An inverse problem for a parabolic equation and fixed-point theorem

Question

I'm reading the paper "The determination of a parabolic equation from initial and final data" http://www.ams.org/journals/proc/1987-099-04/S0002-9939-1987-0877031-4/S0002-9939-1987-0877031-4.pdf. A couple of questions arise. Any idea is welcome, thanks for help!

1. The mapping $a\rightarrow \textbf{T}(a)$ is defined by $\textbf{T}(a)=a+\lambda (u(x,1;a)-g)$ (see (2.5) in the paper), where \begin{equation} a\in\mathcal{S}:= \big\{ h(x) \;\big|\; h\in C^{0+\alpha}(\bar{\Omega}), h(x)\geq0 \big\}. \end{equation} When a fixed-point argument is used to show that $\textbf{T}$ has a fixed point, is it necessary to have the fact, i.e. if $a_0\in\mathcal{S}$, then $T(a_0)\in\mathcal{S}$ too? My doubt arises from the fact that it seems that the sign of $\lambda (u(x,1;a)-g)$ cannot be determined. If we cannot make sure that whenever $a_0\in\mathcal{S}$, $T(a_0)\in\mathcal{S}$, the fixed-point argument still works for $a_0$ and $T(a_0)$ belonging to different function spaces?
2. In (2.13), i.e. (note: there is a typo, $\lambda$ is missing in (2.13)) \begin{equation} ||\textbf{T}'(a)\cdot h||_{\infty}=||h-\lambda\,\hat{u}||_{\infty}<||h||_{\infty} \end{equation} for any $a$, $h\in\mathcal{S}$, when a fixed-point theorem (e.g. Schauder) is used, do we have to prove (2.13) holds for any perturbation $h(x)$, not just for $h(x)\in\mathcal{S}$?
3. In (2.12), i.e. \begin{equation} \phi_t-\Delta \phi=h(x)\,||f||_{\infty}, \end{equation} could we show that $|\phi(x,t)|\le K$ for some constant $K>0$, where $K$ depends only on the initial condition $f$ and the domain $\Omega$, but does not depend on $h(x)$?
4. Due to (2.11), i.e. $0\le \hat{u}(x,t,a,h)\le \phi(x,t)$, if we can show that $|\phi(x,t)|\le K$, where $K$ depends only on the initial condition $f$ and the domain $\Omega$, then can we choose $\lambda>0$ sufficiently small such that $||\textbf{T}'(a)\cdot h||_{\infty}=||h-\lambda\,\hat{u}||_{\infty}<||h||_{\infty}$, where $\hat{u}$ is defined by $\hat{u}=\hat{u}(x,t,a,h)=u(x,t,a)-u(x,t,a+h)$? Notice that here $h(x)\ge0$, so $h(x)\ge0$ could be very small.
5. Even we can show that $||\textbf{T}'(a)\cdot h||_{\infty}=||h-\lambda\,\hat{u}||_{\infty}<||h||_{\infty}$ is true, does this inequality guarantees that $\textbf{T}$ has a fixed point? On the other hand, the inequality $||\textbf{T}'(a)\cdot h||_{\infty}\le\mu\,||h||_{\infty}$ for some constant $\mu>0$ is necessary for a fixed point theorem (e.g. Schauder) to be applicable here?