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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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