Stability of higher-order nonlinear impulsive differential equations

Volume 9, Issue 6, pp 4713--4721 http://dx.doi.org/10.22436/jnsa.009.06.110

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Authors

Shuhong Tang - School of Information and Control Engineering, Weifang University, Weifang, Shandong 261061, P. R. China. Akbar Zada - Department of Mathematics, University of Peshawar, Peshawar 25000, Pakistan. Shah Faisal - Department of Mathematics, University of Peshawar, Peshawar 25000, Pakistan. M. M. A. El-Sheikh - Department of Mathematics, Faculty of Science, Menoufia University, Shebin El-Koom 32511, Egypt. Tongxing Li - LinDa Institute of Shandong Provincial Key Laboratory of Network Based Intelligent Computing, Linyi University, Linyi, Shandong 276005, P. R. China. - School of Informatics, Linyi University, Linyi, Shandong 276005, P. R. China.

Abstract

For a higher-order nonlinear impulsive ordinary differential equation, we present the concepts of Hyers- Ulam stability, generalized Hyers-Ulam stability, Hyers-Ulam{Rassias stability, and generalized Hyers- Ulam-Rassias stability. Furthermore, we prove the generalized Hyers-Ulam-Rassias stability by using integral inequality of Grönwall type for piecewise continuous functions. These results extend related contributions to the corresponding first-order impulsive ordinary differential equation. Hyers-Ulam stability, generalized Hyers-Ulam stability, and Hyers-Ulam-Rassias stability can be discussed by the same methods.

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Keywords

• Hyers-Ulam stability
• generalized Hyers-Ulam stability
• Hyers-Ulam-Rassias stability
• generalized Hyers-Ulam-Rassias stability
• nonlinear impulsive differential equation
• higher-order
• Grönwall inequality.

MSC

•  34A37
•  34D10

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