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Hyers-Ulam-Rassias stability of the Banach space valued linear differential equations $$y'=\lambda y$$. (English) Zbl 1069.34079
The authors consider the Hyers-Ulam-Rassias stability problem for the equation $${\dot y} = \lambda y$$ in a complex Banach space $$X$$, where $$\lambda$$ is a complex number. The main result states that if $$f$$ is a strongly differentiable approximate solution of the above equation, then there exists an exact solution, which approximates $$f$$. The authors deduce interesting consequences and compare the corollaries of the main result with some stability theorems obtained by S.-E. Takahasi, T. Miura and S. Miyajima [Bull. Korean Math. Soc. 39, No. 2, 309–315 (2002; Zbl 1011.34046)] and by S.-M. Jung and K. Lee [Hyers-Ulam-Rasias stability of linear differential equations, to appear].

##### MSC:
 34G10 Linear differential equations in abstract spaces 34D05 Asymptotic properties of solutions to ordinary differential equations
##### Keywords:
Hyers-Ulam-Rassias stability; differential equation
Zbl 1011.34046
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