We study the existence of zero-convergent solutions for the second-order nonlinear difference equation $\Delta \left(a_n{\Phi }_p\left(\Delta x_n\right)\right)=g\left(n,x_{n+1}\right)$, where ${\Phi }_p\left(u\right)={\left|u\right|}^{p-2}u$, $p>1$, $\left\{a_n\right\}$ is a positive real sequence for $n\ge 1$, and $g$ is a positive continuous function on $\mathbb{N}\times \left(0,u_0\right)$, $0<u_0\le \infty$. The effects of singular nonlinearities and of the forcing term are treated as well.

We present basic methods of oscillation theory of symplectic difference systems (SDSs). A particular attention is devoted to the variational principle and to the transformation method. Hyperbolic Prüfer transformation for SDSs is established.

First, we trace the genesis of the canonical form of Heun's biconfluent equation. Second, we present a method which allows us to find an integral expression as a solution to our equation, and finally, using the properties of Meijer $G$-functions, we give an integral representation of a fundamental system of solutions to the biconfluent equation.

Two main results are included in this paper. The first one deals with the leading asymptotic term of the magnetic field outside any conductive medium. In accord with physical reality, it is proved mathematically that the leading approximation is a quadrupole term which means that the conductive brain tissue weakens the intensity of the magnetic field outside the head. The second one concerns the orientation of the silent sources when the geometry of the brain model is not a sphere but an ellipsoid which provides the best possible mathematical approximation of the human brain. It is shown that what characterizes a dipole source as “silent” is not the collinearity of the dipole moment with its position vector, but the fact that the dipole moment lives in the Gaussian image space at the point where the position vector meets the surface of the ellipsoid. The appropriate representation for the spheroidal case is also included.

In 1952, for the wave equation, Protter formulated some boundary value problems (BVPs), which are multidimensional analogues of Darboux problems on the plane. He studied these problems in a $3$D domain ${\Omega }_0,$ bounded by two characteristic cones ${\Sigma }_1$ and ${\Sigma }_{2,0}$ and a plane region ${\Sigma }_0$. What is the situation around these BVPs now after 50 years? It is well known that, for the infinite number of smooth functions in the right-hand side of the equation, these problems do not have classical solutions. Popivanov and Schneider (1995) discovered the reason of this fact for the cases of Dirichlet's or Neumann's conditions on ${\Sigma }_0$. In the present paper, we consider the case of third BVP on ${\Sigma }_0$ and obtain the existence of many singular solutions for the wave equation. Especially, for Protter's problems in ${ℝ}^3$, it is shown here that for any $n\in \mathbb{N}$ there exists a $C^n\left({\overline{\Omega }}_0\right)$ - right-hand side function, for which the corresponding unique generalized solution belongs to $C^n\left({\overline{\Omega }}_0\backslash O\right),$ but has a strong power-type singularity of order $n$ at the point $O$. This singularity is isolated only at the vertex $O$ of the characteristic cone ${\Sigma }_{2,0}$ and does not propagate along the cone.

The paper discusses the asymptotic behaviour of all solutions of the differential equation $\dot{y}\left(t\right)=-a\left(t\right)y\left(t\right)+{\sum }_{i=1}^n{b}_i\left(t\right)y\left({\tau }_i\left(t\right)\right)+f\left(t\right)$, $t\in I=\left[t_0,\infty \right)$, with a positive continuous function $a$, continuous functions $b_i$, $f$, and $n$ continuously differentiable unbounded lags. We establish conditions under which any solution $y$ of this equation can be estimated by means of a solution of an auxiliary functional equation with one unbounded lag. Moreover, some related questions concerning functional equations are discussed as well.

Papkovich and Neuber (PN), and Palaniappan, Nigam, Amaranath, and Usha (PNAU) proposed two different representations of the velocity and the pressure fields in Stokes flow, in terms of harmonic and biharmonic functions, which form a practical tool for many important physical applications. One is the particle-in-cell model for Stokes flow through a swarm of particles. Most of the analytical models in this realm consider spherical particles since for many interior and exterior flow problems involving small particles, spherical geometry provides a very good approximation. In the interest of producing ready-to-use basic functions for Stokes flow, we calculate the PNAU and the PN eigensolutions generated by the appropriate eigenfunctions, and the full series expansion is provided. We obtain connection formulae by which we can transform any solution of the Stokes system from the PN to the PNAU eigenform. This procedure shows that any PNAU eigenform corresponds to a combination of PN eigenfunctions, a fact that reflects the flexibility of the second representation. Hence, the advantage of the PN representation as it compares to the PNAU solution is obvious. An application is included, which solves the problem of the flow in a fluid cell filling the space between two concentric spherical surfaces with Kuwabara-type boundary conditions.