Global Solutions Approaching Lines at Infinity to Second Order Nonlinear Delay Differential Equations

Abstract

Abstract. This article is concerned with second order nonlinear delay, and especially ordinary, di¤erential equations. By the use of the fixed point technique based on the classical Schauder's theorem, for any given line, su‰cient conditions are established in order that there exists at least one global solution which is asymptotic at y to this line. In the special case of ordinary di¤erential equations, via the Banach's Contraction Principle, for any given line, conditions are presented which guarantee that there exists a unique global solution that is asymptotic at y to this line. The application of the results obtained to second order delay, and ordinary, di¤erential equations of Emden-Fowler type is presented, and some examples demonstrating the applicability of the results are given. Finally, some supplementary results are obtained, which provide su‰cient conditions for all global solutions belonging to a suitable class to be asymptotic at y to lines.