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Bounded nonoscillatory solutions of a system of dynamic equations with neutral term on time scales

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Abstract

In this paper we consider a system of k ≥ 3 dynamic equations on time scales such that the first equation has the neutral term

$$\begin{cases}(x_1(t)+p(t)x_1(v(t)))^\Delta=a_1(t)f_1(x_2(u_1(t))),\\{x_i^\Delta}(t)=a_i(t)f_i(x_{i+1}(u_i(t))), & i=2,\ldots,k-1,\\{x_k^\Delta}(t)=a_k(t)f_k(x_1(u_k(t))), & t \in \mathbb{T}.\end{cases}$$

Our purpose is to present sufficient conditions for the existence of eventually positive bounded solutions of the considered system for 1 <α ≤ p(t) ≤ β. The main idea is to apply Krasnoselskii’s fixed point theorem.

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References

  1. M. Bohner and A. Peterson, Dynamic Equations on Time Scales, Birkhäuser Boston, Boston, MA, 2001.

    Book  MATH  Google Scholar 

  2. A. Deniz, Measurable theory on time scales, PhD thesis, The Graduate School of Engineering and Sciences of Izmir Institute of Technology, s2007.

    Google Scholar 

  3. L. H. Erbe, Q. Kong and B. G. Zhang, Oscillation Theory for Functional-Differential Equations, Monographs and Textbooks in Pure and Applied Mathematics, Vol. 190, Marcel Dekker, New York, 1995.

  4. J. Gao and Q. Wang, Existence of nonoscillatory solutions to second-order nonlinear neutral dynamic equations on time scales, Rocky Mountains Journal of Mathematics 43 (2013), 1521–1535.

    Article  MathSciNet  MATH  Google Scholar 

  5. G. Sh. Guseinov, Integration on time scales, Journal of Mathematical Analysis and Applications 285 (2003), 107–127.

    Article  MathSciNet  MATH  Google Scholar 

  6. R. Jankowski, M. Nockowska-Rosiak, E. Schmeidel and B. Łupinska, Monotonic solutions of a higher-order neutral difference system, Discrete and Continuous Dynamical Systems. Series B 23 (2018), 253–261.

    Article  MathSciNet  MATH  Google Scholar 

  7. M. Migda, E. Schmeidel and M. Zdanowicz, Existence of nonoscillatory solutions for system of neutral difference equations, Applicable Analysis and Discrete Mathematics 9 (2015), 271–284.

    Article  MathSciNet  MATH  Google Scholar 

  8. M. Migda, E. Schmeidel and M. Zdanowicz, Periodic solutions of a 2-dimensional system of neutral difference equations, Discrete and Continuous Dynamical Systems. Series B 23 (2018), 359–367.

    Article  MathSciNet  MATH  Google Scholar 

  9. U. Ostaszewska, E. Schmeidel and M. Zdanowicz, Existence of positive bounded solutions of system of three dynamic equations with neutral term on time scales, Tatra Mountains Mathematical Publications 71 (2018), 123–137.

    Article  MATH  Google Scholar 

  10. Y-C. Qiu, Nonoscillatory solutions to third-order neutral dynamic equations on time scales, Advances in Difference Equations 2014 (2014), paper no. 309.

  11. Z. Q. Zhu and Q. R. Wang, Existence of nonoscillatory solutions to neutral dynamic equations on time scales, Journal of Mathematical Analysis and Applications 335 (2007), 751–762.

    Article  MathSciNet  MATH  Google Scholar 

Download references

Acknowledgment

The authors would like to thank the reviewers for their helpful comments and valuable suggestions.

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Authors and Affiliations

  1. Institute of Mathematics, University of Białystok, 15-245, Białystok, Poland

    Urszula Ostaszewska, Ewa Schmeidel & Małgorzata Zdanowicz

Authors
  1. Urszula Ostaszewska
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  2. Ewa Schmeidel
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  3. Małgorzata Zdanowicz
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Corresponding author

Correspondence to Urszula Ostaszewska.

Additional information

This article has received financial support from the Polish Ministry of Science and Higher Education under a subsidy for maintaining the research potential of the Faculty of Mathematics, University of Bialystok.

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Ostaszewska, U., Schmeidel, E. & Zdanowicz, M. Bounded nonoscillatory solutions of a system of dynamic equations with neutral term on time scales. Isr. J. Math. 231, 489–504 (2019). https://doi.org/10.1007/s11856-019-1869-3

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  • DOI: https://doi.org/10.1007/s11856-019-1869-3

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