Hyers-Ulam stability of n ͭ ͪ order linear differential equation

Resumen

In this paper, we investigate the Hyers-Ulam stability and Hyers-Ulam-Rassias stability of the homogeneous linear differential equation of nth order with initial and boundary conditions by using Taylor’s Series formula.

Biografía del autor/a

R. Murali, Sacred Heart College (Autonomous).
PG and Research Department of Mathematics.
A. Ponmana Selvan, Sacred Heart College (Autonomous).
PG and Research Department of Mathematics.

Citas

S. Ulam, A collection of mathematical problems, New York, NY: Interscience Publishers, 1960.

D. Hyers, “On the stability of the linear functional equation”, Proceedings of the National Academy of Sciences, vol. 27, no. 4, pp. 222–224, Apr. 1941, doi: 10.1073/pnas.27.4.222.

T. Rassias, “On the stability of the linear mapping in Banach spaces”, Proceedings of the American Mathematical Society, vol. 72, no. 2, pp. 297–297, Feb. 1978, doi: 10.1090/S0002-9939-1978-0507327-1.

T. Aoki, “On the stability of the linear transformation in Banach spaces.”, Journal of the Mathematical Society of Japan, vol. 2, no. 1-2, pp. 64–66, Sep. 1950, doi: 10.2969/jmsj/00210064.

D. Bourgin, “Classes of transformations and bordering transformations”, Bulletin of the American Mathematical Society, vol. 57, no. 4, pp. 223–238, Jul. 1951, doi: 10.1090/S0002-9904-1951-09511-7.

N. Brillouët-Belluot, J. Brzdęk, and K. Ciepliński, “On some recent developments in Ulams type stability”, Abstract and Applied Analysis, vol. 2012, Art. no. 716936, doi: 10.1155/2012/716936.

M. Burger, N. Ozawa, and A. Thom, “On Ulam stability”, Israel Journal of Mathematics, vol. 193, no. 1, pp. 109–129, May 2012, doi: 10.1007/s11856-012-0050-z.

H. Maeda and A. J. Sommese, “Notes on very ample vector bundles on 3-folds”, Archiv der Mathematik, vol. 85, no. 6, pp. 527–537, Dec. 2005, doi: 10.1007/s00013-005-1445-4.

M. Obloza, “Hyers stability of the linear differential equation”, Rocznik Naukowo-Dydaktyczny. Prace Matematyczne, vol. 13, pp. 259–270, 1993. [On line]. Available: http://bit.ly/2KAUGAp

C. Alsina and R. Ger, “On some inequalities and stability results related to the exponential function”, Journal of Inequalities and Applications, vol. 1998, no. 4, pp. 246-904, 1998, doi: 10.1155/S102558349800023X.

S. Takahasi, T. Miura, and S. Miyajima, “On the Hyers-Ulam stability of the Banach space valued differential equation y’ = λy”, Bulletin of the Korean Mathematical Society, vol. 39, no. 2, pp. 309-315. [On line]. http://bit.ly/2GX6ufr

I. Rus, “Ulam stabilities of ordinary differential equations in Banach space”, Carpathian Journal of Mathematics, vol. 26, no.1, pp. 103-107, 2010. [On line]. Available: http://bit.ly/33onCnR

G. Wang, M. Zhou, and L. Sun, “Hyers–Ulam stability of linear differential equations of first order”, Applied Mathematics Letters, vol. 21, no. 10, pp. 1024–1028, Oct. 2008, doi: 10.1016/j.aml.2007.10.020.

J. Huang and Y. Li, “Hyers–Ulam stability of linear functional differential equations”, Journal of Mathematical Analysis and Applications, vol. 426, no. 2, pp. 1192–1200, Jun. 2015, doi: 10.1016/j.jmaa.2015.02.018.

J. Huang, S. Jung, and Y. Li, “On Hyers-Ulam stability of nonlinear differential equations”, Bulletin of the Korean Mathematical Society, vol. 52, no. 2, pp. 685–697, Mar. 2015, doi:10.4134/BKMS.2015.52.2.685.

Y. Li and Y. Shen, “Hyers–Ulam stability of linear differential equations of second order”, Applied Mathematics Letters, vol. 23, no. 3, pp. 306–309, Mar. 2010, doi: 10.1016/j.aml.2009.09.020.

M. Modebei, O. Olaiya and I. Otaide. “Generalized Hyers-Ulam stability of second order linear ordinary differential equation with initial condition”. Advances in Inequalities and Applications, vol. 2014, no. 36, pp. 1-7, Sep. 2014. [On line]. Available: http://bit.ly/2M95yc8

T. Xu, “On the stability of multi-Jensen mappings in β-normed spaces”, Applied Mathematics Letters, vol. 25, no. 11, pp. 1866–1870, Nov. 2012, doi: 10.1016/j.aml.2012.02.049.

A. Zada, O. Shah, and R. Shah, “Hyers–Ulam stability of non-autonomous systems in terms of boundedness of Cauchy problems”, Applied Mathematics and Computation, vol. 271, pp. 512–518, Nov. 2015, doi: 10.1016/j.amc.2015.09.040.

X. Wang, M. Arif, and A. Zada, “β–Hyers–Ulam–Rassias stability of semilinear nonautonomous impulsive system”, Symmetry, vol. 11, no. 2, Art. No. 231, Feb. 2019, doi: 10.3390/sym11020231.

J. Wang, A. Zada, and W. Ali, “Ulam’s-type stability of first-order impulsive differential equations with variable delay in Quasi–Banach spaces”, International Journal of Nonlinear Sciences and Numerical Simulation, vol. 19, no. 5, pp. 553–560, Jul. 2018, doi: 10.1515/ijnsns-2017-0245.

A. Zada and S. Ali, “Stability analysis of multi-point boundary value problem for sequential fractional differential equations with non-instantaneous impulses”, International Journal of Nonlinear Sciences and Numerical Simulation, vol. 19, no. 7-8, pp. 763–774, Dec. 2018, doi: 10.1515/ijnsns-2018-0040.

A. Zada, W. Ali, and C. Park, “Ulam’s type stability of higher order nonlinear delay differential equations via integral inequality of Grönwall-Bellman-Bihari’s type”, Applied Mathematics and Computation, vol. 350, pp. 60–65, Jun. 2019, doi: 10.1016/j.amc.2019.01.014.

A. Zada and S. Shah, “Hyers-Ulam stability of first order nonlinear delay differential equations with fractional integrable impulses” Hacettepe Journal of Mathematics and Statistics, vol. 47, no. 5, pp. 1196-1205, Oct. 2018. [On line]. Available: http://bit.ly/2YWeGXq

A. Zada, S. Shaleena, and T. Li, “Stability analysis of higher order nonlinear differential equations in β –normed spaces”, Mathematical Methods in the Applied Sciences, vol. 42, no. 4, pp. 1151–1166, Mar. 2018, doi: 10.1002/mma.5419.

A. Zada, M. Yar, and T. Li, “Existence and stability analysis of nonlinear sequential coupled system of Caputo fractional differential equations with integral boundary conditions”, Annales Universitatis Paedagogicae Cracoviensis. Studia Mathematica, vol. 17, no. 1, pp. 103–125, Dec. 2018, doi: 10.2478/aupcsm-2018-0009.

K. Ravi, R.Murali and A. Selvan. “Ulam stability of a general nth order linear differential equation with constant coefficients”. Asian Journal of Mathematics and Computer Research, vol. 11, no. 1, pp. 61-68, Jan. 2016. [On line]. Available: http://bit.ly/2OPqt63

K. Ravi, R. Murali and A. Selvan. “Hyers-Ulam stability of nth order linear differential equation with initial and boundary condition”, Asian Journal of Mathematics and Computer Researc, vol. 11, no. 3, pp. 201-207, Mar. 2016. [On line]. Available: http://bit.ly/2yQwDbb

R. Murali and A. Selvan, “On the generalized Hyers-Ulam stability of linear ordinary differential equations of higher order”, International Journal of Pure and Applied Mathematics, vol. 117, special issue, no. 12 , pp. 317-326, 2017. [On line]. Available: http://bit.ly/2MPLtac

R. Murali and A. Selvan, “Hyers-Ulam stability of nth order differential equation”, Contemporary Studies in Discrete Mathematics, vol. 2, no. 1, pp. 45-50, 2018. [On line]. Available: http://bit.ly/2ONJDZY

R. Murali and A. Selvan, “Hyers-Ulam-Rassias stability for the linear ordinary differential equation of third order”, Kragujevac Journal of Mathematics, vol. 42, no. 4, pp. 579-590, 2018. [On line]. Available: http://bit.ly/31xtgCz

R. Murali and A. Selvan, “Hyers-Ulam stability of linear differential equations”, in Computational Intelligence, Cyber Security and Computational Models. Models and Techniques for Intelligent Systems and Automation. ICC3 2017. (Communications in Computer and Information Science, vol. 844), G. Ganapathi, A. Subramaniam, M. Graña, S. Balusamy, R. Natarajan, P. Ramanathan, Eds. Singapore: Springer, 2018, pp. 183–192, doi :10.1007/978-981-13-0716-4_15.

R. Murali and A. Selvan, “Hyers-Ulam stability of nth order system of differential equation”, American International Journal of Research in Science, Technology, Engineering and Mathematics, vol. 26, no. 1, pp. 71-75, 2019. [On line]. Available: http://bit.ly/2GUUgUn

Publicado
2019-08-14
Cómo citar
[1]
R. Murali y A. P. Selvan, «Hyers-Ulam stability of n ͭ ͪ order linear differential equation», Proyecciones (Antofagasta, En línea), vol. 38, n.º 3, pp. 553-566, ago. 2019.
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