On the transfer of convergence between two sequences in Banach spaces

Description

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.