Método de continuidad

Abstract

It is known that the Implicit Function Theorem for functions of several variables plays important roles in many branches of mathematics (differential manifold, differential geometry, differential topology, etc.). Its extension to infinite-dimensional space is also extremely important in nonlinear analysis, as well as in the study of infinite-dimensional manifolds. The IFT plays an important role in solving nonlinear equations. However, the IFT is only a local statement. If a problem is local, then it is extremely powerful for local solvability. As to global solvability problems, we first solve them locally, and then extend the solutions by continuation. In this work we prove the usefulness of the IFT in the existence of solutions for small perturbations of a given equation which has a known solution. As to large perturbations, the IFT is not enough, we have to add new ingredients. The continuity method is a general principle, which can be applied to prove the existence of solutions for a variety of nonlinear equations.