A Priori Bounds for Periodic Solutions of a Kind of Second Order Neutral Functional Differential Equation with Multiple Deviating Arguments

The main aim of this paper is to use the continuation theorem of coincidence degree theory for studying the existence of periodic solutions to a kind of neutral functional differential equation as follows:\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ {\left( {x{\left( t \right)} - {\sum\limits_{i = 1}^n {c_{i} x{\left( {t - r_{i} } \right)}} }} \right)}^{{\prime \prime }} = f{\left( {x{\left( t \right)}} \right)}{x}\ifmmode{'}\else$'$\fi{\left( t \right)} + g{\left( {x{\left( {t - \tau } \right)}} \right)} + p{\left( t \right)}. $$\end{document} In order to do so, we analyze the structure of the linear difference operator A : C2π →C2π, \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ {\left[ {Ax} \right]}{\left( t \right)} = x{\left( t \right)} - {\sum\nolimits_{i = 1}^n {c_{i} x{\left( {t - r_{i} } \right)}} } $$\end{document} to determine some fundamental properties first, which we are going to use throughout this paper. Meanwhile, we also prove some new inequalities which are useful for estimating a priori bounds of periodic solutions.