A Note on Generalized Contraction Theorems

This work is devoted to estimates of the fixed point of generalized contracting (in the sense of Browder's and Krasnoselskii's definitions) operators $ G $ in a complete metric space $ (X, \rho)$. Upper and lower bounds for the distance $ \rho (x_0, \xi) $ between an arbitrary $ x_0 \in X $ and a fixed point $ \xi $ of the operator $ G $ are obtained. In the case of an “ordinary” $ q $-contraction ($ 0 \le q <1 $), the upper bound obtained in this work yields the inequality
$$ \rho (x_0, \xi) \le{(1-q)} ^{-1}{\rho (x_0, G (x_0))} $$
from Banach's theorem, while the lower bound yields the inequality
$$ \rho (x_0, \xi) \ge{(1 + q)} ^{-1}{\rho (x_0, G (x_0))}. $$
Also, for a generalized contraction operator, we obtain estimates of the distance $ \rho (x_0, x_i) $ from $ x_0 $ to the $ i $th the iteration $ x_i $ (defined by the recurrence relation $ x_j = G (x_{j-1})$, $ j = 1, \dots, i $). Using the obtained estimates, we prove a fixed-point theorem for an operator satisfying a local generalized contraction condition.