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.
[edit] Definition
Let (X, || ||) be a reflexive Banach space. A linear map T : X → X∗ from X into its continuous dual space X∗ is said to be pseudo-monotone if T is a bounded linear operator and if whenever
(i.e. uj converges weakly to u) and
it follows that, for all v ∈ X,
[edit] 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 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.
[edit] References
- Renardy, Michael and Rogers, Robert C. (2004). An introduction to partial differential equations, Second edition, Texts in Applied Mathematics 13, New York: Springer-Verlag, 367. ISBN 0-387-00444-0. (Definition 9.56, Theorem 9.57)
