Explicit criteria for the oscillation of second-order differential equations with several sub-linear neutral coefficients

Abstract

In this work, we present sufficient conditions for oscillation of all solutions of a second-order functional differential equation. We consider two special cases when $\gamma >\beta $ γ > β and $\gamma <\beta $ γ < β . This new theorem complements and improves a number of results reported in the literature. Finally, we provide examples illustrating our results and state an open problem.

Some conditions for the oscillation of second-order differential equations with several mixed delays

“…(3.15) Here, α = 11/3, p(t) = e −t , 0 < q(t) = e −t < 1, ν j (t) = t − (j + 1), P (t) = t 0 e 3s/11 ds = 11 3 (e 3t/11 − 1), g j (t) = t (4j−3)/3 . For β = 7/3, we have δ j = (4j − 3)/3 < β = 7/3 < α = 11/3, and g 1 (t)/t β = t −2 and g 2 (t)/t β = t −2/3 which are both decreasing functions.…”

“…Finally, we refer the interested reader to the following paper and to the references therein for some recent results on the oscillation theory for ordinary differential equations of several orders [11][12][13][14][15][16][17][18].…”

First-order impulsive differential systems: sufficient and necessary conditions for oscillatory or asymptotic behavior

“…Because of Theorem 3.1 [20], ( 27) could be a sufficient and necessary condition for the oscillatory and asymptotic behavior of solutions of system (E1) for different ranges of the neutral coefficient b(ξ ). We guess that ( 27) could be a sufficient and necessary condition for the oscillation of a non-homogeneous counterpart of (E).…”

New Theorems for Oscillations to Differential Equations with Mixed Delays

“…It is interesting to notice that, in the aforementioned works, the authors obtained only sufficient conditions that ensure the oscillation of the solutions of the considered equations. A problem worthy of investigations is the study of necessary and sufficient conditions for oscillation, and some satisfactory answers were given in [11][12][13][14][15][16][17][18]. Finally, the interested readers are referred to the following papers and to the references therein for some recent results on the oscillation theory for ordinary differential equations of several orders [19][20][21][22][23][24][25][26][27].…”