Generalized two point boundary value problems. Existence and uniqueness (Q1199328)

The boundary value problem \(P(t)y'+Q(t)y=f(t)\), \(a\leq t\leq b\), \(My(a)+Ny(b)=d\) where \(P,Q\in[L_ p(a,b)]^{m\times n}\), \(f\in[L_ p(a,b)]^ n\) for some \(p\), \(1\leq p\leq\infty\); \(M,N\in R^{n\times m}\), \(d\in R^ n\), is considered. If this problem is non invertible an algorithm is presented for finding the pseudo-inverse of a rectangular matrix. Using these pseudo-inverses existence and uniqueness of solutions are established. The case of multipoint boundary value problem is considered too. Two illustrative examples are given.