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

Abstract

This article is concerned with second order nonlinear delay, and especially ordinary, differential equations. By the use of the fixed point technique based on the classical Schauder's theorem, for any given line, sufficient conditions are established in order that there exists at least one global solution which is asymptotic at ∞ to this line. In the special case of ordinary differential 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 ∞ to this line. The application of the results obtained to second order delay, and ordinary, differential 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 sufficient conditions for all global solutions belonging to a suitable class to be asymptotic at ∞ to lines.