Existence of solutions for impulsive partial neutral functional differential equation with infinite delay (Q975252)

The existence of mild solutions for first-order impulsive semilinear neutral functional differential equations with infinite delay in Banach spaces \[ \begin{cases} \frac{d}{dt}(x(t)+g(t,x_t))=Ax(t)+f(t,x_t), \,\,\, t\in J=[0,b]\\ x_0=\phi\in {\mathcal B},\\ \Delta x(t_i)=I_i(x_{t_i}), \,\, i=1,2,\dots,n, \end{cases} \] is investigated. Here, \(A\) is the infinitesimal generator of an analytic semigroup of linear operators on a Banach space \(X,\) \(x_t: (-\infty,0]\to X,\) \(x_t(\theta)=x(t+\theta),\) belongs to some abstract space \({\mathcal B}\) defined axiomatically, \(g, f, I_i, i=1,2,\dots, n\) are appropriate functions, \(0<t_1<\dots<t_n<b\) are fixed numbers and \(\Delta \xi(t)\) represent the jump of the functions \(\xi\) at \(t,\) defined by \(\Delta\xi(t)=\xi(t^+)-\xi(t^-).\) Existence results are obtained without the compactness assumption on the associated semigroup, via the Hausdorff measure of noncompactness. An example illustrating the abstract results is also presented.