Asymptotic regularity, fixed points and successive approximations. (English) Zbl 1498.47104

Berinde, Vasile; Rus, Ioan A.

Filomat 34, No. 3, 965-981 (2020).

Summary: Let \((M, d)\) be a metric space. In this paper we survey some of the most relevant results which relate the three concepts involved in the title: (a) the asymptotic regularity; (b) the existence (and uniqueness) of fixed points and (c) the convergence of the sequence of successive approximations to the fixed point(s), for a given operator \(f : M \to M\) or for two operators \(f,g: M \to M\) connected to each other in some sense.

MSC:

47H10	Fixed-point theorems
54H25	Fixed-point and coincidence theorems (topological aspects)
47H09	Contraction-type mappings, nonexpansive mappings, \(A\)-proper mappings, etc.
54E40	Special maps on metric spaces
65J15	Numerical solutions to equations with nonlinear operators
00A27	Lists of open problems

Keywords:

metric space; classes of operators on metric spaces; asymptotic regularity; fixed point; well posed problem of fixed point; orbitally quasinonexpansive operator; weakly Picard operator; equivalent fixed point equations

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