The asymptotic convergence of the forward-backward splitting algorithm
for solving equations of type 0 is an element of T(z) is analyzed, wh
ere T is a multivalued maximal monotone operator in the n-dimensional
Euclidean space, When the problem has a nonempty solution set, and T i
s split in the form T = J + h It with J being maximal monotone and h b
eing co-coercive with modulus greater than 1/2, convergence rates are
shown, under mild conditions, to be linear, superlinear or sublinear d
epending on how rapidly J(-1) and h(-1) grow in the neighborhoods of c
ertain specific points. As a special case, when both J and h are polyh
edral functions, we get R-linear convergence and 2-step e-linear conve
rgence without any further assumptions on the strict monotonicity on 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 get new results, which complement R. T. Rockafellar's and F. J.
Luque's earlier results on the proximal point algorithm.