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Let M be a complete K -metric space with n-dimensional metric ρ(x, y) : M×M → R , where K is the cone of nonnegative vectors in R . 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)−1ρ(Fa, a) for every point a in M . The point x∗ can be obtained by the successive approximation method xk = Fxk−1 , k = 1, 2, . . . , starting from an arbitrary point x0 in M , and the following error estimates hold: ρ(x∗, xk) Q(I −Q)−1ρ(x1, x0) (I −Q)−1Qkρ(x1, x0), k = 1, 2, . . . . Generally, the mappings (I−Q)−1 and Q do not commute. For n = 1, the result is close to M. A. Krasnosel′skii’s generalized contraction principle.

Многомерная версия принципа обобщенного сжатия М. А. Красносельского