James S.W. Wong

On existence of oscillatory solutions of second order Emden–Fowler equations

Abstract We study the second order Emden–Fowler equation (E) y″(t)+a(x)|y| γ sgn y=0, γ>0, where a ( x ) is a positive and absolutely continuous function on (0,∞). Let φ ( x )= a ( x ) x ( γ +3)/2 , γ ≠1, and bounded away from zero. We prove the following theorem. If φ − ′( x )∈ L 1 (0,∞) where φ − ′( x )=−min( φ ′( x ),0), then Eq. xa0(E) has oscillatory solutions . In particular, this result embodies earlier results by Jasny, Kurzweil, Heidel and Hinton, Chiou, and Erbe and Muldowney.

Oscillation and nonoscillation of Hill's equation with periodic damping

We prove new results on the oscillation and nonoscillation of the Hills equation with periodic damping: ny″+p(t)y′+q(t)y=0,t⩾0, nwhere p(t) and q(t) are continuous and periodic. The results show that the equation y″+(sint)y′+(cost)y=0 is nonoscillatory whilst the equation y″+(cost)y′+(sint)y=0 is oscillatory.

A nonoscillation theorem for Emden-Fowler equations

Abstract We study the second order Emden–Fowler equation ( E ) y″+a(x)|y| γ−1 y=0, γ>0 where a(x) is positive and absolutely continuous on (0,∞) . Let ψ(x)=x (γ+3)/2+δ where δ is any positive number. Theorem Let γ≠1. If ψ(x) satisfies (a) lim x→∞ ψ(x)=k>0 and (b) ∫ ∞ |ψ′(x)| dx , then Eq.xa0(E) is nonoscillatory.

Nonoscillation theorems for second order nonlinear differential equations

We prove nonoscillation theorems for the second order EmdenFowler equation (E): y + a(x)lyl-ly = 0, y > 0, where a(x) E C(0,oo) and -y : 1. It is shown that when x(y+3)/2+6a(x) is nondecreasing for any 6 > 0 and is bounded above, then (E) is nonoscillatory. This improves a wellknown result of Belohorec in the sublinear case, i.e. when 0 < 7y < 1 and 0 <6 < (1 -)/2.