Periodicity of solutions to the Cauchy problem for nonautonomous impulsive delay evolution equations in Banach spaces

Abstract

In this paper, we are concerned with the periodicity of solutions to the Cauchy problem for nonautonomous impulsive delay evolution equations with periodic inhomogenous terms in Banach spaces, where the operators in the linear part (possibly unbounded) depend on the time t and generate an evolution family of linear operators. We first establish two new Gronwall-Bellman-type inequalities, and then prove a new and general existence theorem for periodic mild solutions to the nonautonomous impulsive delay evolution equations, which extends essentially some existing results even for the autonomous case as well as for the case when impulsive perturbations or delays are absent.

Existence of positive periodic solutions for abstract evolution equations

“…Thus, compared with the existence results in [7,14,15], our conclusions are new in some respects in some respects.…”

“…The problems concerning periodic solutions of partial differential equations with delay are an important area of investigation since they can take into account seasonal fluctuations occurring in the phenomena appearing in the models, and have been studied by some researchers in recent years. The existence and asymptotic stability of periodic solutions of evolution equation with delay have attracted much attention, see [4,24,16,17,18,12,7,22,14,15].…”

“…Recently, In [22], under suitable assumptions, such as the ultimate boundedness of the solutions of equations, Wang and Zhu established a theorem on periodic solutions to equations of this kind by using the Horn fixed-point theorem. In [14,15], Liang et al also studied nonautonomous evolutionary equations with time delay and impulsive. Under the nonlinear term satisfying continuous and Lipschitzian, the proved the existence theorem for periodic mild solutions to the nonautonomous delay evolution equations by Horn's fixed point theorem or Sadovskii's Fixed Point Theorem.…”

Monotone iterative technique for delayed evolution equation periodic problems in Banach spaces

“…The problem concerning periodic solutions of partial differential equations with delay is an important area of investigation since they can take into account seasonal fluctuations occurring in the phenomena appearing in the models, and have been studied by some researchers in recent years. There has been a significant development in periodic solution of evolution equation with delay in Banach spaces, we refer to the references [8,12,19,20,22,27,31,32,28,29,33].…”

“…In [31], M. Kpoumiè et al studied the existence of a periodic solution for some partial functional differential equations with infinite delay in Banach spaces. Recently, some authors also discussed the periodic solutions for some nonautonomous delay impulsive evolutionary equations (see [29,33,28]). They established some existence results on periodic solutions to the equations under the ultimate boundedness of the solutions of the corresponding initial value problem.…”