Singularly perturbed ordinary differential equations (Q1206912)

Consider abstract nonlinear equations of the form (*) \(Lu+\varepsilon F(u,v)=0\), \(S(v)+\varepsilon V(u,v)=0\) where \(L\) is linear, \(\varepsilon\) is a small parameter. By using the coincidence degree the author derives conditions on the operators such that (*) has a solution. He applies the obtained abstract result to singularly perturbed periodically forced differential systems to establish the existence of a harmonic solution. The systems under consideration have the form \(Bx+x'=\varepsilon [G(t,x,x')x''+ h(t,x,y)]\), \(y'=f(t,y)+\varepsilon k(t,x,y)\), where \(G(t,x,x')=A(t)\), \(A\) symmetric, for \(\dim x>1\).