Abstract

The Baillon–Haddad theorem establishes that the gradient of a convex and continuously differentiable function defined in a Hilbert space is β-Lipschitz if and only if it is 1 / β-cocoercive. In this paper, we extend this theorem to Gâteaux differentiable and lower semicontinuous convex functions defined on an open convex set of a Hilbert space. Finally, we give a characterization of C1 , + convex functions in terms of local cocoercivity.