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

In this paper we extend Mizoguchi–Takahashiʹs fixed point theorem for multi-valued mappings on a metric space endowed with a graph. As an application, we establish a fixed point theorem on an ε-chainable metric space for mappings satisfying Mizoguchi–Takahashi contractive condition uniformly locally. Also, we establish a result on the convergence of successive approximations for certain operators (not necessarily linear) on a Banach space as another application. Consequently, this result yields the Kelisky–Rivlin theorem on iterates of the Bernstein operators on the space C [ 0 , 1 ] and also enables us study the asymptotic behaviour of iterates of some nonlinear Bernstein type operators on C [ 0 , 1 ] .