A characterization of cyclical monotonicity by the Gâteaux derivative (Q1088147)

Let X be a real Banach space, and X' be its dual space. We give an equivalent condition for a weak*-Gâteaux differentiable operator from X to X' with some assumption to be cyclically monotone, by using its Gâteaux derivative. As a corollary, the following is shown. Let A be a Gâteaux differentiable closed operator in a Hilbert space with convex domain D(A), and suppose that A is weakly continuous on every two dimensional subset in D(A). If \(\overline{\delta A(x)}\), the minimal closed extension of the Gâteaux derivative of A at x, is positive self- adjoint for each \(x\in D(A)\), then A is maximal cyclically monotone.