A strong convergence theorem for maximal monotone operators in Banach spaces with applications

An algorithm is constructed to approximate a zero of a maximal monotone operator in a uniformly convex and uniformly smooth real Banach space. The sequence of the algorithm is proved to converge strongly to a zero of the maximal monotone map. In the case where the Banach space is a real Hilbert space, our theorem complements the celebrated proximal point algorithm of Martinet and Rockafellar. Furthermore, our convergence theorem is applied to approximate a solution of a Hammerstein integral equation in our general setting. Finally, numerical experiments are presented to illustrate the convergence of our algorithm.