THE OSCILLATION OF HALF-LINEAR DIFFERENTIAL-EQUATIONS WITH AN OSCILLATORY COEFFICIENT

We derive lower bounds for the distance between consecutive zeros of a solution of the second order half-linear differential equation (\y'(t )\(alpha-1) y'(t))' + q (t) \y (t)\(alpha-1) y(t) = 0, when q(t) : [t( 0), infinity) --> R is locally integrable far some t(0) greater than o r equal to 0. Then we apply these results to the following equations: (p(t) \y'(t)\(alpha-1) y'(t))' + q(t) \y(t)\(alpha-1)y(t) = 0, and [GR APHICS] where p is an element of C([t(0), infinity), (0, infinity)) an d integral(t0)(infinity) p(t)(-1/alpha) dt = infinity; D-i = partial d erivative/partial derivative x(i), D = (D-1,..., D-N), Omega t(0) = {x is an element of R(N) : \x\ greater than or equal to t(0)} is an exte rior domain, and h is an element of C([t(0), infinity), R); and alpha > 0 is a constant; n > 1 and N greater than or equal to 2 are integers .