Erratum to: Fixed point theorems of contractive mappings in cone b-metric spacesand applications
© Xu and Huang; licensee Springer. 2014
Received: 26 November 2013
Accepted: 26 November 2013
Published: 5 March 2014
In this note we correct some errors that appeared in the article (Huang and Xu in FixedPoint Theory Appl. 2013:112, 2013) by modifying some conditions in the main theorems andexamples.
After examining the proofs of the main results in , we can find that there is something wrong with the proof of the Cauchy sequencein [, Theorem 2.1]. This leads to subsequent errors in Theorem 2.3 andrelated examples in . We also find that it is not rigorous to use the corresponding lemmas, and so theproof is inaccurate. The detailed reasons are given in the following.
as . Therefore, it is impossible to utilize [, Lemma 1.8, 1.9] and demonstrate that is a Cauchy sequence.
In this note, we would like to slightly modify only one of the used conditions to achieveour claim.
The following theorem is a modification to [, Theorem 2.1]. The proof is the same as that in  except the proof of the Cauchy sequence. We will attain the desired goal by usingthe new modified condition instead of .
where is a constant. Then T has a unique fixed point in X. Furthermore, the iterative sequence converges to the fixed point.
for all , . So, by [, Lemma 1.9], is a Cauchy sequence in . The proof is completed. □
Hence, by Theorem 2.1, there exists (in fact, it satisfies ) such that is the unique fixed point of T.
For the same reason, we need to use the new condition instead of the original condition in [, Theorem 2.3]. The correct statement is as follows.
where the constant and , . Then T has a unique fixed point in X. Moreover, the iterative sequence converges to the fixed point.
In addition, based on the changes of Theorem 2.1, we need to change the condition into for [, Example 3.1]. Let us give the corrected example.
where is a continuous function.
Set such that , then there exists a unique solution of (2.1).
Proof Let and . Put as with such that . It is clear that is a complete cone b-metric space with.
we speculate that is a contractive mapping.
and Lemma 1.12 in  that , which means , that is, is complete. □
Owing to the above statement, all conditions of Theorem 2.1 are satisfied. HenceT has a unique fixed point . That is to say, there exists a unique solution of (2.1).
The authors thank the referees, the editors and the readers including Prof. SriramBalasubramanian and Prof. Reny George. Special thanks are due to Prof. Ravi P. Agarwal andProf. Ljubomir Ciric, who have made a number of valuable comments and suggestions, whichhave improved  greatly. The research is partially supported by the PhD Start-up Fund ofHanshan Normal University, Guangdong Province, China (No. QD20110920).
- Huang H, Xu S: Fixed point theorems of contractive mappings in cone b -metric spaces andapplications. Fixed Point Theory Appl. 2013., 2013: Article ID 112Google Scholar
- Jovanović M, Kadelburg Z, Radenović S: Common fixed point results in metric-type spaces. Fixed Point Theory Appl. 2010., 2010: Article ID 978121 10.1155/2010/978121Google Scholar
- Khamsi MA: Remarks on cone metric spaces and fixed point theorems of contractive mappings. Fixed Point Theory Appl. 2010., 2010: Article ID 315398 10.1155/2010/315398Google Scholar
- Shah MH, Smić S, Hussain N, Sretenović A, Radenović S: Common fixed points theorems for occasionally weakly compatible pairs on cone metrictype spaces for b -metric spaces. J. Comput. Anal. Appl. 2012, 14(2):290–297.MathSciNetMATHGoogle Scholar
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