Fixed Point Theory for Lipschitzian-type Mappings with Applications

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Approximating fixed points of enriched nonexpansive mappings by Krasnoselskij iteration in Hilbert spaces (2019, background): “x(T) = Fix(Tµ) 6= ∅. Since H is a Hilbert space, Fix(T) is also convex, see for example Theorem 6 in [27]. We include here a proof of the fact that Fix(Tµ) is convex, i.e., when x,y ∈ Fix(Tµ) and λ ∈ [0,1], we have uλ = (1− λ)x+ λy ∈ Fix(Tµ). Indeed, since Tµ is nonexpansive, we have kTµuλ −xk = kTµ”
Data dependence results of new multi-step and S-iterative schemes for contractive-like operators (2012, positive): “6)    x0 ∈ E, xn+1 = (1− αn)Txn + αnTyn, yn = (1− βn)xn + βnTxn, n ∈ N, is known as the S-iteration process (see [9, 26, 27]).”