New construction and proof techniques of projection algorithm for countable maximal monotone mappings and weakly relatively non-expansive mappings in a Banach space.

In a real uniformly convex and uniformly smooth Banach space, some new monotone projection iterative algorithms for countable maximal monotone mappings and countable weakly relatively non-expansive mappings are presented. Under mild assumptions, some strong convergence theorems are obtained. Compared to corresponding previous work, a new projection set involves projection instead of generalized projection, which needs calculating a Lyapunov functional. This may reduce the computational labor theoretically. Meanwhile, a new technique for finding the limit of the iterative sequence is employed by examining the relationship between the monotone projection sets and their projections. To check the effectiveness of the new iterative algorithms, a specific iterative formula for a special example is proved and its computational experiment is conducted by codes of Visual Basic Six. Finally, the application of the new algorithms to a minimization problem is exemplified.