On Eventually Positive Solutions of Quasilinear Second-Order Neutral Differential Equations

Abstract

We study the second-order neutral delay differential equation [ ( )Φ ( ( ))] + ( )Φ ( ( ( ))) = 0, where Φ ( ) = | | −1 , ≥ 1 and ( ) = ( ) + ( ) ( ( )). Based on the conversion into a certain first-order delay differential equation we provide sufficient conditions for nonexistence of eventually positive solutions of two different types. We cover both cases of convergent and divergent integral ∫ ∞ −1/ ( )d . A suitable combination of our results yields new oscillation criteria for this equation. Examples are shown to exhibit that our results improve related results published recently by several authors. The results are new even in the linear case.

New Criteria for Sharp Oscillation of Second-Order Neutral Delay Differential Equations

“…The methods mostly used in investigating the oscillatory behavior of (1) have been based on a reduction of order and comparison with oscillation of first-order delay differential equations, or on reducing (1) to a first-order Riccati inequality, based on a suitable Riccati type substitution, see e.g., [17] for more details. We note that none of the related results [3][4][5][6][7]10,[12][13][14][15][16][17][18][20][21][22]26,[28][29][30][31][32][33][34][35][36]39,42,46] involving (1) with α = 1, r(t) = 1, p(t) = 0, gives a sharp result when applied to the Euler linear delay differential equation…”

“…required by the other methods based on the initial shift of (1) from σ(t) to σ −1 (t), which were used e.g., in the works [13][14][15][16]. On the other hand, we recall the two main disadvantages associated with the lower bound estimation method:…”