A variational principle for equations and inequalities with maximal monotone operators (Q1087819)

Let E be a real locally convex Hausdorff space with dual \(E^*\) and pairing denoted by \(<\cdot,\cdot >\). Let A be a monotone operator from E into \(E^*\) and let M denote a convex subset of the domain of A. The author describes the set of solutions of the variational inequality \[ (VI)\quad x\in M,\quad \tilde Ax\in Ax,\quad <\tilde Ax,x-v>\leq 0\text{ for all } v\in M, \] by means of a variational principle related to the operator A. Specifically, define the function \(J_ A:M\times M\to {\mathbb{R}}\) by \(J_ A(x,y):=\int^{1}_{0}<\tilde A(x+t(y-x)),y-x>dt\), where \(\tilde A\) is an arbitrary single-valued section of A. After preliminary results justifying this definition and noting that \(J_ A\) is a skew-symmetric saddle function, the author proves Theorem 2: Let \(A+\partial I_ M\) be maximal monotone. Then \(x\in M\) is a solution of (VI) \(\Leftrightarrow [x,x]\in M\times M\) is a saddle point of \(J_ A\Leftrightarrow\) there exists \(y\in M\) such that [x,y] is a saddle point of \(J_ A.\) Skew-symmetry of \(J_ A\) plays an important role in the proof. We note also that A need not be the subdifferential of a convex function.