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Abstract and Applied Analysis
Volume 2014 (2014), Article ID 342687, 8 pages
Research Article

On Sufficient Conditions for the Existence of Past-Present-Future Dependent Fixed Point in the Razumikhin Class and Application

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 17 September 2013; Revised 19 December 2013; Accepted 20 December 2013; Published 5 February 2014

Academic Editor: Hichem Ben-El-Mechaiekh

Copyright © 2014 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.


We introduce the new type of nonself mapping and study sufficient conditions for the existence of past-present-future (for short PPF) dependent fixed point for such mapping in the Razumikhin class. Also, we apply our result to prove the PPF dependent coincidence point theorems. Finally, we use PPF dependence techniques to obtain solution for a nonlinear integral problem with delay.