The authors prove existence and uniqueness of periodic solutions to the even-order differential equation in \(\mathbb{R}^N\) \[ u^{(2k)} (t)+\sum^k_{j=1}A_j u^{(2j-1)}(t)+(-1)^{k-1}\nabla G(u,t)=f(t) \] under the assumptions (1) \(A_1, A_2,\dots,A_k\) are symmetric and there exist symmetric matrices \(B_1\) and \(B_2\) and orthogonal matrices \(P_1\) and \(P_2\) such that \(P^T_1B_1P_1\), \(P^T_2 B_2P_2\) and \(P^T_1A_jP_2\), \(j=1,2,\dots,k\), are diagonal, (2) \(G\) is continuous in \((u,t)\) and twice continuously differentiable in \(u\), (3) \(f\) is continuous, (4) \(B_1+\alpha(\|u\|)I\leq \nabla^2 G(u,t)\leq B_2-\beta (\|u\|)I\) where \(\alpha,\beta\) are positive continuous functions such that \(\int^\infty_1\min \{\alpha(s),\beta(s)\}ds=\infty\), (5) the eigenvalues of \(B_1\) and \(B_2\) are \(N_i^{2k}\) and \((N_i+1)^{2k}\), \(i=1,2,\dots,n\), where \(N_i\) represents a nonnegative integer. Several corollaries are stated and examples are given to which this result can be applied but other theorems from the literature cannot.