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

Abstract

In this paper, we investigate the existence and uniqueness of (omega ,Q)-periodic mild solutions for the following problem x′(t)=Ax(t)+f(t,x(t)),t∈R,\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\begin{aligned} x'(t)=Ax(t)+f(t,x(t)),\\quad t\\in \\mathbb {R}, \\end{aligned}$$\\end{document}on a Banach space X. Here, A is a closed linear operator which generates an exponentially stable 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)-periodic mild solutions to the Hopfield-type neural network model.