Oscillatory behavior of even-order nonlinear differential equations with a sublinear neutral term

Abstract

Abstract. The authors present a new technique for the linearization of even-order nonlinear differential equations with a sublinear neutral term. They establish some new oscillation criteria via comparison with higher-order linear delay differential inequalities as well as with first-order linear delay differential equations whose oscillatory characters are known. Examples are provided to illustrate the theorems.

Oscillation of second-order nonlinear difference equations with sublinear neutral term

“…where we also used the decreasing nature of y n in the last estimate. Now (17), in view of (15), leads to…”

“…In view of this motivation, our aim in this paper is to present sufficient conditions which ensure that all solutions of (1) are oscillatory. For related results concerning second-order differential equations with sublinear neutral term, we refer the reader to [3,16,17,23]. Some related results concering second-order dynamic equations on time scales can be found in [6-8, 13, 14, 19, 22].…”

“…Nonlinear difference equations with sublinear neutral term Now proceeding as in the proof of Theorem 2.1, we obtain (17), which with (23) yields 0 ≥ ∆(a n ∆y n ) + q n P β n+1−m y β n+1 = ∆(a n ∆y n ) + q n P β n+1−m y n+1 y 1−β n+1 ≥ ∆(a n ∆y n ) + q n P β n+1−m y n+1 K 1−β = ∆(a n ∆y n ) + Lq n P β n+1−m y n+1 for n ≥ n 3 , with some n 3 ≥ n * 2 . The remainder of the proof is similar to that of Theorem 2.1 and hence is omitted.…”

Oscillation results for nonlinear second order difference equations with mixed neutral terms

“…In the qualitative analysis of such systems, indeed, the oscillatory behavior of solutions of equations, where the rate of the growth depends not only on the current and the past states but also on rate of change in the past, play an important role [22][23][24]. In the light of this motivation and justification, different results have been reported regarding the asymptotic behavior of second order difference equations with neutral terms [25][26][27][28][29][30][31][32][33][34][35][36][37][38][39][40]. For relevant results on the application of oscillation theory, the reader can consult [18,41,42].…”

New criteria for oscillation of nonlinear neutral differential equations

“…Over this decade, a great amount of work has been done on development the oscillation theory of second order delay and advanced equations, see [3, 4, 9, 11-14, 16, 17, 22], and the oscillation theory of higher order delay equations, see [8,10,18,19,21,[24][25][26]29].…”