Fixed-Point Theorems

Abstract

We begin by the well-known Banach contraction principle. A mapping f: X → Y from a metric space (X, ρ ) into a metric space (Y, d) is said to be a contraction if there is a number 0 ≤ γ < 1 such that inequality $d\left( {f\left( x \right),f\left( {x'} \right)} \right) \leqslant \gamma \cdot \rho \left( {x,x'} \right)$ holds, for every pair of points x, x′ ∈ X. The Banach fixed-point theorem states that every contraction f: X → X of a complete metric space (X, ρ) into itself has a point x ∈ X such that f (x) = = x. Such a point x is called a fixed point of the mapping f. Moreover, if x = f (x) and x′ = f (x′), then

$$ d\left( {x,x'} \right) = d\left( {f\left( x \right),f\left( {'x} \right)} \right) \leqslant \gamma d\left( {x,x'} \right). $$