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.
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|||H. Gajewski, K. Gröger, K. Zacharias, "Nichtlineare Operatorgleichungen und Operatordifferentialgleichungen" , Akademie Verlag (1974)|
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|||M.A. Krasnosel'skii, P.P. Zabreiko, "Geometric methods of non-linear analysis" , Springer (1983) (Translated from Russian)|
|||S. Lang, "Introduction to differentiable manifolds" , Interscience (1967) pp. App. III|
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|||L. Nirenberg, "Topics on nonlinear functional analysis" , New York Univ. Inst. Math. Mech. (1974)|
|||E. Hille, R.S. Phillips, "Functional analysis and semi-groups" , Amer. Math. Soc. (1957)|
|[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)|
Non-linear functional analysis. V.I. Sobolev (originator), Encyclopedia of Mathematics. URL: http://www.encyclopediaofmath.org/index.php?title=Non-linear_functional_analysis&oldid=15212