Strong convergence theorems for maximal monotone mappings in Banach spaces

Let E be a uniformly convex and 2-uniformly smooth real Banach space with dual E ∗. Let A:E ∗→E be a Lipschitz continuous monotone mapping with A −1(0) = ∅. For given u, x1 ∈ E, let {xn} be generated by the algorithm xn+1 := βnu + (1 − βn)× (xn − αnAJxn), n 1, where J is the normalized duality mapping from E into E ∗ and {λn} and {θn} are real sequences in (0, 1) satisfying certain conditions. Then it is proved that, under some mild conditions, {xn} converges strongly to x ∗ ∈ E where Jx ∗ ∈ A −1(0). Finally, we apply our convergence theorems to the convex minimization problems. © 2008 Elsevier Inc. All rights reserved.