Some questions on the solubility as a whole of singularly perturbed nonlinear problems (Q1077614)

The problem being discussed here is the solution of the system \[ \epsilon z'=A(x)z+F_ 0(x,y)+\epsilon F_ 1(x,y,\epsilon),\quad z(0,\epsilon)=z^ 0,\quad y'=f(x,y,z),\quad y(0,\epsilon)=y^ 0. \] \(\epsilon\) is a positive small real number, \(x\in [0,a]\), z and y are m and a vectors consecutively. \(F_ 0\), \(F_ 1\), f are certain nonlinear functions satisfying many strong conditions such as that they take the form of special series. A(x) is a square matrix of order n whose spectrum may contain purely imaginary points. The author establishes that the existence of bounded, on the whole interval [0,a], solution as \(\epsilon \to 0^+\) of the above system is equivalent to the solubility of some Cauchy problem, obtained from the orthogonality conditions by the method of Lomov regularization, not containing a small parameter. In addition, he shows that the inverse is also true i.e., if there exists a bounded solution as \(\epsilon \to 0^+\) of the above system on the interval [0,a], then the system of equations generated by the orthogonality conditions of the method of regularization is also soluble.