A Resonance Problem for a Second-Order Vector Differential Equation

Abstract

We shall consider the boundary value problem (BVP)

$$ \left\{ \begin{gathered} \left( {\begin{array}{*{20}c} {x_1 ^{\prime \prime } } \\ {x_2 ^{\prime \prime } } \\ \end{array} } \right) + A\left( {\begin{array}{*{20}c} {x_1 } \\ {x_2 } \\ \end{array} } \right) = \left( {\begin{array}{*{20}c} {f_1 (ax_1 + bx_2 )} \\ {f_2 (cx_1 + dx_2 )} \\ \end{array} } \right) + \left( {\begin{array}{*{20}c} {b_1 (t)} \\ {b_2 (t)} \\ \end{array} } \right) \hfill \\ x_1 (0) = x_2 (0) = x_1 (\pi ) = x_2 (\pi ) = 0. \hfill \\ \end{gathered} \right., $$

(35.1)

Here f ₁, f ₂, b ₁ and b ₂ are continuous and bounded and a, b, c and d are real numbers. The matrix A is diagonalizable and such that the corresponding homogeneous problem has nontrivial solutions, i.e., we have a BVP at resonance. Problems of this type arise in mechanics (coupled oscillators) or in coupled circuits theory [1].