1. Let $f(x)$ be a decreasing function on $\mathbf{R}$, for any $x \in \mathbf{R}$, we have
$$
f(x+2013)=2013 f(x) .
$$

Then a function that satisfies this condition is $\qquad$
1. $-2013^{\frac{x}{2013}}$ (the answer is not unique).

Consider the function $f(x)=-a^{x}(a>1)$.
By the problem, we have
$$
\begin{array}{l}
-a^{x+2013}=-2013 a^{x} \\
\Rightarrow a^{2013}=2013 \Rightarrow a=2013^{\frac{1}{2013}} .
\end{array}
$$

Thus, $f(x)=-2013^{\frac{x}{2013}}$ meets the condition.
Moreover, for any negative number $k, f(x)=2013^{\frac{x}{2013}} k$ meets the condition.