A variety of differentiability results for a multi-point boundary value problem

The goal of this work is to address differentiability of solutions of a nonlinear dynamic system with multi-point boundary conditions described byxΔ=f(t,x),∑m=1k Mmx(tm)=r,where r∈Rn and, for each 1⩽m⩽k, Mm is an n×n constant matrix and tm∈T. The solution of this system is defined on a measure chain, T, and xΔ denotes a generalized derivative known as the delta derivative.