A Hartmann-Nagumo type condition for a class of contractible domains (Q378463)

The authors consider Dirichlet and periodic boundary value problems associated to a system of ODEs of the form NEWLINE\[NEWLINEx''=f(t,x,x'),NEWLINE\]NEWLINE where \(f:[0,1]\times \mathbb R^n \times\mathbb R^n\to \mathbb R^n\) and \(\langle f(t,x,y),n_x\rangle > {\mathbf I}_x(y)\) for all \((t,x,y) \in I \times T\partial D.\) Here, \(D\) is a subset of \({\mathbb R}^n\) with \(C^2\)-boundary, \({\mathbf I}_x(y)\) is the second fundamental form of \(\partial D\) and \(n_x\) is the outer-pointing normal unit vector field. This condition generalizes the one introduced in the 60s by Hartman in the particular case \(D=B(0,R).\) A first result guarantees the existence of at least one solution to the Dirichlet problem associated to the given equation under the assumption that \(|f(t,x,y)| \leq \gamma|y|^2 +C,\) for every \(x \in \bar D,\) where \(C,\gamma\) are constants s.t. \(\gamma R_D <1\) and \(R_D\) is the radius of the smallest ball containing \(D\). In a second result (assuming that \(f\) is \(T\)-periodic in the first variable) it is proved the existence of at least one solution to the periodic problem associated to the given equation. In this second result, besides the above descrived condition of Nagumo-Hartman type, the function \(f\) may satisfy a different technical condition which generalizes similar assumptions available in the literature. The proofs are performed by applying the Leray-Schauder continuation principle.