On nonlinear singular BVPs with nonsmooth data. Part 1: Analytical results

We study boundary value problems for systems of nonlinear ordinary differential equations with a time singularity,x′(t)=M(t)tx(t)+f(t,x(t))t,t∈(0,1],b(x(0),x(1))=0, where M:[0,1]→Rn×n and f:[0,1]×Rn→Rn are continuous matrix-valued and vector-valued functions, respectively. Moreover, b:Rn×Rn→Rn is a continuous nonlinear mapping which is specified according to a spectrum of the matrix M(0). For the case that M(0) has eigenvalues with nonzero real parts, we prove new results about existence of at least one continuous solution on the closed interval [0,1] including the singular point, t=0. We also formulate sufficient conditions for uniqueness. The theory is illustrated by a numerical simulation based on the collocation method.