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<p>In this work, we present a result for the existence and localization of periodic solutions in impulsive second-order generalized systems, using a variant of the method of lower and upper solutions. The main advantage of our approach is the lack of requirements of periodicity for the nonlinearities and of order for the upper and lower solutions. We use Green’s functions to construct the operator of the system and a Nagumo-type condition for the control of the first derivatives. The existence is guaranteed by Shauder’s fixed-point theorem, and the localization is ensured using our method of orderless lower and upper solutions, involving translations. Only a few monotonicity criteria must be met by nonlinearities and impulse functions. We apply our result to a dissipative Liénard-type system with state-dependent impulses.</p>

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Existence and Localization Result to Periodic Second-Order Systems with Generalized Impulse Conditions

  • 期刊论文

  • 出版模式：

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  • 发表日期：

    2025年7月2日

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  • Feliz Minhós,
  • Sara Perestrelo

摘要

In this work, we present a result for the existence and localization of periodic solutions in impulsive second-order generalized systems, using a variant of the method of lower and upper solutions. The main advantage of our approach is the lack of requirements of periodicity for the nonlinearities and of order for the upper and lower solutions. We use Green’s functions to construct the operator of the system and a Nagumo-type condition for the control of the first derivatives. The existence is guaranteed by Shauder’s fixed-point theorem, and the localization is ensured using our method of orderless lower and upper solutions, involving translations. Only a few monotonicity criteria must be met by nonlinearities and impulse functions. We apply our result to a dissipative Liénard-type system with state-dependent impulses.