Heinz‐type inequality and bi‐Lipschitz continuity for quasiconformal mappings satisfying inhomogeneous biharmonic equations

Let be the class of all sense‐preserving homeomorphic self‐mappings of . The aim of this paper is twofold. First, we obtain Heinz‐type inequality for (K,K′)‐quasiconformal mappings satisfying inhomogeneous biharmonic equation Δ(Δω) = g in unit disk with associated boundary value conditions and . Second, we establish biLipschitz continuity for (K,K′)‐quasiconformal mappings satisfying aforementioned inhomogeneous biharmonic equation when and are small enough.     

Adaptive Fourier decomposition in

In this paper, we study decomposition of functions in Hardy spaces . First, we will give a direct application of adaptive Fourier decomposition (AFD) of to functions in . Then, we study adaptive decomposition by the system (1) where A_a,p is the normalization factor making e_a(z) to be of unit p‐norm. Under the proposed decomposition procedure, we show that every can be effectively expressed by a linear combination of . We give a maximal selection principle of at the nth step and prove the convergence.     

Summability factor relations between absolute weighted and Cesàro means

By , we denote the set of all sequences such that Σ?_na_n is summable V whenever Σa_n is summable U, where U and V are two summability methods. Recently, Sar?göl has characterized the set for k > 1,α > ?1 and arbitrary positive sequences Now, in the present paper, we characterize the sets , k > 1 and , k ≥ 1 for arbitrary positive sequences Hence we extend these results to the range α≥ ? 1. In this way, some open problems in this topic are also completed.     

The Hardy space decomposition on multiangular domain

In this paper, we consider the problem of Hardy space decomposition on multiangular domain. By using rational approximation, we achieve that a function f in can be decomposed into a sum in the sense of , where are the boundary limits of functions in .     

Sharp oscillation and nonoscillation tests for delay dynamic equations

In this paper, we give new sufficient conditions for both oscillation and nonoscillation of the delay dynamic equation where and satisfy τ(t) ≤ σ(t) for all large t and . As an important corollary, we obtain the time scale invariant integral condition for nonoscillation: for all large t. Also, with some examples, we show that newly presented results are sharp.     

Fractional powers of the noncommutative Fourier's law by the S‐spectrum approach

Let e_?, for ? = 1,2,3, be orthogonal unit vectors in and let be a bounded open set with smooth boundary ?Ω. Denoting by a point in Ω, the heat equation, for nonhomogeneous materials, is obtained replacing the Fourier law, given by the following: into the conservation of energy law, here a, b, are given functions. With the S‐spectrum approach to fractional diffusion processes we determine, in a suitable way, the fractional powers of T. Then, roughly speaking, we replace the fractional powers of T into the conservation of energy law to obtain the fractional evolution equation. This method is important for nonhomogeneous materials where the Fourier law is not simply the negative gradient. In this paper, we determine under which conditions on the coefficients a, b, the fractional powers of T exist in the sense of the S‐spectrum approach. More in general, this theory allows to compute the fractional powers of vector operators that arise in different fields of science and technology. This paper is devoted to researchers working in fractional diffusion and fractional evolution problems, partial differential equations, and noncommutative operator theory.     

On uniform exponential stability of linear switching system

We prove that the linear switching system , where is bounded valued square matrices and ?:[0,1,2,…)→Ω is an arbitrary switching signals, is uniformly exponentially stable if the sequence is bounded, where s(k) is bounded valued sequence.     

More on a hyperbolic‐cotangent class of difference equations

We present a natural method for solving the difference equation where , parameter a, and initial values x_?j, , , are real or complex numbers. As a concrete example, the case k = 1, l = 2 is studied in detail, and several interesting formulas for solutions and objects used in the analysis of the equation are given. A useful remark on solvability of difference equations on complex domains is presented and used here.     

Positive ground state solutions for fractional Kirchhoff type equations with critical growth

We study the existence of positive ground state solutions for the following fractional Kirchhoff type equation where a,b > 0 are constants, μ is a positive parameter, with and s ∈ (0,1). Under suitable assumptions on V(x), by using a monotonicity trick and a global compactness principle, we prove that the equation admits a positive ground state solution if and μ > 0 large enough.     

Two‐dimensional solvable system of difference equations with periodic coefficients

We show that the following two‐dimensional system of difference equations: where , , , and are periodic sequences, is solvable, considerably extending some results in the literature. In the case when all these four sequences are periodic with period 2 or with period 3, we present closed‐form formulas for the general solutions to the corresponding systems of difference equations. Some comments regarding theoretical and practical solvability of the system, connected to the value of the period of the sequences, are given.     

Multiple positive solutions for Schrödinger‐Poisson system involving singularity and critical exponent

In this paper, the existence and multiplicity of positive solutions is established for Schrödinger‐Poisson system of the form where 0 ∈ Ω is a smooth bounded domain in , , and λ > 0 is a real parameter. Combining with the variational method and Nehari manifold method, two positive solutions of the system are obtained.     

Reciprocity formulae for generalized Dedekind‐Vasyunin‐cotangent sums

Recently, several works are done on the generalized Dedekind‐Vasyunin sum where and q are positive coprime integers, and ζ(a,x) denotes the Hurwitz zeta function. We prove explicit formula for the symmetric sum which is a new reciprocity law for the sum . Our result is a complement to recent results dealing with the sum studied by Bettin‐Conrey and then by Auli‐Bayad‐Beck. Accidentally, when a = 0, our reciprocity formula improves the known result in a previous study.     

On the limit Sobolev regularity for Dirichlet and Neumann problems on Lipschitz domains

We construct a bounded C¹ domain Ω in for which the regularity for the Dirichlet and Neumann problems for the Laplacian cannot be improved, that is, there exists f in such that the solution of in Ω and either on or on is contained in but not in for any . An analogous result holds for Sobolev spaces with .     

A note on the eigenvalues of p‐fractional Hardy–Sobolev operator with indefinite weight

In this article, we study the eigenvalues of p‐fractional Hardy operator where , , , and Ω is an unbounded domain in with Lipschitz boundary containing 0. The weight function V may change sign and may have singular points. We also show that the least positive eigenvalue is simple and it is uniquely associated to a nonnegative eigenfunction. Moreover, we proved that there exists a sequence of eigenvalues as .     

A Keller‐Segel‐fluid system with singular sensitivity: Generalized solutions

In bounded smooth domains , N ∈ {2,3}, we consider the Keller‐Segel‐Stokes system and prove global existence of generalized solutions if These solutions are such that blow‐up into a persistent Dirac‐type singularity is excluded.     

A complete solution to existence of H designs

An H design is a triple , where is a set of points, a partition of into disjoint sets of size , and a set of ‐element transverses of , such that each ‐element transverse of is contained in exactly one of them. In 1990, Mills determined the existence of an H design with . In this paper, an efficient construction shows that an H exists for any integer with . Consequently, the necessary and sufficient conditions for the existence of an H design are , , and , with a definite exception .     

A T1 theorem and Calderón–Zygmund operators in Campanato spaces on domains

Given a Lipschitz domain , a Calderón–Zygmund operator T and a modulus of continuity , we solve the problem when the truncated operator sends the Campanato space into itself. The solution is a T1 type sufficient and necessary condition for the characteristic function of D: To check the hypotheses of T1 theorem we need extra restrictions on both the boundary of D and the operator T. It is proved that the truncated Calderón–Zygmund operator with an even kernel is bounded on , provided D is a ‐smooth domain.     

Boundedness and compactness characterizations of Cauchy integral commutators on Morrey spaces

Let C_Γ be the Cauchy integral operator on a Lipschitz curve Γ. In this article, the authors show that the commutator [b,C_Γ] is bounded (resp, compact) on the Morrey space for any (or some) p ∈ (1,∞) and λ ∈ (0,1) if and only if (resp, ). As an application, a factorization of the classical Hardy space in terms of C_Γ and its adjoint operator is obtained.