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Continuous Dependence and Differentiability of Solutions of Second-Order Impulsive Differential Equations on Initial Values and Impulsive Points

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Abstract

In this paper, we consider the continuous dependence and differentiability of solutions of second-order impulsive differential equations on initial values and impulsive points. By constructing a sequence of iterations, we show the existence of solutions with the perturbation of initial values and impulsive points and the continuous dependence of solution on initial values and impulsive points. Moreover, we also give a further result under some strong conditions. Based on these results, we present the result of differentiability of solutions on initial point and impulsive points. Finally, an example is provided to illustrate the practicability of our results.

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Authors and Affiliations

  1. Department of Mathematics, Guizhou University, Guiyang, 550025, Guizhou, China

    Qian Wen

  2. School of Mathematical Sciences, Qufu Normal University, Qufu, 273165, Shandong, China

    JinRong Wang

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  1. Qian Wen
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  2. JinRong Wang
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Correspondence to JinRong Wang.

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This work is partially supported by the National Natural Science Foundation of China (12161015); Training Object of High Level and Innovative Talents of Guizhou Province [(2016)4006]; the Major Research Project of Innovative Group in Guizhou Education Department [(2018)012]; the Guizhou Data Driven Modeling Learning and Optimization Innovation Team, China [(2020)5016]

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Wen, Q., Wang, J. Continuous Dependence and Differentiability of Solutions of Second-Order Impulsive Differential Equations on Initial Values and Impulsive Points. Qual. Theory Dyn. Syst. 20, 85 (2021). https://doi.org/10.1007/s12346-021-00525-2

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