Multidimensional Version of M.~A.~Krasnosel'skii's Generalized Contraction Principle

A. I. Perov

Funkcionalʹnyj analiz i ego priloženiâ, Tome 44 (2010) no. 1, pp. 83-87

Résumé

Let $M$ be a complete $K$-metric space with $n$-dimensional metric $\rho (x,y)\colon M\times M\to\mathbb{R}^n$, where $K$ is the cone of nonnegative vectors in $\mathbb{R}^n$. A mapping $F\colon M\to M$ is called a $Q$-contraction if $\rho (Fx,Fy)\le Q\rho (x,y)$, where $Q\colon K\to K$ is a semi-additive absolutely stable mapping. A $Q$-contraction always has a unique fixed point $x^*$ in $M$, and $\rho (x^*,a)\le (I-Q)^{-1}\rho(Fa,a)$ for every point $a$ in $M$. The point $x^*$ can be obtained by the successive approximation method $x_k=Fx_{k-1}$, $k=1,2,\dots$, starting from an arbitrary point $x_0$ in $M$, and the following error estimates hold: $\rho(x^*,x_k)\le Q^k(I-Q)^{-1}\rho(x_1,x_0)\le (I-Q)^{-1}Q^k\rho(x_1,x_0)$, $k=1,2,\dots$ . 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.

Détail

Keywords: $K$-metric space, semi-additive mapping, contraction mapping principle.
Mots-clés : $Q$-contraction