Multidimensional version of M. A. KrasnoseA3/4skii's generalized contraction principle

作者：Perov A I^*

来源：Functional Analysis and Its Applications, 2010, 44(1): 69-72.

DOI：10.1007/s10688-010-0008-z

• K-metric space
• semi-additive mapping
• Q-contraction
• contraction mapping principle

Let M be a complete K-metric space with n-dimensional metric rho(x, y): M x M -> R (n) , where K is the cone of nonnegative vectors in R (n) . A mapping F: M -> M is called a Q-contraction if rho (Fx,Fy) a (c) 1/2 Q rho (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 rho(x*,a) a (c) 1/2 (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,..., starting from an arbitrary point x (0) in M, and the following error estimates hold: rho (x*, x (k) ) a (c) 1/2 Q (k) (I - Q)(-1)rho(x (1), (x) (0)) a (c) 1/2 (I - Q)(-1) Q (k) rho(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.

• 出版日期2010-3