Abstract
We study the bifurcation of positive solutions of generalized nonlinear undamped pendulum problems u"+f(u)=0, -L<x<L, u(-L)=o by refining the "time map" techniques of J.Smoller and A. Wasserman (1981).We are able to count the exact number of the time maps and hence are able to count the exact number of positive solutions for these sublinear nonlinearities f satisfying(i) f(0)=f(1)=0, (ii) f(x)>0 in (0,1), and (iii) f" changing sign at most twice in (0,1). We study the monotonicity as well as the convexity if possible of the time maps in (0,1).