Abstract In this paper, we study the existence of mild periodic solutions of abstract semilinear equations in a setting that includes several other types of equations such as delay differential equations, first-order hyperbolic partial differential equations, and reaction-diffusion equations. Under different assumptions on the linear operator and the nonhomogeneous function, sufficient conditions are derived to ensure the existence of mild periodic solutions in the abstract semilinear equations. When the semigroup generated by the linear operator is not compact, Banach fixed point theorem is used whereas when the semigroup generated by the linear operator is compact, Schauder fixed point theorem is employed. In applications, we apply the main results to establish the existence of periodic solutions in delayed red-blood cell models, age-structured models with periodic harvesting, and the diffusive logistic equation with periodic coefficients.

中文翻译：

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抽象半线性方程中周期解的存在及其在生物模型中的应用

摘要 在本文中，我们研究了抽象半线性方程的温和周期解在包括延迟微分方程、一阶双曲偏微分方程和反应扩散方程等几种其他类型方程的环境中的存在性。在对线性算子和非齐次函数的不同假设下，推导出充分条件来保证抽象半线性方程中存在温和周期解。当线性算子产生的半群不紧时，采用Banach不动点定理，而当线性算子产生的半群紧时，采用Schauder不动点定理。在应用中，我们应用主要结果来建立延迟红细胞模型中周期解的存在性，