11. (15 points) As shown in Figure 2, in the rectangular prism $A B C D-$ $A_{1} B_{1} C_{1} D_{1}$, it is known that $A D=1, A B=2, A A_{1}=c$. If there exists a point $P$ on the diagonal $B D_{1}$ such that $P B_{1} \perp P C_{1}$, find the range of values for $c$.
II. 11. Taking $D$ as the origin, and $\overline{D A}, \overline{D C}, \overline{D D_{1}}$ as the positive directions of the $x, y, z$ axes, respectively, we establish a spatial rectangular coordinate system. Thus,
$$
B(1,2,0), B_{1}(1,2, c), C_{1}(0,2, c), D_{1}(0,0, c) \text {. }
$$

Let $\overrightarrow{D_{1} P}=\lambda \overrightarrow{D_{1} B}=(\lambda, 2 \lambda,-c \lambda)$. Then
$$
\begin{array}{l}
\overrightarrow{P C_{1}}=(-\lambda, 2-2 \lambda, c \lambda), \\
\overrightarrow{P B_{1}}=(1-\lambda, 2-2 \lambda, c \lambda) .
\end{array}
$$

By the problem statement,
$$
\begin{array}{l}
\overrightarrow{P C_{1}} \cdot \overrightarrow{P B_{1}}=-\lambda(1-\lambda)+(2-2 \lambda)^{2}+c^{2} \lambda^{2} \\
=\left(c^{2}+5\right) \lambda^{2}-9 \lambda+4=0 .
\end{array}
$$

Thus, $\Delta=81-16\left(c^{2}+5\right)=1-16 c^{2} \geqslant 0$
$$
\Rightarrow 0<c \leqslant \frac{1}{4} \text {. }
$$