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.

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

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

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

${\displaystyle \limsup _{j\to \infty }\langle T(u_{j}),u_{j}-u\rangle \leq 0,}${\displaystyle \limsup _{j\to \infty }\langle T(u_{j}),u_{j}-u\rangle \leq 0,}

it follows that, for all v ∈ X,

${\displaystyle \liminf _{j\to \infty }\langle T(u_{j}),u_{j}-v\rangle \geq \langle T(u),u-v\rangle .}${\displaystyle \liminf _{j\to \infty }\langle T(u_{j}),u_{j}-v\rangle \geq \langle T(u),u-v\rangle .}

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.

• 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)

• Banach spaces
• Calculus of variations
• Operator theory