Electronic Journal of Qualitative Theory of Differential Equations (Mar 2023)

Ulam–Hyers stability and exponentially dichotomic equations in Banach spaces

  • Adriana Buică

DOI
https://doi.org/10.14232/ejqtde.2023.1.8
Journal volume & issue
Vol. 2023, no. 8
pp. 1 – 10

Abstract

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For finite-dimensional linear differential systems with bounded coefficients, we prove that their exponential dichotomy on $\mathbb{R}$ is equivalent to their Ulam–Hyers stability on $\mathbb{R}$ with uniqueness. We also consider abstract non-autonomous evolution equations which are exponentially bounded and exponentially dichotomic and prove that Ulam–Hyers stability with uniqueness is maintained when perturbing them with a nonlinear term having a sufficiently small Lipschitz constant.

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