Simple proof of stationary phase method and application to oscillatory bifurcation problems

We consider the nonlinear eigenvalue problem -u '' (t) = lambda f(u(t)), u(t) > 0, t is an element of I := (-1,1), u(+/- 1) = 0, where f(u) = f(1)(u) = u(3) sin(u(3))/u, f(u) = f(2) (u) = u + u(p) sin(u(q)) (0 <= p < 1, 1 < q <= p + 2) and lambda > 0 is a bifurcation parameter. It is known that, in this case, A is parameterized by the maximum norm alpha = parallel to u(lambda)parallel to(infinity) of the solution u(lambda) associated with lambda and is written as lambda = lambda(alpha). We simplify the argument of the stationary phase method and show the asymptotic formulas for lambda(alpha) for f(1)(u) and f(2) (u) as alpha -> infinity and alpha -> 0. In particular, the shape of bifurcation diagram of lambda(alpha) for f(1)(u) seems to be new. (C) 2019 Elsevier Ltd. All rights reserved.