**Chengjun Guo, Donal O'Regan, Chengjiang Wang, and Ravi P. Agarwal**

##
Periodic Solutions of Superquadratic Nonautonomous Differential Systems with a
Delay

**abstract:**

The nonautonomous delay differential system

$$ x^{\prime}(t)=f(t, x(t-\tau)), $$

is considered, where $\tau>0$, $f:R\times R^n\to R^n$ is a continuous vector
function such that

$$ f(t+4\tau,x)=f(t,x), \quad f(t,x)=\nabla_xF(t,x). $$

Using the critical point theory, the conditions ensuring the existence of a
nontrivial $4\tau$-periodic solution of that system are established in the case,
where $F(t,x)$ is superquadratic in $x$.

**Mathematics Subject Classification:**
34K13, 34K18, 58E50

**Key words and phrases:** Delay differential equations, critical point
theory, linking theorem, superquadratic growth condition