Volume 31, Issue 6

In this paper, we consider a fractional p -Laplacian elliptic Dirichlet problem that possesses one control parameter and has a Lipschitz nonlinearity order of p - 1 {p-1} . The multiplicity of the weak solutions is proved by means of the variational method and critical point theory. We investigate the existence of at least three solutions to the problem.

We present necessary and sufficient conditions under which the sum of two group invertible elements in a ring is group invertible. As applications, we establish the existence of group inverses of certain 2 × 2 {2\times 2} block-operator matrices over a Banach space. These generalize the known results, e.g., Zhou, Chen and Zhu ( Comm. Algebra 48 (2020), 676–690) and Benítez, Liu and Zhu ( Linear Multilinear Algebra 59 (2011), 279–289).

In this paper, we introduce the notion of σ-symmetric amenability of Banach algebras and investigate some hereditary properties of them. We also apply our results to several abstract Segal algebras and group algebras.

The spherical mean operator has been widely studied and has seen remarkable development in many areas of harmonic analysis. In this paper, we consider the Stockwell transform related to the spherical mean operator. Since the study of time-frequency analysis is both theoretically interesting and practically useful, we will study several problems for the generalized Stockwell transform. Firstly, we explore the Shapiro uncertainty principle for this transformation. Next, we will study the boundedness and then the compactness of localization operators related to the generalized Stockwell transform, and finally we will introduce and study its scalogram.

In this work, we establish a theorem concerning the extension of positive weak solutions for a stationary fractional Laplacian problem featuring weight functions that change sign. Additionally, we introduce novel conditions to ensure the existence of a positive solution for the given problem. These conditions are derived utilizing the approach of sub-super solutions, thereby extending and complementing existing results in the literature.

We prove a lower bound estimate for capacities in Hajłasz–Besov, Hajłasz–Triebel–Lizorkin and Hajłasz–Sobolev spaces with generalized smoothness defined on metric spaces in terms of Netrusov–Hausdorff content or Hausdorff content. These results are improvements of the results obtained in [Z. Li, D. Yang and W. Yuan, Lebesgue points of Besov and Triebel–Lizorkin spaces with generalized smoothness, Mathematics 9 2021, 10.3390/math9212724].

In this paper, we study the following elliptic problem involving the ( p ⁢ ( y ) , q ⁢ ( y ) {p(y),q(y)} )-Laplacian operator: { - div ⁡ ( a ⁢ ( y ) ⁢ | ∇ ⁡ v | p ⁢ ( y ) - 2 ⁢ ∇ ⁡ v ) + b ⁢ ( y ) ⁢ | v | p ⁢ ( y ) - 2 ⁢ v - div ⁡ ( | ∇ ⁡ v | q ⁢ ( y ) - 2 ⁢ ∇ ⁡ v ) = g ⁢ ( y , v ) , y ∈ Ω , v = 0 on  ⁢ ∂ ⁡ Ω , \left\{\begin{aligned} \displaystyle{}{-}\operatorname{div}(a(y)|\nabla v|^{p(% y)-2}\nabla v)+b(y)|v|^{p(y)-2}v-\operatorname{div}(|\nabla v|^{q(y)-2}\nabla v% )&\displaystyle=g(y,v),&&\displaystyle y\in\Omega,\\ \displaystyle v&\displaystyle=0&&\displaystyle\phantom{}\text{on }\partial% \Omega,\end{aligned}\right. with Dirichlet boundary condition in an exterior domain Ω ( ⊂ ℝ n ) {(\subset\mathbb{R}^{n})} with smooth boundary, where 1 < q ⁢ ( y ) < p ⁢ ( y ) < n 1<q(y)<p(y)<n . We prove the existence of solutions in W 0 1 , p ⁢ ( y ) ⁢ ( Ω ) {W^{1,p(y)}_{0}(\Omega)} for the superlinear case by using the Mountain Pass Theorem.

Given a linear constant coefficient ODE depending on a parameter, when this parameter approaches zero, the solution set converges to the solution set of the limit differential equation if the leading coefficient does not vanish. The situation is very subtle in the singular case, i.e., in the case when this coefficient becomes zero. The solution set then may even collapse completely. In this note, a formalism is developed in which the solution set of a linear constant coefficient ODE always depends continuously on the equation coefficients.

Weak type estimates for genuine Calderón–Zygmund operators are established on the local Morrey spaces associated with ball quasi-Banach function spaces by two different methods. Above all, we obtain weak type estimates for the operator on the local weak Morrey spaces with variable exponents.

In this study, the commutativity conditions with generalized derivations, which have not been examined before, are discussed. Under these conditions, the subject of generalized derivations in (semi)prime Banach algebras is studied. New fundamental results will be provided for researchers in this field and generalize some of the results found in the literature.

In this paper, we study the existence of multiple solutions for the singular problem { a ⁢ ( x , u , ∇ ⁡ u ) - div ⁢ ( b ⁢ ( x , u , ∇ ⁡ u ) ) = u - α + λ ⁢ c ⁢ ( x , u ) in  ⁢ Ω , u > 0 in  ⁢ Ω , u = 0 on  ⁢ ℝ n ∖ Ω , \left\{\begin{aligned} \displaystyle{}a(x,u,\nabla u)-{\rm div}(b(x,u,\nabla u% ))&\displaystyle=u^{-\alpha}+\lambda c(x,u)&&\displaystyle\phantom{}\text{in }% \Omega,\\ \displaystyle u&\displaystyle>0&&\displaystyle\phantom{}\text{in }\Omega,\\ \displaystyle u&\displaystyle=0&&\displaystyle\phantom{}\text{on }{\mathbb{R}}% ^{n}\setminus\Omega,\end{aligned}\right. where Ω ⊂ ℝ n {\Omega\subset\mathbb{R}^{n}} ( n ≥ 3 ) {(n\geq 3)} is a bounded domain with C 1 {C^{1}} boundary, λ is a positive parameter, 0 < α ≤ 1 < p ≤ n {0<\alpha\leq 1<p\leq n} and p * = n ⁢ p n - p {p^{*}=\frac{np}{n-p}} is the critical exponent for Sobolev embedding. Using the fibering maps and the Nehari manifold, we prove the existence of at least two positive solutions for all values of the parameter λ belonging to an open bounded interval of ℝ + {\mathbb{R}_{+}} .

This paper is concerned with constructions and characterizations of matrix equalities that involve mixed products of Moore–Penrose inverses and group inverses of two matrices. We first construct a mixed reverse-order law ( A ⁢ B ) † = B ∗ ⁢ ( A ∗ ⁢ A ⁢ B ⁢ B ∗ ) # ⁢ A ∗ {(AB)^{{\dagger}}=B^{\ast}(A^{\ast}ABB^{\ast})^{\#}A^{\ast}} , and show that this matrix equality always holds through the use of a special matrix rank equality and some matrix range operations, where A and B are two matrices of appropriate sizes, ( ⋅ ) ∗ {(\,\cdot\,)^{\ast}} , ( ⋅ ) † {(\,\cdot\,)^{{\dagger}}} and ( ⋅ ) # {(\,\cdot\,)^{\#}} mean the conjugate transpose, the Moore–Penrose inverse, and the group inverse of a matrix, respectively. We then give a diverse range of variation forms of this equality, and derive necessary and sufficient conditions for them to hold. Especially, we show an interesting fact that the two reverse-order laws ( A ⁢ B ) † = B † ⁢ A † {(AB)^{{\dagger}}=B^{{\dagger}}A^{{\dagger}}} and ( A ∗ ⁢ A ⁢ B ⁢ B ∗ ) # = ( B ⁢ B ∗ ) # ⁢ ( A ∗ ⁢ A ) # {(A^{\ast}ABB^{\ast})^{\#}=(BB^{\ast})^{\#}(A^{\ast}A)^{\#}} are equivalent.

In the paper, the authors establish several integral inequalities of the Ostrowski type for s -logarithmically convex functions. These integral inequalities modify the conditions and correct errors in two main theorems of the paper [A. O. Akdemir and M. Tunç, Ostrowski type inequalities for s -logarithmically convex functions in the second sense with applications, Georgian Math. J. 22 (2015), no. 1, 1–7].

Yu, Wu and Leng defined the quasi L p {L_{p}} -intersection bodies. In this paper, we consider the polar dualities and star dualities of quasi L p {L_{p}} -intersection bodies and establish some related inequalities.

In this article, we improve some Berezin number inequalities concerning a Hilbert space. It is shown that if T is a bounded linear operator on a Hilbert space, then for any r ≥ 1 {r\geq 1} , 𝐛𝐞𝐫 2 ⁢ r ( T ) ≤ 1 2 𝐛𝐞𝐫 r ( | T * | 2 ⁢ ( 1 - t ) | T | 2 ⁢ t ) + 1 4 ∥ | T | 4 ⁢ r ⁢ t + | T * | 4 ⁢ r ⁢ ( 1 - t ) ∥ 𝐛𝐞𝐫   ( 0 ≤ t ≤ 1 ) , \mathbf{ber}^{2r}(T)\leq\frac{1}{2}\mathbf{ber}^{r}({{|{{T}^{*}}|}^{2(1-t)}}{{% |T|}^{2t}})+\frac{1}{4}{{\|{{|T|}^{4rt}}+{{|{{T}^{*}}|}^{4r(1-t)}}\|}_{\mathbf% {ber}}}\quad(0\leq t\leq 1), where | T | = ( T * ⁢ T ) 1 2 {|T|={{({{T}^{*}}T)}^{\frac{1}{2}}}} .

This is an addendum to [R. Frankiewicz, J. Jureczko and B. Weglorz, On Kuratowski partitions in the Marczewski and Laver structures and Ellentuck topology, Georgian Math. J. 26 2019, 4, 591–598].