The author studies the large-time behavior of solutions to the following semi-linear non-instantaneous impulsive evolution equations
\[ \begin{aligned} u'(t)=A\,u(t)+f(t,u),\,\,t\in [s_i,t_{i+1}],\,\, i=1,2,\dots,\\ u(t_{i}^{+})=\big(E+B_i)\,u(t_{i}^{-})+J_i\big(u(t_{i}^{-})\big),\,\,i=1,2,\dots,\\ u(t)=\big(E+B_i)\,u(t_i^-)+J_i\big(u(t_i^-)\big),\,\,t\in(t_i,s_i),\,\,i=1,2,\dots,\\ u(s_i^+)=u(s_i^-),\,\,i=1,2,\dots, \end{aligned}\tag{1} \]
where \(A: D(A)\subseteq X\mapsto X\) is the infinitesimal generator of a \(C_0\) semigroup \(\{T(t): t\geq 0\}\) on a Banach space \((X,||\cdot||)\), \(B_i:X\mapsto X\) are bounded linear operators, \(f:[0,\infty)\times X\mapsto X\), \(J_i:X\mapsto X\), \(t_i<s_i<t_{i+1},\,i=1,2,\dots,\), \(t_0=s_0=0\), and \(E\) is the identity operator.
Supposing that \(\{T(t): t\geq 0\}\) is exponentially stable (\(||T(t)||\leq L\,e^{\omega\,t},\,t\geq 0,\,\omega<0)\), under some conditions on the number \(r(t)\) of impulsive points on \((0,t)\) and on impulsive jump operators \(B_i\), the author establishes the asymptotical stability of mild solutions of the system (1) in the linear case \((f=0, J_i=0)\).
In the case when the associated to (1) linear system is exponentially stable; \(\displaystyle ||f(u)||\leq C\,||u||,\, ||J_i(u)||\leq C\,||u||,\,i=1,2,\dots,\) \(\forall u\in X\); \(\displaystyle ||f(u)-f(v)||\leq C\,||u-v||,\, ||J_i(u)-J(v)||\leq C\,||u-v||,\,. i=1,2,\dots,\), \(\forall u, v\in X, i=1,2,\dots\) and \(\rho(t)\leq c_0\,t, \forall t>0\), the author proves that the mild solution of the system (1) is globally asymptotically stable in the semi-linear case too.