Erratum to: Some new theorems of expanding mappings without continuity in cone metric spaces
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CORREC TION
Open Access
Correction: Some new theorems of expanding mappings without continuity in cone metric spaces Shaoyuan Xu1* , Suyu Cheng2 and Yan Han3 *
Correspondence: [email protected] 1 Department of Mathematics and Statistics, Hanshan Normal University, Chaozhou, 521041, China Full list of author information is available at the end of the article
Abstract In this note we correct some errors that appeared in the article (Han and Xu in Fixed Point Theory Appl. 2013:3, 2013) by modifying some conditions in the main theorems and corresponding corollaries. MSC: 47H10; 54H25 Keywords: cone metric space; expanding mapping; common fixed point
Correction Upon critical examination of the main results and their proofs in [], we note several critical errors in the conditions of the main theorems in []. These errors lead to subsequent errors in the corresponding corollaries in []. In this note, we would like to supplement several conditions, which are used in their proofs but not referred to in the conditions of the main results, to achieve our claim. The following theorem is a modification to [, Theorem .]. The proof is the same as that in []. We will attain the desired goal by adding two conditions to that in [, Theorem .]. We state that Theorem . in [] is replaced by the following theorem. Theorem . Let (X, d) be a complete cone metric space. Suppose that the mapping f : X → X is onto and such that d(fx, fy) ≥ a d(x, y) + a d(x, fx) + a d(y, fy) + a d(x, fy) + a d(y, fx),
(.)
for all x, y ∈ X, where ai (i = , , , , ) satisfy a ≥ , a ≥ , a +a +a > and a ≤ +a . Then f has a fixed point. Remark . Compared to Theorem . in [], Theorem . mentioned above possesses the conditions a ≥ and a ≥ while, unluckily, Theorem . in [] does not. The reason for supplementing these conditions is the fact that in the proof of [, Theorem .] we have used the conditions a ≥ and a ≥ to ensure that the two deductions (i) d(xn+ , xn– ) ≥ d(xn+ , xn ) – d(xn– , xn ) implies a d(xn+ , xn– ) ≥ a d(xn+ , xn ) – a d(xn– , xn ) and © 2014 Xu et al.; licensee Springer. This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
Xu et al. Fixed Point Theory and Applications 2014, 2014:178 http://www.fixedpointtheoryandapplications.com/content/2014/1/178
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(ii) d(p, q) ≥ d(p, xn+ ) – d(q, xn– ) implies a d(p, q) ≥ a d(p, xn+ ) – a d(q, xn– ) must be valid. Similarly, the following theorem is a modification to [, Theorem .]. The proof is the same as that in []. We state that Theorem . in [] is replaced by the following theorem. Theorem . Let (X, d) be a complete cone metric space. Suppose the mappings f , g : X → X are onto and satisfy d(fx, gy) ≥ a d(x, y) + a d(x, fx) + a d(y, gy) + a d(x, gy) + a d(y, fx),
(.)
for all x, y ∈ X
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