A note on partial functional differential equations with state-dependent delay. (English) Zbl 1109.34060
From the introduction: We establish the existence of mild solutions for a abstract Cauchy problem described in the form
\[
x'(t)=Ax(t)+f\bigl(t,x_{\rho(t,x_t)} \bigr),\quad t\in I= [0,a],\quad x_0=\varphi\in{\mathcal B},
\]
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 and \(f:I\times {\mathcal B}\to X\), \(\rho:I\times{\mathcal B}\to(-\infty,a]\) are appropriate functions.
MSC:
| 34K30 | Functional-differential equations in abstract spaces |
| 35R10 | Partial functional-differential equations |
| 47D06 | One-parameter semigroups and linear evolution equations |
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