Optimality for boundary value problems for Lipschitz equations (Q1825340)

The authors consider the boundary value problems (1) \(x^{(n)}=f(t,x',...,x^{(n-1)})\), (2) \(x^{(i-1)}(t_{j_ s(r)})=x_{i,j_ s(r)},\) \(r\leq i<r+n_ s(r)\), \(1\leq s\leq p(r)\), \(1\leq r\leq \ell\), where f is continuous on slab \((a,b)\times R^ n\) and satisfies the Lipschitz condition \((3)\quad | f(t,x_ 1,...,x_ n)- f(t,y_ 1,...,y_ n)| \leq \sum^{n}_{i=1}k_ i| x_ i-y_ i|\)on the slab, \(a<t_ 1\leq t_ 2\leq...\leq t_ n<b\), \(\tau =(\tau_ 1;...;\tau_{\ell})\) is an increasing partition of \((t_ 1,...,t_ n)\) with \(| \tau_ i| =m_ i\), \(1\leq i\leq \ell\), \(\sum^{\ell}_{i=1}m_ i=n\), the indices \(j_ 1(r)<...<j_{p(r)}(r)\) are such that the distinct components of \(\tau_ r\) are \(t_{j_ 1(r)}<...<t_{j_{p(r)}(r)};\) \(n_ s(r)=multiplicity\) of \(t_{j_ s(r)}\) in \(\tau_ r\), \(1\leq s\leq p(r)\); and \(x_{i,j_ s(r)}\in R\). They determine the optimal length of subintervals of (a,b), in terms of Lipschitz coefficients \(k_ i\), \(1\leq i\leq n\), on which solution of the boundary value problems (1) and (2) are unique.