Abstract
In this paper, we study a system governed by impulsive semilinear nonautonomous differential equations. We present the β –Ulam stability, β –Hyers–Ulam stability and β –Hyers–Ulam–Rassias stability for the said system on a compact interval and then extended it to an unbounded interval. We use Grönwall type inequality and evolution family as a basic tool for our results. We present an example to demonstrate the application of the main result.
Highlights
Differential equations are the key tools for modeling the physical problems in nature
Our main objective of this work is to discuss the uniqueness of solution for the given system and analyze the β–Hyers–Ulam–Rassias stability of semilinear nonautonomous system (2) with the help of evolution family
In the last few decades, many mathematicians showed their interests in the qualitative theory of impulsive differential equations
Summary
Differential equations are the key tools for modeling the physical problems in nature. Physical problems which have rapid changes are blood flows, biological systems such as heart beats, theoretical physics, engineering, control theory, population dynamics, mechanical systems with impact, pharmacokinetics, biotechnology processes, mathematical economy, chemistry, medicine and many more. These problems can be modeled by systems of differential equations with impulses. In 2012, Ulam type stability of impulsive differential equations were discussed by Wang et al [19]. Our main objective of this work is to discuss the uniqueness of solution for the given system and analyze the β–Hyers–Ulam–Rassias stability of semilinear nonautonomous system (2) with the help of evolution family. Different researchers are working to discuss stability analysis of different systems using evolution family. For more details of evolution family we prefer [20,28,37,38,39,40,41,42,43,44]
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