Oscillation of damped second-order linear mixed neutral differential equations
- PDF / 372,610 Bytes
- 20 Pages / 439.37 x 666.142 pts Page_size
- 20 Downloads / 230 Views
Oscillation of damped second-order linear mixed neutral differential equations John R. Graef1
· Orhan Özdemir2 · Adil Kaymaz2 · Ercan Tunç2
Received: 25 March 2019 / Accepted: 17 September 2020 © Springer-Verlag GmbH Austria, part of Springer Nature 2020
Abstract This article concerns the oscillatory behavior of solutions to damped second-order linear functional differential equations with a mixed neutral term. The authors present new oscillation criteria that improve and extend some existing ones in the literature. Examples to illustrate the main results are included. Keywords Oscillation · Neutral differential equations of mixed type · Second-order · Damping term Mathematics Subject Classification 34C10 · 34K11 · 34K40
1 Introduction This paper is concerned with oscillatory behavior of solutions of the damped secondorder linear neutral differential equation of mixed type (1.1) r (t)z (t) + p(t)z (t) + q(t)x(σ (t)) = 0, t ≥ t0 > 0,
Communicated by Adrian Constantin.
B
John R. Graef [email protected] Orhan Özdemir [email protected] Adil Kaymaz [email protected] Ercan Tunç [email protected]
1
Department of Mathematics, University of Tennessee at Chattanooga, Chattanooga, TN 37403, USA
2
Department of Mathematics, Faculty of Arts and Sciences, Gaziosmanpasa University, 60240 Tokat, Turkey
123
J. R. Graef et al.
where z(t) = x(t) + p1 (t)x(τ1 (t)) + p2 (t)x(τ2 (t)). We will make use of the following conditions: (i) p, q, r : [t0 , ∞) → R are real-valued continuous functions with p(t) ≥ 0, r (t) > 0, q(t) ≥ 0 with q(t) not identically zero for large t, and
∞ t0
t 1 exp − p(s)/r (s)ds dt = ∞; r (t) t0
(1.2)
(ii) τ1 , τ2 , σ : [t0 , ∞) → R are real-valued continuous functions such that τ1 and τ2 are strictly increasing, τ1 (t) < t, τ2 (t) > t, and limt→∞ τ1 (t) = limt→∞ τ2 (t) = limt→∞ σ (t) = ∞; and, either (iiia ) p1 , p2 : [t0 , ∞) → R are real-valued continuous functions such that p1 (t) > 0, and p2 (t) ≥ 1 with p2 (t) ≡ 1 for large t; or (iiib ) p1 , p2 : [t0 , ∞) → R are real-valued continuous functions such that p2 (t) > 0, and p1 (t) ≥ 1 with p1 (t) ≡ 1 for large t. By a solution of Eq. (1.1), we mean a function x ∈ C ([tx , ∞), R) for some tx ≥ t0 that has the properties z ∈ C 1 ([tx , ∞), R), r z ∈ C 1 ([tx , ∞), R), and satisfies (1.1) on [tx , ∞). We only consider those solutions of (1.1) that exist on some half-line [tx , ∞) and satisfy the condition sup {|x(t)| : T1 ≤ t < ∞} > 0 for any T1 ≥ tx ; moreover, we tacitly assume that (1.1) possesses such solutions. Such a solution x(t) of (1.1) is said to be oscillatory if it has arbitrarily large zeros on [tx , ∞), i.e., for any t1 ∈ [tx , ∞) there exists t2 ≥ t1 such that x(t2 ) = 0; otherwise it is called nonoscillatory, i.e., if it is eventually positive or eventually negative. Equation (1.1) itself is termed oscillatory if all its solutions are oscillatory. The oscillatory behavior of solutions of second order neutral differential equations with a delayed or advanced argument has been widely studied by m
Data Loading...
