On iterative equivalence of some classes of mappings
Abstract
We introduce the notion of iterative equivalence of two classes of mappings on metric spaces and we demonstrate its utility in metric fixed-point theory. In particular, we show that the fixed-point theorem for Matkowski's contractions can be derived from the corresponding theorem for Browder's contractions, though the first class of mappings is essentially wider than the second one.
References
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2. F.E. Browder, On the convergence of successive approximations for nonlinear functional equations, Indag. Math. 30 (1968), 27-35.
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8. J. Jachymski, Equivalence of some contractivity properties over metrical structures, Proc. Amer. Math. Soc. 125 (1997), 2327-2335.
9. J. Jachymski, An extension of A. Ostrovski's theorem on the round-off stability of iterations, Aequationes Math. 53 (1997), 242-253.
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11. M. Kuczma, B. Choczewski, R. Ger, Iterative Functional Equations, Encyclopedia of Mathematics and its Applications, Vol. 32, Cambridge Univ. Press, Cambridge, 1990.
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16. A. Meir, E. Keeler, A theorem on contraction mappings, J. Math. Anal. Appl. 28 (1969), 326-329.
2. F.E. Browder, On the convergence of successive approximations for nonlinear functional equations, Indag. Math. 30 (1968), 27-35.
3. J. Dugundji, A. Granas, Fixed Point Theory, Polish Scientific Publishers, Warszawa 1982.
4. R. Engelking, General Topology, Polish Scientific Publishers, Warszawa 1977.
5. E. Hille, R.S. Philips, Functional Analysis and Semigroups, Vol. 31, Amer. Math. Soc. Colloq. Publ., Providence, R. I., 1957.
6. J. Jachymski, An iff fixed point criterion for continuous self-mappings on a complete metric space, Aequationes Math. 48 (1994), 163-170.
7. J. Jachymski, Equivalent conditions and the Meir-Keeler type theorems, J. Math. Anal. Appl. 194 (1995), 293-303.
8. J. Jachymski, Equivalence of some contractivity properties over metrical structures, Proc. Amer. Math. Soc. 125 (1997), 2327-2335.
9. J. Jachymski, An extension of A. Ostrovski's theorem on the round-off stability of iterations, Aequationes Math. 53 (1997), 242-253.
10. J. Jachymski, Some consequences of the Tarski-Kantorovitch ordering theorem in metric fixed point theory, Quaestiones Math. 21 (1998), 89-99.
11. M. Kuczma, B. Choczewski, R. Ger, Iterative Functional Equations, Encyclopedia of Mathematics and its Applications, Vol. 32, Cambridge Univ. Press, Cambridge, 1990.
12. S. Leader, Equivalent Cauchy sequences and contractive fixed points in metric spaces, Studia Math. 66 (1983), 63-67.
13. J. Matkowski, Integrable solutions of functional equations, Diss. Math. 127 (1975).
14. J. Matkowski, J. Miś, Examples and remarks to a fixed point theorem, Facta Univ. (Niš) Ser. Math. Inform. 1 (1986), 53-56.
15. J. Matkowski, T. Świątkowski, On subadditive functions, Proc. Amer. Math. Soc. 119 (1993), 187-197.
16. A. Meir, E. Keeler, A theorem on contraction mappings, J. Math. Anal. Appl. 28 (1969), 326-329.
JachymskiJ. (1999). On iterative equivalence of some classes of mappings. Annales Mathematicae Silesianae, 13, 149-165. Retrieved from https://journals.us.edu.pl/index.php/AMSIL/article/view/14145
Jacek Jachymski
jachym@ck-sg.p.lodz.pl
Instytut Matematyki, Politechnika Łódzka Poland
Instytut Matematyki, Politechnika Łódzka Poland
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