$$(\omega ,Q)$$-periodic mild solutions for a class of semilinear abstract differential equations and applications to Hopfield-type neural network model
Edgardo Alvarez, Stiven Díaz, Rogélio Grau · Zeitschrift für angewandte Mathematik und Physik · 2023
Abstract In 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 ) = A x ( t ) + f ( t , x ( t ) ) , t ∈ R , on a Banach space X. Here, A is a closed linear operator which generates an exponentially stable $$C_0$$ C 0 -semigroup and the nonlinearity f satisfies 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.