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Hyers-Ulam-Rassias stability of the Banach space valued linear differential equations y ' =λy. (English) Zbl 1069.34079
The authors consider the Hyers-Ulam-Rassias stability problem for the equation y ˙=λy in a complex Banach space X, where λ 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].
34G10Linear ODE in abstract spaces
34D05Asymptotic stability of ODE