Hyers-Ulam stability of linear differential equations of first order. II. (English) Zbl 1125.34328

Jung, S.-M.

Appl. Math. Lett. 19, No. 9, 854-858 (2006).

Summary: Let \(X\) be a complex Banach space and let \(I\) be an open interval. For given functions \(g : I \rightarrow \mathbb C,\;h : I \rightarrow X\) and \(\varphi : I \rightarrow [0,\infty )\), we will solve the differential inequality \(\| y^{\prime}(t) + g(t)y(t)+h(t)\| \leq \varphi (t)\) for the class of continuously differentiable functions \(y : I \rightarrow X\) under some integrability conditions.
Part I, cf. Appl. Math. Lett. 17, No. 10, 1135–1140 (2004; Zbl 1061.34039); Part III, cf. J. Math. Anal. Appl. 311, No. 1, 139–146 (2005; Zbl 1087.34534).

MSC:

34G10	Linear differential equations in abstract spaces
34A40	Differential inequalities involving functions of a single real variable

Keywords:

Hyers-Ulam stability; Hyers-Ulam-Rassias stability; differential equation

Citations:

Zbl 1087.34534; Zbl 1061.34039

References:

[1]	Hyers, D. H.; Isac, G.; Rassias, Th. M., Stability of Functional Equations in Several Variables (1998), Birkhäuser: Birkhäuser Boston · Zbl 0894.39012
[2]	Jung, S.-M., Hyers-Ulam-Rassias Stability of Functional Equations in Mathematical Analysis (2001), Hadronic Press: Hadronic Press Palm Harbor · Zbl 0980.39024
[3]	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
[4]	Takahasi, S.-E.; Miura, T.; Miyajima, S., On the Hyers-Ulam stability of the Banach space-valued differential equation \(y^\prime = \lambda y\), Bull. Korean Math. Soc., 39, 309-315 (2002) · Zbl 1011.34046
[5]	Miura, T.; Jung, S.-M.; Takahasi, S.-E., Hyers-Ulam-Rassias stability of the Banach space valued linear differential equations \(y^\prime = \lambda y\), J. Korean Math. Soc., 41, 995-1005 (2004) · Zbl 1069.34079
[6]	Miura, T.; Miyajima, S.; Takahasi, S.-E., A characterization of Hyers-Ulam stability of first order linear differential operators, J. Math. Anal. Appl., 286, 136-146 (2003) · Zbl 1045.47037
[7]	Jung, S.-M., Hyers-Ulam stability of linear differential equations of first order, Appl. Math. Lett., 17, 1135-1140 (2004) · Zbl 1061.34039