The existence and uniqueness of local and global solutions for the impulsive functional differential equations with delay terms (Q2846550)

The authors consider the initial value problem for first-order semilinear impulsive functional differential equations NEWLINE\[NEWLINE\begin{aligned} & x'(t)=Ax(t)+f(t,x_t)\quad t\in [0,b]\setminus\{t_1,\dots,t_m\},\\ & x(t)=\phi(t), \quad t\in (-\infty,0],\\ & I_i(x_{t_{i}}) =x(t_i^+)-x(t_i^-), \quad i=1,2,\dots,m,\end{aligned}NEWLINE\]NEWLINE where \(f:[0,b]\times B\to X\) is a given function, \(A: D(A)\to X\) is the infinitesimal generator of the strongly continuous semigroup \(T(t)\), \(t\geq 0,\) \(B\) a phase space, \(\phi\in B\) and \(x(t_i^+),\) \(x(t_i^-)\) are the left and right limits of \(x(t)\) at \(t=t_i\), \(i=1,\dots,m\), respectively. Existence and uniqueness of local and global solutions are proved via the Leray-Schauder nonlinear alternative.