On the other hand, $T$ is demicompact, which is a consequence of the fact that $T$ is Lipschitzian, hence continuous, and of the generalized pseudocontractivity property, according to Lemma 2 in [@Pet], where are presented some classeses of demicontractive operators.

Now the conclusion of the corollary follows by Theorem \[th1\].

It is important to note that the uniqueness of the fixed point is in this case a direct consequence of the generalized pseudocontractivity property. Indeed, suppose that $T$ possesses two distinct fixed points, i.e., $p,q\in Fix\,(T)$ and $p\neq q$. Then, by , we get $$\langle x-y, x-y\rangle \leq  r \|x-y\|^2 \Leftrightarrow  \|x-y\|^2\leq r \|x-y\|^2,$$ a contradiction, since $r<1$.

We close this section by reminding that, under the assumptions of Corollary \[cor2\], if additionally, one has $$r\leq s,$$ where $r$ and $s$ are the constant of generalised pseudocontractivity and Lipschitz constant, respectively, then it is possible to identify the fastest Krasnoselskij iterations amongst those given by , see [@Ber02], [@Ber02a] and [@Ber07], which is obtained for $$\lambda=\frac{1-r}{1-2 r+s^2}.$$ This result has been at the origin of introducing a notion of rate of convergence for comparing two fixed point iterations in [@Ber04a], a concept that turn out to be very useful, see 