Weak and Strong Convergence Theorems for Maximal Monotone Operators in a Banach Space

In this paper, we introduce an iterative sequence for finding a solution of a maximal monotone operator in a uniformly convex Banach space. Then we first prove a strong convergence theorem, using the notion of generalized projection. Assuming that the duality mapping is weakly sequentially continuous, we next prove a weak convergence theorem, which extends the previous results of Rockafellar SIAM J. Control Optim. 14 (1976), 877–898] and Kamimura and Takahashi J. Approx. Theory 106 (2000), 226–240]. Finally, we apply our convergence theorem to the convex minimization problem and the variational inequality problem.