Let $G$ be a compact Lie group and $\mathcal{C}\_G$ the category of $G$-spaces (ie. topological spaces endowed with continuous left $G$-actions). Is there a model category structure on $\mathcal{C}\_G$ for which


(i) weak equivalences are the morphisms $f:X\rightarrow Y$ such that for all closed subgroups $H$ of $G$, $f^{H}:X^H\rightarrow Y^H$ is a weak homotopy-equivalence, and


(ii) cofibrations are the morphisms $f:X\rightarrow Y$ that have the expected $G$-homotopy extension property?


What I have read thus far strongly suggests that this is the case. Nevertheless, I would appreciate a reference that makes this explicit. 



Is the category of $G$-spaces a model category?
Yes, assuming that by "expected $G$-homotopy extension property" you mean the Serre $G$-cofibrations, not the Hurewicz $G$-cofibrations. A very explicit reference is Proposition A.1.18 in Schwede's [Global homotopy theory](http://www.math.uni-bonn.de/~schwede/global.pdf).


