$\mu_i^n(\mathcal{L}_i+\lambda_i m_i)$ is called [*the principal eigenvalue*]{} of if with $\mu_i=\mu_i^n(\mathcal{L}_i+\lambda_i m_i)$ has a positive solution in $\mathcal{X}_i$.

-   A real number $\lambda_i^p(m_i)$ is called a [*principal spectrum point*]{} of the weighted eigenvalue problem if $\mu_i^n(\mathcal{L}_i+\lambda_i^p(m_i) m_i)=0$. When $\lambda_i^p(m_i)$ is a principal spectrum point of and $\mu_i^n(\mathcal{L}_i+\lambda_i^p(m_i) m_i)$ is a principal eigenvalue of , $\lambda_i^p(m_i)$ is also called a [*principal eigenvalue*]{} of .

If not confusion occurs, we may write $\mu_i^n(\mathcal{L}_i+\lambda_i m_i)$ and $\lambda_i^p(m_i)$ as $\mu_i^n(\lambda_i)$ and $\lambda_i^p$, respectively.

Let $$\label{x-d-space}
X_1=X_2=\{u\in C(\bar D, \R)\}$$ with norm $\|u\|_{X_i}=\sup_{x\in\bar D}|u(x)|$ ($i=1,2$), $$\label{x-p-space}
X_3=\{u\in C({\ensuremath{\mathbb{R}}}^N,{\ensuremath{\mathbb{R}}})\,|\, u(x+p_j{\bf e_j})=u(x),\quad x\in\R^N,\, j=1,2,\cdots,N\}$$ with norm $\|u\|_{X_3}=\max_{x\in{\ensuremath{\mathbb{R}}}^N}|u(x)|$. Set $$\label{x-d-positive-cone}
X_i^+=\{u\in X_i\,|\, u\geq 0\},$$ and $$\label{x-d-positive-interior}
X_i^{++}=
\begin{cases}
\{u\in X_i^+\,|\, u(x)>0,\quad x\in \bar D\},\quad & i=1,2,\\
\{u\in X_i^+\,|\, u(x)> 0,\quad x\in{\ensuremath{\mathbb{R}}}^N\},\quad & i=3.
\end{cases}$$ In the case that $m_i(t, x)\equiv m_i(x)$, consider $$\label{autonomous-weight-pev}
K_iu-B_iu+\lambda_i m_iu=0,$$ and $$\label{autonomous-pev}
K_iu-B_iu+\lambda_i m_iu=\mu_i u,$$ in $X_i$ ($i=1,2,3$). Similarly, define [*the principal spectrum point*]{} $\mu_i^n(\lambda_i)$ of the regular eigenvalue problem to be the largest real part of the spectrum set of . We then call a real number $\lambda_i^p$ [*the principal spectrum point*]{} of the weighted eigenvalue problem if $\mu_i^n(\lambda_i^p)=0$.

Note that when $m_i(t,x)\equiv m_i(x)$, the principal spectrum point of and the principal 