# Non-linear functional analysis

The branch of functional analysis in which one studies non-linear mappings (operators, cf. Non-linear operator) between infinite-dimensional vector spaces and also certain classes of non-linear spaces and their mappings. The basic divisions of non-linear functional analysis are the following.

1) Differential calculus of non-linear mappings between Banach, topological vector and certain more general spaces, including theorems on the local inversion of a differentiable mapping and the implicit-function theorem.

2) The search for conditions on the action, such as continuity and compactness, of a non-linear operator acting from one specific infinite-dimensional space into another.

3) Fixed-point principles for various classes of non-linear operators (contractive, compact, compressing, monotone, and others); application of these principles to existence proofs for solutions of various non-linear equations.

4) The study of non-linear operators such as monotone, concave, convex, having a monotone minorant, and others, in spaces endowed with the structure of an ordered vector space.

5) The study of spectral properties of non-linear operators (bifurcation points, continuous branches of eigen vectors, etc.) in infinite-dimensional vector spaces.

6) The approximate solution of non-linear operator equations.

7) The study of spaces that are locally linear and of Banach manifolds — global analysis.

8) The investigation of extrema of non-linear functionals and variational methods for studying non-linear operators.

How to Cite This Entry:
Non-linear functional analysis. V.I. Sobolev (originator), Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Non-linear_functional_analysis&oldid=15212
This text originally appeared in Encyclopedia of Mathematics - ISBN 1402006098