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An extension of A. Ostrowski’s theorem on the round-off stability of iterations. (English) Zbl 0885.47023

Summary: A. M. Ostrowski established the stability of the procedure of successive approximations for Banach contractive maps. In this paper, we generalize the above result by using a more general contractive definition introduced by F. Browder. Further, we study the case of maps on metrically convex metric spaces and compact metric spaces, obtaining results relative to fixed point theorems of D. W. Boyd and J. S. W. Wong, and M. Edelstein. Finally, as a by-product of our basic lemma, we extend a recent result of T. Vidalis concerning the convergence of an iteration procedure involving an infinite sequence of maps.

MSC:

47H10 Fixed-point theorems
54H25 Fixed-point and coincidence theorems (topological aspects)
65D15 Algorithms for approximation of functions
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References:

[1] Boyd, D. W. andWong, J. S. W.,On nonlinear contractions. Proc. Amer. Math. Soc.20 (1969), 458–464. · Zbl 0175.44903
[2] Browder, F. E.,On the convergence of successive approximations for nonlinear functional equation. Indag. Math. (N.S.)30 (1968), 27–35. · Zbl 0155.19401
[3] Browder, F. E.,Nonlinear operators and nonlinear equations of evolution in Banach spaces. [Proc. Symp. Pure Math. Vol. 18], AMS, New York, 1976. · Zbl 0327.47022
[4] Dugundji, J. andGranas, A.,Fixed point theory. Polish Scientific Publishers, Warszawa, 1982.
[5] Edelstein, M.,On fixed and periodic points under contractive mappings. J. London Math. Soc. (2)37 (1962), 74–79. · Zbl 0113.16503
[6] Engelking, R.,General topology. Polish Scientific Publishers, Warszawa, 1977. · Zbl 0373.54002
[7] Harder, A. M. andHicks, T. L.,A stable iteration procedure for nonexpansive mappings. Math. Japon.33 (1988), 687–692. · Zbl 0655.47046
[8] Harder, A. M. andHicks, T. L.,Stability results for fixed point iteration procedures. Math. Japon.33 (1988), 693–706. · Zbl 0655.47045
[9] Hille, E. andPhilips.R. S.,Functional analysis and semi-groups. [Amer. Math. Soc. Colloq. Publ., Vol. 31] AMS, Providence, R.I., 1957. · Zbl 0078.10004
[10] Isac, G.,Fixed point theory, coincidence equations on convex cones and complementarity problems. Contemp. Math.72 (1985),Fixed point theory and its applications, ed. R. F. Brown, pp. 139–155.
[11] Istratescu, V. I.,Fixed point theory. An introduction. [Mathematics and Its Applications], D. Reidel Publishing Company, Dordrecht, 1981. · Zbl 0465.47035
[12] Jachymski, J. R.,An iff fixed point criterion for continuous self-mappings on a complete metric space. Aequationes Math.48 (1994), 163–170. · Zbl 0805.47051
[13] Leader, S.,Fixed points for operators on metric spaces with conditional uniform equivalence of orbits. J. Math. Anal. Appl.61 (1977), 466–474. · Zbl 0368.54016
[14] Matkowski, J. andWegrzyk, R.,On equivalence of some fixed point theorems for selfmappings of metrically convex space. Boll. Un. Mat. Ital. A(5)15 (1978), 359–369. · Zbl 0401.54039
[15] Ostrowski, A. M.,The round off stability of iterations. Z. Angew. Math. Mech.47 (1967), 77–81. · Zbl 0149.36601
[16] Ostrowski, A. M.,Solutions of equations in Euclidean and Banach spaces. Academic Press, New York, 1973. · Zbl 0304.65002
[17] Rakotch, E.,A note on contractive mappings. Proc. Amer. Math. Soc.13 (1962), 459–465. · Zbl 0105.35202
[18] Rhoades, B. E.,Fixed point theorems and stability results for fixed point iteration procedures II. Indian J. Pure Appl. Math.24 (1993), 691–703. · Zbl 0794.54048
[19] Vidalis, T.,Sequences of mappings converging to a contraction mapping. Indian J. Pure Appl. Math.20 (1989), 549–553. · Zbl 0666.54026
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