Periodic mild solutions of infinite delay not instantaneous impulsive evolution inclusions
The authors consider the following first-order problem with infinite delay and non-instantaneous impulses: \[ u^{\prime}(t)+A(t)u(t)\in F\left( t,u(t),u_{t}\right) \text{ for }t\in I_{k}\text{, }k=0,1,\dots \] \[ u(t)=g_{k}\left( t,u\left( t_{k}^{-}\right) \right) \text{ for }t\in J_{k}\text{, }k=1,2,\dots \] \[ u(t)=\phi(t)\text{ for }t\in\left( -\infty,0\right] \] in a real Banach space \(E\), where \(I_{0}=[0,t_{1}]\), \(I_{k}=\left( s_{k},t_{k+1}\right] \), \(J_{k}=\left( t_{k},s_{k}\right] \), \(0<t_{1} <s_{1}<t_{2}<{\dots}<s_{m-1}<t_{m}<s_{m}<t_{m+1}=T<s_{m+1}<t_{m+2}<{\dots}<\infty\) and \(u_{t}\left( \theta\right) =u(t+\theta)\) for \(\theta\in\left( -\infty,0\right] \). \ It is proven that there exists at least one \(T\)-periodic mild solution on \(\left[ 0,\infty\right) \), where \(\left\{ A(t)\right\} \) is a \(T\)-periodic family of closed, densely defined linear unbounded operators from \(E\) to \(E\), \(F:I_{k}\times E\times\mathcal{B} \rightarrow2^{E}\) is compact and convex valued, \(T\)-periodic in \(t\) and \(L^{1}\)-Carathéodory, maps bounded sets to bounded sets and satisfies a condition involving Kuratowski's measure of noncompactness, where \(\mathcal{B}\) is a seminormed linear phase space of functions mapping \(\left( -\infty,0\right] \) into \(E\), \(g_{k}:J_{k}\times E\rightarrow E\) is continuous, is \(T\)-periodic in \(t\), maps bounded sets into bounded sets, satisfies a noncompactness condition and are uniformly integrably bounded and \(\phi:\left( -\infty,0\right] \rightarrow E\). \ The proof applies the fixed point theorems due to Darbo and Sadovskii. \ An example is given to finish the paper.
scientific article; zbMATH DE number 7438398
scientific article