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Abstract and Applied Analysis
Volume 2013 (2013), Article ID 165434, 8 pages
http://dx.doi.org/10.1155/2013/165434
Research Article

The Existence of Fixed Point Theorems via -Distance and -Admissible Mappings and Applications

1Department of Mathematics, King Abdulaziz University, P.O. Box 80203, Jeddah 21589, Saudi Arabia
2Department of Mathematics and Statistics, Faculty of Science and Technology, Thammasat University, Rangsit Center, Pathumthani 12121, Thailand

Received 23 June 2013; Revised 10 September 2013; Accepted 11 September 2013

Academic Editor: Wei-Shih Du

Copyright © 2013 Marwan Amin Kutbi and Wutiphol Sintunavarat. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

We introduce the concept of the generalized -contraction mappings and establish the existence of fixed point theorem for such mappings by using the properties of -distance and -admissible mappings. We also apply our result to coincidence point and common fixed point theorems in metric spaces. Further, the fixed point theorems endowed with an arbitrary binary relation are also derived from our results. Our results generalize the result of Kutbi, 2013, and several results in the literature.