An existence theorem for second order ordinary differential systems with nonlinear mixed boundary conditions
We consider the following differential system with mixed boundary conditions: \(\ddot x(t)=f(t,x(t),\dot x(t)),\quad t\in]0,1[,\quad x(0)=r_ 0(x(0),x(1),\dot x(0),\dot x(1)),\quad x(1)=r_ 1(x(0),x(1),\dot x(0),\dot x(1)),\) where the mapping f (resp.: \(r_ 0,r_ 1)\) is continuous from \([0,1]\times {\mathbb{R}}^ p\times {\mathbb{R}}^ p\) (resp.: \(({\mathbb{R}}^ p)^ 4)\) to \({\mathbb{R}}^ p\), \(p\geq 1\). Under Lipschitz conditions on f, \(r_ 0\) and \(r_ 1\), we prove the existence and uniqueness of a solution by a fixed point method. To do so, we study the T.P.B.V.P.: \[ \ddot x=f(t,x,\dot x),\quad x(0)=x_ 0,\quad x(1)=x_ 1, \] and the regularity in \(x_ 0\), \(x_ 1\) of its solution.
scientific article; zbMATH DE number 4040328