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Stability Analysis for a General Class of Non-instantaneous Impulsive Differential Equations

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Abstract

For a new linear non-instantaneous impulsive differential equations, we introduce the notation of non-instantaneous impulsive Cauchy matrix and analyze its exponential structure in terms of eigenvalues of matrix via the distance between impulsive points and junction points. Many useful criteria for asymptotic stability of linear non-instantaneous impulsive problems are derived, which allow us to establish a uniform framework to deal with asymptotic stability of linear non-instantaneous impulsive differential equations with perturbation under mild sufficient conditions. In particular, two examples are given to demonstrate the application of theoretical results for linear part. In addition, we study nonlinear non-instantaneous impulsive equations and investigate existence, uniqueness of their solutions and Ulam–Hyers–Rassias stability under the restriction of exponential growth or stable conditions for non-instantaneous impulsive Cauchy matrix, respectively, which provide an approach to find approximate solution to nonlinear non-instantaneous impulsive equations in the sense of Ulam–Hyers–Rassias stability. The obtained results cover the standard results for instantaneous impulsive differential equations, which are essentially extend the theory in the previous literatures.

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Authors and Affiliations

  1. Department of Mathematics, Guizhou University, Guiyang, 550025, Guizhou, People’s Republic of China

    JinRong Wang & Ying Tian

  2. Department of Mathematical Analysis and Numerical Mathematics, Faculty of Mathematics, Physics and Informatics, Comenius University in Bratislava, Mlynská dolina, 842 48, Bratislava, Slovakia

    Michal Fečkan

  3. Mathematical Institute of Slovak Academy of Sciences, Štefánikova 49, 814 73, Bratislava, Slovakia

    Michal Fečkan

Authors
  1. JinRong Wang
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  2. Michal Fečkan
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  3. Ying Tian
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Corresponding author

Correspondence to Michal Fečkan.

Additional information

J.W. acknowledges NNSF of China (11661016), Training Object of High Level and Innovative Talents of Guizhou Province ((2016)4006), Unite Foundation of Guizhou Province ([2015]7640), and Graduate Course of Guizhou University (ZDKC[2015]003). M.F. acknowledges the Slovak Grant Agency VEGA No. 2/0153/16 and No. 1/0078/17 and the Slovak Research and Development Agency under the contract No. APVV-14-0378.

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Wang, J., Fečkan, M. & Tian, Y. Stability Analysis for a General Class of Non-instantaneous Impulsive Differential Equations. Mediterr. J. Math. 14, 46 (2017). https://doi.org/10.1007/s00009-017-0867-0

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  • DOI: https://doi.org/10.1007/s00009-017-0867-0

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