AN EXTENSION OF THE FITZPATRICK THEORY

作者：Visintin Augusto^*

来源：Communications on Pure and Applied Analysis, 2014, 13(5): 2039-2058.

DOI：10.3934/cpaa.2014.13.2039

• Maximal monotone operators
• representative functions
• variational formulation
• minimization

In the seminal work [MR 10095941], Fitzpatrick proved that for any maximal monotone operator alpha : V -%26gt; p(V%26apos;) (V being a real Banach space) there exists a lower semicontinuous, convex representative function f(alpha) : V x V%26apos; -%26gt; RU {+infinity} such that %26lt;br%26gt;f(alpha)(v, v%26apos;) %26gt;= (v%26apos;, v) for all(v, v%26apos;), f(alpha)(v, v%26apos;) = v, v%26apos; double left right arrow v%26apos; is an element of alpha(v). %26lt;br%26gt;Here we assume that alpha(v), is a maximal monotone operator for any v is an element of V, and extend the Fitzpatrick theory to provide a new variational formulation for either stationary or evolutionary (nonmonotone) inclusions of the form alpha(v)(v) v%26apos;. For any v%26apos; is an element of V%26apos;, we prove existence of a solution via the classical minimax theorem of Ky Fan. Applications include stationary and evolutionary pseudo-monotone operators, and variational inequalities.

• 出版日期2014-9