# 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.

#### References

 [1] M.M. Vainberg, "Variational method and method of monotone operators in the theory of nonlinear equations" , Wiley (1973) (Translated from Russian) [2] H. Gajewski, K. Gröger, K. Zacharias, "Nichtlineare Operatorgleichungen und Operatordifferentialgleichungen" , Akademie Verlag (1974) [3] J. Eells, "The foundations of global analysis" Uspekhi Mat. Nauk , 24 : 3 (1969) pp. 157–210 (In Russian) [4] M.A. Krasnosel'skii, "Positive solutions of operator equations" , Wolters-Noordhoff (1964) (Translated from Russian) [5] M.A. Krasnosel'skii, P.P. Zabreiko, "Geometric methods of non-linear analysis" , Springer (1983) (Translated from Russian) [6] S. Lang, "Introduction to differentiable manifolds" , Interscience (1967) pp. App. III [7] L.A. Lyusternik, V.I. Sobolev, "Elemente der Funktionalanalysis" , Akademie Verlag (1968) (Translated from Russian) [8] L. Nirenberg, "Topics on nonlinear functional analysis" , New York Univ. Inst. Math. Mech. (1974) [9] E. Hille, R.S. Phillips, "Functional analysis and semi-groups" , Amer. Math. Soc. (1957)