Asymptotic behavior of the nonoscillatory solutions of differential equations with integrable coefficients. (English) Zbl 0637.34046

Graef, John R.; Spikes, Paul W.

Publ. Math. Debr. 32, 211-221 (1985).

The author deals with the nonoscillatory solutions of the differential equation \((1)\quad (a(t)x')'+q(t)f(x)=h(t)\) where a,h,q: [t\({}_ 0,\infty)\to R\) and \(f: R\to R\) are continuous with \(a(t)>0\). Using techniques motivated by the work of I. V. Kamenev [Differ. Uravn. 13, 2141-2148 (1977; Zbl 0386.34031)] the authors obtain conditions which ensure that any nonoscillatory solution x(t) of (1) satisfies either \(\liminf_{t\to \infty}| x(t)| =0\) or x(t)\(\to 0\) as \(t\to \infty\). Some of the results of the paper are extend to the functional differential equation \((a(t)x')'+Q(t,x(t),x(g(t)))=h(t)\) with g(t)\(\to \infty\) as \(t\to \infty\).

Reviewer: J.Ohriska

MSC:

34E05

Asymptotic expansions of solutions to ordinary differential equations

Keywords:

asymptotic expansions; nonoscillatory solutions; functional differential equation

Citations:

Zbl 0386.34031