Hyers-Ulam-Rassias stability of some perturbed nonlinear second order ordinary differential equations
DOI:
https://doi.org/10.22199/issn.0717-6279-5906Keywords:
Hyers-Ulam-Rassias Stability, Euler type differential equation, forcing term, integral inequalityAbstract
In this paper we investigate the Hyers-Ulam-Rassias stability of a perturbed nonlinear second order ordinary differential equation using Gronwall-Bellman-Bihari type integral inequalities. Further, the paper also investigates the Hyers-Ulam-Rassias stability of four different cases of a perturbed nonlinear second order differential equation.
References
C. Alsina and R. Ger, “On Some Inequalities and Stability Result Related to the Exponential Function”, Journal of Inequalities and Applications, vol. 2, pp. 373-380, 1988. https://doi.org/10.1155/S102558349800023X
E. Bicer and C. Tunc, “New theorems for Hyers-Ulam Stability of Lienard Equation with Variable Time Lags”, International Journal of Mathematics and Computer Science, vol. 2, no. 3, pp. 231-242, 2018.
U. D. Dhongade and S. G. Deo, “Some Generalisations of Bellman-Bihari Integral Inequalities”, Journal of Mathematical Analysis and Applications, vol. 44, pp. 218-226, 1973. https://doi.org/10.1016/0022-247X(73)90039-5
I. Fakunle and P. O. Arawomo, “Hyers-Ulam Stability of Certain Class of Nonlinear Second Order Differential Equations”, International Journal of Pure and Applied Mathematical Sciences, vol. 11, no. 1, pp. 55-65, 2018.
I. Fakunle and P.O. Arawomo, “On Hyers-Ulam Stability of Nonlinear Second Order Ordinary and Functional Differential Equations”, International Journal of Differential Equations and Applications, vol. 17, no. 1, pp. 77-88, 2018.
I. Fakunle and P.O. Arawomo, “Hyers-Ulam Stability of a Perturbed Generalised Lienard Equation”, International Journal of Applied Mathematics, vol. 32, no. 3, pp. 479-489, 2019. https://doi.org/10.12732/ijam.v32i3.9
P. Gavruta, S. M. Jung and Y. Li, “Hyers-Ulam Stability for Second Order Liner Differential Equations with Boundary Conditions, 2011”, Electronic Journal of Differential Equations, vol. 2011, no. 80, pp. 1-7, 2011.
D. H. Hyers, “On the Stability of the Linear functional equation”, Proceedings of the National Academy of Science of the United States of America, vol. 27, pp. 222-224, 1941. https://doi.org/10.1073/pnas.27.4.222
E. L. Ince, Ordinary differential Equations. Longmans Green and co., 1926.
S. M. Jung, “Hyers-Ulam Stability of Linear Differential Equations of First Order”, Applied Mathematics Letters, vol. 17, pp. 1135-1140, 2004.https://doi.org/10.1016/j.aml.2003.11.004
S. M. Jung, “Hyers-Ulam Stability of Linear Differential Equations of First Order”, Journal of Mathematics Analysis and Applications, vol. 33, pp. 139-146, 2005. https://doi.org/10.1016/j.jmaa.2005.02.025
S. M. Jung, “Hyers-Ulam Stability of Linear Differential Equations of First Order, II”, Applied Mathematics Letters, vol. 19, pp. 854-858, 2006. https://doi.org/10.1016/j.aml.2005.11.004
Y. Li, “Hyers-Ulam Stability of Linear Differential Equations of First Order”, Thai Journal of Mathematics, vol. 8, 2, pp. 215-219, 2010.
Y. Li and Y. Shen, “Hyers-Ulam Stability of Nonhomogenous Linear Differential Equations of Second Order”, International Journal of Mathematics and Mathematical Sciences, Article ID576852, pp. 7, 2009. https://doi.org/10.1155/2009/576852
T. Miura, S. M. Jung and S. E. Takahasi, “Hyers-Ulam-Rassias Stability of the Banach Space Valued Linear Differential Equations y0 = λy”, Journal of the Korean Mathematics Society, vol. 41, pp. 99-1005, 2004. https://doi.org/10.4134/JKMS.2004.41.6.995
M. Obloza, Hyers-Ulam Stability of the Linear Differential Equations”, Rocznik Naukowo-Dydaktyczny. Prace Matematyczne, vol. 13, pp. 259-270, 1993. [On line]. Available: http://bit.ly/2KAUGAp
M. N. Qarawani, “Hyers-Ulam Stability of Linear and Nonlinear Differential Equations of Second Order”, International Journal of Applied Mathematical Research, vol. 1, no. 4, pp. 422-432, 2012. https://doi.org/10.14419/ijamr.v1i4.160
M. N. Qarawani, “Hyers-Ulam Stability of a Generalised Second Order Nonlinear Differential Equations”, Applied Mathematics, vol. 3, pp. 1857-1861, 2012. https://doi.org/10.4236/am.2012.312252
M. N. Qarawani, “On Hyers-Ulam-Rassias Stability for Bernoulli’s and First Order Linear and Nonlinear Differential Equations”, British Journal of Mathematics and Computer Science, vol. 4, 11, pp. 1628-1628, 2014. https://doi.org/10.9734/BJMCS/2014/9071
H. A. Qusausy and S. M. Jung, “On the Hyers-Ulam Stability of Differential Equations of Second Order”, Abstract and Appliedd Analysis, pp. 1-8, 2014. https://doi.org/10.1155/2014/483707
T. M. Rassias, “On the Stability of the Linear Mapping in Banach Spaces”, Proceedings of the American Mathematical Society, vol. 72, no. 2, pp. 297-300, 1978. https://doi.org/10.2307/2042795
K. Ravi, R. Murali, A. Ponmanaselvan and R. Veeraisivsji, “Hyers-Ulam Stabilty of nth Order Nonlinear Differential Equations with Initial Conditions”, International Journal of Mathematics And Its Applications, vol. 4, pp. 121-132, 2016.
I. A. Rus, “Ulam Stability of Ordinary Differential Equation”, Studia Universities Babes-Bolyal Mathematical, vol. 54, no. 4, pp. 306-309, 2010.
I. A. Rus, “Ulam stability of Ordinary Differential Equations in a Banach Space”, Carpathian Journal of Mathematics, vol. 126, pp. 103-107, 2010.
S. Takahasi, T. Miura and S. Miyajima, “On the Hyers-UIam Stability of the Banach Space-Valued Differential Equation y0 = λy”, Bulletin of the Korean Mathematical Society, vol. 392, pp. 309-315, 2002. https://doi.org/10.4134/bkms.2002.39.2.309
S. M. Ulam, Problems in Modern Mathematics Science Editions. New York, USA: Wily, 1960.
G. Wang, M. Zhou and L. Sun, “Hyers-Ulam Stability of Linear Differential Equations of First Order”, Applied Mathematics Letters, vol. 21, pp. 1024-1028, 2008. https://doi.org/10.1016/j.aml.2007.10.020
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