Strong convergence theorems for Bregman W-mappings with applications to convex feasibility problems in Banach spaces

AbstractIn this paper we introduce new modified Mann iterative processes for computing fixed points of an infinite family of Bregman W-mappings in reflexive Banach spaces. Let $W_{n}$ W n be the Bregman W-mapping generated by $S_{n},S_{n-1},\ldots,S_{1}$ S n , S n − 1 , … , S 1 and $\beta_{n,n},\beta_{n,n-1},\ldots,\beta_{n,1}$ β n , n , β n , n − 1 , … , β n , 1 . We first express the set of fixed points of $W_{n}$ W n as the intersection of fixed points of $\{S_{i}\}_{i=1}^{n}$ { S i } i = 1 n . As a consequence, we show that $W_{n}$ W n is a Bregman weak relatively nonexpansive mapping if $S_{i}$ S i is a Bregman weak relatively nonexpansive mapping for each $i=1,2,\ldots,n$ i = 1 , 2 , … , n . When specialized to the fixed point set of a Bregman nonexpansive type mapping T, the required sufficient condition $\tilde{F}(T)=F(T)$ F ˜ ( T ) = F ( T ) is less restrictive than the usual condition $\hat{F}(T)=F(T)$ F ˆ ( T ) = F ( T ) which is based on the demiclosedness principle. We then prove some strong convergence theorems for these mappings. Some application of our results to convex feasibility problem is also presented. Our results improve and generalize many known results in the current literature.