Vol. 83, No. 2, 1979

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ISSN: 0030-8730
On the nonoscillation of perturbed functional-differential equations

John R. Graef, Yuichi Kitamura, Takaŝi Kusano, Hiroshi Onose and Paul Winton Spikes

Vol. 83 (1979), No. 2, 365–373
Abstract

We study the behavior of the solutions of the second order nonlinear functional differential equation

(a(t)x′)′ = f(t,x(t),x(g(t)))
(1)

where a, g : [t0,) R and f : [t0,) × R2 R are continuous, a(t) > 0, and g(t) →∞ as t →∞. We are primarily interested in obtaining conditions which ensure that certain types of solutions of (1) are nonoscillatory. Conditions which guarantee that oscillatory solutions of (1) converge to zero as t →∞ are also given. We apply these results to the equation

(a(t)x′)′ + q(t)r(x(g(t))) = e(t,x)
(2)

where q : [t0,) R, r : R R, e : [t0, ) × R R are continuous and a and g are as above. We compare our results to those obtained by others. Specific examples are included.

Mathematical Subject Classification
Primary: 34K15
Milestones
Received: 18 October 1978
Revised: 28 February 1979
Published: 1 August 1979
Authors
John R. Graef
Yuichi Kitamura
Takaŝi Kusano
Hiroshi Onose
Paul Winton Spikes