Volume 24, Issue 3

In this paper, we establish several results about the existence of second-order impulsive differential inclusion with periodic conditions. By using critical point theory, several new existence results are obtained. We also provide an example in order to illustrate the main abstract results of this paper.

A part of an exponential system with degenerate coefficients is considered. The frame properties (completeness, minimality, basicity, atomic decomposition) of this system in Hardy classes are studied in the case where the coefficients may not satisfy the Muckenhoupt condition.

In this paper we discuss generalized two-variable Chebyshev polynomials and their relevant relations; in particular, by using their integral representations, we prove some operational identities. The approach is based on the generalized two-variable Hermite polynomials and the integral representations of ordinary Chebyshev polynomials of first and second kind. In addition, we discuss how the families of generalized Chebyshev polynomials can be used to prove some interesting properties related to ordinary Chebyshev polynomials of first and second kind. A fundamental role, as we see, is played by the powerful operational techniques verified by the families of generalized Hermite polynomials.

We study fractional Sturm–Liouville operators. We give some basic definitions and properties of fractional calculus. Using the method of Pavlov [31, 30, 32], we prove a theorem on the completeness of the system of eigenvectors and associated vectors of dissipative fractional Sturm–Liouville operators.

A mapping T:A∪B→A∪B{T\colon A\cup B\to A\cup B} such that T⁢(A)⊆A{T(A)\subseteq A} and T⁢(B)⊆B{T(B)\subseteq B} is called a noncyclic mapping, where A and B are two nonempty subsets of a Banach space X . A best proximity pair (p,q)∈A×B{(p,q)\in A\times B} for such a mapping T is a point such that p=T⁢p,q=T⁢q{p=Tp,q=Tq} and ∥p-q∥=dist⁡(A,B){\|p-q\|=\operatorname{dist}(A,B)}. In the current paper, we establish some existence results of best proximity pairs in strictly convex Banach spaces. The presented theorems improve and extend some recent results in the literature. We also obtain a generalized version of Markov–Kakutani’s theorem for best proximity pairs in a strictly convex Banach space setting.

We study Besov and Triebel–Lizorkin spaces with Muckenhoupt weights which have their singularities at some distance from the set where the trace is taken. We exemplify our results for special weights of type w⁢(x)∼|x|α{w(x)\sim|x|^{\alpha}}. The approach is based on atomic decomposition results.

Let M be a prime Γ-ring with center Z⁢(M){Z(M)}, and let θ be an automorphism of M . An additive map d:M→M{d:M\to M} is called a skew derivation if d⁢(x⁢α⁢y)=d⁢(x)⁢α⁢y+θ⁢(x)⁢α⁢d⁢(y){d(x\alpha y)=d(x)\alpha y+\theta(x)\alpha d(y)} for all x,y∈M{x,y\in M}, α∈Γ{\alpha\in\Gamma}. An additive map F:M→M{F:M\to M} is called a generalized skew derivation if there exists a skew derivation d:M→M{d:M\to M} such that F⁢(x⁢α⁢y)=F⁢(x)⁢α⁢y+θ⁢(x)⁢α⁢d⁢(y){F(x\alpha y)=F(x)\alpha y+\theta(x)\alpha d(y)} holds for all x,y∈M{x,y\in M}, α∈Γ{\alpha\in\Gamma}. In the present paper, our main objective is to prove some commutativity results for prime Γ-rings M admitting a generalized skew derivation F satisfying anyone of the properties: (i) F⁢(x⁢α⁢y)±x⁢α⁢y∈Z⁢(M){F(x\alpha y)\pm x\alpha y\in Z(M)}, (ii) F⁢(x⁢α⁢y)±y⁢α⁢x∈Z⁢(M){F(x\alpha y)\pm y\alpha x\in Z(M)}, (iii) F⁢(x)⁢α⁢F⁢(y)±x⁢α⁢y∈Z⁢(M){F(x)\alpha F(y)\pm x\alpha y\in Z(M)}, (iv) F⁢([x,y]α)±[x,y]α=0{F([x,y]_{\alpha})\pm[x,y]_{\alpha}=0}, (v) F⁢(〈x,y〉α)±〈x,y〉α=0{F(\langle x,y\rangle_{\alpha})\pm\langle x,y\rangle_{\alpha}=0} for all x,y∈I{x,y\in I} and α∈Γ{\alpha\in\Gamma}. In fact, we obtain rather more general results which unify, extend and complement several well-known results proved in [3, 4, 5, 6, 32].

In this paper, we introduce and study some properties of the sets, namely Δμ{\Delta_{\mu}}-sets, ∇μ{\nabla_{\mu}}-sets and Δμ*{\Delta_{\mu}^{*}}-closed sets in a generalized topological space.

Problems with linear initial-boundary conditions for higher order nonlinear hyperbolic equations are investigated. The concept of strong well-posedness of an initial-boundary value problem is introduced, and conditions guaranteeing solvability and strong well-posedness of the problem under consideration are established.

Archimedes knew that for a point P on a parabola X and a chord AB of X parallel to the tangent of X at P , the area of the region bounded by the parabola X and chord AB is four thirds of the area of the triangle △A⁢B⁢P{\bigtriangleup ABP}. Recently, the first two authors have proved that this fact is the characteristic property of parabolas. In this paper, we study strictly locally convex curves in the plane ℝ2{{\mathbb{R}}^{2}}. As a result, generalizing the above mentioned characterization theorem for parabolas, we present two conditions, which are necessary and sufficient, for a strictly locally convex curve in the plane to be an open arc of a parabola.

Let X be a nonempty space. In the family of all topologies on X we can consider different kinds of relations. We examine almost semi-correspondence which is between known relations of similarity and semi-correspondence. We study this relation in the family of density-type topologies.

The paper presents the generalized form of the Bernstein operator associated with any real-valued function f:[a,b]→ℝ{f\colon[a,b]\to\mathbb{R}}. For this generalized Bernstein operator, we study the qualitative and quantitative aspects concerning uniform convergence, order of approximation and asymptotic behavior.

In this paper, we prove some fixed point theorems of Ćirić type for mappings having non-unique fixed points by employing rational-type contractive conditions and the notion of comparison functions. Our results are generalizations, extensions and improvements of some previous and corresponding results in the literature.

Methods of summability with variable order are considered. A connection of series convergence with series summability is established. A non-strengthening in a certain sense of some results is proved.

The problems of absolute convergence of multiple Fourier series with respect to both general orthonormal systems and multiple Haar series are studied.