Periodic boundary value problem for impulsive evolution equations with noncompact semigroup (Q6155159)

In this paper the authors study the following periodic boundary value problem of a first order semilinear impulsive evolution equation in a Banach space \(E\) \[ \begin{cases} u'(t)+Au(t)=f(t,u(t)), ~~ t\in J=[0,\omega], ~~ t\ne t_k,\\ \Delta u|_{t=t_k}=I_k(u(t_k)), ~~ k=1,2,\dots, m,\\ u(0)=u(\omega), \end{cases} \] where \(A : D(A)\subset E\to E\) is a linear operator and \(-A\) generates a \(C_0\)-semigroup \(T(t)\), \((t\ge 0)\), \(0=t_0<t_1<\cdots<t_m<t_{m+1}=\omega\), \(f : J \times E \to E \) and \(I_k : E \to E\) \((k = 1,2,\dots, m)\) are continuous, \(\Delta u|_{t=t_k}=u(t_k^+) - u(t_k^-)\), where \(u(t_k^+)\) and \(u(t_k^-)\) represent the right and left limits of \(u(t)\) at \(t =t_k\), respectively. Existence and uniqueness results of mild solution are obtained via Sadovskii's fixed point theorem under growth conditions and measure of noncompactness conditions on \(f\) and \(I_k\), without assuming the compactness of the semigroup. An example illustrating the obtained results is also presented.