On Fixed Points of Generalized α-φ Contractive Type Mappings in Partial Metric Spaces

Abstract

Recently, Samet et al. (B. Samet, C. Vetro and P. Vetro, Fixed point theorem for $\alpha$-$\psi$ contractive type mappings, Nonlinear Anal. 75 (2012), 2154--2165) introduced a very interesting new category of contractive type mappings known as $\alpha$-$\psi$ contractive type mappings. The results obtained by Samet et al. generalize the existing fixed point results in the literature, in particular the Banach contraction principle. Further, Karapinar and Samet (E. Karapinar and B. Samet, Generalized $\alpha$-$\psi$-contractive type mappings and related fixed point theorems with applications, Abstract and Applied Analysis 2012 Article ID 793486, 17 pages doi:10.1155/2012/793486) generalized the $\alpha$-$\psi$ contractive type mappings and established some fixed point theorems for this generalized class of contractive mappings. In (G. S. Matthews, Partial metric topology, Ann. New York Acad. Sci. 728 (1994), 183--197), the author introduced and studied the concept of partial metric spaces, and obtained a Banach type fixed point theorem on complete partial metric spaces. In this paper, we establish the fixed point theorems for generalized $\alpha$-$\psi$ contractive mappings in the context of partial metric spaces. As consequences of our main results, we obtain fixed point theorems on partial metric spaces endowed with a partial order and that for cyclic contractive mappings. Our results extend and strengthen various known results. Some examples are also given to show that our generalization from metric spaces to partial metric spaces is real.