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A Counterexample to "An Extension of Gregus Fixed Point Theorem"
Fixed Point Theory and Applications volume 2011, Article number: 484717 (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.
Let be a mapping satisfying the inequality
for all , where , and . Then has a unique fixed point.
for all , where , and .
Let be a topological vector space on . The mapping is said to be an such that for all
(iv) for all with ,
(v)if and , then .
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.
Let be a closed convex subset of a complete metrizable topological vector space and a mapping that satisfies
for all , where is an on , , and . Then has a unique fixed point.
Here, we give an example to show that the above mentioned theorem is not correct.
Let endowed with the Euclidean metric and . Let defined by . Let and such that . Then for all such that , we have that
We have two cases, or .
If , then , and hence inequality (2.1) is true. If , then , and so , and hence inequality (2.1) is true. So condition (1.3) holds for and , but has not fixed point.
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Moradi, S. A Counterexample to "An Extension of Gregus Fixed Point Theorem". Fixed Point Theory Appl 2011, 484717 (2011). https://doi.org/10.1155/2011/484717
- Banach Space
- Vector Space
- Point Theorem
- Differential Geometry
- Convex Subset