An implicit algorithm for finding a fixed point of a $Q$-nonexpansive mapping in locally convex spaces

TL;DR

This paper introduces an implicit iterative method to find fixed points of $Q$-nonexpansive mappings in locally convex spaces, proving its convergence under the topology determined by a family of seminorms.

Contribution

It defines $q$-duality mappings in locally convex spaces and develops a new implicit algorithm with proven convergence for fixed point problems.

Findings

The implicit scheme converges to a fixed point in the topology $ au_{Q}$.

The method extends fixed point theory to locally convex spaces with $Q$-nonexpansive mappings.

The paper provides a theoretical foundation for iterative fixed point algorithms in generalized spaces.

Abstract

Suppose that $Q$ is a family of seminorms on a locally convex space $E$ which determines the topology of $E$. In this paper, first we define the notation of the $q$-duality mappings in locally convex spaces. Then we introduce an implicit method for finding an element of the set of fixed points of a $Q$-nonexpansive mapping. Then we prove the convergence of the proposed implicit scheme to a fixed point of the $Q$-nonexpansive mapping in $\tau_{Q}$.