Oscillatory bifurcation for semilinear ordinary differential equations

We consider the nonlinear eigenvalue problem u '' (t) + lambda f (u(t)) = 0, u(t) > 0, t is an element of I : = (-1, 1), u (1) = u (-1) = 0, where f (u) = u + (1/2)sin(k) u (k >= 2) and lambda > 0 is a bifurcation parameter. It is known that lambda is parameterized by the maximum norm alpha = vertical bar vertical bar u(lambda)vertical bar vertical bar(infinity) of the solution u(lambda) associated with lambda and is written as lambda = lambda (k, alpha). When we focus on the asymptotic behavior of lambda (k, alpha) as alpha -> infinity, it is natural to expect that lambda (k, alpha) -> pi(2)/4, and its convergence rate is common to k. Contrary to this expectation, we show that lambda(2n(1) + 1,alpha) tends to pi(2)/4 faster than lambda(2n(2), alpha) as alpha -> infinity, where n(1) >= 1, n(2) >= 1 are arbitrary given integers.