Periodic solutions to nonlinear nonautonomous system of differential equations (Q2399458)

The paper is devoted to the investigation of the periodic problem for a nonautonomous nonlinear system of differential equations of the following form \[ \dot{x}=(A+B(t,\lambda)+X(t,\varphi,x,\varepsilon,\lambda))x, \] \[ \dot{\varphi}=\mu(t,\varepsilon)+\Phi(t,\varphi,x,\varepsilon,\lambda), \] where \(x,\varphi\) are \(n\)-, \(p\)-dimensional vectors respectively; \(\varepsilon, \lambda\) are \(l\)-, \(q\)-dimensional vector parameters; \(A, B(t,\lambda), X(t,\varphi,x,\varepsilon,\lambda)\) are \((n\times n)\)-matrices; \(\mu(t,\varepsilon), \Phi(t,\varphi,x,\varepsilon,\lambda)\) are \(p\)-dimensional vector-functions. Applying the fixed point theory to a nonlinear operator defined on a topological product of two compact sets, the authors prove the existence of nonzero periodic solutions for the above system. Moreover, in the last section of the paper, a corresponding example is considered.