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Hindawi Publishing Corporation Fixed Point Theory and Applications Volume 2011, Article ID 484717, 3

Hindawi Publishing Corporation Fixed Point Theory and Applications Volume 2011, Article ID 484717, 3 pages doi:10.1155/2011/484717 Letter to the Editor A Counterexample to “An Extension of Gregus Fixed Point Theorem” Sirous Moradi Department of Mathematics, Faculty of Science, Arak University, Arak 38156-8-8349, Iran Correspondence should be addressed to Sirous Moradi, sirousmoradi@gmail.com Received 29 November 2010; Accepted 21 February 2011 Copyright q 2011 Sirous Moradi. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. In the paper by Olaleru and Akewe 2007 ,. | Hindawi Publishing Corporation Fixed Point Theory and Applications Volume 2011 Article ID 484717 3 pages doi 10.1155 2011 484717 Letter to the Editor A Counterexample to An Extension of Gregus Fixed Point Theorem Sirous Moradi Department of Mathematics Faculty of Science Arak University Arak 38156-8-8349 Iran Correspondence should be addressed to Sirous Moradi sirousmoradi@gmail.com Received 29 November 2010 Accepted 21 February 2011 Copyright 2011 Sirous Moradi. This is an open access article distributed under the Creative Commons Attribution License which permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited. 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. 1. Introduction Let X be a Banach space and C be a closed convex subset of X. In 1980 Gregus 1 proved the following results. Theorem 1.1. Let T C C be a mapping satisfying the inequality Tx - Ty a x - y b x - Tx c y - Ty M for all x y e C where 0 a 1 b c 0 and a b c 1. Then T has a unique fixed point. Several papers have been written on the Gregus fixed point theorem. For example see 2-6 . We can combine the Gregus condition by the following inequality where T is a mapping on metric space X d d Tx Ty ad x y bd x Tx cd y Ty ed y Tx fd x Ty 1.2 for all x y e X where 0 a 1 b c e f 0 and a b c e f 1. 2 Fixed Point Theory and Applications Definition 1.2. Let X be a topological vector space on K C or R . The mapping F X R is said to be an F-norm such that for all x y e X i F x 0 ii F x 0 x 0 iii F x y F x F y iv F Ax F x for all A e K with A 1 v if An 0 and An e K then FfAnx 0. In 2007 Olaleru and Akewe 7 considered the existence of fixed point of T when T is defined on a closed convex subset C of a complete metrizable topological vector space X and satisfies condition 1.2 and extended the Gregus fixed point. Theorem 1.3. Let C be a closed .