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On the Convergence of Solutions to Dynamic Equations on Time Scales

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Abstract

In this paper, we study the convergence of solutions to dynamic equations \(x^{\Delta }=f(t,x)\) on time scales \(\{\mathbb {T}_n\}_{n=1}^{\infty }\) when this sequence converges to the time scale \(\mathbb {T}\). The convergent rate of solutions is estimated when f satisfies the Lipschitz condition in both variables. By a general view, we derive a new approach to the approximation of dynamic equations on time scales, especially the Euler method for differential equations. Some examples are given to illustrate our results.

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Acknowledgments

This work was supported financially by Vietnam National Foundation for Science and Technology Development (NAFOSTED) 101.03-2014.58.

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Authors and Affiliations

  1. Department of Basic Science, Electric Power University, 235 Hoang Quoc Viet Str., Hanoi, Vietnam

    Nguyen Thu Ha

  2. Department of Mathematics, Mechanics and Informatics, Vietnam National University, 334 Nguyen Trai Str., Hanoi, Vietnam

    Nguyen Huu Du & Le Cong Loi

  3. School of Applied Mathematics and Informatics, Hanoi University of Science and Technology, 1 Dai Co Viet Str., Hanoi, Vietnam

    Do Duc Thuan

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  1. Nguyen Thu Ha
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  2. Nguyen Huu Du
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  3. Le Cong Loi
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  4. Do Duc Thuan
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Correspondence to Nguyen Huu Du.

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Ha, N.T., Du, N.H., Loi, L.C. et al. On the Convergence of Solutions to Dynamic Equations on Time Scales. Qual. Theory Dyn. Syst. 15, 453–469 (2016). https://doi.org/10.1007/s12346-015-0166-8

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  • DOI: https://doi.org/10.1007/s12346-015-0166-8

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