Existence of small periodic solutions to nonlinear systems of ordinary differential equations (Q2768828)

The author deals with the system NEWLINE\[NEWLINE\dot x=A(t,\lambda)x+F(t,x,\lambda)x,\tag{1}NEWLINE\]NEWLINE with \(\dim x=n\), \(A(t,\lambda)\) and \(F(t,x,\lambda)\) \(n\times n\)-matrix-valued \(\omega \)-periodic functions in \(t\) and \(\lambda \) an \(r\)-dimensional parameter. The problem is to establish sufficient conditions for the existence of a small \(\omega \)-periodic solution, i.e. a nontrivial \(\omega \)-periodic solution \(x(t,\lambda)\) depending on the parameter \(\lambda \) and vanishing together with \(\|\lambda \|\).NEWLINENEWLINENEWLINEThe case is studied where the fundamental matrix \(X(t)\) of the linear system \(\dot x=A_0(t)x\), \(A_0=A(t,0)\), contains \(n-k\) \(\omega \)-periodic columns. The author derives an equation for the initial value of the desired solution, represents this equation in a form convenient for local analysis and obtains a number of sufficient conditions for its solvability. The fulfillment of these conditions depends on properties of the matrix \(Q(\lambda)=X(\omega)\int_{0}^{\omega }X^{-1}(t)[A(t,\lambda)-A_0(t)]dt\) as well as of some nonlinearities generated by \(F(t,x,\lambda)\).