The asymptotic behavior of nonoscillatory solutions of some nonlinear differential equations
The authors deal with the differential equation \[ (r(t)g(y'(t)))'-p(t)f(y(t))=0,\quad t\geq t_0 \tag{1} \] with continuous and sufficiently smooth functions \(r, g, p\) and \(f\). By a solution to (1) it is meant a function \(y \in C^1([t_0,\infty),\mathbb{R})\) such that \(r(t)g(y'(t))\) is differentiable and \(y\) satisfies (1) for all \(t\geq t_0\). The authors study the asymptotic behavior of the so-called extremal solutions and they give their characterization in four theorems. This extends earlier results concerning the asymptotic behavior of solutions of special cases to (1) contained in the works listed in references of the paper under review. Further, using the fixed-point theorem on ordered sets by \textit{A. Tarski} [Pac. J. Math. 5, 285-309 (1955; Zbl 0064.26004)], the authors prove the existence and estimates of nonextremal solutions to (1). Illustrated examples are shown here, as well.
scientific article; zbMATH DE number 1513298
scientific article