Browder–Minty theorem explained

In mathematics, the Browder–Minty theorem (sometimes called the Minty–Browder theorem) states that a bounded, continuous, coercive and monotone function T from a real, separable reflexive Banach space X into its continuous dual space X^∗ is automatically surjective. That is, for each continuous linear functional g ∈ X^∗, there exists a solution u ∈ X of the equation T(u) = g. (Note that T itself is not required to be a linear map.)

The theorem is named in honor of Felix Browder and George J. Minty, who independently proved it.^[1]

See also

• Pseudo-monotone operator; pseudo-monotone operators obey a near-exact analogue of the Browder–Minty theorem.

References

1. Browder. Felix E.. Existence and perturbation theorems for nonlinear maximal monotone operators in Banach spaces. Bulletin of the American Mathematical Society. 73. 3. 1967. 322–328. 0002-9904. 10.1090/S0002-9904-1967-11734-8. free.

• Book: Renardy, Michael . Rogers, Robert C. . amp . An introduction to partial differential equations. Texts in Applied Mathematics 13. Second. Springer-Verlag. New York. 2004. 364. 0-387-00444-0. (Theorem 10.49)