The first order linear non-instantaneous impulsive evolution equations is given as;

$u'(t)=Au(t)~~ t\in[s_i,t_{i+1}],\,i\in\mathbb{N_0}:=\{0,1,2,...\}$

$u(t_i^+)=(E+B_i)u(t_i^-),\,\,i\in\mathbb{N}:=\{1,2,...\},$

$u(t)=(E+B_i)u(t_i^-),\,\, t\in(t_i,s_i],\,\, i\in\mathbb{N},$

$u(s_i^+)=u(s_i^-),\,\, i\in\mathbb{N}$

where $A:D(A)\subseteq X\rightarrow X$ is the Generator of a $C_0$-semigroup $\{T(t)\}_{t\geq 0}$ on a Banach space $X$ with a norm $\|.\|,\,B_i:X\rightarrow X,\,i\in\mathbb{N}$ is bounded linear operator,the sequences $\{s_i\}_{i\in\mathbb{N_0}}$ and $\{t_i\}_{i\in\mathbb{N_0}}$ are satisfied with the relation $t_i<s_i<t_{i+1},\,i\in\mathbb{N}$ and set $t_o=s_0=0$. Moreover $E$ denotes the standard identity.

Question: I want to know if any such second order non-instantaneous impulsive evolution equation exists in the literature? If yes, please refer me to that. TIA

The first order linear non-instantaneous impulsive evolution equations is given as;

$$u'(t)=Au(t)~~ t\in[s_i,t_{i+1}],\,i\in\mathbb{N_0}:=\{0,1,2,...\}$$

$$u(t_i^+)=(E+B_i)u(t_i^-),\,\,i\in\mathbb{N}:=\{1,2,...\},$$

$$u(t)=(E+B_i)u(t_i^-),\,\, t\in(t_i,s_i],\,\, i\in\mathbb{N},$$

$$u(s_i^+)=u(s_i^-),\,\, i\in\mathbb{N}$$

where $A:D(A)\subseteq X\rightarrow X$ is the generator of a $C_0$-semigroup $\{T(t)\}_{t\geq 0}$ on a Banach space $X$ with a norm $\|.\|,\,B_i:X\rightarrow X,\,i\in\mathbb{N}$ is bounded linear operator,the sequences $\{s_i\}_{i\in\mathbb{N_0}}$ and $\{t_i\}_{i\in\mathbb{N_0}}$ are satisfied with the relation $t_i<s_i<t_{i+1},\,i\in\mathbb{N}$ and set $t_o=s_0=0$. Moreover $E$ denotes the standard identity.

Question: I want to know if any such second order non-instantaneous impulsive evolution equation exists in the literature? If yes, please refer me to that. TIA