A Counterexample to "An Extension of Gregus Fixed Point Theorem"
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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, [email protected] 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, 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 Greguˇs 1 proved the following results. Theorem 1.1. Let T : C → C be a mapping satisfying the inequality T x − T y ≤ ax − y bx − T x cy − T y,
1.1
for all x, y ∈ 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 T x, T y ≤ ad x, y bdx, T x cd y, T y ed y, T x fd x, T y ,
for all x, y ∈ X, where 0 < a < 1, b, c, e, f ≥ 0, and a b c e f 1.
1.2
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 ∈ X i Fx ≥ 0, ii Fx 0 → x 0, iii Fx y ≤ Fx Fy, iv Fλx ≤ Fx for all λ ∈ K with |λ| ≤ 1, v if λn → 0 and λn ∈ K, then Fλn x → 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 convex subset of a complete metrizable topological vector space X and T : C → C a mapping that satisfies F T x − T y ≤ aF x − y bFx − T x cF y − T y eF y − T x fF x − T y
1.3
for all x, y ∈ X, where F is an F-norm on X, 0 < a < 1, b, c, e, f ≥ 0, and a b c e f 1. Then T has a unique fixed point. Here, we give an example to show that the above mentioned theorem is not correct.
2. Counterexample Example 2.1. Let X R endowed with the Euclidean metric and C X. Let T : C → C defined by T x x 1. Let 0 < a < 1 and e > 0 such that a 2e 1. Then for all x ∈ C such that y > x, we have that T x − T y ≤ ax − y ey − T x ex − T y ⇐⇒ y − x ≤ a y − x ey − x − 1 ex − y − 1 ⇐⇒ y − x ≤ a y − x ey − x − 1 e y 1 − x ⇐⇒ e y − x 1 − a − e y − x ≤ ey − x − 1 e ⇐⇒ y − x ≤ y − x − 1 1.
2.1
We have two cases, y > x 1 or y ≤ x 1. If y > x 1
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