\,\,k>0,\,\,k\neq1,\,\,p_0<0.$

Keywords: quasilinear differential equation, power nonlinearity, oscillatory solution, oscillatory criterion, periodic solutions, quasi-periodic solutions.

UDC: 517.9

Received: 18.07.2015

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