| Contents |
| In the recent join work with professor Irena Rachunkova (Palacky University Olomouc) we investigate one class of solutions to the second order differential equation
$$
\left( p(t)u'(t) \right)' + q(t)f\left( u(t) \right)=0,
$$
on the interval $[a,\infty), a\geq 0$, where $p$ and $q$ are regularly varying functions.
Our aim is to describe asymptotic behaviour of non-oscillatory solutions satisfying one of the following conditions
\begin{eqnarray*}
u(a)=u_0\in (0,L), \ 0\leq u(t)\leq L, \ t\in [a,\infty),\\
u(a)=u_0\in (L_0,0), \ L_0\leq u(t)\leq 0,\ t\in [a,\infty),
\end{eqnarray*}
where the interval $[L_0, L]$ is determinated by function $f$. Asymptotic formulas are derived for solutions of the above problem and for their first derivatives.
We prove the existence of Kneser solutions on $[a,\infty), a>0$ and we also discuss the singular case $a=0$ and $p(0)=0$ provided $q\equiv p$. |
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