A nonlinear integral transform and a global inverse bifurcation theory

Abstract

We consider a nonlinear integral transform and show that the transform acts as a homeomorphism between certain metric spaces of positive functions. We apply the result to the inverse bifurcation problem of determining the nonlinear term of a certain nonlinear Sturm-Liouville problem from its first bifurcating branch, and we establish the well-posedness of the inverse problem. An application to an inverse problem of determining a restoring force from a timemap is also given. Mathematics Subject Classification (2010): 44A15 (primary); 34A55, 45P05 (secondary). 1. The main result This paper studies the nonlinear integral transform K defined by (K f )(x) = ∫ 1 0 dt (∫ 1 t sβ−1 f (xs)ds )1−δ , x ∈ I, (1.1) where 0 0, and I is a bounded, closed interval containing 0. Our objective is to show that the transform K is a homeomorphism of an appropriate metric space onto a twin metric space reflecting the smoothing property of K and, as its application, that an inverse problem to determine a nonlinear term of a certain nonlinear Sturm-Liouville problem from its first bifurcating branch is globally wellposed. Throughout the paper, θ denotes the Euler differential operator θ = x d dx . To state the main result explicitly, for a nonnegative integer k, a number α ∈ (0, 1], Supported by Grant-in-Aid for Scientific Research No. 21540165, Japan Society for Promotion of Science. Received February 12, 2009; accepted in revised form July 29, 2010.