Hyers-Ulam Stability of a Free and Forced Vibrations
Download PDF
Authors: R. MURALI AND A. P. SELVAN
DOI: 10.46793/KgJMat2002.299M
Abstract:
In this paper, we study the Hyers-Ulam stability and Hyers-Ulam- Rassias stability of the general differential equation of the Free Damped Vibrations, Undamped Vibrations and Forced Vibrations by using initial conditions.
Keywords:
Hyers-Ulam stability, Hyers-Ulam-Rassias stability, free damped vibrations, undamped vibrations, forced vibrations and initial conditions.
References:
[1] M. Almahalebi, A. Chahbi and S. Kabbaj, A fixed point approach to the stability of a bi-cubic functional equations in 2-Banach spaces, Palest. J. Math. 5(2) (2016), 220–227.
[2] C. Alsina and R. Ger, On some inequalities and stability results related to the exponential function, J. Inequal. Appl. 2 (1998), 373–380.
[3] T. Aoki, On the stability of the linear transformation in Banach spaces, J. Math. Soc. Japan 2 (1950), 64–66.
[4] D. G. Bourgin, Classes of transformations and bordering transformations, Bull. Amer. Math. Soc. 57 (1951), 223–237.
[5] N. Brillouet-Belluot, J. Brzdek and K. Cieplinski, On some recent developements in Ulam’s type stability, Abstr. Appl. Anal. 2012 (2012), Article ID 716936, 41 pages.
[6] M. Burger, N. Ozawa and A. Thom, On Ulam stability, Israel J. Math. 193 (2013), 109–129.
[7] L. Cadariu, L. Gavruta and P. Gavruta, Fixed points and generalized Hyers-Ulam stability, Abstr. Appl. Anal. 2012 (2012), Article ID 712743, 10 pages.
[8] J. Chung, Hyers-Ulam stability theorems for Pexiders equations in the space of Schwartz distributions, Arch. Math. 84 (2005), 527–537.
[9] J. Huang and Y. Li, Hyers-Ulam stability of linear differential equations, J. Math. Anal. Appl. 426 (2015), 1192–1200.
[10] J. Huang, S. M. Jung and Y. Li, On Hyers-Ulam stability of non-linear differential equations, Bull. Korean Math. Soc. 52(2) (2015), 685–697.
[11] D. H. Hyers, On the stability of the linear functional equation, in: N. Raikhel (Ed.), Proceeding of the National Academy of Sciences of the United States of America, 27 (1941), 222–224.
[12] S. M. Jung, Hyers-Ulam-Rassias Stability of Functional Equation in Nonlinear Analysis, Springer Optimization and Its Applications 48, Springer, New York, 2011.
[13] Y. Li and Y. Shen, Hyers-Ulam stability fo linear differential equations of second order, Appl. Math. Lett. 23 (2010), 306–309.
[14] M. I. Modebei, O. O. Olaiya and I. Otaide, Generalized Hyers-Ulam stability of second order linear ordinary differential equation with initial condition, Advances in Inequalities and Applications 2014(36) (2014), 1–7.
[15] R. Murali, M. J. Rassias and V. Vithya, The general solution and stability of nonadecic functional equation in matrix normed spaces, Malaya Journal of Matematik 5(2) (2017), 416–427.
[16] R. Murali and A. P. Selvan, On the generalized Hyers-Ulam stability of linear ordinary differential equations of higher order, International Journal of Pure and Applied Mathematics 117(12) (2017), 317–326.
[17] R. Murali and A. P. Selvan, Hyers-Ulam stability of nth order differential equation, Contemporary Studies in Discrete Mathematics (CSDM) 2(1) (2018), 45–50.
[18] R. Murali and A. P. Selvan, Hyers-Ulam-Rassias stability for the linear ordinary differential equation of third order, Kragujevac J. Math. 42(4) (2018), 579–590.
[19] M. Obloza, Connections between Hyers and Lyapunov stability of the ordinary differential equations, Rocznik Nauk.-Dydakt. Prace Matematyczne 14 (1997), 141–146.
[20] M. Obloza, Hyers stability of the linear differential equation, Rocznik Nauk.-Dydakt. Prace Matematyczne 13 (1993), 259–270.
[21] Th. M. Rassias, On the stability of the linear mapping in Banach spaces, Proc. Amer. Math. Soc. 72 (1978), 297–300.
[22] K. Ravi, R. Murali and A. P. Selvan, Ulam stability of a general nth order linear differential equation with constant coefficients, Asian Journal of Mathematics and Computer Research 11(1) (2016), 61–68.
[23] K. Ravi, R. Murali and A. P. Selvan, Hyers-Ulam stability of nth order linear differential equation with initial and boundary condition, Asian Journal of Mathematics and Computer Research 11(3) (2016), 201–207.
[24] K. Ravi, J. M. Rassias and B. V. S. Kumar, Ulam-Hyers stability of undecic functional equation in quasi-beta-normed spaces fixed point method, Tbilisi Math. J. 9(2) (2016), 83–103.
[25] K. Ravi, J. M. Rassias, S. Pinelas and S. Suresh, General solution and stability of quattuordecic functional equation in quasi-beta-normed spaces, Advances in Pure Mathematics 6 (2016), 921–941.
[26] I. A. Rus, Ulam stabilities of ordinary differential equations in Banach space, Carpathian J. Math. 26(1) (2010), 103–107.
[27] S. Takahasi, T. Miura and S. Miyajima, On the Hyers-Ulam stability of the Banach space valued differential equation y′(t) = λy(t), Bull. Korean Math. Soc. 39 (2002), 309–315.
[28] S. M. Ulam, A Collection of Mathematical Problems, Interscience Publishers, New York, 1960.
[29] T. Z. Xu, On the stability of Multi-Jensens mappings in β-normed spaces, Appl. Math. Lett. 25(11) (2012), 1866–1870.
[30] A. Zada, O. Shah and R. Shah, Hyers-Ulam stability of non-autonomous systems in terms of boundedness of cauchy problems, Appl. Math. Comput. 271 (2015), 512–518.