## Existence MD Estimates for Solutions of Nonlinear Equations Near a Branch Point

##### Abstract

Consider the equation (1.1) [see pdf for notation]
on a Hilbert space H. Here n is a scalar and [see pdf for notation] is a linear Fredholm operator. That is:
(a) L is closed; (b) The domain, D(L) is dense in H;
(c) The range, R(L) is closed in H; (d) The dimension of the null space, dim n(L) <= (e) The dimension of the null space of the adjoint dim n(L*) <= The operator N, which may be nonlinear, is defined for sufficiently small and appropriately restricted [see pdf for notation], and [see pdf for notation] Using the method of Lyapunov-Schmidt (cf., e.g. [4] or [5]) we express w in the form (1.2) [see pdf for notation] where [see pdf for notation] denotes the orthogonal complement of n(L) in H. Suppose (u,v) satisfy the simultaneous equations (1.3) [see pdf for notation] (1.4) [see pdf for notation] where P is the orthogonal projection operator of H onto R(L), I is
the identity operator on H, and J is a right inverse of L on R(L), i.e.