Necessary and sufficient condition for oscillation of nonlinear neutral first-order differential equations with several delays

Second-Order Differential Equation: Oscillation Theorems and Applications

“…e obtained oscillation theorems complement the well-known oscillation results present in the literature. is work, as well as [31][32][33][34][35][36][37][38][39][40][41], leads us to pose an open problem: Can we find necessary and sufficient conditions for the oscillation of solutions to second-order differential equation r(t) (y(t) + p(t)y(τ(t))) ′ c ′ + m i�1 q i (t)y α i τ i (t) ) � 0, for p ∈ C R + , R ?.…”

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].…”

Oscillation of Emden–Fowler-Type Neutral Delay Differential Equations

“…For further work on the oscillation of the solutions to this type of equation, we refer the readers to [16][17][18][19][20][21][22][23][24][25][26][27]. Note that the majority of publications consider only sufficient conditions, and merely a few consider necessary and sufficient conditions.…”