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 ⇀ u in X as j → ∞ {\displaystyle u_{j}\rightharpoonup u{\mbox{ in }}X{\mbox{ as }}j\to \infty }
(i.e. uj converges weakly to u) and
lim sup j → ∞ ⟨ T ( u j ) , u j − u ⟩ ≤ 0 , {\displaystyle \limsup _{j\to \infty }\langle T(u_{j}),u_{j}-u\rangle \leq 0,}
it follows that, for all v ∈ X,
lim inf j → ∞ ⟨ T ( u j ) , u j − v ⟩ ≥ ⟨ T ( u ) , u − v ⟩ . {\displaystyle \liminf _{j\to \infty }\langle T(u_{j}),u_{j}-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 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)
