The periodic solutions of a nonlinear nonautonomous second-order system of equations (Q700026)

The authors study periodic solutions to the boundary value problem \[ \frac{d^2x}{dt^2}=f(x,y)+e_1(t),\;x(0)=x(\omega),\quad \frac{d^2y}{dt^2}=g(x,y)+e_2(t),\;x(0)=x(\omega), \tag{*} \] where \(e_1(t)\), \(e_2(t)\) are \(\omega\)-periodic functions of \(t\in\mathbb{R}\) such that \(\int_{0}^{\omega}e_i(t) dt=0\), \(i=1,2\). It is supposed that there are constants \(b\geq 0\) and \(D\) such that, for \(|x|\geq{}b\), \(|y|\geq{}b\), and ``all \(t\in[0,\omega]\)'', \(xf(x,y)\geq 0\), \(yg(x,y)\geq 0\) and that \[ \max\{\omega^2(\max_{|x|\leq{}D,|y|\leq{}D}|f(x,y)|+\max_{0\leq{}t\leq\omega}|e_1(t)|), \omega^2(\max_{|x|\leq{}D,|y|\leq{}D}|g(x,y)|+\max_{0\leq{}t\leq\omega}|e_2(t)|) \}+b\leq{}D. \] No conditions on the smoothness of \(f\) or \(g\) are given. The main result is the existence of a nontrivial \(\omega\)-periodic solution to (*).