Existence results for impulsive neutral functional integro-differential equations in Banach space (Q2792068)

In this paper, the following impulsive initial value problem for neutral functional integro-differential equations NEWLINE\[NEWLINE\left\{\begin{aligned} & \frac{d}{dt}[x(t)-g(t,x_t)]=Ax(t)+f\left(t,x_t,\int_0^tk(t,s,x_s)ds\right), ~~ t\in J:=[0,b],\;t\neq t_k,\\ &\Delta |_{t=t_k}=I_k(x(t_k^-)), ~~ k=1, 2, \ldots, m,\\ &x_0=\phi, \end{aligned}\right. NEWLINE\]NEWLINE is considered, where \(A\) is the infinitesimal generator of an analytic semigroup of bounded linear operators \(\{T(t), t\geq 0\}\) on a Banach space \(X,\) \(g: J\times {\mathcal B}_h\to X,\) \(k: J\times J\times {\mathcal B}_h\to X\) and \(f: J\times {\mathcal B}_h\times X\to X\) are given functions, \({\mathcal B}_h\) an abstract phase space, \(0=t_0<\ldots<t_m<t_{m+1}=b,\) \(I_k\in C(X, X) ~ (k=1,2,\ldots,m)\) are bounded functions, \(\Delta x|_{t=t_k}=x(t_k^+)-x(t_k^-)\) and \(x(t_k^-), x(t_k^+)\) represent the left and right limits of \(x(t)\) at \(t=t_k\), respectively.NEWLINENEWLINEThe existence of mild solutions is proved via the Krasnoselskii-Schaefer fixed point theorem.NEWLINENEWLINEFor the entire collection see [Zbl 1325.35002].