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

$u_j\rightharpoonup u\text{ in }X\text{ as }j\to \mathrm{\infty }$

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

$\underset{j\to \mathrm{\infty }}{\mathrm{lim\; sup}}⟨T(u_j),u_j-u⟩\le 0,$

it follows that, for all v ∈ X,

$\underset{j\to \mathrm{\infty }}{\mathrm{lim\; inf}}⟨T(u_j),u_j-v⟩\ge ⟨T(u),u-v⟩.$

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

• Template:Cite book (Definition 9.56, Theorem 9.57)