PROCEEDINGS OF THE AMERICAN MATHEMATICAL SOCIETY Volume 30, No. 1, September 1071 NONOSCILLATION PROPERTIES OF A NONLINEAR DIFFERENTIAL EQUATION MICHAEL E. HAMMETT Abstract. Sufficient conditions are given for the approach to zero of all nonoscillatory solutions of ipit)x')' +q(t)g(x) =/(/). The conditions are related to an oscillation theorem of N. P. Bhatia concerning the equation (pit)x')'-\-qit)gix) =0. Call a function on [a, 4-oo) oscillatory if it has arbitrarily large zeros. Otherwise call it nonoscillatory. Consider the differential equation (1) ipit)x')' + qiOgix) = 0 where pit), qit)EC[0, +«>), pit)>0, gix)ECi- <»,+ <*>)- Bhatia [l] has proved the following result. Theorem. All solutions of (1) defined on [0, 4- co) are oscillatory on [0, 4- «o ) provided thefollowing conditions hold: /■+» i — dt = +cc, o Pit) (3) f q(t)dt=+«>, J o (4) xgix) > 0 if x 9± 0, (5) g'ix) ^ 0. Consider now the equation (6) ipiDx')' + qit)gix) = f(t) where fit) E C [0, 4- °° ) ■ Clearly the solutions of (6) are not necessarily all oscillatory even if conditions (2)-(5) are satisfied and fit) is small in a strong sense, such as /J"" \f(t)\dt< + <*>. The equation x"4~x = 1er1 satisfies (2)-(5) and /0+" 2trx dt < + ». However, the solution x(t)=e~l is nonoscillatory. Note that x(/) = e~'—>0 as I—>+ ». This simple example illustrates the main result of this paper. Received by the editors November 30, 1970. AMS 1970 subject classifications. Primary 34C10, 34C15; Secondary 34D05, 34E0S. Key words and phrases. Oscillatory, nonoscillatory, nonlinear, differential equation, solution, monotonie, approach zero. Copyright © 1971, American Mathematical Society 92 License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use