On periodic solutions for the periodic quasilinear ordinary differential system containing a parameter (Q1107002)

This paper deals with the dependence on a parameter \(\lambda\) of \(T\)-periodic solutions for the \(T\)-periodic quasilinear ordinary differential system: \[ \dot x=A(t,x,\lambda)x+\lambda F(t,x,\lambda)+f(t), \] where \(A\) is a real \(n\times n\) matrix and \(F\) is an \(\mathbb R^n\)-valued function. It is assumed that \(A\) and \(F\) are defined on \(\mathbb R\times \mathbb R^n\times [-\lambda_ 0,+\lambda_ 0]\), continuous in \((t,x,\lambda)\) and \(T\)-periodic in \(t\), where \(\lambda_ 0>0\). It is assumed that \(f\) is an \(\mathbb R^n\)-valued function continuous on \(\mathbb R\) and \(T\)-periodic.