Ran-Reurings fixed point theorem is an immediate consequence of the Banach contraction principle

Banach fixed point theorem and applications

A Discrete Fixed Point Theorem of Eilenberg as a Particular Case of the Contraction Principle

Theorem 1.1 (Eilenberg). Let X be an abstract set and let (Rn)n=0 be a sequence of equivalence relations in X such that (i) X ×X = R0 ⊇ R1 ⊇ ··· ; (ii) ⋂∞ n=0Rn = ∆, the diagonal in X ×X ; (iii) given a sequence (xn)n=0 such that (xn,xn+1) ∈ Rn for all n ∈ N0, there is an x ∈ X such that (xn,x) ∈ Rn for all n∈ N0. If F is a self-map of X such that given n∈ N0 and x, y ∈ X , (x, y) ∈ Rn =⇒ (Fx,F...

Common Fixed-point Theorems for Nonlinear Weakly Contractive Mappings

The Banach contraction principle is one of the pivotal results in the metric fixed-point theory. It is a popular tool for the solution of existence problems in various fields of mathematics. There are several generalizations of the Banach contraction principle in the related literature on the metric fixed-point theory. Ran and Reurings [15] extended the Banach contraction principle in partially...

A Fixed Point Theorem for Iterative Random Contraction Operators over Banach Spaces

Consider a contraction operator T over a Banach space X with a fixed point x. Assume that one can approximate the operator T by a random operator T̂ using N ∈ N independent and identically distributed samples of a random variable. Consider the sequence (X̂ k )k∈N, which is generated by X̂ N k+1 = T̂ (X̂ k ) and is a random sequence. In this paper, we prove that under certain conditions on the random...

A generalized metric space and related fixed point theorems

We introduce a new concept of generalized metric spaces for which we extend some well-known fixed point results including Banach contraction principle, Ćirić’s fixed point theorem, a fixed point result due to Ran and Reurings, and a fixed point result due to Nieto and Rodríguez-López. This new concept of generalized metric spaces recover various topological spaces including standard metric spac...