The authors consider the second-order nonlinear differential equation \[ x''(t)+a(t)x'(t)+f(t,x(t))=0, \] where \(a(t)\in C(\mathbb{R},\mathbb{R})\), \(a(t+\omega)\equiv a(t)\), \(f(t,u)\in C(\mathbb{R}\times\mathbb{R},\mathbb{R})\), \(f(t+\omega,u)\equiv f(t,u)\) and investigate existence of positive periodic solutions using a fixed point theorem for cones [see e.g. \textit{D. Guo} and \textit{V. Lakshmikantham}, Nonlinear problems in abstract cones. Boston, MA: Academic Press (1988; Zbl 0661.47045) or \textit{K. Deimling}, Nonlinear Functional Analysis. Berlin etc.: Springer (1985; Zbl 0559.47040)]. They extend some of results achieved by \textit{Y. Li} [Chin. Ann. Math., Ser. B 25, No. 3, 413--420 (2004; Zbl 1073.34041)], where criteria for existence of positive periodic solutions of the equation \[ x''(t)+f(t,x(t))=0, \] are proved.