Asymptotic integration of second-order nonlinear differential equations via principal and nonprincipal solutions

摘要

Let u and v denote respectively the principal and nonprincipal solutions of the second-order linear equation (p(t)x′)′+q(t)x=0 defined on some half-line of the form [t∗,∞).It is shown that for any given real numbers a and b the nonlinear equation(p(t)x′)′+q(t)x=f(t,x)has a solution x(t) asymptotic to av(t)+bu(t) as t→∞ under some reasonable conditions on the function f. Examples are given to illustrate the results.