Boundary value problems for \(n\)th order Lipschitz equations
It is assumed that \(f:(a,b)\times R^ n\to R\) is continuous and satisfies a Lipschitz condition \(| f(t,y_ 1,...,y_ n)-f(t,z_ 1,...,z_ n)| \leq \sum^{n}_{i=1}k_ i| y_ i-z_ i|\) on \((a,b)\times R^ n\). Then, optimal length subintervals (\(\alpha\),\(\beta)\) of (a,b) are determined, in terms of the Lipschitz coefficients, on which solutions are unique for the boundary value problems, (1) \(y^{(n)}=f(t,y,y',...,y^{(n-1)}),\) \(y^{(i)}(t_ i')=y_{i+1},\) \(0\leq i\leq n-h+k-1\), \(y^{(i)}(t_ i)=y_{n-h+i+1},\) \(k\leq i\leq h-1\) where \(\alpha <t_ i'<t_ k\leq...\leq t_{h-1}<\beta\), \(0\leq k<h\leq n\), and \(y_ i\in R\), \(1\leq i\leq n\). The techniques employed involve a linearization of (1) and an application of the Pontryagin maximum principle to a related class of two-point boundary value problems.
scientific article; zbMATH DE number 4077731
scientific article