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Hindawi Publishing Corporation Fixed Point Theory and Applications Volume 2010, Article ID 234717,

Hindawi Publishing Corporation Fixed Point Theory and Applications Volume 2010, Article ID 234717, 15 pages doi:10.1155/2010/234717 Research Article On a Suzuki Type General Fixed Point Theorem with Applications S. L. Singh,1, 2 H. K. Pathak,1, 3 and S. N. Mishra1 1 2 Department of Mathematics, Walter Sisulu University, Mthatha 5117, South Africa 21 Govind Nagar, Rishikesh 249201, India 3 School of Studies in Mathematics, Pt. Ravishankar Shukla University, Raipur 492010, India Correspondence should be addressed to S. N. Mishra, smishra@wsu.ac.za Received 29 October 2010; Accepted 2 December 2010 Academic Editor: A. T. M. Lau Copyright q 2010 S. L. Singh et al. This is. | Hindawi Publishing Corporation Fixed Point Theory and Applications Volume 2010 Article ID 234717 15 pages doi 10.1155 2010 234717 Research Article On a Suzuki Type General Fixed Point Theorem with Applications S. L. Singh 1 2 H. K. Pathak 1 3 and S. N. Mishra1 1 Department of Mathematics Walter Sisulu University Mthatha 5117 South Africa 2 21 Govind Nagar Rishikesh 249201 India 3 School of Studies in Mathematics Pt. Ravishankar Shukla University Raipur 492010 India Correspondence should be addressed to S. N. Mishra smishra@wsu.ac.za Received 29 October 2010 Accepted 2 December 2010 Academic Editor A. T. M. Lau Copyright 2010 S. L. Singh et al. 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. The main result of this paper is a fixed-point theorem which extends numerous fixed point theorems for contractions on metric spaces and recently developed Suzuki type contractions. Applications to certain functional equations and variational inequalities are also discussed. 1. Introduction The classical Banach contraction theorem has numerous generalizations extensions and applications. In a comprehensive comparison of contractive conditions Rhoades 1 recognized that Ciric s quasicontraction 2 see condition C below is the most general condition for a self-map T of a metric space which ensures the existence of a unique fixed point. Pal and Maiti 3 proposed a set of conditions see PM.1 - PM.4 below as an extension of the principle of quasicontraction C under which T may have more than one fixed point see Example 2.7 below . Thus the condition C is independent of the conditions PM.1 - PM.4 see also Rhoades 4 page 42 . On the other hand Suzuki 5 recently obtained a remarkable generalization of the Banach contraction theorem which itself has been extended and generalized on various settings see e.g 6-15 . With a view of .