The asymptotic convergence of the forward–backward splitting algorithm for solving equations of type O∈T(x) is analysed, where T is a multivalued maximal monotone operator in the n-dimensional Euclidean space. When the problem has a nonempty solution set, and T is split in the form T=J+h It, with J being maximal monotone and h being coercive with modulus greater than 1/2, convergence rates are shown, under mild conditions to be linear, superlinear or sublinear depending on how J−1 and h−1 grow in the neighborhoods of certain specific points. As a special case, when both J and h are polyhedral functions we get R-linear convergence and 2-step ε-convergence without any further assumptions on the strict monotonicity of T or on the uniqueness of the solution. As another special case, when h=0, the splitting algorithm reduces to the proximal point algorithm and we obtain new results which complement Rockafellar's and Luque's earlier results on the proximal point algorithm.
