These days Hyers-Ulam stability is an active area of research. The notion was arise from the answer of Hyers to the famous question of Ulam. Many research articles deal with Hyers-Ulam stability of functional equations, differential equations, difference equations, partial differential equations, fractional differential equation etc. But these results did't catch attention of applied mathematicians. They still using the classical notions of stability (e.g asymptotic stability etc), like people working in mathematical biology, mathematical physics, etc.
Dear Sir, I think that these notions Hyers-Ulam stability and asymptotic stability are two different things. The first is about static property, so it is studied an equation and its solution and the stability of the existence of solution under perturbation. This could be interesting for applied numerical experts: stability of a numerical scheme. The asymptotic stability is a dynamic matter: how solutions of evolution equations behave in time passing to infinity.
Dear Sir, I think that these notions Hyers-Ulam stability and asymptotic stability are two different things. The first is about static property, so it is studied an equation and its solution and the stability of the existence of solution under perturbation. This could be interesting for applied numerical experts: stability of a numerical scheme. The asymptotic stability is a dynamic matter: how solutions of evolution equations behave in time passing to infinity.
I endorse Professor Michal Feckan answer. Hyers Ulam Stability concept is a general notion which gives the difference/bounds between the exact and its perturb equations. In numerical analysis (applied mathematics) researchers analyze the error analysis with some test problems, which is the sub case of HU-stability. So they are using it but this is a fact that, most of the researchers working with HU-stability concepts are experts in qualitative theory. In my opinion, with passage of time people working in numerical analysis will familiar with the notion and they first will do HU-stability results and then as particular examples they will take the error analysis, as a sub case. In short, HU-stability concept will be/is a bridge between qualitative analysis and numerical analysis
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