Then $m_{i, \epsilon}\in{\mathcal X}_2$, $$\|m_i-m_{2,\epsilon}\|\leq \|m_i-\overline m_{i, \epsilon}\|+\|\widehat h_{i, \epsilon}-\overline h_{i, \epsilon}\|\leq\|m_i-\overline m_{i, \epsilon}\|+\|\widehat h_{i, \epsilon}-\widetilde h_{i, \epsilon}\|+\|\widetilde h_{i, \epsilon}-\overline h_{i, \epsilon}\|\leq \epsilon,$$ and $$-b_i(x)+\widehat m_{i,\epsilon}(x)=\widehat h_{i, \epsilon}(x).$$ Therefore, $-b_i+\widehat m_{i, \epsilon}$ is $C^N$, attains its maximum at some point $x_0\in {\rm Int}D_i$, and the partial derivatives of $-b_i+\widehat m_{i, \epsilon}$ up to order $N-1$ at $x_0$ are zero. The lemma is thus proved.

Comparison Principle
--------------------

In this subsection, we present the comparison principle for the solutions to the following evolution equations associated to the eigenvalue problem with $\lambda_i\ge 0$, $$\label{dirichlet-ori-evol}
 \p_t u(t, x)=\int_D \kappa(y-x)u(t, y)dy-u(t, x)+ \lambda_1 m_1(t,x)u(t, x) \quad \text{ in } \bar D,$$ $$\label{neumann-ori-evol}
 \p_t u(t, x)=\int_D \kappa(y-x)[u(t, y)-u(t, x)]dy+ \lambda_2 m_2(t,x)u(t, x) \quad \text{ in } \bar D,$$ and $$\label{periodic-ori-evol}
 \p_t u(t, x)=\int_{\R^N} \kappa(y-x)[u(t, y)-u(t, x)]dy+\lambda_3 m_3(t, x)u(t, x) \quad \text{ in } \R^N,$$ where $m_i(t, x)\in \mathcal X_i(i=1, 2, 3)$.

By general semigroup theory, (resp. , ) generates evolution families $\{\Phi_1(t, s; m_1)\}$ (resp. $\{\Phi_2(t, s; m_2), \Phi_3(t, s; m_3)\}$) on $X_1$ (resp. $X_2$, $X_3$), that is, for any $u_0\in X_1$ (resp. $u_0\in X_2, u_0\in X_3$), $u(t, x; s, u_0, m_1):=(\Phi_1(t, s; m_1)u_0)(x)$ (resp. $u(t, x; s, u_0, m_2):=(\Phi_2(t, s; m_2)u_0)(x)$, $u(t, x; s, 