Asymptotic analysis of positive solutions of first-order cyclic functional differential systems

Abstract

AbstractThe structure and the asymptotic behavior of positive increasing solutions of functional differential systems of the form$x^{\prime}(t)=p(t)\varphi_{\alpha}\bigl{(}y(k(t))\bigr{)},\quad y^{\prime}(t)=% q(t)\varphi_{\beta}\bigl{(}x(l(t))\bigr{)}$are investigated in detail, where α and β are positive constants,${p(t)}$and${q(t)}$are positive continuous functions on${[0,\infty)}$,${k(t)}$and${l(t)}$are positive continuous functions on${[0,\infty)}$tending to${\infty}$witht, and${\varphi_{\gamma}(u)=\lvert u\rvert^{\gamma}\operatorname{sgn}u}$,${\gamma>0}$,${u\in\mathbb{R}}$. An extreme class of positive increasing solutions, calledrapidly increasing solutions, of the system above is analyzed by means of regularly varying functions. The results obtained find applications to systems of the form$x^{\prime}(g(t))=p(t)\varphi_{\alpha}\bigl{(}y(k(t))\bigr{)},\quad y^{\prime}(% h(t))=q(t)\varphi_{\beta}\bigl{(}x(l(t))\bigr{)},$and to scalar equations of the type$\Bigl{(}p(t)\varphi_{\alpha}\bigl{(}x^{\prime}(g(t))\bigr{)}\Bigr{)}^{\prime}=% p(t)\varphi_{\beta}\bigl{(}x(l(t))\bigr{)}.$