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Semilinear delay evolution equations with measures subjected to nonlocal initial conditions. (English) Zbl 1409.34064

Summary: We prove a global existence result for bounded solutions to a class of abstract semilinear delay evolution equations with measures subjected to nonlocal initial data of the form \[ \begin{aligned} \mathrm du(t)=\{Au(t)+f(t,u_t)\}\mathrm dt+\mathrm dg(t),\quad & t\in\mathbb R_+,\\ u(t)=h(u)(t),\quad & t\in[-\tau,0],\end{aligned} \] where \(\tau\geq 0\), \(A:D(A)\subseteq X\to X\) is the infinitesimal generator of a \(C_0\)-semigroup, \(f:\mathbb R_+\times\mathcal R([-\tau,0];X)\to X\) is continuous, \(g\in BV_{\mathrm{loc}}(\mathbb R_+;X)\), and \(h:\mathcal R_b(\mathbb R_+;X)\to\mathcal R([-\tau,0];X)\) is nonexpansive.

MSC:

34K30 Functional-differential equations in abstract spaces
47D06 One-parameter semigroups and linear evolution equations
34K12 Growth, boundedness, comparison of solutions to functional-differential equations
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