We propose and analyze methods for the numerical solution of an integral equation which arises in statistical physics and spatial statistics. Instances of this equation include the Mean Field, Poisson-Boltzmann and Emden equations for the density of a molecular gas, and the Poisson saddlepoint approximation for the intensity of a spatial point process. Conditions are established under which the Picard iteration and the under relaxation iteration converge. Numerical validation is included.

In this paper, the problem of the order of approximation for the multivariate sampling Kantorovich operators is studied. The cases of uniform approximation for uniformly continuous and bounded functions/signals belonging to Lipschitz classes and the case of the modular approximation for functions in Orlicz spaces are considered. In the latter context, Lipschitz classes of Zygmund-type which take into account of the modular functional involved are introduced. Applications to $L^p(\R^n)$, interpolation and exponential spaces can be deduced from the general theory formulated in the setting of Orlicz spaces. The special cases of multivariate sampling Kantorovich operators based on kernels of the product type and constructed by means of Fej\'er's and B-spline kernels have been studied in details.

It is shown that the weakly singular integral operator~$\int_{-1}^{1}\big(\phi(\tau)/|\tau -t|^{\gamma}\big)\,d\tau$, where $0\lt \gamma\lt 1$, maps the weighted Sobolev space~$W_{p;\alpha,\beta}^{(n)}(\Omega)$ compactly into itself for $1\lt p\lt \infty$, $0\lt \alpha+1/q, \beta+1/q\lt 1$~and $n\in \mathbb{N}_0$.

We consider various initial-value problems for ordinary integro-differential equations of first order that are characterized by convolution-terms, where all factors depend on the solutions of the equations. Applications of such problems are descriptions of certain glass-transition phenomena based on mode-coupling theory, for instance. We will prove results concerning well-posedness of such problems and the asymptotic behaviour of their solutions.

In this paper, we study control systems governed by impulsive Riemann-Liouville fractional differential equations in Banach spaces. Firstly, we introduce $PC_{1-\alpha}$-mild solutions for impulsive Riemann-Liouville fractional differential equations. Then, we make a set of assumptions to guarantee the existence and uniqueness of mild solutions. Finally, approximate controllability of the associated impulsive Riemann-Liouville fractional evolution control systems is also formulated and proved.

In this paper we extend the results proposed in \cite{Onofrei-S} and study the problem of active control in the context of a scalar Helmholtz equation. Given a source region $D_a$ and $\{v_0,v_1,\ldots,v_n\}$, a set of solutions of the homogeneous scalar Helmholtz equation in $n$ mutually disjoint ``control" regions $\{D_0,D_1,\ldots,D_n\}$ of $\RR^2$ or $\RR^3$, respectively, the main objective of this paper is to characterize the necessary boundary data on $\partial D_a$ so that the solution to the corresponding exterior scalar Helmholtz problem will closely approximate $v_i$ in $D_i$, respectively, for each $i\in\{0,\ldots,n\}$. Building up on the previous ideas presented in \cite{Onofrei-S} we show the existence of a class of solutions to the problem, provide numerical support of the results in 2D and 3D and discuss the existence of a minimal energy solution and its stability.

Employing the generalized Parseval equality for the Mellin transform and elementary trigonometric formulas, the iterated Hartley transform on the nonnegative half-axis (the iterated half-Hartley transform) is investigated in $L_2$. Mapping and inversion properties are discussed, its relationship with the iterated Stieltjes transform is established. Various compositions with the Fourier cosine and sine transforms are obtained. The results are applied to the uniqueness and universality of the closed form solutions for certain new singular integral and integro-functional equations. \bigskip