Solutions of second order homogeneous Dirichlet systems (Q1375408)

The following homogeneous system \[ y''= f(t,y,y'),\;y(0) =y(1)=0, \tag{1} \] with \(f:[0,1] \times \mathbb{R}^m \times \mathbb{R}^m \to\mathbb{R}^m\) continuous, is considered. For different initial conditions for second order nonlinear systems of the form (1) five existence theorems and two uniqueness theorems are obtained. Proofs are given in detail. The important theorems are presented in seven examples.