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Mild Solutions of Second-Order Semilinear Impulsive Differential Inclusions in Banach Spaces

Author

Listed:
  • Martina Pavlačková

    (Department of Informatics and Mathematics, Moravian Business College Olomouc, tř. Kosmonautů 1288/1, 77900 Olomouc, Czech Republic)

  • Valentina Taddei

    (Department of Sciences and Methods for Engineering, University of Modena and Reggio Emilia, Via G. Amendola, 2-pad. Morselli, 42122 Reggio Emilia, Italy)

Abstract

In this paper, the existence of a mild solution to the Cauchy problem for impulsive semilinear second-order differential inclusion in a Banach space is investigated in the case when the nonlinear term also depends on the first derivative. This purpose is achieved by combining the Kakutani fixed point theorem with the approximation solvability method and the weak topology. This combination enables obtaining the result under easily verifiable and not restrictive conditions on the impulsive terms, the cosine family generated by the linear operator and the right-hand side while avoiding any requirement for compactness. Firstly, the problems without impulses are investigated, and then their solutions are glued together to construct the solution to the impulsive problem step by step. The paper concludes with an application of the obtained results to the generalized telegraph equation with a Balakrishnan–Taylor-type damping term.

Suggested Citation

  • Martina Pavlačková & Valentina Taddei, 2022. "Mild Solutions of Second-Order Semilinear Impulsive Differential Inclusions in Banach Spaces," Mathematics, MDPI, vol. 10(4), pages 1-25, February.
    • Handle: RePEc:gam:jmathe:v:10:y:2022:i:4:p:672-:d:754753
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    References listed on IDEAS

    as
    1. Liu, X. & Zeng, Y.M., 2019. "Analytic and numerical stability of delay differential equations with variable impulses," Applied Mathematics and Computation, Elsevier, vol. 358(C), pages 293-304.
    2. Henríquez, Hernán R. & Pierri, Michelle & Rolnik, Vanessa, 2016. "Pseudo S-asymptotically periodic solutions of second-order abstract Cauchy problems," Applied Mathematics and Computation, Elsevier, vol. 274(C), pages 590-603.
    3. G. Arthi & K. Balachandran, 2012. "Controllability of Damped Second-Order Impulsive Neutral Functional Differential Systems with Infinite Delay," Journal of Optimization Theory and Applications, Springer, vol. 152(3), pages 799-813, March.
    4. Mouffak Benchohra & Noreddine Rezoug & Bessem Samet & Yong Zhou, 2019. "Second Order Semilinear Volterra-Type Integro-Differential Equations with Non-Instantaneous Impulses," Mathematics, MDPI, vol. 7(12), pages 1-20, November.
    Full references (including those not matched with items on IDEAS)

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