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New Results for Some Damped Dirichlet Problems with Impulses

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Abstract

This paper investigate the following damped nonlinear impulsive differential equations.

$$ \begin{gathered} - u''(t) + p(t)u'(t) + q(t)u(t) = f(t,u(t)),\;a.e.\;t \in [0,T], \hfill \\ \Delta u'(t_{j} ) = I_{j} (u(t_{j} )),\begin{array}{*{20}c} {} & {j = 1,2, \ldots ,m,} \\ \end{array} \hfill \\ u(0) = u(T) = 0. \hfill \\ \end{gathered} $$

Applying fountain theorem and a new analytical approach, we obtain that the aforementioned problem has infinitely many solutions under the local superlinear condition \(\mathop {\lim }\nolimits_{\left| u \right| \to + \infty } \frac{{\int_{0}^{u} {f(t,s)ds} }}{{u^{2} }} = + \infty\) uniformly in \(t \in (\widetilde{a},\widetilde{b})\) for some \((\widetilde{a},\widetilde{b}) \subset [0,T]\) instead of the global superlinear condition \(\mathop {\lim }\nolimits_{\left| u \right| \to + \infty } \frac{{\int_{0}^{u} {f(t,s)ds} }}{{u^{2} }} = + \infty\) uniformly in \(t \in [0,T]\).

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Authors and Affiliations

  1. School of Mathematics and Physics, University of South China, Hengyang, 421001, Hunan, China

    Huiwen Chen, Zigen Ouyang & Maoxin Liao

  2. School of Mathematics and Statistics, Central South University, Changsha, 410083, Hunan, China

    Zhimin He

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  1. Huiwen Chen
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  2. Zhimin He
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  3. Zigen Ouyang
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  4. Maoxin Liao
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Correspondence to Huiwen Chen.

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This work is supported by the National Natural Science Foundation of China (No. 11601222), Hunan Provincial Natural Science Foundation of China (Nos. 2019JJ50487, 2019JJ40240), and Doctor Priming Fund Project of University of South China (2014XQD13).

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Chen, H., He, Z., Ouyang, Z. et al. New Results for Some Damped Dirichlet Problems with Impulses. Qual. Theory Dyn. Syst. 21, 36 (2022). https://doi.org/10.1007/s12346-022-00559-0

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