Optimal Hyers-Ulam’s constant for the linear differential equations. (English) Zbl 1351.34065
Summary: We obtain the optimal Hyers-Ulam’s constant for the first-order linear differential equation
\[
p(t)y'(t) - q(t)y(t) - r(t) = 0.
\]
MSC:
| 34D10 | Perturbations of ordinary differential equations |
| 34A30 | Linear ordinary differential equations and systems |
References:
| [1] | Ulam, SM: A Collection of Mathematical Problems. Interscience, New York (1960) · Zbl 0086.24101 |
| [2] | Hyers, DH: On the stability of the linear functional equation. Proc. Natl. Acad. Sci. USA 27, 222-224 (1941) · Zbl 0061.26403 · doi:10.1073/pnas.27.4.222 |
| [3] | Brillouët-Belluot, N, Brzdȩk, J, Ciepliński, K: On some recent developments in Ulam’s type stability. Abstr. Appl. Anal. 2012, Article ID 716936 (2012) · Zbl 1259.39019 · doi:10.1155/2012/716936 |
| [4] | Aoki, T: On the stability of the linear transformation in Banach spaces. J. Math. Soc. Jpn. 2, 64-66 (1950) · Zbl 0040.35501 · doi:10.2969/jmsj/00210064 |
| [5] | Rassias, TM: On the stability of the linear mapping in Banach spaces. Proc. Am. Math. Soc. 72, 297-300 (1978) · Zbl 0398.47040 · doi:10.1090/S0002-9939-1978-0507327-1 |
| [6] | Găvruţa, P: A generalization of the Hyers-Ulam-Rassias stability of approximately additive mappings. J. Math. Anal. Appl. 184, 431-436 (1994) · Zbl 0818.46043 · doi:10.1006/jmaa.1994.1211 |
| [7] | Obloza, M: Hyers stability of the linear differential equation. Rocz. Nauk.-Dydakt., Pr. Mat. 13, 259-270 (1993) · Zbl 0964.34514 |
| [8] | Obloza, M: Connections between Hyers and Lyapunov stability of the ordinary differential equations. Rocz. Nauk.-Dydakt., Pr. Mat. 14, 141-146 (1997) · Zbl 1159.34332 |
| [9] | Alsina, C, Ger, R: On some inequalities and stability results related to the exponential function. J. Inequal. Appl. 2, 373-380 (1998) · Zbl 0918.39009 |
| [10] | Miura, T, Takahasi, SE, Choda, H: On the Hyers-Ulam stability of real continuous function valued differential map. Tokyo J. Math. 24, 467-476 (2001) · Zbl 1002.39039 · doi:10.3836/tjm/1255958187 |
| [11] | Miura, T, Miyajima, S, Takahasi, SE: A characterization of Hyers-Ulam stability of first order linear differential operators. J. Math. Anal. Appl. 286, 136-146 (2003) · Zbl 1045.47037 · doi:10.1016/S0022-247X(03)00458-X |
| [12] | Takahasi, SE, Miura, T, Miyajima, S: On the Hyers-Ulam stability of the Banach space valued differential equation y′=λy \(y' = \lambda y\). Bull. Korean Math. Soc. 39, 309-315 (2002) · Zbl 1011.34046 · doi:10.4134/BKMS.2002.39.2.309 |
| [13] | Takahasi, SE, Takagi, H, Miura, T, Miyajima, S: The Hyers-Ulam stability constant of first order linear differential operators. J. Math. Anal. Appl. 296, 403-409 (2004) · Zbl 1074.47022 · doi:10.1016/j.jmaa.2003.12.044 |
| [14] | Jung, S-M: Hyers-Ulam stability of linear differential equations of first order. Appl. Math. Lett. 17, 1135-1140 (2004) · Zbl 1061.34039 · doi:10.1016/j.aml.2003.11.004 |
| [15] | Miura, T, Hirasawa, G, Takahasi, SE: Note on the Hyers-Ulam-Rassias stability for the first order linear differential equation y′(t)+p(t)y(t)+q(t)=\(0y'(t) + p(t)y(t) + q(t) = 0\). Int. Math. Sci. 22, 1151-1158 (2004) · Zbl 1079.47050 · doi:10.1155/S0161171204304333 |
| [16] | Jung, S-M: Hyers-Ulam stability of linear differential equations of first order (II). Appl. Math. Lett. 19, 854-858 (2006) · Zbl 1125.34328 · doi:10.1016/j.aml.2005.11.004 |
| [17] | Jung, S-M: Hyers-Ulam stability of linear differential equations of first order (III). J. Math. Anal. Appl. 311, 139-146 (2005) · Zbl 1087.34534 · doi:10.1016/j.jmaa.2005.02.025 |
| [18] | Wang, G, Zhou, M, Sun, L: Hyers-Ulam stability of linear differential equations of first order. Appl. Math. Lett. 21, 1024-1028 (2008) · Zbl 1159.34041 · doi:10.1016/j.aml.2007.10.020 |
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