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On the periodic boundary value problem for fully nonlinear differential equations with finite delay in Banach spaces. (English) Zbl 1419.34082

For the following nonlinear evolution differential inclusion with a finite delay in a real Banach space \(E\): \[ \begin{aligned} & x^{\prime}(t) \in - A x(t) + F(t, x_t), \\ & x_0 = x_T, \end{aligned} \] with accretive operator \(A\) and the Caratheodory function \(F\) (\(x_t(\theta) = x(t + \theta), \theta\in [-r,0]\)), it is found a sufficient condition for existence of an integral solution in the space of continuous functions with value in \(E\). It also shown that under certain assumptions the set of integral solutions is nonempty and compact in \({\mathcal C}([-r, T]; E)\).

MSC:

34A60 Ordinary differential inclusions
34G25 Evolution inclusions
47H08 Measures of noncompactness and condensing mappings, \(K\)-set contractions, etc.
47H06 Nonlinear accretive operators, dissipative operators, etc.
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