On solvability of boundary value problems for systems of differential equations (Q1097387)

In some recent papers H. W. Knobloch and L. W. Erbe have established existence and uniqueness theorems for the nonlinear boundary value problem \[ (1)\quad x''=g(t,x),\quad 0\leq t\leq 1,\quad (2)\quad x(0)=x_ 0,\quad x(1)=x_ 1, \] where x, g(t,x) are n-dimensional column vectors. The authors are interested in solutions to the differential equation system (1) satisfying periodic-like boundary conditions of the form \[ (3)\quad x(0)=Qx(1),\quad x'(0)=Qx'(1) \] or \[ (4)\quad B_ 1x(0)-B_ 2x'(0)=0,\quad C_ 1x(1)+C_ 2x'(1)=0 \] and which also stay in a certain region \(\Omega\) in (t,x)-space. Here Q, \(B_ i\), \(C_ i\), \(i=1,2\), denote \(n\times n\) matrices with Q nonsingular and \(B_ 2\), \(C_ 2\) positive definite. Different conditions in terms of the Jacobian matrix \(g_ x(t,x)\) and an auxiliary positive definite symmetric matrix P(t) are given which yield the existence of at least one solution to the above boundary value problem (1), (3) or (1), (4).