Ion channels are small holes embedded in cell membranes. They open and close to control the flow of ions. In this way, cells function properly. Poisson-Nernst-Planck (PNP) model is a mathematical model commonly used to study ionic flow. In this talk, a classical PNP model is considered. We show that the system has a unique solution. The proof is constructive. With singular perturbation analysis the problem is reduced to an inverse problem of a first order linear system with boundary conditions. Then with the help of matrix theory, it is further reduced to a root problem of a meromorphic function, which is solved by Cauchy Argument Principle. We show how matrix eigenvalue theory is used in solving the problem.

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