Periodic solutions of first order systems

Let $f\in C(%%TCIMACRO{\U{211d} }%%BeginExpansion\mathbb{R}%EndExpansion^{m},%%TCIMACRO{\U{211d} }%%BeginExpansion\mathbb{R}%EndExpansion^{m})$ and $p\in C([0,T],%%TCIMACRO{\U{211d} }%%BeginExpansion\mathbb{R}%EndExpansion^{m})$ be continuous functions. We consider the $T$ periodic boundary valueproblem (*) $u^{\prime}(t)=f(u(t))+p(t),$ $u(0)=u(T).$ It is shown that when$f$ is a coercive gradient function, or the bounded perturbation of a coercivegradient function, and the Brouwer degree $d_{B}(f,B(0,r),0)\neq0$ for large$r$, there is a solution for all $p.$ A result for bounded $f$ is also obtained.