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Research Article New Results on the Nonoscillation of Solutions of Some Nonlinear Differential Equations of Third Order | Hindawi Publishing Corporation Journal of Inequalities and Applications Volume 2009 Article ID 896934 11 pages doi 10.1155 2009 896934 Research Article New Results on the Nonoscillation of Solutions of Some Nonlinear Differential Equations of Third Order Ercan Tunic Department of Mathematics Faculty of Arts and Sciences Gaziosmanpa a University 60240 Tokat Turkey Correspondence should be addressed to Ercan Tunc ercantunc72@yahoo.com Received 27 July 2009 Accepted 6 November 2009 Recommended by Patricia J. Y. Wong We give sufficient conditions so that all solutions of differential equations r t y t q t kfy t ptr tgay fif t t0 and r t y t qOKy tty p t h y glt f t t t0 are nonoscillatory. Depending on these criteria some results which exist in the relevant literature are generalized. Furthermore the conditions given for the functions k and h lead to studying more general differential equations. Copyright 2009 Ercan Tunc. This is an open access article distributed under the Creative Commons Attribution License which permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited. 1. Introduction This paper is concerned with study of nonoscillation of solutions of third-order nonlinear differential equations of the form hawa qtaHy a p ẹ ya g 0 f fh t t0 1.1 ra y a qa Hy 0 pa Hy ga far t t0 Ơ.2 where t0 0 is a fixed real number f p q r and g e C 0 to R such that r t 0 and f t 0 for all t e 0 to . k h e C R R are nondecreasing such that h y y 0 kty iy 0 for all y 0 y f 0. Throughout the paper it is assumed for all g f and a appeared in 1.1 and 1.2 that g t t for all t t0 limtTOg t to a 0 is a quotient of odd integers. It is well known from relevant literature that there have been deep and thorough studies on the nonoscillatory behaviour of solutions of second- and third-order nonlinear differential equations in recent years. See for instance 1-37 as some related papers or 2 Journal of Inequalities and Applications books on
