ON RECTIFIABLE OSCILLATION OF EMDEN-FOWLER EQUATIONS

We are interested in the oscillatory behavior of solutions of Emden Fowler equation y" f(x) broken vertical bar y broken vertical bar(gamma-1) y= 0, (1) where gamma > 0 and gamma not equal 1, f (x) is an element of C-1(0, 1] and f (x) > 0 for x is an element of (0, 1]. A solution y(x) is rectifiable oscillatory if the solution curve {(x, y(x)) : x is an element of (0, 1]} has a finite are-length. When the arc -length of the solution curve is infinite, the solution y(x) is said to be unrectifiable oscillatory. We prove integral criteria in terms of f(x) which are necessary and sufficient for both rectifiable and unrectifiable oscillations of all solutions of (1). For a discussion on rectifiable oscillation of the linear differential equation, i.e. the equation (1) when gamma = 1, we refer to Pasic [15], Wong [17].