By Hille E.

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Additional info for An Integral Equality and its Applications

Example text

Let n = 2 in (II) and R+°, g :R - R is assume that H(t,u) a p(t)g(u) with p : RT Moreover, let there increasing, and ug(u) > 0 for u ¢ 0. , on which Q1(t) > 6. Then the conditions Tp(t)dt = +c and frr QI(s)dsl = o IJ t JTp(s)ds T are sufficient for (II) to be oscillatory. As Teufel notes in [319], Duffing's equation satisfies the above hypotheses. 2 were extended by True [325; Chapter 3] to functional differential equations. Similar considerations can be found in the papers of True [326], Kusano and Onose [183] and Onose [247].

Amer. Math. Soc. 179 (1973), 193-209. T. A. Burton and C. G. Townsend, On the Generalized Lidnard equation with forcing function, J. Differential Equations 4 (1968), 620-633. 37. 38. 39. G. J. Butler, The oscillatory behaviour of a second order nonlinear differential equation with damping, J. Math. Anal. , to appear. G. J. Butler, Oscillation theorems for a nonlinear analogue of Hill's equation, Quart. J. Math. Oxford, to appear. 40. G. J. Butler, The existence of continuable solutions of a second order differential equation, to appear.

S. p : RT - R_, and Q : RT -} R with uH(t,u) < 0 for every (t,u) x (R - {O}). If w(t) + r(t) > 0 (w(t) + r(t) < 0) for R F- T t > t1 > T, then w(n-2)(t) is of fixed sign on the interval [tl,-). If, moreover, I tp(t)dt then w(n-2) (t) Proof. 14) > 0 (w (n-2) (t) < 0) eventually. Let w(t) + r(t) > 0 for t > t1 > T. Moreover, assume that y(t) a w(n-2)(t) takes positive and negative values for all large t. Then y(t) satisfies the equation y" + p(t)y + q(t) = 0 where q(t) a H(t,w(t) + r(t)) < 0, t > t1.