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

Yıl 2018, Cilt: 8 Sayı: 2, 249 - 254, 31.07.2018

Emel Biçer

https://doi.org/10.17714/gumusfenbil.350309

Öz

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.

Anahtar Kelimeler

Banach Fixed Point Theorem, Delay Differential Equation, Hyers-Ulam Rassias Stability

İkinci Mertebeden Gecikmeli Bir Diferansiyel Denklemin Hyers-Ulam Rassias Kararlılığı

Öz

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. 

Anahtar Kelimeler

Banach Sabit Nokta Teoremi, Gecikmeli Diferansiyel Denklem, Hyers-Ulam Rassias Kararlılık

Kaynakça

• 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