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Existence of solutions for delay evolution equations with nonlocal conditions. (English) Zbl 1381.34096

This paper discusses the existence of mild solutions to the nonlocal Cauchy problem for evolution equations with delay \[ \begin{aligned} u'(t)+Au(t)&=f(t, u(t),u_t), \quad t\in [0,a],\\ u_0&=\phi+g(u), \end{aligned} \] where \(-A\) is the infinitesimal generator of an equicontinuous semigroup in a Banach space \(X\), \(a,q>0\), \(\phi\in C([-q,0],X)\), \(f:[0,a]\times X\times C([-q,0],X)\rightarrow X\) and \(g:C([0,a], X) \rightarrow C([-q,0],X)\). The main tools are the Kuratowski measure of noncompactness and Sadovskii’s fixed point theorem.

MSC:

34K30 Functional-differential equations in abstract spaces
34K10 Boundary value problems for functional-differential equations
47N20 Applications of operator theory to differential and integral equations
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