Oscillation of higher-order forced nonlinear differential equations Y.G. Sun a, * , A.B. Mingarelli b a Department of Mathematics, Qufu Normal University, Qufu, Shandong 273165, PR China b School of Mathematics and Statistics, Carleton University, Ottawa, Ont., Canada K1S 5B6 Abstract Some new criteria are established for the oscillation of higher-order forced nonlinear differential equations by introduc- ing a class of new auxiliary functions. No restriction is imposed on the forcing term as is generally assumed. With the help of the new auxiliary functions, the main results in this paper are different from those in the paper [Y.G. Sun, S.H. Saker, Forced oscillation of higher-order nonlinear differential equations, Appl. Math. Comput. 173 (2006) 1219–1226] and are more effective than many existing results. Ó 2007 Elsevier Inc. All rights reserved. Keywords: Oscillation; Forced oscillation; Nonlinear equation; Young’s inequality 1. Introduction Consider the forced nonlinear nth order differential equation L n xðtÞ þ ð1Þ n qðtÞF ðxðtÞÞ ¼ eðtÞ; ð1Þ where L 0 xðtÞ¼ xðtÞ; L k xðtÞ¼ a k ðtÞðL k1 xðtÞÞ 0 ; k ¼ 1; 2; ... ; n 0 ¼ d dt ; q 2 Cð½t 0 ; 1Þ; ð0; 1ÞÞ, a i 2 C ni ð½t 0 ; 1; ð0; 1ÞÞ, i ¼ 1; 2; ... ; n 1, a n 1, e 2 Cð½t 0 ; 1Þ; RÞ, and F 2 CðR; RÞ with R ¼ ð1; 1Þ. When a i 1 for i ¼ 1; 2; ... ; n 1, (1) reduces to the following equation: x ðnÞ ðtÞ þ ð1Þ n qðtÞF ðxðtÞÞ ¼ eðtÞ: ð2Þ 0096-3003/$ - see front matter Ó 2007 Elsevier Inc. All rights reserved. doi:10.1016/j.amc.2007.01.083 * Corresponding author. E-mail address: yugsun@263.net (Y.G. Sun). Applied Mathematics and Computation 190 (2007) 905–911 www.elsevier.com/locate/amc