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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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