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 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.