On the transfer of convergence between two sequences in Banach spaces

Abstract

"Let $(X,\|\cdot\|)$ be a Banach space and $T:A\to X$ a contraction mapping, where $A\subset X$ is a closed set. Consider a sequence $\{x_n\}\subset A$ and define the sequence $\{y_n\}\subset X$, by $y_n=x_n+T\left(x_{\sigma(n)}\right)$, where $\{\sigma(n)\}$ is a sequence of natural numbers. We highlight some general conditions so that the two sequences $\{x_n\}$ and $\{y_n\}$ are simultaneously convergent. Both cases: 1) $\sigma(n)<n$, for all $n$, and 2) $\sigma(n)\ge n$, for all $n$, are discussed. In the first case, a general Picard iteration procedure is inferred. The results are then extended to sequences of mappings and some appropriate applications are also proposed."