Volume 23, Issue 4

In the present paper, we mainly study quantitative Voronovskaya-type theorems for q -Szász operators defined in [20]. We consider weighted spaces of functions and the corresponding weighted modulus of continuity. We obtain the quantitative q -Voronovskaya-type theorem and the q -Grüss–Voronovskaya-type theorem in terms of the weighted modulus of continuity of q -derivatives of the approximated function. In this way, we either obtain the rate of pointwise convergence of q -Szász operators or we present these results for a subspace of continuous functions, although the classical ones are valid for differentiable functions.

This paper focuses on the analysis of the stationary case of incompressible viscoelastic generalized Oldroyd-B fluids derived in [2] by Bejaoui and Majdoub. The studied model is different from the classical Oldroyd-B fluid model in having a viscosity function which is shear-rate depending, and a diffusive stress added to the equation of the elastic part of the stress tensor. Under some conditions on the viscosity stress tensor and for a large class of models, we prove the existence of weak solutions in both two-dimensional and three-dimensional bounded domains for shear-thickening flows.

We obtain invertibility and Fredholm criteria for the Wiener–Hopf plus Hankel operators acting between variable exponent Lebesgue spaces on the real line. Such characterizations are obtained via the so-called even asymmetric factorization, which is applied to the Fourier symbols of the operators under study.

In the paper we consider three-dimensional Riquier-type and classical mixed boundary value problems for the polymetaharmonic equation (Δ+k12)⁢(Δ+k22)⁢u=0${(\Delta+k^{2}_{1})(\Delta+k^{2}_{2})u=0}$. We investigate these problems by means of the potential method and the theory of pseudodifferential equations. We prove the existence and uniqueness theorems in Sobolev–Slobodetskii spaces, analyse the asymptotic properties of solutions and establish the best Hölder smoothness results for solutions.

We investigate screen type mixed boundary value problems for anisotropic pseudo-Maxwell’s equations. We show that the problems with tangent traces are well posed in tangent Sobolev spaces. The unique solvability results are proven based on the potential method and coercivity result of Costabel on the bilinear form associated with pseudo-Maxwell’s equations.

Theorems on the continuous dependence of a solution of the Cauchy problem with respect to the nonlinear term of the right-hand side and initial data are proved for neutral functional differential equations whose right-hand sides are linear with respect to the prehistory of the phase velocity. Under the initial data we imply a collection of initial moment, variable delays entering in the phase coordinates, initial vector and initial functions. In this paper, an essential novelty is that perturbations of variable delays are taken into account in proving the main theorems.

For higher order nonlinear functional differential equations, sufficient conditions for the solvability and unique solvability of some nonlinear nonlocal boundary value problems are established.

The present paper deals with the Riemann boundary value problem for analytic functions in the framework of the new function spaces introduced by the first two authors, the so-called grand variable exponent Lebesgue spaces which unify two non-standard type function spaces: variable exponent Lebesgue spaces and grand Lebesgue spaces.

In this paper, we establish three kinds of endpoint estimates for a class of multilinear singular integral operators and obtain the boundedness of this kind of multilinear singular integral operators on the product of BMO spaces, product of LMO spaces, and product of λ-central BMO spaces, respectively. Moreover, as special cases, the corresponding results of multilinear Calderón–Zygmund operators can be deduced.

An efficient condition is established ensuring that on any interval of length ω, any nontrivial solution of the equation u′′=p⁢(t)⁢u${u^{\prime\prime}=p(t)u}$ has at most one zero. Based on this result, the unique solvability of a periodic boundary value problem is studied.

In this paper, we present an asymptotic formula for Bateman’s G -function G⁢(x)${G(x)}$ and deduce the double inequality 12⁢x2+3/2<G⁢(x)-1x<12⁢x2,x>0.$\frac{1}{2x^{2}+3/2}<G(x)-\frac{1}{x}<\frac{1}{2x^{2}},\quad x>0.$ We apply this result to find estimates for the error term of the alternating series ∑k=1∞(-1)k-1k+h$\sum_{k=1}^{\infty}\kern-2.0pt\frac{(-1)^{k-1}}{k+h}$, h≠-1,-2,-3,…$h\kern-1.0pt\neq\kern-1.0pt-1,\kern-0.5pt-2,\kern-0.5pt-3,\ldots\,$. Also, we study the monotonicity of some functions involving the function G⁢(x)${G(x)}$. Finally, we propose a sharp double inequality for the function G⁢(x)${G(x)}$ as a conjecture.

In this paper, we introduce and investigate the weighted generalized Drazin inverse in rings. We also introduce and investigate the weighted EP elements

The paper is devoted to the Lp${L^{p}}$-theory of boundary integral operators for boundary value problems described by anisotropic Helmholtz operators with variable coefficients in unbounded domains with unbounded smooth boundary. We prove the invertibility of boundary integral operators for Dirichlet and Neumann problems in the Bessel-potential spaces Hs,p⁢(∂⁡D)${H^{s,p}(\partial D)}$, p∈(1,∞)${p\in(1,\infty)}$, and the Besov spaces Bp,qs⁢(∂⁡D)${B_{p,q}^{s}(\partial D)}$, p,q∈[1,∞]${p,q\in[1,\infty]}$. We prove also the Fredholmness of the Robin problem in these spaces and give the index formula.

In this paper, we give an interpretation of some classical objects of function theory in terms of Banach algebras of linear operators in a Hilbert space. We are especially interested in quasisymmetric homeomorphisms of the circle. They are boundary values of quasiconformal homeomorphisms of the disk and form a group QS⁡(S1)${\operatorname{QS}(S^{1})}$ with respect to composition. This group acts on the Sobolev space H01/2⁢(S1,ℝ)${H^{1/2}_{0}(S^{1},\mathbb{R})}$ of half-differentiable functions on the circle by reparameterization. We give an interpretation of the group QS⁡(S1)${\operatorname{QS}(S^{1})}$ and the space H01/2⁢(S1,ℝ)${H^{1/2}_{0}(S^{1},\mathbb{R})}$ in terms of noncommutative geometry.