APPROXIMATE FIXED POINT THEOREMS FOR PARTIAL GENERALIZED CONVEX CONTRACTION MAPPINGS IN $\alpha$-COMPLETE METRIC SPACES

Abstract

In this paper, we introduce the new concept called partial generalized convex contractions and partial generalized convex contractions of order 2. Also, we establish some approximate fixed point theorems for such mappings in $\alpha$-complete metric spaces. Our results extend and unify the results of Miandaragh et al. [M. A. Miandaragh, M. Postolache, S. Rezapour, Approximate fixed points of generalized convex contractions, Fixed Point Theory and Applications 2013, 2013 :255] and several well-known results in literature. We give some examples of a nonlinear contraction mapping, which is not applied to the existence of approximate fixed point and fixed point by using the results of Miandaragh et al. We also consider approximate fixed point results in metric space endowed with an arbitrary binary relation and approximate fixed point results in metric space endowed with graph.