Local properties of solutions of a differential equation via normed linear spaces

Abstract

The non-linear differential equation Δu + f(u) = 0 arises in many different physical settings, including nuclear physics, phase transitions in Van der Waals fluids, combustion theory and population genetics. Underlying all of the work that has been done with this problem are the facts that a related ordinary differential equation problem, namely can be solved uniquely on a small r-interval of the form [0, δ] for some δ > 0, the variation v := ∂u/∂a uniquely solves: on the same interval [0, δ], and both u and v depend continuously on the initial data a. We have found that instead of using standard contraction arguments, a simpler way to show these results hold is to choose appropriate Banach spaces and linear operators and apply a differentiable fixed-point theorem.