On the equivalence between Perov fixed point theorem and 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...

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...

On the equivalence of the Arrow impossibility theorem and the Brouwer fixed point theorem

We will show that in the case where there are two individuals and three alternatives (or under the assumption of free-triple property) the Arrow impossibility theorem (Arrow (1963)) for social welfare functions that there exists no social welfare function which satisfies transitivity, Pareto principle, independence of irrelevant alternatives, and has no dictator is equivalent to the Brouwer fix...

Transversal spaces and common fixed point Theorem

In this paper we formulate and prove some xed and common xed pointTheorems for self-mappings dened on complete lower Transversal functionalprobabilistic spaces.