Let $X$ be a Banach space and $K$ its non-empty subset. Let $S:K\to X$ be a single-valued mapping. Then $S$ satisfies the condition $(C_\lambda)$ if for $x,y\in K$ and $\lambda\in(0,1)$, $\lambda\|x-Sx\|\leq \|x-y\|\implies \|Sx-Sy\|\leq \|x-y\|$
I understand the second condition to mean $S$ is a nonexpansive mapping. But I don't understand this whole condition.
From the first condition, we can extrapolate and get $\lambda\|y-Sy\|\leq \|x-y\|$. Adding, we get $\lambda(\|x-Sx\|+\|y-Sy\|)\leq 2\|x-y\|$, or $\frac{\lambda}{2}(\|x-Sx\|+\|y-Sy\|)\leq \|x-y\|$. If this condition implies that $\|Sx-Sy\|\leq \|x-y\|$, then is the actual statement of this condition this: $$\|Sx-Sy\|\leq \frac{\lambda}{2}(\|x-Sx\|+\|y-Sy\|)$$
And if the actual meaning of the condition $C_\lambda$ is indeed this, why don't we write it this way?
Thanks in advance!
