New Oscillation Criteria for Second Order Quasilinear Neutral Delay Differential Equations
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New Oscillation Criteria for Second Order Quasilinear Neutral Delay Differential Equations N. Prabaharan1 · C. Dharuman1 · E. Thandapani2 · S. Pinelas3
© Foundation for Scientific Research and Technological Innovation 2020
Abstract The authors present some new sufficient conditions for the oscillation of second order quasilinear neutral delay differential equation
(a(t)(z� (t))𝛽 )� + q(t)x𝛾 (𝜎(t)) = 0, t ≥ t0 > 0, where z(t) = x(t) + p(t)x(𝜏(t)) . Our oscillation results depend on only one condition and essentially improve, complement and simplify many related ones in the literature. Exam‑ ples are provided to illustrate the value of the main results. Keywords Quasilinear · Neutral differential equation · Second order · Oscillation Mathematics Subject Classification 34C10 · 34K11
Introduction This paper deals with the oscillatory behavior of solutions of second order quasilinear neu‑ tral delay differential equation of the form (1.1)
(a(t)(z� (t))𝛽 )� + q(t)x𝛾 (𝜎(t)) = 0, t ≥ t0 > 0, where z(t) = x(t) + p(t)x(𝜏(t)) , subject to the following conditions: * S. Pinelas [email protected] N. Prabaharan [email protected] C. Dharuman [email protected] E. Thandapani [email protected] 1
Department of Mathematics, SRM University, Ramapuram Campus, Chennai 600 089, India
2
Ramanujan Institute for Advanced Study in Mathematics, University of Madras, Chennai 600 005, India
3
Departamento de Ciencias Exatas e Naturais, Academia Militar, 2720‑113 Amadora, Portugal
13
Vol.:(0123456789)
Differential Equations and Dynamical Systems
(H1 ) 𝛽 and 𝛾 are the ratios of odd positive integers; (H2 ) a ∈ C1 ([t0 , ∞), (0, ∞)) and ∞
A(t) =
∫t
− 𝛽1
a
(s)ds with A(t0 ) < ∞;
(1.2)
(H3 ) 𝜎 ∈ C1 ([t0 , ∞), ℝ) and 𝜏 ∈ C([t0 , ∞), ℝ) satisfy 𝜏(t), 𝜎(t) ≤ t, 𝜎 � (t) > 0 and limt→∞ 𝜏(t) = limt→∞ 𝜎(t) = ∞; (H4 ) p, q ∈ C([t0 , ∞), [0, ∞)), p(t) < 1 and q does not vanishes identically on any halfline of the form [t∗ , ∞) for t∗ ≥ t0; A(t) . p(t) ≤ A(𝜏(t)) (H5 ) By a solution of Eq. (1.1), we mean a function x ∈ C([tx , ∞), ℝ) for some tx ≥ t0 , which has the property a(t)(z� (t))𝛽 ∈ C1 ([tx , ∞), ℝ) and satisfies Eq. (1.1) on [tx , ∞). We consider only those solutions of Eq. (1.1) which exist on [tx , ∞) and satisfy sup{|x(t)| ∶ t ≥ T} > 0 for all T ≥ tx . As usual, a solution x of Eq. (1.1) is said to be oscillatory if it is neither even‑ tually positive nor eventually negative, and nonoscillatory otherwise. The equation itself is called oscillatory if all its solutions oscillate. Second order differential equations are derived from many applications in nuclear phys‑ ics, astrophysics, fluid mechanics and gas dynamics, and the oscillation criteria for the equation
x�� (t) + a(t)|x(t)|𝛾 sgnx(t) = 0 was firstly established by Wong [25] in the super-linear case, that is, 𝛾 > 1. Since then a number of researchers established several oscillation criteria on this class of differential equations, see for example [1, 2] and the references cited therein. Delay differential equations play an import
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