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On state-dependent delay partial neutral functional–differential equations. (English) Zbl 1119.35106
Summary: The purpose of this article is establish the existence of mild solutions for a class of abstract neutral functional-differential equations with state-dependent delay described by the form $\frac{d}{dt}D(u_t)=AD(u_t)+F (t,x_{\rho(t,x_t)}), \quad t\in I=[0,a],\tag{1}$
$x_0=\varphi\in{\mathcal B}, \tag{2}$ where $$A$$ is the infinitesimal generator of a compact $$C_0$$-semigroup of bounded linear operators $$(T(t))_{t\geq 0}$$ on a Banach space $$X$$; the function $$x_s:(-\infty,0]\to X$$, $$x_s(\theta)=x(s+ \theta)$$, belongs to some abstract phase space $${\mathcal B}$$ described axiomatically; $$F,G$$ are appropriate functions; and $$D\psi=\psi(0)-G(t, \psi)$$, where $$\psi$$ is in $${\mathcal B}$$.

##### MSC:
 35R10 Partial functional-differential equations 47D03 Groups and semigroups of linear operators
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