Pseudo-monotone operator

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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[edit]

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

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

it follows that, for all v ∈ X,

Properties of pseudo-monotone operators[edit]

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 continuous, 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[edit]

  • 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)