Tomoyuki Tanigawa

Generalized Regularly Varying Functions of Self-Adjoint Functional Differential Equations

abstract:
The sharp sufficient conditions of the existence of generalized regularly varying solutions (in the sense of Karamata) of differential equations of the type
\begin{equation*}
\big(p(t)\varphi (x^{\pr}(t))\big)^{\pr}\pm \sum_{i=1}^{n}\Big[q_{i}(t)\varphi \big(x(g_{i}(t))\big)+r_{i}(t)\varphi \big(x(h_{i}(t))\big)\Big]=0
\end{equation*}
are established. Here, $p,q_{i},r_{i}:[a,\infty)\to(0,\infty)$ are continuous functions, $g_{i},h_i:[a,+\infty)\to R$ are continuous and increasing functions such that $g_{i}(t)<t$, $h_{i}(t)>t$ for $t\geq a$, $\lim\limits_{t\to\infty}g_{i}(t)=\infty$ and $\varphi(\xi)\equiv |\xi|^{\alpha}\hsgn \xi$, $\al>0$.

Mathematics Subject Classification: 34C11, 26A12

Key words and phrases: Second order nonlinear differential equation with deviating arguments, generalized regularly varying solution, asymptotic behavior of a solution