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Hyers-Ulam Rassias Stability of a Second Order Delay Differential Equation

Year 2018, Volume: 8 Issue: 2, 249 - 254, 31.07.2018
https://doi.org/10.17714/gumusfenbil.350309

Abstract

In this study, firstly, we
use Banach fixed point theorem to show that the Hyers-Ulam Rassias stability of
a first order delay differential equation with constant delay of the form

where is continuous function and  is a nonnegative real
constant. By taking advantage of this result, we investigate Hyers-Ulam Rassias
stability of a second order delay differential equation with constant delay of
the form









where  are continuous
functions and  is a nonnegative real
constant. Also we present an example to illustrate the theoretical analysis.

References

  • Alsina, C. ve Ger, R., 1998. On some Inequalities and Stability Results Related to the Exponential Function, J. Inequal. Appl, 4, 373–380.
  • Biçer, E. ve Tunç, C., 2017. On the Hyers-Ulam stability of certain partial differential equations of second order. Nonlinear Dyn. Syst. Theory. 17, 150–157.
  • Cimpean, DS. ve Popa, D., 2011. Hyers-Ulam stability of Euler's equation. Appl. Math. Lett, 9, 1539–1543.
  • Gordji, M. ve Cho, YJ., Ghaemi, MB. ve Alizadeh, B., 2011. Stability of the second order partial differential equations. J. Inequal. Appl, 81, 10 pp.
  • Hyers, Donald H., 1941. On the Stability of the Linear Functional Equation. Proc. Nat. Acad. Sci, U.S.A. 27, 222–224.
  • Huimin, L. ve Xiangkui, Z., 2013. Hyers-Ulam-Rassias stability of second order partial differential equations. Ann. Differential Equations, 29, 430–437.
  • Jung, S.M., 2005. Hyers Ulam stability of linear differential equations of first order (III). J. Math. Anal. Appl, 311, 139–146.
  • Jung, S.M., 2006. Hyers Ulam stability of linear differential equations of first order (II). Appl. Math. Lett, 19, 854–858.
  • Jung, S.M., 2007. Hyers Ulam stability of first order linear partial differential equations with constant coefficients. Math. Inequal. Appl, 10, 261-266.
  • Jung, S.M., 2009. Hyers Ulam stability of linear partial differential equations of first order. Appl. Math. Lett, 22, 70–74.
  • Jung, S.M. ve Brzdȩk, J., 2010. Hyers-Ulam Stability of the Delay Equation Abstr. Appl. Anal, 1-10.
  • Li, Y. and Shen, Y. 2009. Hyers-ulam stability of nonhomogeneous linear differential equations of second order. International Journal of Mathematics and Mathematical Analysis, 1–7.
  • Lungu, N. ve Popa, D., 2012. Hyers-Ulam stability of a first order partial differential equation. J.Math.Anal.Appl, 86-91.
  • Lungu, N. ve Popa, D., 2014. Hyers-Ulam stability of some partial differential equations. Carpathian J. Math, 30, 327–334.
  • Otrocol, D. ve Ilea, V., 2013. Ulam Stability for a Delay Differential Equation. Cent. Eur. J. Math, 7, 1296-1303.
  • Rassias, TM., 1978. On the Stability of the Linear Mapping in Banach Spaces. Proc Amer Math. Soc, 72, 297–300.
  • Tunc, C. ve Bicer, E., 2015. Hyers-Ulam-Rassias stability for a first order functional differential equation. J. Math. Fundam. Sci, 47, 143–153.
  • Ulam, S. M., 1964. Problems in Modern Mathematics. Science Editions John Wiley & Sons, Inc., New York

İkinci Mertebeden Gecikmeli Bir Diferansiyel Denklemin Hyers-Ulam Rassias Kararlılığı

Year 2018, Volume: 8 Issue: 2, 249 - 254, 31.07.2018
https://doi.org/10.17714/gumusfenbil.350309

Abstract

Bu çalışmada, ilk olarak  bir sürekli fonksiyon
ve  negatif olmayan reel
bir sabit olmak üzere

,

şeklindeki birinci mertebeden sabit gecikmeli bir diferansiyel denklemin Banach
sabit nokta teoremi kullanılarak Hyers-Ulam Rassias kararlılığı gösterildi.
Buradan elde edilen sonuçtan faydalanılarak  sürekli fonksiyonlar
olmak üzere









biçimindeki ikinci mertebeden sabit gecikmeli bir diferansiyel denklemin
Hyers-Ulam Rassias kararlılığı araştırıldı. Ayrıca çalışmadaki teorik
analizleri açıklamak için bir örnek verildi. 

References

  • Alsina, C. ve Ger, R., 1998. On some Inequalities and Stability Results Related to the Exponential Function, J. Inequal. Appl, 4, 373–380.
  • Biçer, E. ve Tunç, C., 2017. On the Hyers-Ulam stability of certain partial differential equations of second order. Nonlinear Dyn. Syst. Theory. 17, 150–157.
  • Cimpean, DS. ve Popa, D., 2011. Hyers-Ulam stability of Euler's equation. Appl. Math. Lett, 9, 1539–1543.
  • Gordji, M. ve Cho, YJ., Ghaemi, MB. ve Alizadeh, B., 2011. Stability of the second order partial differential equations. J. Inequal. Appl, 81, 10 pp.
  • Hyers, Donald H., 1941. On the Stability of the Linear Functional Equation. Proc. Nat. Acad. Sci, U.S.A. 27, 222–224.
  • Huimin, L. ve Xiangkui, Z., 2013. Hyers-Ulam-Rassias stability of second order partial differential equations. Ann. Differential Equations, 29, 430–437.
  • Jung, S.M., 2005. Hyers Ulam stability of linear differential equations of first order (III). J. Math. Anal. Appl, 311, 139–146.
  • Jung, S.M., 2006. Hyers Ulam stability of linear differential equations of first order (II). Appl. Math. Lett, 19, 854–858.
  • Jung, S.M., 2007. Hyers Ulam stability of first order linear partial differential equations with constant coefficients. Math. Inequal. Appl, 10, 261-266.
  • Jung, S.M., 2009. Hyers Ulam stability of linear partial differential equations of first order. Appl. Math. Lett, 22, 70–74.
  • Jung, S.M. ve Brzdȩk, J., 2010. Hyers-Ulam Stability of the Delay Equation Abstr. Appl. Anal, 1-10.
  • Li, Y. and Shen, Y. 2009. Hyers-ulam stability of nonhomogeneous linear differential equations of second order. International Journal of Mathematics and Mathematical Analysis, 1–7.
  • Lungu, N. ve Popa, D., 2012. Hyers-Ulam stability of a first order partial differential equation. J.Math.Anal.Appl, 86-91.
  • Lungu, N. ve Popa, D., 2014. Hyers-Ulam stability of some partial differential equations. Carpathian J. Math, 30, 327–334.
  • Otrocol, D. ve Ilea, V., 2013. Ulam Stability for a Delay Differential Equation. Cent. Eur. J. Math, 7, 1296-1303.
  • Rassias, TM., 1978. On the Stability of the Linear Mapping in Banach Spaces. Proc Amer Math. Soc, 72, 297–300.
  • Tunc, C. ve Bicer, E., 2015. Hyers-Ulam-Rassias stability for a first order functional differential equation. J. Math. Fundam. Sci, 47, 143–153.
  • Ulam, S. M., 1964. Problems in Modern Mathematics. Science Editions John Wiley & Sons, Inc., New York
There are 18 citations in total.

Details

Primary Language Turkish
Subjects Engineering
Journal Section Articles
Authors

Emel Biçer

Publication Date July 31, 2018
Submission Date November 9, 2017
Acceptance Date March 9, 2018
Published in Issue Year 2018 Volume: 8 Issue: 2

Cite

APA Biçer, E. (2018). İkinci Mertebeden Gecikmeli Bir Diferansiyel Denklemin Hyers-Ulam Rassias Kararlılığı. Gümüşhane Üniversitesi Fen Bilimleri Dergisi, 8(2), 249-254. https://doi.org/10.17714/gumusfenbil.350309