Multidimensional version of M. A. Krasnoseľskii’s generalized contraction principle

Abstract

Let M be a complete K-metric space with n-dimensional metric ρ(x, y): M × M → Rⁿ, where K is the cone of nonnegative vectors in Rⁿ. A mapping F: M → M is called a Q-contraction if ρ (Fx,Fy) ⩽ Qρ (x,y), where Q: K → K is a semi-additive absolutely stable mapping. A Q-contraction always has a unique fixed point x* in M, and ρ(x*,a) ⩽ (I - Q)^-1 ρ(Fa, a) for every point a in M. The point x* can be obtained by the successive approximation method x _k = Fx _k-1, k = 1, 2,..., starting from an arbitrary point x ₀ in M, and the following error estimates hold: ρ (x*, x _k ) ⩽ Q ^k(I - Q)^-1ρ(x ₁, _x ⁰) ⩽ (I - Q)^-1 Q ^kρ(x ₁, x ₀), k = 1, 2,.... Generally the mappings (I - Q)^-1 and Q ^k do not commute. For n = 1, the result is close to M. A. Krasnosel’skii’s generalized contraction principle.