Abstract
Differential equations of second order appear in a wide variety of applications in physics, mathematics, and engineering. In this paper, necessary and sufficient conditions are established for oscillations of solutions to second-order half-linear delay differential equations of the form ς y u ′ y a ′ + p y u c ϑ y = 0 , for y ≥ y 0 , under the assumption ∫ ∞ ς η − 1 / a = ∞ . Two cases are considered for a < c and a > c , where a and c are the quotients of two positive odd integers. Two examples are given to show the effectiveness and applicability of the result.
Highlights
Differential equations (DEs) have received a lot of attention, and it is an active research area among scientists and engineers [1,2,3,4,5,6,7,8]. e DEs have ability to formulate many complex phenomena in various fields such as biology, fluid mechanics, plasma physics, fluid mechanics, and optics; many exact and numerical schemes have been being derived such as [9,10,11,12,13,14,15]
Differential equation of second order appears in models as well as in physical applications such as fluid dynamics, electromagnetism, acoustic vibrations and quantum mechanics, biological, physical and chemical phenomena, optimization, mathematics of networks, and dynamical systems
Where a and c are the quotient of two positive odd integers, and the functions p, ς, and θ are continuous that satisfy the conditions stated below: (A1) θ ∈ C([0, ∞), R), θ(y) < y, limy⟶∞θ(y) ∞. (A2) ς ∈ C1([0, ∞), R), p ∈ C([0, ∞), R); 0 < ς(y), 0 ≤ p(y) for all y ≥ 0; p(y) is not identically zero in any interval [b, ∞). (A3) Υ(y) yy1 ς− 1/a(η)dη with limy⟶∞Υ(y) ∞. (A4) the existence of a differentiable function θ0 such that 0 < θ0(y) ≤ θ(y), for θ0′(y) ≥ θ0 > 0, for y ≥ y0
Summary
Differential equations (DEs) have received a lot of attention, and it is an active research area among scientists and engineers [1,2,3,4,5,6,7,8]. e DEs have ability to formulate many complex phenomena in various fields such as biology, fluid mechanics, plasma physics, fluid mechanics, and optics; many exact and numerical schemes have been being derived such as [9,10,11,12,13,14,15]. Dzurina and Dzurina [27] have studied the oscillatory behavior of the solutions of (2) under the assumptions 0 ≤ q(y) < ∞ and limy⟶∞Υ(y) ∞. In [28], Bohner et al have studied the oscillatory behavior of solutions of (2) under a c, limy⟶∞Υ(y) < ∞, and 0 ≤ q(y) < 1. In [30], Ali has studied the oscillatory behavior of the solutions of (2), under the assumptions limy⟶∞Υ(y) < ∞ and q(y) ≥ 0.
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