Fredholm boundary value problems with a singular perturbation
The authors consider the singularly perturbed system \[ \varepsilon \dot x = Ax+\varepsilon A_1(t)x + \varphi(t), \quad t\in [a,b],\;0<\varepsilon\ll 1,\tag{1} \] with the boundary conditions \[ lx(\cdot) = h,\quad h\in \mathbb{R}^m, \tag{2} \] where \(A \) is a constant \(n\times n\)-matrix, \(A_1(t) \) is an \(n\times n\)-matrix, \(\varphi(t) \) is an \(n \)-vector function, and \(l \) is a linear bounded functional. They assume that the matrix \(A \) has \(k \) zero eigenvalues \((k<n) \), all other eigenvalues have negative real parts. The degenerate system \[ Ax + \varphi(t) =0 \] has a solution \(x_0 \). The aim is to derive conditions for the existence of a solution to (1), (2) and find an asymptotic representation for it. The solution \(x(t,\varepsilon) \) tends to the solution of the degenerate problem, \(x_0 \), as \(\varepsilon \to 0 \).
scientific article; zbMATH DE number 1915928