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Existence of solutions for impulsive partial neutral functional differential equation with infinite delay. (English) Zbl 1198.34177
The existence of mild solutions for first-order impulsive semilinear neutral functional differential equations with infinite delay in Banach spaces $$\cases \frac{d}{dt}(x(t)+g(t,x_t))=Ax(t)+f(t,x_t), \,\,\, t\in J=[0,b]\\ x_0=\phi\in {\Cal B},\\ \Delta x(t_i)=I_i(x_{t_i}), \,\, i=1,2,\dots,n, \endcases$$ 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 ${\Cal B}$ 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.

##### MSC:
 34K30 Functional-differential equations in abstract spaces 34K40 Neutral functional-differential equations 34K45 Functional-differential equations with impulses 47D03 (Semi)groups of linear operators
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##### References:
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