Semilinear equations at resonance with the kernel of arbitrary dimension (Q1283234)

The paper deals with a nonlinear system of first-order functional-differential equations of mixed type \[ \begin{aligned} \dot x_i(t) & =B_ix_i(t)+ F_i\bigl(t,x(t+ \cdot),y(t+ \cdot)\bigr) +p_i(t),\;i=1, \dots,n_1,\\ \dot y_j(t) & =f_j \bigl(t,x (t+\cdot), y(t+\cdot) \bigr)+E_j(t),\;j=1,\dots, n_2,\tag{1}\end{aligned} \] where \(n_1,n_2\) are nonnegative integers with \(n_1+n_2\geq 1\), \(x_i(t)\in \mathbb{R}^2\), \(y_j(t)\in \mathbb{R}\), \(B_i\in\mathbb{R}^4\). Further, \(p_i,E_j, F_i, f_j\) are supposed to be \(2\pi\)-periodic in \(t\), continuous and bounded and \[ B_i=\left( \begin{matrix} 0 & m_i\\ -m_i & 0\end{matrix} \right),\quad i=1,\dots, n_1, \] where \(m_i\) are some positive integers. So, the dimension of the kernel of the linear operator associated to (1) with the \(2\pi\)-periodic conditions is equal to \(2n_1+n_2\). The main result are sufficient conditions for the existence of a \(2\pi\)-periodic solution to (1). The proofs are based on coincidence topological degree theory, particularly on the Borsuk theorem. Some examples are presented.