On linear ODEs with a time singularity of the first kind and unsmooth inhomogeneity (Q481451)

The solvability of the boundary value problem \[ y'(t)=\frac{M}{t}y(t)+\frac{f(t)}{t},\;\;t\in(0,1], \] \[ B_0y(0)+B_1y(1)\beta \] is studied in this paper. Here \(M\in \mathbb R^{n\times n},\;B_0,B_1\in \mathbb R^{m\times n}, m\leq n, \beta\in \mathbb R^m,\) and \(f\in C[0,1]\) but \(f(t)/t\) may not be integrable on \([0,1].\) A result guaranteeing existence and uniqueness of a \(C[0,1]\)-solution is obtained as a sequence of results which infer that the considered differential equation has a unique solution satisfying the initial condition \(B_0y(0)=\beta\) or the terminal condition \(B_1y(1)=\beta.\) The results for the initial value problem are obtained under the assumption that all eigenvalues of \(M\) are zero or have negative real parts, while those for the terminal value problem are obtained under the assumtion that all eigenvalues of \(M\) are zero or have positive real parts. Collocation schemes are proposed also for the numerical treatment of the considered problems.