We show that the eigenvalues of the Neumann-Poincare operator on analytic boundaries of simply connected bounded planar domains tend to zero exponentially fast, and the exponential convergence rate is determined by the maximal Grauert radius of the boundary. We present a few examples of boundaries to show that the estimate is optimal.

We show a new and unexpected application of integral equations and their systems to the problem of the unique identification of continuous probability distributions based on the knowledge of exactly one regression function of ordered statistical data. The most popular example of such data are the order statistics which are obtained by non-decreasing ordering of elements of the sample according to their magnitude. However, our considerations are conducted in the abstract setting of so-called generalized order statistics. This model includes order statistics and other interesting models of ordered random variables. We prove that the uniqueness of characterization is equivalent to the uniqueness of the solution to the appropriate system of integral equations with non-classical initial conditions. This criterion for uniqueness is then applied to give new examples of characterizations.

A numerical method for the Dirichlet initial boundary value problem for the elastic equation in the exterior and unbounded region of a smooth, closed, simply connected two-dimensional domain, is proposed and investigated. This method is based on a combination of a Laguerre transformation with respect to the time variable and a boundary integral equation approach in the spatial variables. Using the Laguerre transformation in time reduces the time-dependent problem to a sequence of stationary boundary value problems, which are solved by a boundary layer approach resulting in a sequence of boundary integral equations of the first kind. The numerical discretization and solution are obtained by a trigonometrical quadrature method. Numerical results are included.

In this paper, we investigate the existence of stable and center-stable manifolds of admissible classes for mild solutions to partial functional differential equations of the form $\dot {u}(t)=A(t)u(t)+f(t,u_t)$, $t\ge 0$. These manifolds are constituted by trajectories of the solutions belonging to admissible function spaces which contain wide classes of function spaces like $L_p$-spaces and many other function spaces occurring in interpolation theory such as the Lorentz spaces $L_{p,q}$. Results in this paper are the generalization and development for our results in \cite {HD1}. The existence of these manifolds obtained in the case that the family of operators $(A(t))_{t\ge 0}$ generate the evolution family $(U(t,s))_{t\ge s\ge 0}$ having an exponential dichotomy or trichotomy on the half-line and the nonlinear forcing term $f$ satisfies the $\varphi $-Lipschitz condition, i.e., $\| f(t,u_t) -f(t,v_t)\| \le \varphi (t)\|u_t -v_t\|_{\mathcal {C}}$, where $u_t,\ v_t \in \mathcal{C} :=C([-r, 0], X)$, and $\varphi (t)$ belongs to some admissible Banach function space and satisfies certain conditions.

In this paper, we give new oscillation criteria for discrete Volterra equations having different non-linearities such as super-linear and sub-linear cases. We also present some new sufficient conditions for oscillation under the effect of the oscillatory forcing term.