# 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 map *T* : *X* → *X*^{∗} from *X* into its continuous dual space *X*^{∗} is said to be **pseudo-monotone** if *T* is a bounded operator (not necessarily continuous) and if whenever

(i.e. *u*_{j} converges weakly to *u*) and

it follows that, for all *v* ∈ *X*,

## 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 bounded, 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

- Renardy, Michael & Rogers, Robert C. (2004).
*An introduction to partial differential equations*. Texts in Applied Mathematics 13 (Second ed.). New York: Springer-Verlag. p. 367. ISBN 0-387-00444-0. (Definition 9.56, Theorem 9.57)