Approximation of common fixed points of a countable family of nonexpansive mappings in a Banach space

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Strong convergence of an iterative algorithm involving nonlinear mappings of nonexpansive and accretive type (2018, background): “7 ([23]): Let {an} be a sequence of nonnegative real numbers such that an+1 ≤ (1 − tn)an + bn + cn, ∀n ≥ 0, where {cn} is a sequence of nonnegative real numbers, {tn} ⊂ (0, 1) and {bn} is a sequence of real numbers.”
Effective metastability of Halpern iterates in CAT(0) spaces (2011, background): “The following lemma about sequences of real numbers was proved in [2].”
Strong Convergence of the Halpern Subgradient Extragradient Method for Solving Variational Inequalities in Hilbert Spaces (2014, neutral): “3 [23, 24] Let {an} be a sequence of non-negative real numbers, {αn} a sequence in ]0,1[ with ∞n=1 αn =∞, {bn} a sequence of non-negative real numbers with ∑∞ n=1 bn <∞ and {γn} a sequence of real numbers with lim supn→∞ γn ≤ 0.”
TWO GENERALIZED STRONG CONVERGENCE THEOREMS OF HALPERN’S TYPE IN HILBERT SPACES AND APPLICATIONS (2012, neutral): “([2, 28]).”