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Abstract and Applied Analysis
Volume 2013 (2013), Article ID 375604, 8 pages
http://dx.doi.org/10.1155/2013/375604
Research Article

The Existence Theorems of an Optimal Approximate Solution for Generalized Proximal Contraction Mappings

1Department of Mathematics and Statistics, Faculty of Science and Technology, Thammasat University Rangsit Center, Pathumthani 12121, Thailand
2Department of Mathematics, Faculty of Science, King Mongkut's University of Technology Thonburi (KMUTT), Bangkok 10140, Thailand

Received 3 June 2013; Accepted 31 July 2013

Academic Editor: Calogero Vetro

Copyright © 2013 Wutiphol Sintunavarat and Poom Kumam. 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

Recently, Basha (2011) established the best proximity point theorems for proximal contractions of the first and second kinds which are extension of Banach's contraction principle in the case of non-self-mappings. The aim of this paper is to extend and generalize the notions of proximal contractions of the first and second kinds which are more general than the notion of self-contractions, establish the existence of an optimal approximate solution theorems for these non-self-mappings, and also give examples to validate our main results.