A fixed point theorem for (\phi, \shi)-convex contraction in metric spaces

Abstract

In the present paper, we introduce the notion of (\phi, \shi)-convex contraction mapping of order m and establish a fixed point theorem for such mappings in complete metric spaces. The present result extends and generalizes the well known result of Dutta and Choudhary (Fixed Point Theory Appl. 2008 (2008), Art. ID 406368), Rhoades (Nonlinear Anal., 47(2001), 2683-2693), Istratescu (Ann. Mat. Pura Appl., 130(1982), 89-104) and besides many others in the existing literature. An illustrative example is also provided to exhibit the utility of our main results.