Let

$(X,d)$ be a metric space and

${\Delta}$ denote the diagonal of

$X\times X$. Let

$G$ be a directed graph such that the set of its vertices coincides with

$X$ and the set

$E\left(G\right)$ of its edges contains all loops, i.e.,

$E\left(G\right)\supseteq {\Delta}$. A map

$f:X\to X$ is called

$G$-contraction if it preserves the edges of

$G$, i.e.,

$(x,y)\in E\left(G\right)$ implies

$(fx,fy)\in E\left(G\right)$ and

$f$ decreases weights of edges of

$G$, i.e.,

$(x,y)\in E\left(G\right)$ implies

$d(fx,fy)\le \alpha d(x,y)$ for some

$\alpha \in (0,1)$. The author presents some fixed point results for

$G$-contractions, being a hybrid of the Banach and Knasterâ€“Tarski theorems and generalizing a number of known assertions. As an application, the convergence of successive approximations for some linear operators on Banach spaces is considered.