  We consider delay differential equations of the form $
y'(t)=-ay(t)+bf(y(t-1)) $ with positive parameters $a,b$ and a unimodal
$f:[0,\infty)\to [0,1]$. It is assumed that the nonlinear $f$ is close to a
function $g:[0,\infty)\to [0,1]$ with $g(\xi)=0$ for all $\xi>1$. The fact
$g(\xi)=0$ for all $\xi>1$ allows to construct stable periodic orbits for the
equation $x'(t)=-cx(t)+dg(x(t-1))$ with some parameters $d>c>0$. Then it is
shown that the equation $ y'(t)=-ay(t)+bf(y(t-1)) $ also has a stable periodic
orbit provided $a,b,f$ are sufficiently close to $c,d,g$ in a certain sense.
The examples include $f(\xi)=\frac{\xi^k}{1+\xi^n}$ for parameters $k>0$ and
$n>0$ together with the discontinuous $g(\xi)=\xi^k$ for $\xi\in[0,1)$, and
$g(\xi)=0$ for $\xi>1$. The case $k=1$ is the famous Mackey--Glass equation,
the case $k>1$ appears in population models with Allee effect, and the case
$k\in(0,1)$ arises in some economic growth models. The obtained stable periodic
orbits may have complicated structures.
