Inertial Iterative Process for Fixed Points of Certain Quasi-nonexpansive Mappings

Abstract

This paper deals with a general formalism which consists in approximating a point in a nonempty set $S$, in a real Hilbert space $H$, by a sequence $(x_n) \subset H$ such that $x_{{n + 1}} : = {\user1{\mathcal{T}}}_{n} {\left( {x_{n} + \theta _{n} {\left( {x_{n} - x_{{n - 1}} } \right)}} \right)}$, where ${\left( {\theta _{n} } \right)} \subset \left[ {0,} \right.\left. 1 \right)$, $x_0$ $x_1$ are in $H$ and ${\left( {{\user1{\mathcal{T}}}_{n} } \right)}_{{n \geqslant 0}}$ is a sequence included in a certain class of self-mappings on $H$, such that every fixed point set of ${\user1{\mathcal{T}}}_{n}$ contains $S$. This iteration method is inspired by an implicit discretization of the second order ‘heavy ball with friction’ dynamical system. Under suitable conditions on the parameters and the operators ${\left( {{\user1{\mathcal{T}}}_{n} } \right)}$, we prove that this scheme generates a sequence which converges weakly to an element of $S$. In particular, by appropriate choices of ${\left( {{\user1{\mathcal{T}}}_{n} } \right)}$, this algorithm works for approximating common fixed points of infinite countable families of a wide class of operators which includes $\alpha$-averaged quasi-nonexpansive mappings for $\alpha \in {\left( {0,\;1} \right)}$.