Ulam–Hyers stability and exponentially dichotomic equations in Banach spaces

Abstract

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.

Keywords

• ulam–hyers stability
• evolution family
• nonautonomous
• mild solution
• exponential dichotomy
• small lipschitz constant