Nagumo conditions for systems of second-order differential equations (Q1083593)

The existence of solutions for the boundary value problem \[ (1)\quad x''(t)+f(t,x(t),x'(t))=0,\quad t\in [a,b],\quad x(a)=A,\quad x(b)=B \] where x takes values in a Hilbert space H and f:[a,b]\(\times H\times H\times H\to H\) is a compact mapping, is studied. By transforming the problem (1) into a fixed point problem for an operator \(T:{\bar \Omega}_{\phi,\rho}\to C^ 1([a,b],H),\) where \[ \Omega_{\phi,\rho}=\{x\in C^ 1([a,b],H);\quad | x(t)| <\phi (t),\quad \| x'\| <\rho \} \] the author obtains an a priori estimate for \(| x'(t)|\), when x(t) has a known bound. For this aim, the author proves two results in the Nagumo's conditions on the \(<x,x''>\), \(<x',x''>.\) Theorem 1 is a generalization of the results obtained by Hartman, Opial, Schmidt and Thomson. The results established by Theorem 2 are strongly connected to those obtained by Lasota, Yorke, Fabry and Habets. The main result is the proof of the existence of the solutions of problem (1) (Theorem 3). Also the restrictive conditions, but easy to verify, for the Theorem 3, are indicated.