Fixed points and their approximation in Banach spaces for certain commuting mappings

Khan, M. S.

Glasgow mathematical journal, Tome 23 (1982) no. 1, pp. 1-6

Résumé

1. Let X be a Banach space. Then a self-mapping A of X is said to be nonexpansive provided that ‖AX − Ay‖≤‖X − y‖ holds for all x, y ∈ X. The class of nonexpansive mappings includes contraction mappings and is properly contained in the class of all continuous mappings. Keeping in view the fixed point theorems known for contraction mappings (e.g. Banach Contraction Principle) and also for continuous mappings (e.g. those of Brouwer, Schauderand Tychonoff), it seems desirable to obtain fixed point theorems for nonexpansive mappings defined on subsets with conditions weaker than compactness and convexity. Hypotheses of compactness was relaxed byBrowder [2] and Kirk [9] whereas Dotson [3] was able to relax both convexity and compactness by using the notion of so-called star-shaped subsets of a Banach space. On the other hand, Goebel and Zlotkiewicz [5] observed that the same result of Browder [2] canbe extended to mappings with nonexpansive iterates. In [6], Goebel-Kirk-Shimi obtainedfixed point theorems for a new class of mappings which is much wider than those of nonexpansive mappings, and mappings studied by Kannan [8]. More recently, Shimi [12] used the fixed point theorem of Goebel-Kirk-Shimi [6] to discuss results for approximating fixed points in Banach spaces.

Détail

DOI : 10.1017/S0017089500004717

Bibliographie

[1] 1.Brosowski, B., Fixpunktsätze in der Approximationstheorie, Mathematica (Cluj), 11 (1969), 195–220. Google Scholar

[2] 2.Browder, F. E., Nonexpansive nonlinear operators in Banach space, Proc. Nat. Acad. Sci. U.S.A. 54 (1965), 1041–1044. Google Scholar PubMed | DOI

[3] 3.Dotson, W. G. Jr, Fixed point theorems for nonexpansive mappings onstarshpaed subsets of Banach spaces, J. London Math. Soc., (2) 4 (1972), 408–410. Google Scholar | DOI

[4] 4.Edelstein, M., On fixed and periodic points under contractive mappings, J. London Math. Soc., 37 (1962), 74–79. Google Scholar | DOI

[5] 5.Goebel, K. and Zlotkiewicz, E., Some fixed point theorems in Banach spaces, Colloq. Math., 23 (1971), 103–106. Google Scholar | DOI

[6] 6.Goebel, K., Kirk, W. A. and Shimi, T. N., A fixed point theorem in uniformly convex spaces, Boll. Un. Mat. Ital., (4) 7 (1973), 67–75. Google Scholar

[7] 7.Jungck, G., An iff fixed point criterion, Math. Mag., 49 (1976), 32–34. Google Scholar | DOI

[8] 8.Kannan, R., Some results on fixed points III, Fund. Math., 70 (1971), 169–177. Google Scholar | DOI

[9] 9.Kirk, W. A., A fixed point theorem for mappings which do not increase distances, Amer. Math. Monthly, 72 (1965), 1004–1006. Google Scholar | DOI

[10] 10.Krasnoselskii, M. A., Two remarks about the method of successive approximations, Uspehi Mat. Nauk., 63 (1955), 123–127. Google Scholar

[11] 11.Park, S., Fixed points of f-contractive maps, Rocky Mountain J. Math., 8 (1978), 743–750. Google Scholar | DOI

[12] 12.Shimi, T. N., Approximation of fixed points of certain nonlinear mappings, J. Math. Anal. Appl., 65 (1978), 565–571. Google Scholar | DOI

[13] 13.Singh, S. P., An application of a fixed-point theorem to approximation theory, J. Approximation Theory, 25 (1979), 89–90. Google Scholar | DOI