Strong Convergence of Halpern Iteration for Products of Finitely Many Resolvents of Maximal Monotone Operators in Banach Spaces

Abstract

In this paper, using Bregman functions, we introduce a new Halpern-type iterative algorithm
for finding common zeros of finitely many maximal monotone operators and obtain a strongly convergent
iterative sequence to the common zeros of these operators in a reflexive Banach space. Furthermore, we
study Halpern-type iterative schemes for finding common solutions of a finite system of equilibrium problems
and null spaces of a -inverse strongly monotone mapping in a 2-uniformly convex Banach space. Some
applications of our results to the solution of equations of Hammerstein-type are presented. Our scheme has
an advantage that we do not use any projection of a point on the intersection of closed and convex sets which
creates some difficulties in a practical calculation of the iterative sequence. So the simple construction of
Halpern iteration provides more flexibility in defining the algorithm parameters which is important from the
numerical implementation perspective. Presented results improve and generalize many known results in the
current literature.