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Hyers-Ulam stability of linear differential equations of first order. II. (English) Zbl 1125.34328
Summary: Let $X$ be a complex Banach space and let $I$ be an open interval. For given functions $g : I \rightarrow \Bbb C,\ h : I \rightarrow X$ and $\varphi : I \rightarrow [0,\infty )$, we will solve the differential inequality $\Vert y^{\prime}(t) + g(t)y(t)+h(t)\Vert \leq \varphi (t)$ 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).

34G10Linear ODE in abstract spaces
34A40Differential inequalities (ODE)
Full Text: DOI
[1] Hyers, D. H.; Isac, G.; Rassias, Th.M.: Stability of functional equations in several variables. (1998) · Zbl 0907.39025
[2] Jung, S. -M.: Hyers--Ulam--rassias stability of functional equations in mathematical analysis. (2001) · 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’=${\lambda}$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’=${\lambda}$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