An Oscillation Criterion for Nonlinear Third-order Differential Equations

نویسنده

  • M. GREGUŠ
چکیده

Sufficient conditions for oscillation of a certain class of nonlinear third-order differential equations are found. In this paper we considear a nonlinear third-order differential equation of the form y′′′ + q(t)y′ = f(t, y, y′, y′′) (1) where (i) q, q′ ∈ C ( (a,∞) ) for some a with 0 < a < ∞, (ii) f ∈ C ( (a,∞)×R3 ) , t ∈ (a,∞), and y1, y2, y3 ∈ R and f(t, y1, y2, y3)y1 < 0 for all t ∈ (a,∞) and all y1, y2, y3 ∈ R with y1 6= 0. By a solution of (1) (proper solution) we mean a function y defined on an interval [t0,∞), t0 > a, which has a continuous third derivative with sup(|y(s)| : s > t) > 0 for any t ∈ [t0,∞), and satisfies equation (1). By an oscillatory solution we mean a solution y of (1) that has arbitrarily large zeros. Otherwise the solution is said to be nonoscillatory. The aim of this paper is to study the oscillatory properties of proper solutions of equation (1) when the operator on the left-hand side of equation (1) is oscillatory. I. T. Kiguradze investigated in [1] the equation u(n) + u(n−2) = f ( t, u, u′, . . . , u(n−1) ) (2) and our aim is to generalize one of his results concerning equation (1). Papers [2], [3] investigated similar problems. The equation in [1] is a special case of (1) and in [3] there is an equation with a more general linear operator on the left-hand side, but with a special case of nonlinearity. 1991 Mathematics Subject Classification. 34C15.

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تاریخ انتشار 2001