Let us consider the operator $T$ defined by$$\eqalign{ & T:{L^2}((a,b) \times (c,d)) \to {L^2}((c,d)) \cr & Tf(s,x) \mapsto \int\limits_{q(x)}^{p(x)} {f(\alpha (s,x),s)ds} \cr} $$ where $c \leqslant q(x) \leqslant p(x) \leqslant d$ for all $x\in (c,d)$ and $\alpha$ has a range in $(a,b)$ fo all $(s,x)\in ((a,b) \times (c,d))$.
What are the assumptions on the functions $p$,$q$ and $\alpha$ to ensure that $T$ is onto. Thank you.