Fixed point theorems for contractions of Reich type on a metric space with a graph

In a metric space with a directed graph G, Jachymski (Proc Am Math Soc 1(136):1359–1373, 2008) introduced the concept of Banach G-contraction and proved two fixed point theorems for such mappings. Bojor (Nonlinear Anal 75:3895–3901, 2012) generalized this concept to Reich G-contraction and obtain a fixed point theorem. Note that Bojor’s theorem is established under the additional type of connectedness of G and it does not include Jachymski’s results as a special case. Moreover, there are some mistakes in several corollaries. Some examples and counterexamples are illustrated. It is our purpose to improve Bojor’s theorem and to present two fixed point theorems for Reich G-contractions. Our results are extensions of the two Jachymski’s theorems. Finally, we also discuss some priori error estimates.