A Counterexample on a Theorem by Khojasteh, Goodarzi, and Razani
© Ivan D. Arandelović and Dragoljub J. Kečkić. 2010
Received: 27 June 2010
Accepted: 1 September 2010
Published: 7 September 2010
In the paper by Khojasteh et al. (2010), the authors tried to generalize Branciari's theorem, introducing the new integral type contraction mappings. In this note we give a counterexample on their main statement (Theorem 2.9). Also we give a comment explaining what the mistake in the proof is, and suggesting what conditions might be appropriate in generalizing fixed point results to cone spaces, where the cone is taken from the infinite dimensional space.
In the paper , Branciari proved the following fixed point theorem with integral-type contraction condition.
There are many generalizations of fixed point results to the so-called cone metric spaces, introduced by several Russian authors in mid-20th. These spaces are re-introduced by Huang and Zhang . In the same paper, the notion of convergent and Cauchy sequences are given.
In the paper  Khojasteh et al. tried to generalize Branciari fixed point result to the cone metric spaces. They introduce the concept of integration along the interval as a limit of Cauchy sums.
Definition 1.5 (see ).
Using this concept, they stated the following statement (Theorem in ).
Theorem 1.6 (see ).
However, the last statement is not true. This will be proved in the next section.
2. Constructing the Counterexample
and (b) obvious.
which completes the proof of the first statement.
On the other hand, has no fixed point. Namely, if we suppose that is a fixed point for , it means that for all , and moreover for all , and also for all , by induction. By continuity of , it follows that implying !
3. A Comment
and the existence of the total ordering on . However, in infinite dimensional case, such conclusion invokes continuity of the function inverse to . Even for the linear mappings this is not always true, but only under additional assumption that initial mapping is bijective. This asserts the well known Banach open mapping theorem. In the absence of some generalization of the open mapping theorem to nonlinear case, it is necessary to include continuity of the inverse function in the assumptions, as it was done in .
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