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 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 and if whenever
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 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 = g.