Hyers-Ulam Stability of First Order Nonlinear Delay Difference Equations

Arun Kumar Tripathy; Binayak Dihudi

doi:doi:10.11648/j.ijtam.20220803.12

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Hyers-Ulam Stability of First Order Nonlinear Delay Difference Equations

Arun Kumar Tripathy, Binayak Dihudi

Published in International Journal of Theoretical and Applied Mathematics (Volume 8, Issue 3)

Received: 23 June 2022 Accepted: 5 August 2022 Published: 28 September 2022

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Abstract

Initially Ulam’s stability problem has originated for functional equations of both linear and nonlinear types. Because of Hyers and Rassias, the Ulam’s stability problem has come to different shapes over different spaces. Slowly, it has got its name as Hyers-Ulam stability and Hyers-Ulam-Rassias stability. Meanwhile, the Hyers-Ulam stability has been extended to special functional equations like differential and difference equations. In this study, we examine the Hyers-Ulam stability and Hyers-Ulam-Rassias stability of a class of first-order nonlinear delay difference equations with real coefficients on Banach space. Also, its nonhomogeneous counterpart has been studied for the same. Why we are interested in this study is that this is a special type of stability unlike the so called stability of differential or difference equations. As soon as we locate a solution in a Banach space, it is in the -neighbourhood while the concerned difference inequality is in an -neighbourhood. Lipschitz condition and Banach’s fixed point theorem are our state of art to apply. Main results are illustrated by the examples.

Published in	International Journal of Theoretical and Applied Mathematics (Volume 8, Issue 3)
DOI	10.11648/j.ijtam.20220803.12
Page(s)	58-64
Creative Commons	This is an Open Access article, distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution and reproduction in any medium or format, provided the original work is properly cited.
Copyright	Copyright © The Author(s), 2022. Published by Science Publishing Group

Keywords

Hyers-Ulam Stability, Difference Equation, Nonlinear, Delay

References

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[2]	J. Brzdek, D. Popa, B. Xu (2010). Remarks on stability of linear recurrence of higher order. Appl. Math. Lett. 23: 1459-1463.
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[4]	K. W. Jun, Y. H. Lee (1999). A generalization of the Hyers-Ulam-Rassias stability of the Jenson’s equation. J. Math.Anal. Appl. 238:305-315.
[5]	S. M. Jung (2004). Hyers-Ulam stability of linear differential equations of first order. Appl. Math. Lett. 17: 1135-1140.
[6]	S. M. Jung (2005). Hyers-Ulam stability of linear differential equations of first order (III). J. Math. Anal. Appl. 311: 139-146.
[7]	S. M. Jung (2006). Hyers-Ulam stability of linear differential equations of first order (II). Appl. Math. Lett. 19: 854-858.
[8]	S. M. Jung, Th. M. Rassias; Ulam’s problem for approximate homomorphisms in connection with Bernoulli’s differential equation, Appl. Math. Comp. 187: 223-227.
[9]	S. M. Jung (2015). Hyers -Ulam stability of the first order matrix difference equations, Adv. Differ. Equs. 2015: 1- 13.
[10]	Y. Li (2010). Hyers-Ulam stability of linear differential equations y’’= λ²y. Thai J. Math. 8: 215-219.
[11]	T. Miura, S. E. Takahasi, H. Choda (2001). On the Hyers-Ulam stability of real continuous function valued differentiable map. Tokyo J. Math. 467-476.
[12]	T. Miura (2002). On the Hyers -Ulam stability of a differentiable map. Sci. Math. Japan. 55: 17-24.
[13]	T. Miura, S. Miyajima, S. E. Takahasi (2003). A characterization of Hyers-Ulam stability of first order linear differential operators J. Math. Anal. Appl. 286: 136-146.
[14]	T. Miura, S. M. Jung, S. E. Takahasi (2004). Hyers-Ulam stability of the banach space valued linear differential equations y’= λy. Korean Math. Soc. 41: 995-1005.
[15]	M. Obloza (1993). Hyers stability of the linear differential equation. Rocznik Nauk-Dydakt. Proce. Mat. 13: 259-270.
[16]	M. Obloza (1997). Connections between Hyers and Lyapunov stability of the ordinary differential equations. Rocznik Nauk-Dydakt. Proce. Mat. 14: 141-146.
[17]	D. Popa (2005). Hyers-Ulam-Rassias stability of a linear recurrence. J. Math. Anal. Appl. 309: 591-597.
[18]	Th. M. Rassias (1978). On the stability of linear mapping in Banach spaces. Proc. Amer. Math. Soc. 72: 297-300.
[19]	Th. M. Rassias (2000). On the stability of functional equations and a problem of Ulam. Acta Appl. Math. 62: 23-30.
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Arun Kumar Tripathy, Binayak Dihudi. (2022). Hyers-Ulam Stability of First Order Nonlinear Delay Difference Equations. International Journal of Theoretical and Applied Mathematics, 8(3), 58-64. https://doi.org/10.11648/j.ijtam.20220803.12

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Arun Kumar Tripathy; Binayak Dihudi. Hyers-Ulam Stability of First Order Nonlinear Delay Difference Equations. Int. J. Theor. Appl. Math. 2022, 8(3), 58-64. doi: 10.11648/j.ijtam.20220803.12

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AMA Style

Arun Kumar Tripathy, Binayak Dihudi. Hyers-Ulam Stability of First Order Nonlinear Delay Difference Equations. Int J Theor Appl Math. 2022;8(3):58-64. doi: 10.11648/j.ijtam.20220803.12

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@article{10.11648/j.ijtam.20220803.12,
  author = {Arun Kumar Tripathy and Binayak Dihudi},
  title = {Hyers-Ulam Stability of First Order Nonlinear Delay Difference Equations},
  journal = {International Journal of Theoretical and Applied Mathematics},
  volume = {8},
  number = {3},
  pages = {58-64},
  doi = {10.11648/j.ijtam.20220803.12},
  url = {https://doi.org/10.11648/j.ijtam.20220803.12},
  eprint = {https://article.sciencepublishinggroup.com/pdf/10.11648.j.ijtam.20220803.12},
  abstract = {Initially Ulam’s stability problem has originated for functional equations of both linear and nonlinear types. Because of Hyers and Rassias, the Ulam’s stability problem has come to different shapes over different spaces. Slowly, it has got its name as Hyers-Ulam stability and Hyers-Ulam-Rassias stability. Meanwhile, the Hyers-Ulam stability has been extended to special functional equations like differential and difference equations. In this study, we examine the Hyers-Ulam stability and Hyers-Ulam-Rassias stability of a class of first-order nonlinear delay difference equations with real coefficients on Banach space. Also, its nonhomogeneous counterpart has been studied for the same. Why we are interested in this study is that this is a special type of stability unlike the so called stability of differential or difference equations. As soon as we locate a solution in a Banach space, it is in the -neighbourhood while the concerned difference inequality is in an -neighbourhood. Lipschitz condition and Banach’s fixed point theorem are our state of art to apply. Main results are illustrated by the examples.},
 year = {2022}
}

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TY - JOUR
T1 - Hyers-Ulam Stability of First Order Nonlinear Delay Difference Equations
AU - Arun Kumar Tripathy
AU - Binayak Dihudi
Y1 - 2022/09/28
PY - 2022
N1 - https://doi.org/10.11648/j.ijtam.20220803.12
DO - 10.11648/j.ijtam.20220803.12
T2 - International Journal of Theoretical and Applied Mathematics
JF - International Journal of Theoretical and Applied Mathematics
JO - International Journal of Theoretical and Applied Mathematics
SP - 58
EP - 64
PB - Science Publishing Group
SN - 2575-5080
UR - https://doi.org/10.11648/j.ijtam.20220803.12
AB - Initially Ulam’s stability problem has originated for functional equations of both linear and nonlinear types. Because of Hyers and Rassias, the Ulam’s stability problem has come to different shapes over different spaces. Slowly, it has got its name as Hyers-Ulam stability and Hyers-Ulam-Rassias stability. Meanwhile, the Hyers-Ulam stability has been extended to special functional equations like differential and difference equations. In this study, we examine the Hyers-Ulam stability and Hyers-Ulam-Rassias stability of a class of first-order nonlinear delay difference equations with real coefficients on Banach space. Also, its nonhomogeneous counterpart has been studied for the same. Why we are interested in this study is that this is a special type of stability unlike the so called stability of differential or difference equations. As soon as we locate a solution in a Banach space, it is in the -neighbourhood while the concerned difference inequality is in an -neighbourhood. Lipschitz condition and Banach’s fixed point theorem are our state of art to apply. Main results are illustrated by the examples.
VL - 8
IS - 3
ER -

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Author Information

Arun Kumar Tripathy

Department of Mathematics, Sambalpur University, Sambalpur, India
Binayak Dihudi

Department of Mathematics, Sambalpur University, Sambalpur, India

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APA Style

Arun Kumar Tripathy, Binayak Dihudi. (2022). Hyers-Ulam Stability of First Order Nonlinear Delay Difference Equations. International Journal of Theoretical and Applied Mathematics, 8(3), 58-64. https://doi.org/10.11648/j.ijtam.20220803.12

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ACS Style

Arun Kumar Tripathy; Binayak Dihudi. Hyers-Ulam Stability of First Order Nonlinear Delay Difference Equations. Int. J. Theor. Appl. Math. 2022, 8(3), 58-64. doi: 10.11648/j.ijtam.20220803.12

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AMA Style

Arun Kumar Tripathy, Binayak Dihudi. Hyers-Ulam Stability of First Order Nonlinear Delay Difference Equations. Int J Theor Appl Math. 2022;8(3):58-64. doi: 10.11648/j.ijtam.20220803.12

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@article{10.11648/j.ijtam.20220803.12,
  author = {Arun Kumar Tripathy and Binayak Dihudi},
  title = {Hyers-Ulam Stability of First Order Nonlinear Delay Difference Equations},
  journal = {International Journal of Theoretical and Applied Mathematics},
  volume = {8},
  number = {3},
  pages = {58-64},
  doi = {10.11648/j.ijtam.20220803.12},
  url = {https://doi.org/10.11648/j.ijtam.20220803.12},
  eprint = {https://article.sciencepublishinggroup.com/pdf/10.11648.j.ijtam.20220803.12},
  abstract = {Initially Ulam’s stability problem has originated for functional equations of both linear and nonlinear types. Because of Hyers and Rassias, the Ulam’s stability problem has come to different shapes over different spaces. Slowly, it has got its name as Hyers-Ulam stability and Hyers-Ulam-Rassias stability. Meanwhile, the Hyers-Ulam stability has been extended to special functional equations like differential and difference equations. In this study, we examine the Hyers-Ulam stability and Hyers-Ulam-Rassias stability of a class of first-order nonlinear delay difference equations with real coefficients on Banach space. Also, its nonhomogeneous counterpart has been studied for the same. Why we are interested in this study is that this is a special type of stability unlike the so called stability of differential or difference equations. As soon as we locate a solution in a Banach space, it is in the -neighbourhood while the concerned difference inequality is in an -neighbourhood. Lipschitz condition and Banach’s fixed point theorem are our state of art to apply. Main results are illustrated by the examples.},
 year = {2022}
}

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