Strong convergence theorems for maximal monotone mappings in Banach spaces

Abstract

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 , x 1 ∈ 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.