Oscillation criteria of a certain class of third order nonlinear delay differential equations with damping

Abstract

In this paper, we are concerned with the oscillation of third order nonlinear delay differential equations of the formBy using a generalized Riccati transformation and integral averaging technique, we establish some new sufficient conditions which insure that any solution of this equation oscillates or converges to zero. In particular, several examples are given to illustrate the importance of our results.

Oscillation criteria for third order nonlinear delay differential equations with damping

“…3. Finally, we note that the results in [15] are applicable to equation (1.1) if g(t) ≤ t, while our oscillation results are applicable to equation (1.1) if g(t) < t. Thus, as is well known, it is the delay in equation (1.1) that can generate the oscillations.…”

“…We note that there are many criteria in the literature for the oscillation of second order dynamic equations, and so by applying these results to equation (1.1) and (2.25), we can obtain many oscillation results which are of similar types to these in [1,15] or else, of different types. The formulations of such results are left to the reader.…”

“…In the literature there are some papers and books, for example Agarwal et al [2], Grace and Lalli [5], Parhi and Das [10,11], Parhi and Padhi [12], Skerlik [13], and Tiryaki and Yaman [14], which deal with the oscillatory and asymptotic behavior of solutions of functional differential equations. In [1,15], the authors used a generalized Riccati transformation and an integral averaging technique for establishing some sufficient conditions which insure that any solution of equation (1.1) oscillates or converges to zero. The purpose of this note is to improve and unify the results in [1,15] and present some new sufficient conditions which insure that any solution of equation (1.1) oscillates when equation ( * * ) is nonoscillatory, or oscillatory.…”

“…In [1,15], the authors used a generalized Riccati transformation and an integral averaging technique for establishing some sufficient conditions which insure that any solution of equation (1.1) oscillates or converges to zero. The purpose of this note is to improve and unify the results in [1,15] and present some new sufficient conditions which insure that any solution of equation (1.1) oscillates when equation ( * * ) is nonoscillatory, or oscillatory.…”

“…It is easy that to check that all hypotheses of Theorem 2.9 are satisfied and hence every solution y of equation (2.24) is oscillatory or y is oscillatory. One such solution is y(t) = sin t. We note that none of the results in [3,8,[10][11][12][13][14][15] are applicable to equation (2.24).…”