Pseudo-monotone operator

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)


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