On the transfer of convergence between two sequences in Banach spaces

"Let (X,\|·\|) be a Banach space and T:A→ X a contraction mapping, where A⊂ X is a closed set. Consider a sequence ₙ\⊂ A and define the sequence ₙ\⊂ X, by yₙ=xₙ+T(xσ(n)), where \σ(n)\ is a sequence of natural numbers. We highlight some general conditions so that the two sequences ₙ\ and ₙ\ are simultaneously convergent. Both cases: 1) $σ(n)