Abstract
In this paper, we study oscillation and asymptotic properties for half-linear second order differential equations with mixed argument of the form r(t)(y′(t))α′=p(t)yα(τ(t)). Such differential equation may possesses two types of nonoscillatory solutions either from the class N0 (positive decreasing solutions) or N2 (positive increasing solutions). We establish new criteria for N0=∅ and N2=∅ provided that delayed and advanced parts of deviating argument are large enough. As a consequence of these results, we provide new oscillatory criteria. The presented results essentially improve existing ones even for a linear case of considered equations.
Highlights
We consider the half-linear second order differential equations with mixed deviating argument of the following form.Equations with Mixed Deviating r (t)(y0 (t))αArguments
Agarwal and Omer San (H1 ) p, r ∈ C ([t0, ∞)), p(t) > 0, r (t) > 0, α is the ratio of two positive odd integers; (H2 ) τ (t) ∈ C ([t0, ∞)), τ 0 (t) > 0; lim τ (t) = ∞
There are numerous papers devoted to oscillation theory of differential equations, see, e.g., [1,2,3,4,5,6,7,8,9,10,11,12]
Summary
We consider the half-linear second order differential equations with mixed deviating argument of the following form. There are numerous papers devoted to oscillation theory of differential equations, see, e.g., [1,2,3,4,5,6,7,8,9,10,11,12] It is known (see, e.g., [7]) that if y(t) is a nonoscillatory solution of (1), eventually either:. Our results are of high generality and, what is more, they hold for all α > 0, and our technique does not require discussing cases α ∈ (0, 1) and α > 1 separately as it is common, see [1,2,3,4,5,6,7,8,9,10]
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