On the transfer of convergence between two sequences in Banach spaces

Marinescu, Dan Ştefan and Păltănea, Eugen

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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.


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Marinescu, Dan Ştefan, Păltănea, Eugen