Existence of solutions of a boundary value problem on the half-line to second order nonlinear delay differential equations (Q818741)

The authors study, by means of Schauder's fixed-point theorem, the existence of solutions of some boundary value problem on the half-line. Namely, they prove existence of at least one solution of the second-order nonlinear delay differential equation \[ x''(t)+f(t,x_{t},x'(t))=0, \; t \in [0,\infty), \] subject to the initial and asymptotic conditions \[ x(t)=\phi(t) \;\; \text{for} \;\; -r\leq t \leq 0 \;\; \text{and} \;\; \lim_{t\to \infty} x'(t)=\xi, \] where \(\xi\) is a given real number and \(\phi(t) \in C([-r,0], \mathbb{R})\) is a given function with \(\phi(0)=0\). The authors apply their results to second-order ordinary differential equations of the type \[ x''(t)+g(t,x(t),x'(t))=0, \; t \in [0,\infty), \] subject to \[ x(0)= 0 \;\; \text{and} \;\; \lim_{t\to \infty} x'(t)=\xi \] and also to a specific class of second-order nonlinear delay differential equations. Finally, as an example, they illustrate the applicability of their theory to nonlinear differential equations of Emden-Fowler type.