Structure of the set of bounded solutions for a class of nonautonomous second order differential equations (Q618978)

This paper studies the structure of the set of bounded solutions of the second order non-autonomous differential equation \[ x''+g(x)x'+f(\theta_t h,x)=0. \] If \(f,g\) are locally Lipschitz, \(g\) is nonnegative and \(f\) is strictly decreasing in \(x\), it is proved that the existence of a bounded solution on \((0,+\infty )\) implies the existence of a unique bounded solution on the whole real line. Besides, any bounded solution on \((0,+\infty )\) is attracted by such a solution, and the set of initial conditions of the bounded solutions on \((0,+\infty )\) is a non-increasing curve. The results are applicable to singular nonlinearities and generalize former results by \textit{P. Cieutat} [Nonlinear Anal., Theory Methods Appl. 58, No.~7--8, A, 885--898 (2004; Zbl 1059.34030)] and the reviewer and collaborators [\textit{P. Martínez-Amores} and \textit{P. J. Torres}, J. Math. Anal. Appl. 202, No.~3, 1027--1039 (1996; Zbl 0865.34030); \textit{J. Campos} and \textit{P. J. Torres}, Proc. Am. Math. Soc. 127, No.~5, 1453--1462 (1999; Zbl 0920.34045)].