View source for Non-linear functional analysis

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



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====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  J.T. Schwartz,   "Nonlinear functional analysis" , Gordon &amp; Breach  (1969)</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  E. Zeidler,   "Nonlinear functional analysis and its applications" , '''1–3''' , Springer  (1986)  (Translated from Russian)</TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top">  M.S. Berger,   "Nonlinearity and functional analysis" , Acad. Press  (1977)</TD></TR></table>