Volume 30, Issue 2

The paper is devoted to the study of the relative B-essential spectra of 2 × 2 {2\times 2} unbounded operator matrix pencils defined with nondiagonal domain. The main tool used to prove the results is the Frobenuis–Schur factorization associated with this kind of operator matrix which has proven very useful in dealing with stability analysis in the theory of B-Fredholm operators pencils. Therefore, the theoretical results are tested in pencil transport integro-differential equation with specific boundary conditions.

In this research paper, we develop some aspects of the theory of Fredholm linear operators acting in p-adic (or non-Archimedean) Banach spaces. In this regard, we establish sufficient conditions for the p-adic Fredholmness of the algebraic sum of unbounded linear operators. Next, we study the perturbation of p-adic upper semi-Fredholm operators under strictly singular operators. Moreover, we give some results concerning quasi-compact and p-adic lower semi-Fredholm operators.

In this paper, we discuss the rate of the approximation by matrix transform means of Fourier series with respect to the character system of the group of 2-adic integers in L p ⁢ ( I ) {L^{p}(I)} spaces ( 1 ≤ p < ∞ {1\leq p<\infty} ) and in C ⁢ ( I ) {C(I)} . Moreover, we give an application for functions in Lipschitz classes Lip ⁢ ( α , p , I ) {\mathrm{Lip}(\alpha,p,I)} ( α > 0 {\alpha>0} , 1 ≤ p < ∞ {1\leq p<\infty} ) and Lip ⁢ ( α , C ⁢ ( I ) ) {\mathrm{Lip}(\alpha,C(I))} ( α > 0 {\alpha>0} ).

A notion of pentaction of any object in the category 𝐫𝐆𝐫 ∙ {\mathbf{rGr}^{\bullet}} of reduced groups with action is introduced. The operations are defined in the set 𝖯𝖾𝗇𝗍𝖺𝖼𝗍 ⁢ ( A ) {\mathsf{Pentact}(A)} of pentactions of an object A of 𝐫𝐆𝐫 ∙ {\mathbf{rGr}^{\bullet}} . It is proved that if an object A is perfect with zero weak stabilizer in the sense defined in this paper, then 𝖯𝖾𝗇𝗍𝖺𝖼𝗍 ⁢ ( A ) {\mathsf{Pentact}(A)} is an object of 𝐫𝐆𝐫 ∙ {\mathbf{rGr}^{\bullet}} , it has a derived action on A , the object A is action representable and 𝖯𝖾𝗇𝗍𝖺𝖼𝗍 ⁢ ( A ) {\mathsf{Pentact}(A)} represents all actions on A .

We show that the necessity part of the main result of [L. Diening, O. Karlovych and E. Shargorodsky, On interpolation of reflexive variable Lebesgue spaces on which the Hardy–Littlewood maximal operator is bounded, Georgian Math. J. 29 2022, 3, 347–352] can be derived easily from its predecessor [2, Theorem 4.1] and its sufficiency part.

Let 𝔸 = ( A O O A ) \mathbb{A}={\bigl{(}\begin{smallmatrix}A&O\\ O&A\\ \end{smallmatrix}\bigr{)}} be a 2 × 2 2\times 2 diagonal operator matrix whose each diagonal entry is a bounded positive (semi-definite) linear operator A acting on a complex Hilbert space ℋ \mathcal{H} . In this paper, we derive several 𝔸 \mathbb{A} -numerical radius inequalities for 2 × 2 2\times 2 operator matrices whose entries are bounded with respect to the seminorm induced by the positive operator A on ℋ \mathcal{H} . Some applications of our inequalities are also given.

In this paper, we introduce a new type of stability for nonlinear impulsive systems of differential equations, namely practical h -stability. By using the Lyapunov stability theory, some sufficient conditions which guarantee practical h -stability are established. Our original results generalize well-known fundamental stability results, practical stability, practical exponential stability and practical asymptotic stability for nonlinear time-varying impulsive systems. Then two classes of nonlinear impulsive systems, namely perturbed and cascaded impulsive systems, are discussed. Furthermore, the problem of practical h -stabilization for certain classes of nonlinear impulsive systems is considered. Finally, two numerical examples are given to show the effectiveness of our theoretical results.

The order of rational Fourier coefficients for functions of Akhobadze variation, B Λ ( p ( n ) ↑ ∞ , φ , 𝕋 ¯ ) {B\Lambda(p(n)\uparrow\infty,\varphi,\mathbb{\overline{T}})} , is estimated. Extending the definition of Akhobadze class for multiple variables, the result of one variable is extended for multiple variables.

The aim of this paper is to investigate some properties, recurrence relations and identities involving degenerate hyperharmonic numbers, hyperharmonic numbers and degenerate harmonic numbers. In particular, we derive an explicit expression of the degenerate hyperharmonic numbers in terms of the degenerate harmonic numbers. This is a degenerate version of the corresponding identity representing the hyperharmonic numbers in terms of harmonic numbers due to Conway and Guy.

We derive a generalized Pohozhaev’s identity for radial solutions of p -Laplace equations, by using the approach in [P. Korman, Global Solution Curves for Semilinear Elliptic Equations, World Scientific Publishing, Hackensack, 2012], thus extending the work [H. Brézis and L. Nirenberg, Positive solutions of nonlinear elliptic equations involving critical Sobolev exponents, Comm. Pure Appl. Math. 36 1983, 4, 437–477], where this identity was implicitly used for the Laplace equation.

This paper studies the transfer of pseudo-almost valuation property (PAVR property for short) to various context of commutative ring extensions such as power series ring, trivial ring extension and amalgamation. Our work is motivated by an attempt to generate new original classes of rings satisfying this property. The obtained results are backed with new and illustrative examples arising as trivial ring extensions, amalgamations and pullback constructions.

In this paper we prove the existence of distributional solutions to certain anisotropic nonlinear weighted elliptic equations with variable exponents, where the weight function belongs to the anisotropic Sobolev space with variable exponents and zero boundary. The functional setting involves anisotropic variable exponents Lebesgue–Sobolev spaces.

We obtain a complete description of the structure of a finite group G in which every non-nilpotent subgroup is a TI-subgroup or has p ′ {p^{\prime}} -order for some fixed prime divisor p of | G | {|G|} .

In the present paper, we address the global solvability and finite-time attractivity of strong solutions for an abstract Rayleigh–Stokes-type equation involving nonlinear perturbations in Hilbert spaces. Based on regularity estimates of resolvent operator, local estimates on Hilbert scales and fixed point arguments, we obtain some results on global existence and finite-time attractivity of strong solutions to our problem.

Suppose that the set of square-integrable solutions of an elliptic equation which have a value at some given point equal to c is not empty. Then there is exactly one element with minimal L 2 {L^{2}} -norm. Moreover, it is shown that this minimal element depends continuously on a domain of integration, i.e., on the set on which our solutions are defined, and on a weight of integration, i.e., on the deformation of an inner product. The theorems are proved using the theory of reproducing kernels and Hilbert spaces of square-integrable solutions of elliptic equations. We prove the existence of such a reproducing kernel using theory of Sobolev spaces. We generalize the well-known Ramadanov theorem. This is done in three different ways. Two of them are similar to the techniques used by I. Ramadanov and M. Skwarczyński(see [11, 14, 13]) , while the third method using weak convergence is new. Moreover, we show that our reproducing kernel depends continuously on a weight of integration. The idea of using the minimal norm property in such a proof is novel and, which is important, it needs the convergence of weights only almost everywhere.