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Some convergence results for the Jungck–Mann and the Jungck–Ishikawa iteration processes in the class of generalized Zamfirescu operators. (English) Zbl 1174.47056
Let \(E\) be a Banach space, \(Y\) an arbitrary set, \(S,T\:Y\to E\), \(TY\subseteq SY\). Take \(SY\) to be complete and \(S\) to be injective. The authors introduce a new (Jungck–Ishikawa) iteration process: \[ Sx_{n+1}=(1-\alpha _n)Sx_n+\alpha _nTy_n\,,\quad Sy_n=(1-\beta _n)Sx_n+\beta _nTx_n\,, \] where \(\alpha _n,\beta _n\in [0,1]\). The main result of this paper is a condition under which the Jungck–Ishikawa process converges to \(p\) satisfying \(Sz=Tz=p\). One of these conditions is the existence of \(z\in Y\) such that \(Sz=Tz\). The results are generalizations of V. Berinde [Acta Math. Univ. Comenian. 73, 119–126 (2004; Zbl 1100.47054)].

MSC:
47J25 Iterative procedures involving nonlinear operators
47H09 Contraction-type mappings, nonexpansive mappings, \(A\)-proper mappings, etc.
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