We prove the existence of bounded and periodic solutions for planar systems by introducing a notion of lower and upper solutions which generalizes the classical one for scalar second order equations. The proof relies on phase plane analysis; after suitably modifying the nonlinearities, the Ważewski theory provides a solution which remains bounded in the future. For the periodic problem, the Massera Theorem applies, yielding the existence result. We then show how our result generalizes some well known theorems on the existence of bounded and of periodic solutions. Finally, we provide some corollaries on the existence of almost periodic solutions for scalar second order equations.

中文翻译：

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平面系统上下解决方案的动力学方法“致Massimo Tarallo的记忆”

我们通过引入下限和上限解的概念来证明平面系统的有界和周期解的存在，该概念推广了经典的标量二阶方程组。证明依赖于相平面分析；在适当地修改了非线性之后，沃夫斯基理论提供了将来仍然有局限性的解决方案。对于周期性问题，应用了Massera定理，得出了存在性结果。然后，我们展示了我们的结果如何推广关于有界和周期解的存在性的一些众所周知的定理。最后，我们提供了关于标量二阶方程的几乎周期解的存在性的一些推论。