Volume 22, Issue 2

An optical vortex can appear when a light beam with nonzero angular momentum propagates in a suitable nonlinear medium. In some situations has been observed that the light intensity vanish at the center of the vortex (where the phase of the electromagnetic field is undefined), while the light beam assumes a ring-shaped structure. In this paper we consider two classical cases in which such kind of phenomena occur: the case of the self focusing cubic nonlinearity, and the case of competing quintic and cubic nonlinearity. In both cases we study the nonlinear Schrödinger equation of the optical field (with various boundary conditions) by means of min-max methods, and we prove the existence of saddle point type solutions, as well as minimum type solutions.

In the present paper, the Sharma–Tasso–Olever (STO) equation is considered by the Lie symmetry analysis. All of the geometric vector fields to the STO equation are obtained, and then the symmetry reductions and exact solutions of the equation are investigated. Our results witness that symmetry analysis is a very efficient and powerful technique in finding the solutions of the proposed equation.

In this paper we study closed subspaces of Banach spaces that admit relative Chebyshev centers for all bounded subsets of the space. We exhibit new classes of spaces which have this property and study stability results similar to the ones studied in the literature in the context of proximinal subspaces and Chebyshev centers. For the space C⁢(K)${C(K)}$ of continuous functions on a compact set K , we show that a closed subspace of finite codimension has relative Chebyshev centers for all bounded sets in C⁢(K)${C(K)}$ if and only if it is a strongly proximinal subspace.

In this paper, a set-valued iteration regularized semigroup, i.e. a family {Ft}t≥0${\{F^{t}\}_{t\geq 0}}$ of set-valued functions for which Fs+t∘C=Fs∘Ft,F0=C,s,t≥0,$F^{s+t}\circ C=F^{s}\circ F^{t},\quad F^{0}=C,\quad s,t\geq 0,$ will be considered, where C is a set-valued function on a closed convex cone in a Banach space. Under some appropriate conditions the generator of a set-valued regularized concave semigroup is introduced and some of its properties are investigated. Also differentiability of the iteration family {C∘Ft}t≥0${\{C\circ F^{t}\}_{t\geq 0}}$ is discussed.

In this paper, an operational method for solving linear and nonlinear systems described by ordinary differential equations is presented. The construction is based on the generalized derivative in the sense of distribution theory. The approach allows the response computation without needing the use of any integral transform as in the Mikusiński operational calculus. A general description of the algorithm is done and some illustrating examples are presented. The algorithm is recursive allowing to add and remove any pole or zero contribution. The extension to nonlinear systems is done by means of the Adomian polynomials.

In the present paper, we introduce the notion of symmetrically porouscontinuous functions. We investigate some properties of symmetric porouscontinuity and its connections with the notion of porouscontinuity, studied by Borsík and Holos in [2]. We prove that there are 2𝔠${2^{\mathfrak{c}}}$ symmetrically porouscontinuous functions, which extends results of [1] concerning ρ-upper continuous functions.

A version of the Arzelà–Ascoli theorem for X being a σ-locally compact Hausdorff space is proved. The result is used in proving compactness of Fredholm, Hammerstein and Urysohn operators. Two fixed point theorems, for Hammerstein and Urysohn operators, are derived on the basis of Schauder fixed point theorem.

We introduce and study s-lq${l_{q}}$-complete and c0⁢s${c_{0s}}$-μ convergences, and we obtain a new result regarding statistical convergences of sequences of measurable functions.

An inverse problem concerning a diffusion equation with source control parameter is considered. The approximation of the problem is based on the Ritz method with satisfier function. The Ritz method together with the least squares approximation (Ritz-least squares method) are utilized to reduce the inverse problem to the solution of algebraic equations. We extensively discuss the convergence of the method and finally present illustrative examples to demonstrate validity and applicability of the new technique.