Stability of Nonlinear Operator Equations and Their Discretizations

In this talk nonlinear operator equations of the form F (x) = 0 (1) are considered. We formulate such problems in a Banach space setting and therefore, F: B1 → B2, were B1, B2 are Banach spaces. A Banach space is complete, which means that all Cauchy series converge in the underlying norm. We first define two important notions, stability and isolated solutions,which are important for this talk. Because of its generality the theory presented here can be applied in many contexts. We will be interested in applying it to boundary value problems in ODEs with a singularity of the first kind, y′(t) − M t y(t) − f(t, y(t)) = 0, t ∈ (0, 1], y(t) ∈ C1[0, 1], b(y(0), y(1)) = 0, where y, f are vector-valued functions of dimension n, M is an n×n matrix, and