Volume 29, Issue 4

In this paper, we derive the convergence of 𝑇 means of Vilenkin–Fourier series with monotone coefficients of integrable functions in Lebesgue and Vilenkin–Lebesgue points. Moreover, we discuss the pointwise and norm convergence in L p L_{p} norms of such 𝑇 means.

In this paper, we introduce a new diophantine equation with prime numbers. Let [ ⋅ ] [\,\cdot\,] be the floor function. We prove that, when 1 < c < 23 21 1<c<\frac{23}{21} and θ > 1 \theta>1 is fixed, then every sufficiently large positive integer 𝑁 can be represented in the form N = [ p 1 c ⁢ tan θ ⁡ ( log ⁡ p 1 ) ] + [ p 2 c ⁢ tan θ ⁡ ( log ⁡ p 2 ) ] + [ p 3 c ⁢ tan θ ⁡ ( log ⁡ p 3 ) ] , N=[p^{c}_{1}\tan^{\theta}(\log p_{1})]+[p^{c}_{2}\tan^{\theta}(\log p_{2})]+[p^{c}_{3}\tan^{\theta}(\log p_{3})], where p 1 , p 2 , p 3 p_{1},p_{2},p_{3} are prime numbers. We also establish an asymptotic formula for the number of such representations.

Using a fixed point theorem due to Schaefer, the existence of solutions for an odd-order m -point impulsive boundary value problem on time scales is obtained. The problem considered is of general form, where both the differential equation and the impulse effects are nonlinear. Illustrative examples are provided.

We show that the incenter of an isosceles triangle is a stable equilibrium point of the electrostatic potential of certain point charges placed at its vertices. To this end, explicit formulas for these charges are given and the hessian of their electrostatic potential is computed. The behavior of this hessian in a family of triangles with the given inscribed and circumscribed circles is investigated and its extremal values are computed. As an application, we prove that each point in the unit disc is a stable equilibrium point of a certain triple of point charges on its boundary, which yields an explicit scenario of robust electrostatic control in Euclidean discs. In conclusion, several examples are presented which show that analogous problems are interesting and feasible for convex quadrilaterals.

The paper deals with the convergence of the series ∑ k = 1 ∞ | c k ⁢ ( f ) | r ⁢ γ k , r ∈ ( 0 , 2 ) , \sum_{k=1}^{\infty}|c_{k}(f)|^{r}\gamma_{k},\quad r\in(0,2), where c k ⁢ ( f ) {c_{k}(f)} are the Fourier coefficients of the function f ∈ L 2 ⁢ ( [ 0 , 1 ] ) {f\in L_{2}([0,1])} with respect to the general orthonormal system, and ( γ k ) {(\gamma_{k})} is the non-negative sequence of numbers satisfying sufficiently general conditions. From the obtained results there follow the theorems of S. B. Stečkin, A. A. Konyuškov, B. I. Golubov, R. A. DeVore and V. N. Temlyakov.

In this paper, we define a generalized resolvent operator involving a 𝒢 ⁢ ( ⋅ , ⋅ ) {\mathcal{G}(\,\cdot\,,\cdot\,)} -co-monotone mapping for solving a generalized variational inclusion problem. We show that the resolvent operator under consideration is single-valued as well as Lipschitz continuous. An iterative algorithm is constructed to obtain the approximate solution of our problem. An example is provided to illustrate the theoretical findings.

It is proved that, for certain types of ( n + 1 ) (n+1) -colorings of the Euclidean space R n \mathbf{R}^{n} , there are continuum many “rainbow” isosceles 𝑛-simplexes in R n \mathbf{R}^{n} .

In this paper, for the fourth order linear ordinary differential equation u ( 4 ) ⁢ ( t ) = p ⁢ ( t ) ⁢ u ⁢ ( t ) - μ ⁢ u ⁢ ( t ) , u^{(4)}(t)=p(t)u(t)-\mu u(t), where p : I → R {p:I\to R} is a Lebesgue integrable function, we study the question of disconjugacy on the interval [ a , b ] {[a,b]} , and the problem of the Green’s functions sign for some two-point boundary problems. The optimal sufficient conditions of the disconjugacy and necessary and sufficient conditions of non-negativity (non-positivity) of the Green’s function are found for the mentioned two-point boundary problems.

In this paper, we give characterization of subclasses of bi-univalent functions and also consider bi-starlike functions on the unit disk in ℂ in terms of Löwner chains.

Some sufficient conditions are presented for the continuity of the mapping f : ⟨ X , τ X * ⟩ → ⟨ Y , τ Y * ⟩ f\colon\langle X,\tau_{X}^{*}\rangle\to\langle Y,\tau_{Y}^{*}\rangle , where τ X * \tau_{X}^{*} and τ Y * \tau_{Y}^{*} are the topologies induced by the local function on 𝑋 and 𝑌, resp. under the assumption that the mapping from ⟨ X , τ X ⟩ \langle X,\tau_{X}\rangle to ⟨ Y , τ Y ⟩ \langle Y,\tau_{Y}\rangle is continuous. Further, we consider open and closed functions in the cases in which the open (or closed) mapping is preserved through the domain and codomain “idealization”. Through several examples, we illustrate that the considered conditions cannot be weakened.

An estimation of the approximation rate by positive linear operators of functions defined on the positive half line that have finite limit at infinity is discussed. For the application of the obtained results, we introduce the positive linear operators which preserve 2 a ⁢ x 2^{ax} , a > 0 a>0 , and obtain their uniform convergence, order of approximation via a certain weighted modulus of continuity, and a quantitative Voronovskaya type theorem.

The purpose of this paper is to study the commutativity of a prime ring ℛ {\mathscr{R}} with anti-automorphism ψ and generalized derivation ( ℱ , ξ ) {(\mathscr{F},\xi)} satisfying certain algebraic identities. Furthermore, we describe the structure of any additive map which is ψ-centralizing on a 2-torsion free prime ring with anti-automorphism ψ. We also provide examples to show that various conditions imposed on the hypotheses of our results are essential.

In this paper, we establish the convolution and product theorems for the quadratic-phase Fourier transform (QPFT) and also deduce the convolution theorem associated with generalized translation. Moreover, the Shannon-type reconstruction formula for band-limited signals in the QPFT domain is formulated.

In this paper, the algebraic structure of connected torsion-free weakly locally compact abelian groups and properties of some weakly locally compact abelian groups are established.

The purpose of this paper is to prove the following assertion. Let S be a ring having a nontrivial idempotent element e which satisfies some conditions. We prove that any multiplicative centrally-extended derivation of S is a centrally-extended derivation.

In this work, the initial-boundary value problem is considered for the dynamic Kirchhoff string equation u t ⁢ t - ( α ⁢ ( t ) + β ⁢ ∫ - 1 1 u x 2 ⁢ d x ) ⁢ u x ⁢ x = f u_{tt}-\bigl{(}\alpha(t)+\beta\int_{-1}^{1}u_{x}^{2}\,\mathrm{d}x\bigr{)}u_{xx}=f . Here α ⁢ ( t ) \alpha(t) is a continuously differentiable function, α ⁢ ( t ) ≥ c 0 > 0 \alpha(t)\geq\mathrm{c}_{0}>0 and 𝛽 is a positive constant. For solving this problem approximately, a symmetric three-layer semi-discrete scheme with respect to the temporal variable is applied, in which the value of a nonlinear term is taken at the middle point. This approach allows us to find numerical solutions per temporal steps by inverting the linear operators. In other words, applying this scheme, a system of linear ordinary differential equations is obtained. The local convergence of the scheme is proved. The results of numerical computations using this scheme for different test problems are given for which the Legendre–Galerkin spectral approximation is applied with respect to the spatial variable.

We study the boundedness and compactness of the Volterra composition operators J g , φ {J_{g,\varphi}} and I g , φ {I_{g,\varphi}} from Dirichlet–Morrey spaces to Bloch-type spaces. Furthermore, the essential norms of these operators are given.