On the oscillation of second-order half-linear functional differential equations with mixed neutral term

Abstract

In this article, the authors establish new sufficient conditions for the oscillation of solutions to a class of second-order half-linear functional differential equations with mixed neutral term. The results obtained improve and complement some known results in the relevant literature. Examples illustrating the results are included.

Second-Order Non-Canonical Neutral Differential Equations with Mixed Type: Oscillatory Behavior

“…On the other hand, the works that studied these equations under the condition (5) obtained two conditions to ensure the oscillation. Therefore, our results are an extension and simplification as well as improvement of previous results in [3][4][5]8,11].…”

“…In [11], Tunc et al studied the oscillatory behavior of the differential Equation (1) under the condition (4). Moreover, they considered the two following cases:…”

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…”

Comparison theorems on the oscillation of even order nonlinear mixed neutral differential equations

“…A great deal of effort has been devoted to investigating the qualitative behavior of solutions of NDEs in recent years; see, for example, previous studies 11–17 . Observe that when $$$ \vartheta (t)=1,\mathcal{B}(t)=0,\gamma =\theta =1 $$$, and $$$ \mathcal{F}\left(\xi \right)=\xi $$$, Equation () reduces to a linear neutral delay DE of even order $$$$ {\left(x(t)+\mathcal{A}(t)x\left(\alpha (t)\right)\right)}^{(n)}+\mathcal{H}(t)x\left(\Psi (t)\right)=0, $$$$ whose oscillatory properties were discussed by a number of researchers, for instance, 18–21 and the references contained therein.…”

“…A great deal of effort has been devoted to investigating the qualitative behavior of solutions of NDEs in recent years; see, for example, previous studies. [11][12][13][14][15][16][17] Observe that when 𝜗(t) = 1, (t) = 0, 𝛾 = 𝜃 = 1, and  (𝜉) = 𝜉, Equation (E1) reduces to a linear neutral delay DE of even order…”