Monotone CQ Algorithm of Fixed Points for Weak Relatively Nonexpansive Mappings and Applications

Abstract:

      Matsushita, Takahashi$^{[4]}$ proved a strong convergence theorem for relatively nonexpansive mappings in a Banach space by using the hybrid method ($CQ$ method) in mathematical programming. The purpose of this paper is to modify the hybrid method of Matsushita, Takahashi by monotone $CQ$ method, and to prove strong convergence theorems for weak relatively nonexpansive mappings and maximal monotone operators in Banach spaces. The convergence rate of monotone $CQ$ method is faster than the hybrid method of Matsushita, Takahashi. In addition, the Cauchy sequence method is used in this paper without using the Kadec-Klee property. The results of this paper modify and improve the results of Matsushita, Takahashi and some others.