Ulam-Hyers stability and exponentially dichotomic evolution equations in Banach spaces

Abstract:

In 1941 D. Hyers gave an answer to the following question of S. Ulam. "Suppose that f satisfies only approximately the equation f(x+y)=f(x)+f(y). Then does there exist a solution of this equation which f approximates?" Since then, this type of stability was studied for functional equations, difference equations, and differential equations, too. The first notable result for differential equations is that $x'=\lambda x$ is Ulam-Hyers
stable on $\mathbb{R}$ if and only if $\lambda\neq 0$. In this talk we prove that the system $X'=AX$ is Ulam-Hyers stable  on $\mathbb{R}$  if and only if $A$ is hyperbolic. We generalize this result for evolution
families in Banach spaces using results from the book Evolution Semigroups in Dynamical Systems and Differential Equations by C. Chicone and Y. Latushkin. The stability is maintained when adding a nonlinear term which is globally Lipschitz and whose Lipschitz constant is sufficiently small.

Video of the talk.