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 {x_n} be generated by the algorithm x_{n + 1} : = β_n u + (1 - β_n) (x_n - α_n A J x_n), n ≥ 1, where J is the normalized duality mapping from E into E^* and {λ_n} and {θ_n} are real sequences in (0, 1) satisfying certain conditions. Then it is proved that, under some mild conditions, {x_n} converges strongly to x^* ∈ E where J x^* ∈ A^-1 (0). Finally, we apply our convergence theorems to the convex minimization problems.