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 &rarr; X&lowast; from X into its continuous dual space X&lowast; is said to be pseudo-monotone if T is a bounded operator (not necessarily continuous) and if whenever


 * $$u_{j} \rightharpoonup u \mbox{ in } X \mbox{ as } j \to \infty$$

(i.e. uj converges weakly to u) and


 * $$\limsup_{j \to \infty} \langle T(u_{j}), u_{j} - u \rangle \leq 0,$$

it follows that, for all v &isin; X,


 * $$\liminf_{j \to \infty} \langle T(u_{j}), u_{j} - v \rangle \geq \langle T(u), u - v \rangle.$$

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 &rarr; X&lowast; is bounded, coercive and pseudo-monotone. Then, for each continuous linear functional g &isin; X&lowast;, there exists a solution u &isin; X of the equation T(u) = g.