$$(\omega ,Q)$$-periodic mild solutions for a class of semilinear abstract differential equations and applications to Hopfield-type neural network model

Abstract

AbstractIn this paper, we investigate the existence and uniqueness of$$(\omega ,Q)$$(ω,Q)-periodic mild solutions for the following problem$$\begin{aligned} x'(t)=Ax(t)+f(t,x(t)),\quad t\in \mathbb {R}, \end{aligned}$$x′(t)=Ax(t)+f(t,x(t)),t∈R,on a Banach spaceX. Here,Ais a closed linear operator which generates an exponentially stable$$C_0$$C0-semigroup and the nonlinearityfsatisfies suitable properties. The approaches are based on the well-known Banach contraction principle. In addition, a sufficient criterion is established for the existence and uniqueness of$$(\omega ,Q)$$(ω,Q)-periodic mild solutions to the Hopfield-type neural network model.