Periodic solutions of semilinear multivalued and functional evolution equations in Banach spaces (Q609420)

The authors consider the semilinear evolution inclusion \[ x^{\prime }(t)+A(t)x(t)\in F(t,x(t))\tag{1} \] in a Banach space. For the first theorem, it is assumed that the linear operators \(\left\{ A(t):t\in [ a,b]\right\} \) have a common domain which is dense and generate a strongly continuous evolution system, and \(F\) is convex- and compact-valued, strongly measurable in \(t\), upper semicontinuous in \(x\), integrably bounded and satisfies the property that there exists a \(k\in L_{+}^{1}([a,b])\) such that \(\chi (F(t,\Omega ))\leq k(t)\chi (\Omega )\) for any bounded set \(\Omega \), where \(\chi \) is the Hausdorff measure of noncompactness, among other conditions. \ It is then proven that the set of mild solutions to (1) with periodic boundary conditions \(x(a)=x(b)\) is non-empty and compact. The proof uses a fixed point theorem and other results for condensing multivalued functions from [\textit{M. Kamenskii, V. Obukhovskii} and \textit{P. Zecca}, Condensing Multivalued Maps and Semilinear Differential Inclusions in Banach Spaces. de Gruyter Series in Nonlinear Analysis and Applications. 7. Berlin: de Gruyter (2001; Zbl 0988.34001)]. Using a different assumption involving the measure of noncompactness, the authors use the same techniques to show that the set of periodic mild conditions is non-empty and compact for the functional problem \[ x^{\prime }(t)+A(t)x(t)\in F(t,x_{t}) \] where for each \(t\in [ a,b]\), \(x_{t}(s)=x(t+s)\) for \(s\in [ -h,0]\).