Summary: The purpose of this article is to prove strong convergence theorems for common fixed points of two closed hemi-relatively nonexpansive mappings in Banach spaces. In order to get the strong convergence theorems, monotone hybrid algorithms are presented and are used to approximate the common fixed points. Finally, a new simplified hybrid algorithm is proposed and a relative convergence theorem is proved by using a new method for the proofs. The results of this article modify and improve the results of S.-Y. Matsushita
and W. Takahashi
[J. Approximation Theory 134, No. 2, 257–266 (2005; Zbl 1071.47063
)], S. Plubtieng
and K. Ungchittrakool
[J. Approximation Theory 149, No. 2, 103–115 (2007; Zbl 1137.47056
)] and many others.