Abstract

Let \(f:\mathbb{R}_+\to \mathbb{R}^n\) and \(K:\mathbb{R}_+\to \mathbb{R}^{n\times n}\). The linear Volterra integral equation \(x(t)=f(t)+\int^t_0K(t-s)x(s)ds\) \((t\geq 0)\) is called positive if for every \(f\in L^1_{\text{loc}}(\mathbb{R}_+, \mathbb{R}^n)\) being nonnegative, the corresponding solution \(x(\cdot,f)\) is also nonnegative. It is shown that such a linear Volterra integral equation is positive if and only if the resolvent of \(K\) is nonnegative. For the case of \(K\geq 0\), a variant of the Paley-Wiener theorem and a Perron-Frobenius type theorem for the equation are given. Moreover, the authors prove that for \(K\in L^1((0,\sigma), \mathbb{R}^{n\times n})\), the initial function semigroup determined by \(K\) on \((0,\sigma)\) is positive if and only if \(K\geq 0\).