Approximating Zero Points of Accretive Operators with Compact Domains in General Banach Spaces

Let E be a real Banach space, let C be a closed convex subset of E, let T be a nonexpansive mapping of C into itself, that is, ‖Tx−Ty‖ ≤ ‖x− y‖ for each x, y ∈ C, and let A⊂ E×E be an accretive operator. For r > 0, we denote by Jr the resolvent of A, that is, Jr = (I + rA)−1. The problem of finding a solution u∈ E such that 0∈ Au has been investigated by many authors; for example, see [3, 4, 7, 16, 26]. We know the proximal point algorithm based on a notion of resolvents of accretive operators. This algorithm generates a sequence {xn} in E such that x1 = x ∈ E and xn+1 = Jrnxn for n= 1,2, . . . , (1.1)