Monotonic operator

A monotonic operator is a term from mathematics from the sub-area of nonlinear functional analysis . They are special (non-linear) operators and are a generalization of the monotonic real functions of a variable.

definition

Let there be a normalized space and a convex subset of . A (non-linear) operator is called monotonic if the inequality for all ${\ displaystyle V}$ ${\ displaystyle M \ subset V}$ ${\ displaystyle V}$ ${\ displaystyle F \ colon M \ rightarrow V ^ {*}}$ ${\ displaystyle x, y \ in M}$

{\ displaystyle \ langle xy, F (x) -F (y) \ rangle \ geq 0}

applies. Here refers to the topological dual space of and the dual pairing . ${\ displaystyle V ^ {*}}$ ${\ displaystyle V}$ ${\ displaystyle \ langle \ cdot, \ cdot \ rangle}$ ${\ displaystyle V \ times V ^ {*} \ rightarrow \ mathbb {K}, \, (x, f) \ mapsto f (x)}$

This term can literally be applied to more general room classes, in particular to locally convex rooms . This term can also be extended to set -valued functions . Such a function is then called monotonic, if for all and the inequality ${\ displaystyle F: M \ rightarrow {\ mathcal {P}} (V ^ {*})}$ ${\ displaystyle x, y \ in M}$ ${\ displaystyle f \ in F (x), g \ in F (y)}$

{\ displaystyle \ langle xy, fg \ rangle \ geq 0}

applies.

application

The concept of the monotonic operator has many applications in nonlinear functional analysis , especially in nonlinear partial differential equations .

literature

Heinz H. Bauschke & Patrick L. Combettes: Convex Analysis and Monotone Operator Theory in Hilbert Spaces . Springer New York, New York, NY 2011, ISBN 978-1-4419-9466-0 .
RE Showalter: Monotone Operators in Banach Space and Nonlinear Partial Differential Equations . American Mathematical Society, 2014, ISBN 978-1-4704-1280-7 (English).

Individual evidence

^ RE Showalter: Monotone Operators in Banach Space and Nonlinear Partial Differential Equations . American Mathematical Society, 2014, ISBN 978-1-4704-1280-7 , pp. 37 (English).
↑ Regina S. Burachik, Alfredo N. Iusem: Set-Valued Mappings and Enlargements of Monotone Operators , Springer-Verlag (2008), ISBN 978-0-387-69755-0
^ Klaus Deimling: Nonlinear Functional Analysis. 1st edition. Springer-Verlag, Berlin / Heidelberg 1985, ISBN 3-540-13928-1 , chapter 3.

[1] RE Showalter: Monotone Operators in Banach Space and Nonlinear Partial Differential Equations . American Mathematical Society, 2014, ISBN 978-1-4704-1280-7 , pp. 37 (English).

[2] Regina S. Burachik, Alfredo N. Iusem: Set-Valued Mappings and Enlargements of Monotone Operators , Springer-Verlag (2008), ISBN 978-0-387-69755-0

[3] Klaus Deimling: Nonlinear Functional Analysis. 1st edition. Springer-Verlag, Berlin / Heidelberg 1985, ISBN 3-540-13928-1 , chapter 3.