Vol. 24, No. 2, 1968

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ISSN: 0030-8730
A stability theorem for a third order nonlinear differential equation

J. L. Nelson

Vol. 24 (1968), No. 2, 341–344
Abstract

A stability theorem and a corollary are proved for a nonlinear nonautonomous third order differential equation. A remark shows that the results do not hold for the linear case. THEOREM. Let p(t) and q(t) be continuous and q(t) 0, p(t) < 0 with p(t) 0. For any A and B suppose

        ∫
t
A +Bt −  t1 q(s)ds < 0

for large t where Q(t) = t0tq(s)ds, then any nonoscillatory solution ixj(t) of the equation

˙x = p(t)˙x+ q(t)x2n+1 = 0,n = 1,2,S,⋅⋅⋅ ,

has the following properties; sgn x = sgn X, sgn ,limt→∞(t)

= lt→im∞ ˙x(t) = 0,t li→m∞ |x(t)| = L ≧ 0,

and x(t)X(t),X(t) are monotone functions. COROLLARY. If q(t) >> 0 for large t, then limt→∞x(t) = 0.

Mathematical Subject Classification
Primary: 34.50
Milestones
Received: 1 February 1967
Published: 1 February 1968
Authors
J. L. Nelson