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
applies. Here refers to the topological dual space of and the dual pairing .
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
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.