Fixed points of Mizoguchi–Takahashi contraction on a metric space with a graph and applications

On Reich fixed point theorem of G -contraction mappings on modular function spaces

“…Motivated by Sultana and Vetrivel example [17], we introduce the following Bernstein operator B n defined by ρ T (f (t)) − (1 − t)T (f (0)) − t T (f (1)) < +∞.…”

A graphical version of Reichs fixed point theorem

“…Moreover, we will define a vector valued Bernstein operator and give a more general version of the Kelisky and Rivlin's theorem. In particular, we will improve the conclusion of [26] regarding the Bernstein operator.…”

“…In their attempt to extend Mizoguchi-Takahashi's fixed point theorem for Reich multivalued contraction mappings to metric spaces endowed with a graph, Sultana and Vetrivel [26] introduced the concept of Reich G-contractions.…”

“…Mizoguchi and Takahashi [18] gave a partial answer to Reich' s problem. Following the publication of Ran and Reuring's fixed point theorem [20] seen as the Banach contraction principle in metric spaces endowed with a partial order, Sultana and Vetrivel [26] tried to extend the main results of [18] to metric spaces endowed with a graph. In particular, they used their ideas to discuss the iterate of the Bernstein operator.…”

“…In this paper, we will revisit the definition of the Reich multivalued mappings given by the authors in [26]. Then we will prove a fixed point theorem of these mappings.…”