Abstract
Differential equations of second order appear in physical applications such as fluid dynamics, electromagnetism, acoustic vibrations, and quantum mechanics. In this paper, necessary and sufficient conditions are established of the solutions to second-order half-linear delay differential equations of the formςyu′ya′+∑j=1mpjyucjϑjy=0 for y≥y0, under the assumption∫∞ςη−1/adη=∞. We consider two cases whena<cjanda>cj, whereaandcjare the quotient of two positive odd integers. Two examples are given to show effectiveness and applicability of the result.
Highlights
Differential equations of second order appear in physical applications such as fluid dynamics, electromagnetism, acoustic vibrations, and quantum mechanics
Necessary and sufficient conditions are established of the solutions to secondorder half-linear delay differential equations of the assumption ∞(ς(η))− 1/adη ∞
Introduction e differential equation of second order appears in models concerning biological, physical, and chemical phenomena, optimization, the mathematics of networks, and dynamical systems, see [1]
Summary
Second-Order Differential Equation with Multiple Delays: Oscillation Theorems and Applications. Necessary and sufficient conditions are established of the solutions to secondorder half-linear delay differential equations of the assumption ∞(ς(η))− 1/adη ∞. Let (A1)–(A3) hold and that u is an eventually positive solution of (1). Since limy⟶∞Y(y) ∞, there exists a positive constant d such that (4) holds. Since ς(y)(u′(y))a is positive and nonincreasing, limy⟶∞ς(y)(u′(y))a exists and is nonnegative. Assume that there exists a constant b1 and the quotient of two positive odd integers, such that 0 < cj < b1 < a. B1/a > 0 and θj(η) < η, and it follows that ucj θj(η) ≥ dcj− b1 Υcj θj(η)wb1/aθj(η) ≥ dcj− b1 Ycj θj(η)wb1/a(η). Is contradicts (25) and completes the proof of sufficiency for eventually positive solutions. Κ 1/a Y(y) u(y) κ1/aY(y), y ≥ y1
Sign-in/Register to view highlights & summary Sign-in/Register to access full text options 
Talk to us
Join us for a 30 min session where you can share your feedback and ask us any queries you have
Schedule a call
