Volume 15, Issue 4

The paper deals with some direct results on ordinary and simultaneous approximations for iterative combinations of a new type of Bernstein–Durrmeyer operators. Gupta and Vasishtha [Math. Comput. Modelling 39: 521–527, 2004] have recently claimed that iterative combinations can be applied only for those operators for which 𝑡 maps exactly to 𝑥. Here we disagree with their claim and state that iterative combinations can be applied for other operators which do not reproduce linear functions either.

The present paper is devoted to the investigation of one dynamical three-dimensional mathematical model of the fluid-solid interaction. The variational formulation of the corresponding initial boundary problem is considered and a problem for abstract second order evolution equation is formulated, which is a generalization of the three-dimensional initial boundary value problem. For the stated abstract problem the existence and uniqueness of solution, and the energy equality are proved, which yield the corresponding result for the dynamical three-dimensional problem of fluid-solid interaction.

We present a simple proof of the existence and uniqueness of a weak solution for a class of quasilinear elliptic reaction-diffusion systems. The proof is based on an iterative process and on some a priori estimates.

We prove an existence result for the Dirichlet problem associated to some degenerate quasilinear elliptic equations in a bounded open set Ω in in the setting of weighted Sobolev spaces .

We use the modification of Krasnoselskii's fixed point theorem due to T. A. Burton ([Proc. Amer. Math. Soc. 124: 2383–2390, 1996]) to show that the scalar nonlinear differential equation with functional delay 𝑥′(𝑡) = –𝑎(𝑡)𝑥 3 (𝑡) + 𝐺(𝑡, 𝑥 3 (𝑡 – 𝑟(𝑡))) has a periodic solution. It is not required that 𝑟(𝑡) be differentiable, while 𝑎 and 𝐺 are continuous with respect to their arguments.

For and β 1 , . . . , β 𝑛 > 1 let be defined by , let 𝐵 be the open unit ball in and let ∑ = {(𝑥, φ (𝑥)) : 𝑥 ∈ 𝐵}. For let 𝑅𝑓 : ∑ → ℂ be defined by where denotes the usual Fourier transform of 𝑓. Let σ be the Borel measure on ∑ defined by σ (𝐴) = ∫ 𝐵 χ 𝐴 (𝑥, φ (𝑥)) 𝑑𝑥 and 𝐸 be the type set for the operator 𝑅, i.e. the set of pairs for which there exists 𝑐 > 0 such that for all . In this paper we obtain a polygonal domain contained in 𝐸. We also give necessary conditions for a pair . In some cases this result is sharp up to endpoints.

We obtain the existence and stability results for a fourth order Dirichlet problems with nonlinearity being convex in a certain interval. A dual variational method is introduced, which relies on investigating the primal and dual action functionals on certain subsets of their domains. The dependence on a functional parameter for a fourth order Dirichlet problem is considered as a consequence of stability.

A version of the precise definition of Euler–Venn diagram for a given family of subsets of a universal set is presented. Certain geometrical properties of such diagrams are discussed and close connections with purely combinatorial problems and with the theory of convex sets are indicated. In particular, some geometrical realizations of uncountable independent families of sets are considered.

For the two-dimensional linear differential system with Lebesgue integrable coefficients 𝑝 𝑖𝑘 : [𝑎, 𝑏] → ℝ (𝑖 = 1, 2), a Beurling–Borg type theorem is proved on an upper estimate of the number of zeros of an arbitrary non-trivial solution.

We study the boundedness of the maximal operator in the weighted variable exponent spaces 𝐿 𝑝(·) (𝑋, ϱ ) on a doubling measure metric space 𝑋. When 𝑋 is bounded, the weight belongs to a version of a Muckenhoupt-type class, which is narrower than the expected Muckenhoupt condition for a variable exponent, but coincides with the usual Muckenhoupt class 𝐴 𝑝 in the case of a constant 𝑝. For the bounded 𝑋 we also consider the class of weights of the form , where the functions 𝑤 𝑘 (𝑟) have finite upper and lower indices 𝑚(𝑤) and 𝑀(𝑤) satisfying the condition , where 𝔡𝔦𝔪 (𝑋) is a version of lower dimension of the space 𝑋. In the case of unbounded 𝑋 we admit weights of the form . Some of the results are new even in the case of a constant 𝑝. We also deal with some new notions of upper and lower local dimensions of measure metric spaces.

We study tests of power one for the following change-point problem. Suppose one observes a process 𝑊 which is either a Brownian motion without drift or a Brownian motion that has zero drift up to a random time τ after which with equal probability the drift becomes either θ or – θ , where the value of θ > 0 is known. The distribution of τ is also assumed to be known. We search for a stopping time 𝑇* that minimizes an appropriate Bayes risk and give a solution that is asymptotically optimal, when the cost of observation tends to zero.

Using the notion of weighted sharing of values which was introduced by I. Lahiri [Nagoya Math. J. 161: 193–206, 2001], we deal with the uniqueness problem of meromorphic functions concerning differential polynomials and obtain some theorems which not only improve a recent result of W. L. Xiong, W. C. Lin and S. Mori [Sci. Math. Jpn. 62: 305–315, 2005], but also improve and supplement the result of W. C. Lin and H. X. Yi [Complex Var. Theory Appl. 49: 793–806, 2004].

We consider analytic and pluriharmonic functions belonging to the classes 𝐵 𝑝 (Ω) and 𝑏 𝑝 (Ω) and defined in the ball . The theorems established in the paper make it possible to obtain some integral representations of functions of the above-mentioned classes. The existence of bounded projectors from the space 𝐿( ρ , Ω) into the space 𝐵 𝑝 (Ω) and from the space 𝐿( ρ , Ω) into the space 𝑏 𝑝 (Ω) is proved. Also, consideration is given to the existence of boundary values of fractional integrals of functions of the spaces 𝐵 𝑝 (Ω) and 𝑏 𝑝 (Ω).

For the second order nonlinear singular differential equation 𝑢″ + 𝑓(𝑡, 𝑢, 𝑢′) = 0, the unimprovable sufficient conditions for the solvability of the problem with the condition at infinity are established.

The following generalization (for α = 2 we suppose that β = 2 and λ > 1/4), of the Riemann–Weber version of Euler differential equation is introduced and it is considered together with a suitable boundary layer condition depending on α near 𝑥 = 0. It is shown that this problem is rectifiable (resp., unrectifiable) oscillatory on (0, 𝑏) provided α ∈ [2, 4) (resp., α ≥ 4). It is a kind of geometrical oscillations on (0, 𝑏) for which the finite (resp., infinite) length of the graphs of all its solutions is proposed.

We investigate compact composition operators acting on generalized Hardy spaces 𝐻 𝑤 . In fact, we prove that if 𝑤 is a differentiable, subharmonic and strictly increasing function defined on [0, ∞), then 𝐶 φ is compact on the generalized Hardy spaces if and only if it is compact on the Hardy space 𝐻 2 .

Let ( ξ 𝑘 ) 𝑘 ≥ 1 be a sequence of independent, identically distributed second order mean zero random elements in a separable Hilbert space 𝐻 and 𝐴 be an element of a certain class of linear continuous operators 𝐻 → 𝐻 such that ‖𝐴‖ < 1. Denote . We prove that if ‖𝐼 – 𝐴‖ tends to zero, where 𝐼 is the identity operator, then the normalized sum (𝐼 – 𝐴 2 ) 1/2 η 𝐴 converges in distribution to a Gaussian random element.

The solvability of the first boundary value problem is studied for a second order elliptic system with degeneration on the entire boundary of a multidimensional domain.

We classify the finite groups satisfying the following property: the orders of every three non-conjugated non-central elements are set-wise relatively prime.