Optimality and existence for Lipschitz equations (Q1107689)

The paper deals with the nth order differential equation \((1)\quad x^{(n)}=f(t,y,...,y^{(n-1)}),\) where f is continuous and Lipschitzian \((\quad (2)\quad | f(t,y_ 1,...,y_ n)-f(t,z_ 1,...,z_ n)| \leq \sum^{n}_{i=1}k_ k| y_ i-z_ i|)\) on \((a,b)\times R^ n\). Let [n/2]\(\leq h\leq n\) and let for any \(\ell\), \(0\leq \ell \leq h\), \(\gamma_{\ell}\) denotes the smallest positive number such that there exists a solution x(t) of the boundary value problem \(x^{(n)}=(- 1)^{h-\ell}[k_ ix+\sum^{n}_{i=1}k_ i| x^{(i- 1)}|],\)x\({}^{(i)}(0)=0\) \((0\leq i\leq n-h+\ell -1)\), \(x^{(i)}(\gamma_{\ell})=0\) (\(\ell \leq i\leq h-1)\) with \(x(t)>0\) on \((0,\gamma_{\ell})\) \((\gamma_{\ell}=\infty\) if no such solution exists). The author shows that if \(\gamma =\min \{\gamma_{\ell}\); \(\ell =0,1,...,h-1\}\), then the boundary value problem (1,1), \(y^{(i)}(t_ 1)=y_{i+1}\) \((0\leq i\leq n-h+\ell -1)\), \(y^{(i)}(t_ 2)=y_{n- h+i+1}\) (\(\ell \leq i\leq h-1)\), where \(a<t_ 1<t_ 2<b\), \(0\leq \ell \leq h\), has a unique solution for any \((y_ 1,y_ 2,...,y_ n)\in R^ n\) provided that \(t_ 2-t_ 1<\gamma\) and that this result is best possible for the class of all differential equations (1) that satisfy (2). A similar assertion concerning more general (multipoint) boundary value problems is stated without proof. (The proof is said to be analogous to the proof of the above assertion.)