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Asymptotic solutions for second order delay differential equations. (English) Zbl 0871.34045

We study asymptotic formulae for the solutions of the second order delay-differential equations \[ u''(t)+ b(t)u(t- r(t))=0 \] and \[ (a(t)u'(t))'+ b(t)u(t-r(t))=0. \] More generally, consider the second order functional differential equation \[ (a(t)u'(t))'+ b(t)f(t,u(t-r(t)))=0, \] where the functions \(a,b: [0,\infty)\to (0,\infty)\), \(f:[0,\infty)\times \mathbb{R}\to \mathbb{R}\) and \(r:[0,\infty)\to [0,r_0]\) (for some \(r_0\)) are continuous.

MSC:

34K25 Asymptotic theory of functional-differential equations
34B27 Green’s functions for ordinary differential equations
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