In this paper, we are devoted to consider the periodic problem for the neutral evolution equation with delay in Banach space. By using operator semigroups theory and fixed point theorem, we establish some new existence theorems of periodic mild solutions for the equation. In addition, with the aid of a new integral inequality with delay, we present essential conditions on the nonlinear function to guarantee that the equation has an asymptotically stable periodic mild solution.

中文翻译：

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时滞中立型发展方程周期解的存在性与渐近稳定性

在本文中，我们致力于考虑Banach空间中带时滞的中立型演化方程的周期问题。利用算子半群理论和不动点定理，建立了该方程周期适解的一些新的存在性定理。另外，借助于新的带延迟的积分不等式，我们给出了非线性函数的必要条件，以确保方程具有渐近稳定的周期温和解。