f_2(t, x, 0)$, and $ f_3(t, x, 0)$ satisfies (D), (N), (P), respectively. Then ( or ) admits a unique positive time-periodic solution $u^*(t, x)$ if and only if $$\lambda_i>\lambda_i^p,$$ where $\lambda_i^p$ is the positive number such that $\mu_i^n(\lambda_i^p)=0$.

Preliminary
===========

In this section, we present some notations and preliminary materials to be used in the proofs of the main results in Section 4. We first present a technical lemma in subsection 3.1. Then we present a comparison principle for solutions of some linear nonlocal evolution equations in subsection 3.2. Finally we present some basic properties of principal spectrum points of the regular eigenvalue problem in subsection 3.3. Throughout this section, for $1\leq i\leq 3$, let \[hi\] h\_i(t, x; \_i)=-b\_i(x)+\_i m\_i(t, x), and \[hathi\] h\_i(x;\_i)=-b\_i(x)+\_im\_i(x), where $\widehat m_i$ is as in . For $D_i$ defined in , set \[hathi-min-max\] h\_[i, ]{}(\_i)=\_[x|D\_i]{} h\_i(x;\_i), h\_[i, ]{}(\_i)=\_[x|D\_i]{}h\_i(x;\_i). If there is no confusion, we may omit $\lambda_i$, and abbreviate the notations in , , and as $h_i(t, x)$, $\widehat h_i(x)$ and $\widehat h_{i, \max}$, respectively.

A technical lemma
-----------------

In this subsection, we provide a useful technical lemma.

\[technical-lm\] Let $1\le i\le 3$. For any fixed $0<\lambda_i\in \R$, any $m_i\in \mathcal X_i$ and any 