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

:$u_ \rightharpoonup u \mbox X \mbox j \to \infty$

(i.e. ''u''_''j'' converges weakly to ''u'') and

:$\limsup_ \langle T(u_), u_ - u \rangle \leq 0,$

it follows that, for all ''v'' ∈ ''X'',

:$\liminf_ \langle T(u_), u_ - v \rangle \geq \langle T(u), u - v \rangle.$

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'' → ''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

* (Definition 9.56, Theorem 9.57)

{{Functional analysis

Banach spaces
Calculus of variations
Operator theory