Muhammad Arif

In this paper, we study a system governed by impulsive semilinear nonautonomous differential equations. We present the $\beta$ –Ulam stability, $\beta$ –Hyers–Ulam stability and $\beta$ –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.

Let $$x_{\mathcal{V},f}$$ be the solution of the discrete Cauchy problem $$\zeta{(n+1)} = \mathcal{B}(n)\zeta{(n)}+ f(n)$$, $$\zeta(0)= 0$$, where $$\mathcal{B}(n)$$ is a sequence of bounded linear operators on a Banach space $$\mathcal{X}$$. We prove that the discrete evolution family $$\mathcal{V}:= \{{\mathbb{V}(n,k): n,k\in \mathcal{Z}_+, n\geq k}\}$$ of bounded linear operators acting on a complex Banach space $$\mathcal{X}$$ is uniformly exponentially stable if and only if for each $$f\in \mathcal{C}_{AAP_0}(\mathcal{Z}_+,\mathcal{X})$$ the solution $$x_{\mathcal{V},f}$$ belongs to $$\mathcal{C}_{AAP_0}(\mathcal{Z}_+,\mathcal{X})$$, where $$\mathcal{C}_{AAP_0}(\mathcal{Z}_+,\mathcal{X})$$ is the space of asymptotically almost periodic sequences. We use the approach of discrete evolution semigroups.