In this paper, we present some results concerning the existence and global asymptotic stability of solutions for a functional integral equation of fractional order. We use Schauder's fixed point theorem for the existence of solutions, and we prove that all these solutions are globally asymptotically stable.

We analyze the existence, uniqueness and regularity of solutions to a class of third-kind Volterra integral equations, including equations with weakly singular kernels. Of particular interest are those integral equations that can be transformed into cordial Volterra integral equations whose underlying integral operator may be non-compact.

In the paper, we prove the existence, uniqueness and differentiable dependence of solutions for some nonlinear Urysohn integral equations on parameters. Some sufficient conditions for the nonlinear integral operator of the Urysohn type to be a diffeomorphism are stated. Global invertibility of the Urysohn operator in a certain Sobolev space is ascertained. Consequently, global solvability of Urysohn equations is claimed. Similar results are obtained for some nonlinear Urysohn integral equations with controls by the use of the global implicit function theorem published in the recent paper by Idczak. The proofs of global diffeomorphisms and global implicit functions theorems, the main tools used in the paper, rely in an essential way on the mountain pass theorem. Applications of results to some specific nonlinear Urysohn integral equations are also presented.

One of the cornerstones of the theory of linear integral equations is the establishment of properties of the resolvent kernel. This theory has consequences in the study of a wide class of nonlinear integral equations, including those of the form \[ x(t)=f(t) -\int^t_0 C(t-s) h(s,x(s))\,ds, \tag{1} \] that commonly arise in applied mathematics. Using such a theory, in this work, we study conditions on (1) that relate to the existence of a solution $x$ that is non-negative on $[0,\infty)$. In the classical literature, one finds amongst sufficient conditions for non-negativity of $x$ on $[0,\infty)$, the requirements \[ \textnormal{$f(t)>0$ on $[0,\infty)$, $C(t) >0$ on $(0,\infty)$, and $\frac{f(T)}{f(t)}\leq \frac{C(T-s)}{C(t-s)}$} \] when $0\leq s \lt T \lt t$, for $t\in [0,\infty)$. Our work begins here when we show that this assumption implies that $\int^{\infty}_0 f(t)\,dt = \infty$ whenever $\int^{\infty}_0 C(t) \,dt =\infty$. This motivated the conjecture that, when $\int^{\infty}_0 C(t)\, dt =\infty$, then a necessary condition for the existence of a non-negative solution is that $\int^{\infty}_0 f(t) dt =\infty$. The proof of that conjecture is the main result of this paper.

Finally, we point out that, while (1) seems very special and possibly of narrow interest, that is not the case. Equation (1) is of prime importance in establishing properties of the resolvent kernel for scalar equations arising in mathematical physics from areas as diverse as heat problems and problems in turbulence as well as fractional differential equations of both Caputo and Riemann-Liouville type. There is a myriad of real-world problems modeled by the latter two types, and such studies are among the very active in mathematics today.

In this paper, we consider the transmission eigenvalue problem for Maxwell's equations corresponding to non-magnetic inhomogeneities with contrast in electric permittivity that changes sign inside its support. We formulate the transmission eigenvalue problem as an equivalent homogeneous system of the boundary integral equation and, assuming that the contrast is constant near the boundary of the support of the inhomogeneity, we prove that the operator associated with this system is Fredholm of index zero and depends analytically on the wave number. Then we show the existence of wave numbers that are not transmission eigenvalues which by an application of the analytic Fredholm theory implies that the set of transmission eigenvalues is discrete with positive infinity as the only accumulation point.

We consider a class of potential problems on a periodic half-space for the modeling of electrified oil films, which are used in the development of novel switchable liquid optical devices (diffraction gratings). A boundary integral formulation which reduces the problem to the study of the oil-air interface alone is derived and solved in a highly efficient manner using the Nystr\"{o}m method. The oil films encountered experimentally are typically very thin and thus an interface-only integral representation is important for avoiding the near-singularity problems associated with boundary integral methods for long slender domains. The super-algebraic convergence of the proposed method is discussed and demonstrated via appropriate numerical experiments.

We propose a new well-conditioned boundary integral equation to solve transmission problems of electromagnetism. This equation is well posed and appears as a compact perturbation of the identity leading to fast iterative solutions without the help of any preconditioner. Some numerical experiments confirm this result.