Oscillation of Second-Order Half-Linear Neutral Advanced Differential Equations
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Oscillation of Second‑Order Half‑Linear Neutral Advanced Differential Equations Shan Shi1 · Zhenlai Han1 Received: 18 September 2019 / Revised: 25 July 2020 / Accepted: 1 August 2020 © Shanghai University 2020
Abstract The purpose of this paper is to study the oscillation of second-order half-linear neutral differential equations with advanced argument of the form
(r(t)((y(t) + p(t)y(𝜏(t)))� )𝛼 )� + q(t)y𝛼 (𝜎(t)) = 0, t ⩾ t0 ,
when ∫ r− 𝛼 (s)ds < ∞ . We obtain sufficient conditions for the oscillation of the studied equations by the inequality principle and the Riccati transformation. An example is provided to illustrate the results. ∞
1
Keywords Oscillation theory · Second-order differential equations · Neutral · Advanced argument · Asymptotic behavior Mathematics Subject Classification 34C10 · 34K40 · 35B40
1 Introduction In this paper, we consider the oscillation of second-order half-linear neutral advanced differential equations of the form (1.1)
(r(t)((y(t) + p(t)y(𝜏(t)))� )𝛼 )� + q(t)y𝛼 (𝜎(t)) = 0, t ⩾ t0 ,
where 𝛼 > 0 is a quotient of odd positive integers, r(t), 𝜎(t) , 𝜏(t) ∈ C1 ([t0 , ∞), (0, ∞)) , q(t) ∈ C([t0 , ∞), (0, ∞)) , and p(t) ∈ C11 ([t0 , ∞), (0, ∞)) . For our further references, we ∞ − denote ( and assume ) that 𝜋(t) = ∫t r 𝛼 (s)ds , and then our results satisfy the condition 𝜋(𝜏(t)) inf 1 − p(t) 𝜋(t) > 0 . We also suppose that, for all t ⩾ t0 , 𝜎(t) ⩾ t , 𝜎 � (t) ⩾ 0 , 𝜏(t) ⩽ t ,
t⩾t0
This research is supported by the Shandong Provincial Natural Science Foundation of China (ZR2017MA043). * Zhenlai Han [email protected] Shan Shi [email protected] 1
School of Mathematical Sciences, University of Jinan, Jinan 250022, China
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Vol.:(0123456789)
Communications on Applied Mathematics and Computation
lim 𝜏(t) = ∞ , and p(t), q(t) do not vanish identically on any half-line of the form [t0 , ∞) . Moreover, we set
t→∞
z(t) = y(t) + p(t)y(𝜏(t)). As is customary, a solution y(t) of (1.1) is said to be oscillatory if it is neither eventually positive nor eventually negative. Otherwise, it is said to be nonoscillatory. The equation itself is termed oscillatory if all its solutions oscillate. Under the solution of (1.1), we mean a real valued function y(t) ∈ C([Tx , ∞), ℝ) , Tx ⩾ t0 , which has the property r(t)(z� (t))𝛼 ∈ C1 ([Tx , ∞), ℝ) and satisfies (1.1) on [Tx , ∞) . We only consider those solutions of (1.1) which exist on some half-line [Tx , ∞) and satisfy the condition sup{|y(t)| ∶ Ty ⩽ t < ∞} > 0 for any Ty ⩾ Tx. Following Trench [17], we shall say that (1.1) is in the canonical form if
∫t 0
t
1
r− 𝛼 (s)ds → ∞, t → ∞.
Conversely, we say that (1.1) is in the noncanonical form if
∫t 0
∞
1
r− 𝛼 (s)ds < ∞.
(1.2)
Differential equations are widely applied in a large number of practical problems in many fields, including physics, biology, and economics. The oscillation of differential equations with deviating arguments has been studied by Fite [10] in 1921. In recent years, the study of oscillatory properties of differential equations with deviating arguments has been an
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