Volume 8, Issue 2

In this paper we consider an oceanic domain in ℝ 3 , in which there exists, at initial time, a current U 0 , a pressure p 0 and a density ρ 0 . The perturbation U , p and ρ of the velocity, the pressure and the density are induced by a perturbation of the mean windstress. The equations are of Navier-Stokes type for the velocity and pressure, of transport-diffusion type for the density. They are linearized around a given mean circulation and modified by the physical assumptions including the Boussinesq approximation and the Hydrostatic approximation with vertical viscosity. The existence and uniqueness of the solution for the variational problem are studied for the three-dimensional problem, and for the two-dimensional cyclic problem derived by assuming a sinusoidal x -dependence for the perturbation of mean flow. The latter corresponds to a modelization of tropical instability waves which are illustrated by El Nino phenomenon.

The density topologies with respect to measure and category are motivation to consider the density topologies with respect to invariant σ -ideals on ℝ. The properties of such topologies, including the separation axioms, are studied.

Singularly perturbed linear Volterra integral equations are solved in this paper. To improve the results which has been published earlier, formal solutions of systems of equations are determined and rigorously proved to be asymptotic to the exact solutions.

In this paper, nonlocal in time problem for abstract evolution equation of second order is studied and theorem on existence and uniqueness of its solution is proved. Some applications of this theorem for hyperbolic partial differential equations and systems are considered and it is proved, that well-posedness of the mentioned problems depends on algebraic properties of ratios between the dimensions of the spatial boundary and the times appearing in the nonlocal in time initial conditions.

In this paper we shall consider the nonlinear neutral delay differential equations with variable coefficients. Some new sufficient conditions for oscillation of all solutions are obtained. Our results extend and improve some of the well known results in the literature. Some examples are considered to illustrate our main results. The neutral logistic equation with variable coefficients is considered to give some new sufficient conditions for oscillation of all positive solutions about its positive steady state.

The order structure of time projections associated with random times in a von Neumann algebra is investigated in the general setup as well as that of the CAR and CCR algebras. In the second case various additional properties (such as e.g. the upper/lower continuity) of the lattice of time projections are also discussed.

Let K be a closed convex cone with the nonempty interior in a real Banach space and cc ( K ) denote the family of all nonempty convex compact subsets of K . If { F t : t ≥ 0} is a concave iteration semigroup of continuous linear set-valued functions F t : K → cc ( K ) with F 0 ( x ) = { x } for x ∈ K , then D t F t ( x ) = F t ( G ( x )) for x ∈ K and t ≥ 0, where D t F t ( x ) denotes the Hukuhara derivative of F t ( x ) with respect to t and for x ∈ K .