This paper concerns the well-posedness and energy decay of a linear wave equation with density, infinite memory and time-varying delay in the whole space $\mathbb {R}^n$ $(n\geq 3)$. We consider the weighted spaces $\mathcal {D}^{1,2}(\mathbb {R}^n)$ and $L^2_{\rho }(\mathbb {R}^n)$ introduced by Karachalios and Stavrakakis (1999) to overcome the difficulty that some operators on $\mathbb {R}^n$ are not compact. We prove the global well-posedness of the Cauchy problem by using Faedo-Galerkin approximation and establish the exponential decay of energy when the amplitude of the time delay term is small by using suitable Lyapunov functional.

We study the classical Volterra integral equation of the second kind on an interval, in which the Lebesgue type integral is replaced by the more general Choquet integral with respect to a monotone, submodular and continuous from below and from above set function, including the so-called distorted Lebesgue measures.

The Neumann-Poincare operator is a boundary-integral operator associated with harmonic layer potentials. This article proves the existence of eigenvalues within the essential spectrum for the Neumann-Poincar\'e operator for certain Lipschitz curves in the plane with reflectional symmetry, when considered in the functional space in which it is self-adjoint. The proof combines the compactness of the Neumann-Poincare operator for curves of class $C^{2,\alpha }$ with the essential spectrum generated by a corner. Eigenvalues corresponding to even (odd) eigenfunctions are proved to lie within the essential spectrum of the odd (even) component of the operator when a $C^{2,\alpha }$ curve is perturbed by inserting a small corner.

We study a new kind of linear integral equations for a relativistic quantum-mechanical two-particle wave function $\psi (x_1,x_2)$, where $x_1,x_2$ are spacetime points. In the case of retarded interaction, these integral equations are of Volterra-type in the in the time variables, i.e., they involve a time integration from 0 to $t_i = x_i^0, i=1,2$. They are interesting not only in view of their applications in physics, but also because of the following mathematical features: (a) time and space variables are more interrelated than in normal time-dependent problems, (b) the integral kernels are singular, and the structure of these singularities is non-trivial, (c) they feature time delay. We formulate a number of examples of such equations for scalar wave functions and prove existence and uniqueness of solutions for them. We also point out open mathematical problems.

This paper deals with a systematic study of the convolution operator defined on the weighted pseudo almost automorphic functions space ($,\mathit{PAA} (\mathbb {X}, \rho )$). The purpose of this, is to ensure the existence and uniqueness of solutions in $,\mathit{PAA} (\mathbb {X},\rho )$ for general abstract neutral integral equations of convolution type. Upon making different assumptions on the kernel $k$ of the convolution operator and the weight $\rho $ we obtain results about the convolution invariance of the operator on $,\mathit{PAA} (\mathbb {X}, \rho )$. Essentially the assumptions are of two type, one is a new condition on $\rho $ valid for every kernel $k$ and the other is a ($k$, $\rho $)-type condition, in which the kernel $k$ helps the weight $\rho $. These conditions are not known in the literature. Explicit examples show the utility of these two different assumptions. Taking advantage of the rich properties of the convolution we have obtained new results about composition which permits study the existence and uniqueness of mild solutions in $,\mathit{PAA} (\mathbb {X}, \rho )$ for general abstract neutral integral equations. The results obtained are directly applied to integro differential equations, partial differential equations, logistic equations and differential equations of first and fractional order, among other. Several examples and concrete classes of differential equations illustrate our results.