Bifurcation of solutions of singular Fredholm boundary value problems (Q647674)

Under the assumption that the unperturbed singular differential system can be reduced to a canonical form, the authors give conditions for the bifurcation of solutions of the boundary value problem \[ B(t){{dx}\over{dt}}=A(t)x+\varepsilon A_1(t)x+f(t),\;\;t\in[a,b], \] \[ lx(.)=\alpha+\varepsilon l_1x, \] where \(\varepsilon\) is a small parameter, \(A(t), B(t)\) and \(A_1(t)\) are \(n\times n\) matrices, \(f(t)\) is a column \(n\)-vector whose components are real functions which are sufficiently many times (depending on the rang of B) continuously differentiable, \(\alpha\) is a column \(m\)-vector of constants, \(l=\text{col}(l^1,\dots,l^m):C[a,b]\to \mathbb{R}^m\) and \(l_1=\text{col}(l_1^1,\dots,l_m^1):C[a,b]\to \mathbb{R}^m,\) \(l^i,l_i^1:C[a,b]\to R,\) are linear vector functionals. This problem is singular since \(\text{det}B(t)=0\) for all \(t\in[a,b].\) It is suggested an algorithm for finding a family of linearly independent solutions. Moreover, the Vishik-Lyusternik method and the technique of Moore-Penrose pseudoinverse matrices are applied.