On a periodic prey-predator system with infinite delays

Based on the theory of coincidence degree, the existence of positive periodic solutions is established for a periodic prey-predator system with infinite delays \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document} $$\begin{gathered} \dot x\left( t \right) = x\left( t \right)\left[ {\alpha \left( t \right) - \gamma \left( t \right)y\left( t \right) - \gamma \left( t \right)\int_0^\infty {K_1 } \left( {t,s} \right)y\left( {t - s} \right)ds - } \right. \hfill \\ \left. {\int_0^\infty {\int_0^\infty {R_1 \left( {t,s,\theta } \right)y\left( {t - s} \right)y\left( {t - \theta } \right)d\theta ds} } } \right], \hfill \\ \dot y\left( t \right) = y\left( t \right)\left[ { - \beta \left( t \right) + \mu \left( t \right)x\left( t \right) + \mu \left( t \right)\int_0^\infty {K_2 } \left( {t,s} \right)x\left( {t - s} \right)ds} \right. + \hfill \\ \left. {\int_0^\infty {\int_0^\infty {R_2 \left( {t,s,\theta } \right)x\left( {t - \theta } \right)x\left( {t - s} \right)d\theta ds} } } \right], \hfill \\ \end{gathered} $$ \end{document} where α, γ, β, μ are positive continuous ω-periodic functions, Ki∈C(R×[0, ∞), (0, ∞)) (i=1, 2) are ω-periodic with respect to their first arguments,. respectively, Ri∈C(R×[0, ∞)×[0, ∞), (0, ∞)) (i=1, 2) are ω-periodic with respect to their first arguments, respectively.