Volume 33, Issue 2

In this paper, we characterize quasi-contractive, stable and convergent weighted conditional-type (WCT) operators on L p ⁢ ( μ ) {L^{p}(\mu)} . Indeed we provide equivalent conditions for quasi-contraction WCT operators. Also, we prove that convergence, uniform stability, strong stability and weak stability of WCT operators are equivalent. Finally, we provide some concrete examples to illustrate our main results.

We prove the following series representation for the Catalan constant G : G = 3 10 - 48 ⁢ ∑ n = 0 ∞ 4 ⁢ n 2 + 8 ⁢ n - 9 ( 4 ⁢ n 2 - 4 ⁢ n + 5 ) ⁢ ( 4 ⁢ n 2 + 20 ⁢ n + 29 ) ⁢ S n 3 , G=\frac{3}{10}-48\sum_{n=0}^{\infty}\frac{4n^{2}+8n-9}{(4n^{2}-4n+5)(4n^{2}+20% n+29)}S_{n}^{3}, where S n = ∑ k = 0 n ( - 1 ) k 2 ⁢ k + 1 S_{n}=\sum_{k=0}^{n}\frac{(-1)^{k}}{2k+1} is the n th partial sum of the classical Leibniz series for π 4 {\frac{\pi}{4}} .

This paper discusses possible new bounds for the singular values and unitarily invariant norms of accretive-dissipative matrices. The obtained bounds are possible generalizations for certain known results from the context of positive matrices to the context of accretive-dissipative ones.

The present paper introduces the notions of strong P p 2 {P_{p}^{2}} -convergence and P p 2 {P_{p}^{2}} -statistical convergence for double sequences with respect to an Orlicz function. Based on these new notions of convergence, new double sequence spaces and some of their properties are given. Furthermore, certain inclusion relations are analyzed. Finally, we observe that, under proper choices, the analogous results can be obtained for modulus function.

Let S be an infinite, finitely generated semigroup, endowed with a probability measure. Based on the work [A. Erschler and A. Karlsson, Homomorphisms to ℝ \mathbb{R} constructed from random walks, Ann. Inst. Fourier (Grenoble) 60 2010, 6, 2095–2113], we construct a real character of S using a random walk approach. In our construction, the word length used in Erschler and Karlsson’s work is replaced by a quasicharacter.

This article explores a new contact problem involving viscoelastic materials locked with a rigid foundation. The constitutive relation in our study is constructed using a fractional Kelvin–Voigt model to describe displacement behavior. The contact is characterized by a variation of the Signorini condition, while friction is modeled by a nonlocal Coulomb’s friction law. We present the mathematical model for the viscoelastic process, provide the variational formulation, and establish the existence of a solution. We present fully discrete finite element schemes for the variational problem and derive error estimates for the approximations.

In the present paper, we introduce certain classes of functions of weighted bounded oscillation on bounded Vilenkin groups. For such classes, we employ the summability methods of the theory of double Vilenkin–Fourier series.

This paper investigates the existence of at least two distinct weak solutions using Bonanno’s theorem [G. Bonanno, Relations between the mountain pass theorem and local minima, Adv. Nonlinear Anal. 1 (2012), no. 3, 205–220, Theorem 3.2], and establishes the existence of no fewer than two nontrivial weak solutions through the Bonanno–D’Aguì theorem [G. Bonanno and G. D’Aguì, Two nonzero solutions for elliptic Dirichlet problems, Z. Anal. Anwend. 35 (2016), no. 4, 449–464, Theorem 2.1] applied to a general class of fourth-order Leray–Lions problems that include an s ⁢ ( x ) {s(x)} -Hardy potential and a nonlocal singular term.

In this paper, the well-posedness of a boundary value problem for an inhomogeneous equation of string vibration with Dirichlet, Neumann and Robin conditions on separate parts of the boundary is proved. In this case, the solution is given in quadratures. If the Robin condition turns into the Neumann condition, then the problem becomes ill-posed. In particular, it is shown that the corresponding homogeneous problem has an infinite number of linearly independent solutions that can be written explicitly.

This manuscript explores various characteristics of generalized fractal measures. We expand the concept of fractal integrals in relation to step functions and examine their numerous properties. Notably, since all step functions are classified as simple functions, we apply the aforementioned generalized measure to introduce Lebesgue-type integrals, referred to as FL -integrals. Additionally, we demonstrate that all F α {F^{\alpha}} -integrable functions are FL -integrals. Lastly, we address the bounded convergence theorem within the context of fractals.

Our aim in this note is to deal with Trudinger-type inequalities for variable Riesz potentials of functions in Musielak–Orlicz spaces over bounded metric measure spaces as an extension of the previous paper by Hurri-Syrjänen and the authors ( Complex Var. Elliptic Equ. 68 (2023), no. 10, 1694–1714). As an application, we prove Trudinger-type inequalities for double phase functionals.

This paper explores the properties of multiplicative Lie algebra structures on a nilpotent group of class 2.

This paper investigates summability properties under nonstandard growth conditions. We consider the boundary value problem of the 𝒜 {\cal A} -harmonic equation { - div ⁡ 𝒜 ⁢ ( x , ∇ ⁡ u ⁢ ( x ) ) = f ⁢ ( x ) in  ⁢ Ω , u ⁢ ( x ) = u ∗ ⁢ ( x ) on  ⁢ ∂ ⁡ Ω , \left\{\begin{aligned} \displaystyle{}{{-}\operatorname{div}\mathcal{A}(x,% \nabla u(x))}&\displaystyle=f(x)&&\displaystyle\mbox{in }\Omega,\\ \displaystyle u(x)&\displaystyle=u_{\ast}(x)&&\displaystyle\mbox{on }{\partial% \Omega},\end{aligned}\right. where f ⁢ ( x ) {f(x)} and D ⁢ u * ⁢ ( x ) {Du_{*}(x)} lie in some Marcinkiewicz spaces. Summability properties of entropy solutions are obtained under nonstandard growth conditions on the operator 𝒜 : Ω × ℝ n → ℝ n {{\cal A}:\Omega\times\mathbb{R}^{n}\rightarrow\mathbb{R}^{n}} . We also consider T -minima of the integral functional 𝒥 ⁢ ( v ) = ∫ Ω j ⁢ ( x , ∇ ⁡ v ⁢ ( x ) ) ⁢ 𝑑 x - ∫ Ω f ⁢ ( x ) ⁢ v ⁢ ( x ) ⁢ 𝑑 x , {\cal J}(v)=\int_{\Omega}j(x,\nabla v(x))\,dx-\int_{\Omega}f(x)v(x)\,dx, where f ⁢ ( x ) {f(x)} lies in some Marcinkiewicz space. We consider T -minima among functions whose boundary values agree with u * ⁢ ( x ) {u_{*}(x)} , and obtain some summability properties under nonstandard growth conditions of the operator j ⁢ ( x , ξ ) : Ω × ℝ n → ℝ {j(x,\xi):\Omega\times\mathbb{R}^{n}\rightarrow\mathbb{R}} .

Let A and G be finite groups such that A acts coprimely on G by automorphisms. For any fixed prime divisor p of | G | {|G|} , we provide a complete characterization of the structure of a group G in which every maximal A -invariant subgroup of order divisible by p is nilpotent.

A way is presented to extend the notion of pseudoanalyticity of the Bers–Vekua theory to differential forms defined on a complex manifold. A sheaf is defined whose sections over an open subset are differential forms whose component functions with respect to any trivializing sections satisfy in each argument the linear Carleman–Bers–Vekua equation. Such a sheaf can be defined as the kernel sheaf of an appropriate sheaf homomorphism. For it, under certain conditions, an analogue of the Poincaré lemma is valid and, under the same conditions, the sequence of such sheaf homomorphisms forms a complex, which makes is possible to prove an analogue of the Dolbeault’s theorem for these sheaves.

We present an elementary construction of a (highly degenerate) Hopf pairing between the universal enveloping algebra U ⁢ ( 𝔤 ) {U(\mathfrak{g})} of a finite-dimensional Lie algebra 𝔤 {\mathfrak{g}} over arbitrary field 𝒌 {{\boldsymbol{k}}} , and the Hopf algebra 𝒪 ⁢ ( Aut ⁡ ( 𝔤 ) ) {\mathcal{O}(\operatorname{Aut}(\mathfrak{g}))} of regular functions on the automorphism group of 𝔤 {\mathfrak{g}} . This pairing induces a Hopf action of 𝒪 ⁢ ( Aut ⁡ ( 𝔤 ) ) {\mathcal{O}(\operatorname{Aut}(\mathfrak{g}))} on U ⁢ ( 𝔤 ) {U(\mathfrak{g})} , which, together with an explicitly given coaction, makes U ⁢ ( 𝔤 ) {U(\mathfrak{g})} into a braided commutative Yetter–Drinfeld 𝒪 ⁢ ( Aut ⁡ ( 𝔤 ) ) {\mathcal{O}(\operatorname{Aut}(\mathfrak{g}))} -module algebra. From these data one constructs a Hopf algebroid structure on the smash product algebra ♯ ⁡ 𝒪 ⁢ ( Aut ⁡ ( 𝔤 ) ) U ⁢ ( 𝔤 ) {\mathcal{O}(\operatorname{Aut}(\mathfrak{g}))\mathbin{\sharp}U(\mathfrak{g})} , retaining essential features from earlier constructions of a Hopf algebroid structure on infinite-dimensional versions of the Heisenberg double of U ⁢ ( 𝔤 ) {U(\mathfrak{g})} , including a noncommutative phase space of Lie algebra type, while avoiding the need of completed tensor products. We prove a slightly more general result, where the algebra 𝒪 ⁢ ( Aut ⁡ ( 𝔤 ) ) {\mathcal{O}(\operatorname{Aut}(\mathfrak{g}))} is replaced by 𝒪 ⁢ ( Aut ⁡ ( 𝔥 ) ) {\mathcal{O}(\operatorname{Aut}(\mathfrak{h}))} and where 𝔥 {\mathfrak{h}} is any finite-dimensional Leibniz algebra having 𝔤 {\mathfrak{g}} as its maximal Lie algebra quotient.

We present a rigorous proof of the time evolution integration theorems that suitably extend differential geometry to the calculus of moving surfaces. This proof demonstrates that the calculus of moving surfaces is a complete extension of differential geometry to the dynamic manifolds.

In this paper, we study the generalized Volterra integral operators T g n , m ⁢ f ⁢ ( z ) = ∫ 0 z f ( n ) ⁢ ( w ) ⁢ g ( m ) ⁢ ( w ) ⁢ 𝑑 w T_{g}^{n,m}f(z)=\int_{0}^{z}f^{(n)}(w)g^{(m)}(w)\,dw acting on the weighted Fock spaces F α , ω p {F_{\alpha,\omega}^{p}} , where ω is a weight satisfying certain restricted A ∞ {A_{\infty}} -conditions. Using a unified approach, the boundedness, compactness, and essential norm of the generalized Volterra integral operators T g n , m : F α , ω 1 p → F β , ω 2 q {T_{g}^{n,m}:F_{\alpha,\omega_{1}}^{p}\rightarrow F_{\beta,\omega_{2}}^{q}} for all 0 < p , q < ∞ {0<p,q<\infty} are completely characterized. As an application, we generalize our results to the Fock–Sobolev-type spaces.

In this paper, the Duhamel product is used to endow analytic tent spaces with a Banach algebra structure. The Hadamard–Bergman convolution on these spaces is also described. Additionally, the Deddens algebras of composition operators are studied.