Namespaces
Variants
Actions

Non-linear functional analysis

From Encyclopedia of Mathematics
Revision as of 17:11, 7 February 2011 by 127.0.0.1 (talk) (Importing text file)
(diff) ← Older revision | Latest revision (diff) | Newer revision → (diff)
Jump to: navigation, search
The printable version is no longer supported and may have rendering errors. Please update your browser bookmarks and please use the default browser print function instead.

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)


Comments

References

[a1] J.T. Schwartz, "Nonlinear functional analysis" , Gordon & Breach (1969)
[a2] E. Zeidler, "Nonlinear functional analysis and its applications" , 1–3 , Springer (1986) (Translated from Russian)
[a3] M.S. Berger, "Nonlinearity and functional analysis" , Acad. Press (1977)
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