Abstract
A new modified Halpern‐Mann type iterative method is constructed. Strong convergence of the scheme to a common element of the set of fixed points of a relatively nonexpansive mapping and the set of common solutions to a system of equilibrium problems in a uniformly convex real Banach space which is also uniformly smooth is proved. The results presented in this work improve on the corresponding ones announced by many others.
Highlights
Throughout this paper, we denote by N and R the sets of positive integers and real numbers, respectively
Motivated by Nilsrakoo and Saejung 15 and Nilsrakoo 35, we present a strong convergence theorem of a new modified Halpern-Mann iterative scheme to find a common element of the set of fixed points of a relatively nonexpansive mapping and the set of common solutions to a system of equilibrium problems in a uniformly convex real Banach space which is uniformly smooth
We introduce a modified Halpern-Mann type iteration without using the generalized metric projection and prove a strong convergence theorem for finding a common element of the set of fixed points of a relatively nonexpansive mapping and the set of solutions to a system of equilibrium problems in a uniformly convex and uniformly smooth Banach space
Summary
Throughout this paper, we denote by N and R the sets of positive integers and real numbers, respectively. 1.3 where x1 ∈ C is arbitrary, {αn} is an appropriate sequence in 0, 1 , S is a relatively nonexpansive mapping, and ΠC denotes the generalized projection from E onto a closed convex subset C of E They proved that the sequence {xn} converges weakly to a fixed point of T. Motivated by Nilsrakoo and Saejung 15 and Nilsrakoo 35 , we present a strong convergence theorem of a new modified Halpern-Mann iterative scheme to find a common element of the set of fixed points of a relatively nonexpansive mapping and the set of common solutions to a system of equilibrium problems in a uniformly convex real Banach space which is uniformly smooth. The results in this work improve on the corresponding ones announced by many others
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