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Erratum to: Fixed point theorems of contractive mappings in cone b-metric spacesand applications
Fixed Point Theory and Applications volume 2014, Article number: 55 (2014)
- The original article was published in Fixed Point Theory and Applications 2013 2013:112
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.
On p.5 in , we conclude that
as for any . This is incorrect. Indeed, note that taking and leads to
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 .
Theorem 2.1 Let be a complete cone b-metric space with the coefficient . Suppose that the mapping satisfies the contractive condition
where is a constant. Then T has a unique fixed point in X. Furthermore, the iterative sequence converges to the fixed point.
Proof In order to show that is a Cauchy sequence, we only need the following calculations.For any , , it follows that
Let be given. Notice that as for any p. Making full use of [, Lemma 1.8], we find such that
for each . Thus,
for all , . So, by [, Lemma 1.9], is a Cauchy sequence in . The proof is completed. □
Example 2.2 Let , and let be a constant. Take . We define as
Then is a complete cone b-metric space with. Let us define as
Thus, for all , we have
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.
Theorem 2.3 Let be a complete cone b-metric space with the coefficient . Suppose that the mapping satisfies the contractive condition
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.
We now apply Theorem 2.1 to the first-order periodic boundary problem
where is a continuous function.
Example 2.4 Consider boundary problem (2.1) with the continuous function F,and suppose that satisfies the local Lipschitz condition, i.e., if, , it induces
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.
Note that (2.1) is equivalent to the integral equation
Define a mapping by . If
we speculate that is a contractive mapping.
Finally, we prove that is complete. In fact, suppose that is a Cauchy sequence in . Then is also a Cauchy sequence in X. Since is complete, there is such that (). So, for each , there exists N, whenever , we obtain . Thus, it follows from
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).
Huang H, Xu S: Fixed point theorems of contractive mappings in cone b -metric spaces andapplications. Fixed Point Theory Appl. 2013., 2013: Article ID 112
Jovanović M, Kadelburg Z, Radenović S: Common fixed point results in metric-type spaces. Fixed Point Theory Appl. 2010., 2010: Article ID 978121 10.1155/2010/978121
Khamsi MA: Remarks on cone metric spaces and fixed point theorems of contractive mappings. Fixed Point Theory Appl. 2010., 2010: Article ID 315398 10.1155/2010/315398
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.
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).
The online version of the original article can be found at 10.1186/1687-1812-2013-112
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Huang, H., Xu, S. Erratum to: Fixed point theorems of contractive mappings in cone b-metric spacesand applications. Fixed Point Theory Appl 2014, 55 (2014). https://doi.org/10.1186/1687-1812-2014-55