Asymptotical structure of resolvent of the unstable Volterra equation with a difference kernel (Q1589204)

The author studies the asymptotic structure of the resolvent of the convolution Volterra equation \[ x(t)= \int_0^t K(t-s)x(s) ds +f(t) \] in the unstable case when its symbol \(1-\widehat{K}(z)\) has zeros in the closed half-plane \[ 1-\widehat{K}(z)=\prod\limits_j^k (z-i\gamma_j)^{m_j+\alpha_j} \prod\limits_{j>k} (z-\lambda_j)^{m_j} w(z), \] where \(w (z)\neq 0\), \(\Re z \geq 0\), \(m_j \in Z_+\), \(\alpha_j \in (0,1).\) Here \(\widehat{K}(z)\) is the Fourier transform of the kernel \(K(t).\) The asymptotic structure of the resolvent of the unstable Volterra equation in terms connecting with the zeros (and their orders) of the symbol \(1-\widehat{K}(z)\) is given.