A Note on Generalized Contraction Theorems

Abstract

This work is devoted to estimates of the fixed point of a generalized contracting (in the sense of Browder’s and Krasnoselskii’s definitions) operator $G$ 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.