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Fixed points of quasi-nonexpansive mappings

  • W. G. Dotson (a1)
Abstract

A self-mapping T of a subset C of a normed linear space is said to be non-expansive provided ║TxTy║ ≦ ║xy║ holds for all x, yC. There has been a number of recent results on common fixed points of commutative families of nonexpansive mappings in Banach spaces, for example see DeMarr [6], Browder [3], and Belluce and Kirk [1], [2]. There have also been several recent results concerning common fixed points of two commuting mappings, one of which satisfies some condition like nonexpansiveness while the other is only continuous, for example see DeMarr [5], Jungck [8], Singh [11], [12], and Cano [4]. These results, with the exception of Cano's, have been confined to mappings from the reals to the reals. Some recent results on common fixed points of commuting analytic mappings in the complex plane have also been obtained, for example see Singh [13] and Shields [10].

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This list contains references from the content that can be linked to their source. For a full set of references and notes please see the PDF or HTML where available.

[1] L. P. Belluce and W. A. Kirk , ‘Fixed-point theorems for families of contraction mappings’, Pacific J. Math. 18 (1966), 213217.

[3] F. E. Browder , ‘Nonexpansive nonlinear operators in a Banach space’, Proc. Nat. Acad. Sci. U.S.A. 54 (1965), 10411044.

[4] J. Cano , ‘On commuting mappings by successive approximations’, Amer. Math. Monthly 75 (1968), 393394.

[5] R. DeMarr , ‘A common fixed point theorem for commuting mappings’, Amer. Math. Monthly 70 (1963), 535537.

[6] R. DeMarr , ‘Common fixed points for commuting contraction mappings’, Pacific J. Math. 13 (1963), 11391141.

[7] J. B. Diaz and F. T. Metcalf , ‘On the structure of the set of subsequential limit points of successive approximations’, Bull. Amer. Math. Soc. 73 (1967), 516519.

[8] G. Jungck , ‘Commuting mappings and common fixed points’, Amer. Math. Monthly 73 (1966), 735738.

[9] W. A. Kirk , ‘A fixed point theorem for mappings which do not increase distances’, Amer. Math. Monthly 72 (1965), 10041006.

[10] A. L. Shields , ‘On fixed points of commuting analytic functions’, Proc. Amer. Math. Soc. 15 (1964), 703706.

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Journal of the Australian Mathematical Society
  • ISSN: 1446-7887
  • EISSN: 1446-8107
  • URL: /core/journals/journal-of-the-australian-mathematical-society
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