Abstract
The objective in this work is to study oscillation criteria for second-order quasi-linear differential equations with an advanced argument. We establish new oscillation criteria using both the comparison technique with first-order advanced differential inequalities and the Riccati transformation. The established criteria improve, simplify and complement results that have been published recently in the literature. We illustrate the results by an example.
Highlights
1 Introduction In this work, we study sufficient conditions for the oscillation of the solutions of secondorder nonlinear differential equations with an advanced argument of the form r u α (t) + p(t)f u g(t) = 0, (1.1)
This work aims at further developing the oscillation theory of second-order quasi-linear equations with advanced argument
We may assume that u(t) > 0, u(g(t)) > 0 for t ≥ t1 ≥ t0
Summary
We study sufficient conditions for the oscillation of the solutions of secondorder nonlinear differential equations with an advanced argument of the form r u α (t) + p(t)f u g(t) = 0, (1.1). Where we assume that the following conditions hold:. (H1) α and β are quotients of odd positive integers; (H2) r ∈ C1([t0, ∞), (0, ∞)), satisfies μ(t0) := ∞ t0 1 r1/α (s) ds < ∞;. (H3) g ∈ C1([t0, ∞), R), and we suppose that, for all t ≥ t0, g(t) ≥ t, g (t) ≥ 0 and p ∈ C[t0, ∞), [0, ∞) does not vanish identically. (H4) f ∈ (R, R) is such that uf (u) > 0 for u = 0 and satisfies the following condition: There exists a constant κ > 0 such that f (u) > κuβ for all u = 0.
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