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Coupled coincidence point and common coupled fixed point theorems lacking the mixed monotone property

Ravi P Agarwal12, Wutiphol Sintunavarat3* and Poom Kumam3*

Author Affiliations

1 Department of Mathematics, Texas A&M University Kingsville, Kingsville, TX, 78363, USA

2 Department of Mathematics, Faculty of Science, King Abdulaziz University, Jeddah, 21589, Saudi Arabia

3 Department of Mathematics, Faculty of Science, King Mongkut’s University of Technology Thonburi (KMUTT), Bang Mod, Thrung Kru, Bangkok, 10140, Thailand

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Fixed Point Theory and Applications 2013, 2013:22  doi:10.1186/1687-1812-2013-22

Published: 31 January 2013

Abstract

In this paper, we prove the coupled coincidence point theorems for a <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/22/mathml/M2','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/22/mathml/M2">View MathML</a>-compatible mapping in partially ordered cone metric spaces over a solid cone without the mixed g-monotone property. In the case of a totally ordered space, these results are automatically obvious under the assumption given. Therefore, these results can be applied in a much wider class of problems. We also prove the uniqueness of a common coupled fixed point in this setup and give some example which is not applied to the existence of a common coupled fixed point by using the mixed g-monotone property but can be applied to our results.

MSC: 47H10, 54H25.

Keywords:
cone metric spaces; common coupled fixed point; coupled coincidence point; <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/22/mathml/M2','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/22/mathml/M2">View MathML</a>-compatible mappings; mixed g-monotone property