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.     

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.     

On a parabolic‐elliptic chemotaxis system with periodic asymptotic behavior

We study a parabolic‐elliptic chemotactic PDEs system, which describes the evolution of a biological population “u” and a chemical substance “v” in a bounded domain . We consider a growth term of logistic type in the equation of “u” in the form μu(1 ? u + f(t,x)). The function “f,” describing the resources of the systems, presents a periodic asymptotic behavior in the sense where f ^? is independent of x and periodic in time. We study the global existence of solutions and its asymptotic behavior. Under suitable assumptions on the initial data and f ^?, if the constant chemotactic sensitivity χ satisfies we obtain that the solution of the system converges to a homogeneous in space and periodic in time function.     

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.     

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.     

Existence of nontrivial solutions for fractional Schrödinger equations with critical or supercritical growth

In this paper, we study the following fractional Schrödinger equation with critical or supercritical growth where 0 < s < 1, N > 2s, λ > 0, , , ( ? Δ)^s denotes the fractional Laplacian of order s and f is a continuous superlinear but subcritical function. Under some suitable conditions, we prove that the equation has a nontrivial solution for small λ > 0 by variational methods. Our main contribution is related to the fact that we are able to deal with the case .     

Spatially almost periodic complex‐valued solutions of the Boussinesq equations

We extend several ideas developed in E. Dinaburg, Ya. G. Sinai and D. Li's papers (see Lemma 2.2 , Remark   2.1 , Lemma 2.4 , Remark   2.2 , Remark 2.3 ) and construct a unique mild solution (u,θ) of the system 1.1 satisfying 1.3 with spatially almost periodic initial data . Moreover, (u,θ) is also a spatially almost periodic complex‐valued mild solution. Assume further that (see Definition 1.6 ). Then, we overcome the difficulty in the work of Y. Giga et al. (see Introduction below) and show that the above mild solution also satisfies 1.4 for all .     

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.     

Blow‐up dynamics of L2−critical inhomogeneous fractional nonlinear Schrödinger equation

The purpose of this work is to investigate the blow‐up dynamics of L²?critical focusing inhomogeneous fractional nonlinear Schrödinger equation: with 0<b<1. For this, we establish a new compactness lemma related to the equation. By applying this lemma, we study the dynamical behavior for blow‐up solutions for initial data satisfying , where Q is the ground state solution of our problem.     

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 .     

The inverse backscattering for Schrödinger operators for potentials with noncompact support

We consider the inverse backscattering problem for the Schrödinger operator H = ?Δ + V on , n ≥ 3, as well as the higher‐order Schrödinger operator ( ? Δ)^m + V, m = 2,3,…. We show that in some suitable Banach spaces, the map from the potential to the backscattering amplitude is a local diffeomorphism. This kind of problem (for m = 1) was studied by Eskin and Ralston [Comm. Math. Phys., 124(2), 169‐215 (1989)], where they assumed that . In this paper, we replace the assumption on V with certain decay assumption at infinity.     

Lower bound for the ratios of eigenvalues of Schrödinger with nonpositive single‐barrier potentials

Horváth and Kiss (Proc. Amer. Math. Soc., 2005) proved the upper bound estimate for Dirichlet eigenvalue ratios of the Schrödinger problem ?y^′′ + q(x)y = λy with nonnegative and single‐well potential q. In this paper, we prove that if q(x) is a nonpositive, continuous, and single‐barrier potential, then for λ_n > λ_m≥ ? 2q^?, where . In particular, if q(x) satisfies the additional condition , then λ₁ > 0 and for n > m ≥ 1. For this result, we develop a new approach to study the monotonicity of the modified Prüfer angle function.     

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.     

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.     

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.     

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.     

Schrödinger‐Poisson system with Hardy‐Littlewood‐Sobolev critical exponent

In this paper, we consider the following Schrödinger‐Poisson system: where parameters α,β∈(0,3),λ>0, , , and are the Hardy‐Littlewood‐Sobolev critical exponents. For α<β and λ>0, we prove the existence of nonnegative groundstate solution to above system. Moreover, applying Moser iteration scheme and Kelvin transformation, we show the behavior of nonnegative groundstate solution at infinity. For β<α and λ>0 small, we apply a perturbation method to study the existence of nonnegative solution. For β<α and λ is a particular value, we show the existence of infinitely many solutions to above system.     

An Osgood type regularity criterion for the 2D Oldroyd‐B model

By means of the Littlewood‐Paley decomposition and the div‐curl Theorem by Coifman‐Lions‐Meyer‐Semmes, we prove an Osgood type regularity criterion for the 2D incompressible Oldroyd‐B model, that is, where denotes the Fourier localization operator whose spectrum is supported in the shell {|ξ|≈2^j}.     

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.