Nonlinear boundary-value problems unsolved with respect to the derivative (Q2218766)

The authors consider the boundary value problem: \[ z' = A(t)z + f(t) + \epsilon Z(z,z',t,\epsilon), \\ \ell z(\cdot,\epsilon) = \alpha + \epsilon J(z(\cdot,\epsilon),z'(\cdot,\epsilon),\epsilon), \] where \(A : [a,b] \to {\mathbb R}^{n\times n}\), \(f : [a,b] \to {\mathbb R}^n\), \(Z\) and \(J\) have certain regularity properties, \(\ell : C([a,b]) \to {\mathbb R}^{m}\) is a linear bounded functional, \(\alpha \in {\mathbb R}^m\). They obtain necessary and sufficient conditions for the existence of a solution of the problem. Also they suggest convergent iterative schemes for finding an approximate solutions. The results are illustrated by an example describing the motion of satellites on elliptic orbits.