ABSTRACT
Let (X, d) be a complete metric space and f : X → X $ f:X\rightarrow X $ https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781351243377/a2fb70c4-ea85-4ffd-9347-9273d5aafc73/content/inline-math3_2.tif"/> be a mapping. If there exists a real number k ∈ [ 0 , 1 ) $ k\in [0,1) $ https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781351243377/a2fb70c4-ea85-4ffd-9347-9273d5aafc73/content/inline-math3_3.tif"/> such that for all x , y ∈ X $ x,y\in X $ https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781351243377/a2fb70c4-ea85-4ffd-9347-9273d5aafc73/content/inline-math3_4.tif"/> , the following inequality holds: d ( f x , f y ) ≤ k d ( x , y ) , $$ \begin{aligned} d(fx,fy) \le kd(x,y), \end{aligned} $$ https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781351243377/a2fb70c4-ea85-4ffd-9347-9273d5aafc73/content/math3_1.tif"/>
