A Counterexample to "An Extension of Gregus Fixed Point Theorem"
© Sirous Moradi. 2011
Received: 29 November 2010
Accepted: 21 February 2011
Published: 14 March 2011
In the paper by Olaleru and Akewe (2007), the authors tried to generalize Gregus fixed point theorem. In this paper we give a counterexample on their main statement.
Let be a Banach space and be a closed convex subset of . In 1980 Greguš  proved the following results.
In 2007, Olaleru and Akewe  considered the existence of fixed point of when is defined on a closed convex subset of a complete metrizable topological vector space and satisfies condition (1.2) and extended the Gregus fixed point.
Here, we give an example to show that the above mentioned theorem is not correct.
- Greguš M Jr.: A fixed point theorem in Banach space. Unione Matematica Italiana. Bollettino. A 1980,17(1):193–198.MathSciNetMATHGoogle Scholar
- Ćirić LjB: On a generalization of a Greguš fixed point theorem. Czechoslovak Mathematical Journal 2000,50(3):449–458. 10.1023/A:1022870007274MathSciNetView ArticleMATHGoogle Scholar
- Fisher B, Sessa S: On a fixed point theorem of Greguš. International Journal of Mathematics and Mathematical Sciences 1986,9(1):23–28. 10.1155/S0161171286000030MathSciNetView ArticleMATHGoogle Scholar
- Jungck G: On a fixed point theorem of Fisher and Sessa. International Journal of Mathematics and Mathematical Sciences 1990,13(3):497–500. 10.1155/S0161171290000710MathSciNetView ArticleMATHGoogle Scholar
- Mukherjee RN, Verma V: A note on a fixed point theorem of Greguš. Mathematica Japonica 1988,33(5):745–749.MathSciNetMATHGoogle Scholar
- Murthy PP, Cho YJ, Fisher B: Common fixed points of Greguš type mappings. Glasnik Matematički. Serija III 1995,30(2):335–341.MathSciNetMATHGoogle Scholar
- Olaleru JO, Akewe H: An extension of Gregus fixed point theorem. Fixed Point Theory and Applications 2007, 2007:-8.MathSciNetView ArticleMATHGoogle Scholar
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