Pseudo-monotone operator

In mathematics, a pseudo-monotone operator from a reflexive Banach space into its continuous dual space is one that is, in some sense, almost as well-behaved as a monotone operator. Many problems in the calculus of variations can be expressed using operators that are pseudo-monotone, and pseudo-monotonicity in turn implies the existence of solutions to these problems.

Definition

Let ("X", || ||) be a reflexive Banach space. A linear map "T" : "X" → "X"^∗ from "X" into its continuous dual space "X"^∗ is said to be pseudo-monotone if "T" is a bounded linear operator and if whenever

:$u\_\{j\}\; ightharpoonup\; u\; mbox\{\; in\; \}\; X\; mbox\{\; as\; \}\; j\; o\; infty$

(i.e. "u"_"j" converges weakly to "u") and

:$limsup\_\{j\; o\; infty\}\; langle\; T(u\_\{j\}),\; u\_\{j\}\; -\; u\; angle\; leq\; 0,$

it follows that, for all "v" ∈ "X",

:$liminf\_\{j\; o\; infty\}\; langle\; T(u\_\{j\}),\; u\_\{j\}\; -\; v\; angle\; geq\; langle\; T(u),\; u\; -\; v\; angle.$

Properties of pseudo-monotone operators

Using a very similar proof to that of the Browder-Minty theorem, one can show the following:

Let ("X", || ||) be a real, reflexive Banach space and suppose that "T" : "X" → "X"^∗ is continuous, coercive and pseudo-monotone. Then, for each continuous linear functional "g" ∈ "X"^∗, there exists a solution "u" ∈ "X" of the equation "T"("u") = "g".

References

* cite book
author = Renardy, Michael and Rogers, Robert C.
title = An introduction to partial differential equations
series = Texts in Applied Mathematics 13
edition = Second edition
publisher = Springer-Verlag
location = New York
year = 2004
pages = 367
id = ISBN 0-387-00444-0 (Definition 9.56, Theorem 9.57)