Note on the stability for linear systems of differential equations
In this paper, by applying the fixed point alternative method, we give a necessary and sufficient condition in order that the first order linear system of differential equations $ \dot z(t) + A(t)z(t) + B(t) = 0 $ has the Hyers-Ulam-Rassias stability and find Hyers-Ulam stability constant under those conditions. In addition to that we apply this result to a second order differential equation $ \ddot y(t) + f(t)\dot y(t) + g(t)y(t) + h(t) = 0 $.
Keywords: Fixed point method; Hyers-Ulam-Rassias stability; System of dierential equations.
C.Alsina and R.Ger, On some inequalities and stability results related to the exponential function, J.of Inequal and Appl.. 2, (1998),373-380.
C. and Chicone , Ordinary differential equations with applications, Springer,New York, (2006).
C. and Corduneanu, Principles of differential and integral equations, Chelsea Publ.Company, New York,(1971).
D. H. Hyers, On the stability of the linear functional equation, Proc. Natl. Soc. USA 27 (1941), 222-224.
D. Popa, Hyers-Ulam-Rassias stability of a linear recurrence, J. Math. Anal. Appl. 309 (2005), 591-597.
H. and Amann, Ordinary differential equations, Walter de Gruyter,Berlin , (1990) .
H. Rezaei, S.-M. Jung and Th. M. Rassias, Laplace transform and Hyers-Ulam stability of linear differential equations, J. Math. Anal. Appl. 403 (2013), 244-251.
Hsu and,S.-B., Ordinary differential equations with applications, World Scientific, New Jersey, (2006).
J.B.Diaz and B.Margolis, A fixed point theorem of the alternative for contractions on a generalized complete metric space, Bull.Amer.Math.Soc., 74, (1968), 305-309.
J.Jost, Postmodern Analysis,Third Edition ,Springer (2005).
P. Gavruta and L. Gavruta, A new method for the generalized Hyers-Ulam-Rassias stability, Internat. J. Nonlinear Anal.2 (2010), 11-18.
R. Bellman, Stability Theory of Differential Equations, New York (1953).
S.-M. Jung, Hyers-Ulam stability of first order linear differential equations with constant coefficients, Math.Anal.Appl., 320 (2006),549-561.
S.-M. Jung,and Rassias, Generalized Hyers-Ulam stability of Riccati differential equation,Math.Ineq.Appl., 11(2008),No.4,777-782.
S.-M. Jung, Hyers-Ulam stability of linear differential equations of first order, Appl. Math. Lett. 17 (2004), 1135-1140.
S.-M. Jung, Hyers-Ulam stability of linear differential equations of first order, III, J. Math. Anal. Appl. 311 (2005),139-146.
S.-M. Jung, Hyers-Ulam stability of linear differential equations of first order, II, Appl. Math. Lett. 19 (2006), 854-858.
S. M. Ulam, Problems in Modern Mathematics, Chapter VI, Scince Editors, Wiley, New York, 1960.
T.Mura,S-E.Takahasi,H.Choda, On the Hyers-Ulam stability for real continuous function valued differentiable map, Tokyo J.Math. 24, (2001), 467-478.
T.Mura,S-E.Takahasi, H.Choda, On the Hyers-Ulam stability for real continuous function valued differentiable map,Tokyo J.Math. 24, (2001),467-478.
Th. M. Rassias, On the stability of the linear mapping in Banach spaces, Proc. Amer. Math. Soc. 72 (1978), 297-300.
Y.-H. Lee and K.-W. Jun, A generalization of Hyers-Ulam-Rassias stability of Jensen's equation, J. Math. Anal. Appl. 238 (1999), 305-315.
Y. Li and Y. Shen, Hyers-Ulam stability of nonhomogeneous linear differential equations of second order, Internat. J. Math. Math. Sci. 2009 (2009), Article ID 576852, 7 pages.