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Generalized Dichotomies and Hyers–Ulam Stability

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Abstract

We consider a semilinear and nonautonomous differential equation

$$\begin{aligned} x'=A(t)x+f(t,x) \quad t\ge 0, \end{aligned}$$
(1)

acting on an arbitrary Banach space X. Provided that the linear part \(x'=A(t)x\) exhibits a very general form of dichotomic behaviour and that the nonlinear term f is Lipschitz in the second variable (with a suitable Lipshitz constant), we prove that (1) admits two different forms of a generalized Hyers–Ulam stability. Moreover, we obtain the converse result which shows that under suitable additional assumptions, the presence of these two forms of a generalized Hyers–Ulam stability for the linear equation \(x'=A(t)x\) implies that it exhibits this general dichotomic behaviour.

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Data sharing not applicable to this article as no datasets were generated or analysed during the current study.

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Funding

D. D. was supported in part by Croatian Science Foundation under the project IP-2019-04-1239 and by the University of Rijeka under the projects uniri-prirod-18-9 and uniri-prprirod-19-16.

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  1. Faculty of Mathematics, University of Rijeka, Rijeka, 51000, Croatia

    Davor Dragičević

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  1. Davor Dragičević
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Correspondence to Davor Dragičević.

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Dragičević, D. Generalized Dichotomies and Hyers–Ulam Stability. Results Math 79, 37 (2024). https://doi.org/10.1007/s00025-023-02071-6

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