Periodic solutions for second order damped boundary value problem with nonnegative Green’s functions

In this article, we study the existence of positive periodic solutions of second order damped boundary value problem u″+p(t)u′+q(t)u=g(t,u,u′)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$u'' + p(t)u'+q(t)u = g(t,u,u')$\end{document}, u(0)=u(T)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$u(0) = u(T )$\end{document}, u′(0)=u′(T)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$u'(0) = u'(T )$\end{document}. The main tools are the nonlinear alternative principle of Leray–Schauder and Schauder’s fixed point theorem. We emphasize that the damped term and nonnegative Green’s functions are the key points. We also apply the results to examples for testing. Some recent results in the literature are improved and generalized.