Some approximation theorems for modified Bernstein-Durrmeyer operators

The modified Bernstein-Durrmeyer operators discussed in this paper are given byM_nf≡M_n(f,x)=(n+2)P_(n,k)∫_0~1p_n+1.k(t)f(t)dt,whereWe will show,for 0<α<1 and 1≤p≤∞     

Rate of convergence bounds of difference schemes for fourth-order elliptical equations in regions of arbitrary shape

The fictitious domain method is applied to construct a difference scheme for the first boundary-value problem for fourth-order elliptical equations in regions of arbitrary shape. The rate of convergence bound $$\left\| {y - \bar u} \right\|_{W_2^k (\omega )} \leqslant Mh^{\tfrac{{3 - k}}{2}} \left\| f \right\|_{L_2 (\Omega )} ,k = 1,2,$$ is proved. Here y is the solution of the difference problem, — u is the solution of the original problem continued as zero Ω₁, where Ω₁ is the complement of the region Ω to the rectangle Ω₀.     

Exact Non-Self-Similar Solutions of the Equation $$u_t = \Delta \ln u$$

In this paper, we obtain new exact non-self-similar solutions of the nonlinear diffusion equation $$\begin{gathered} {\text{ }}u_t = \Delta \ln u, \hfill \\ u \triangleq u\left( {x,t} \right):\Omega \times \mathbb{R}^ + \to \mathbb{R},{\text{ }} x \in \mathbb{R}^n , \hfill \\ \end{gathered} $$ where $\Omega \subset \mathbb{R}^n $ is the domain and $\mathbb{R}^ + = \left\{ {t:0 \leqslant t < + \infty } \right\},{\text{ }}u\left( {x,t} \right) \geqslant 0$ is the temperature of the medium.     

Strong approximation of Riesz means at critical index on Hp(T) (0<p≤1)

This paper is a continuation of [3]. Suppose f∈H^p(T), 0σ _r ^σ f,σ=1/p?1. When p=1, it is just the partial Fourier sums S_kf. In this paper we establish the sharp estimations on the degree of approximation: $$\left\{ { - \frac{1}{{logR}}\int\limits_1^R {\left\| {\sigma _r^\delta f - f} \right\|_{H^p (T)}^p \frac{{dr}}{r}} } \right\}^{1/p} \leqq C{\mathbf{ }}{}_p\omega \left( {f,{\mathbf{ }}( - \frac{1}{{logR}})^{1/p} } \right)_{H^p (T)} ,0< p< 1,$$ and \(\frac{1}{{\log L}}\sum\limits_{k - 1}^L {\frac{{\left\| {S_k f - f} \right\|_H 1_{(T)} }}{k} \leqq Cp\omega (f; - \frac{1}{{\log L}})_H 1_{(T)} } \) Where $$\omega (f,{\mathbf{ }}h)_{H^p (T)} \begin{array}{*{20}c} { = Sup} \\ {0 \leqq \left| u \right| \leqq h} \\ \end{array} \left\| {f( \cdot + u) - f( \cdot )} \right\|_{H^p (T).} $$ .     

THE BEST APPROXIMATION CONSTANT FOR THE JACKSON'S TYPE OPERATOR J_(n,3) (f;x)

In this paper we obtain the best approximation constant of function f(x)(∈C_(2π))by theJackson's type operator J_(π3)(f;x),i.e.‖J_(n,3)(f,x)-f(x)‖_c≤(4-6/π)ω(f,1/n),‖J_(n,3)(f,x)-f(x)‖_c≤(8-17/π)ω_2(f,1/n)     

Bang-bang property for Bolza problems in two dimensions

Consider the following Bolza problem: $$\begin{gathered} \min \int {h(x,u) dt,} \hfill \\ \dot x = F(x) + uG(x), \hfill \\ \left| u \right| \leqslant 1, x \in \Omega \subset \mathbb{R}^2 , \hfill \\ x(0) = x_0 , x(1) = x_1 . \hfill \\ \end{gathered} $$ We show that, under suitable assumptions onF, G, h, all optimal trajectories are bang-bang. The proof relies on a geometrical approach that works for every smooth two-dimensional manifold. As a corollary, we obtain existence results for nonconvex optimization problems.     

Regularity for solutions to anisotropic obstacle problems

For Ω a bounded subset of R n,n 2,ψ any function in Ω with values in R∪{±∞}andθ∈W1,(q i)(Ω),let K(q i)ψ,θ(Ω)={v∈W1,(q i)(Ω):vψ,a.e.and v-θ∈W1,(q i)0(Ω}.This paper deals with solutions to K(q i)ψ,θ-obstacle problems for the A-harmonic equation-divA(x,u(x),u(x))=-divf(x)as well as the integral functional I(u;Ω)=Ωf(x,u(x),u(x))dx.Local regularity and local boundedness results are obtained under some coercive and controllable growth conditions on the operator A and some growth conditions on the integrand f.     

Remarks on eigenvalues and eigenfunctions of a special elliptic system

LetΛ ₁(Ω) be the first eigenvalue of the vector-valued problem $$\begin{gathered} \Delta u + \alpha grad div u + \Delta u = 0 in \Omega , \hfill \\ u = 0 in \partial \Omega , \hfill \\ \end{gathered} $$ , withα>0. Letλ ₁(Ω) be the first eigenvalue of the scalar problem $$\begin{gathered} \Delta u + \lambda u = 0 in \Omega , \hfill \\ u = 0 on \partial \Omega . \hfill \\ \end{gathered} $$ . The paper contains a proof of the inequality $$\left( {1 + \frac{\alpha }{n}} \right)\lambda _1 \left( \Omega \right) > \Lambda _1 \left( \Omega \right) > \left( \Omega \right)$$ and improves recent estimates of Sprössig [15] and Levine and Protter [11]. Moreover we show, ifΩ is a ball, that an eigensolution u₁, associated withΛ ₁(Ω) is not unique and that the eigensolutions for this and higher eigenvalues are never rotationally invariant. Finally we calculate some eigensolutions explicitly.     

Three Topological Problems about Integral Functionals on Sobolev Spaces

In this paper, I propose some problems, of topological nature, on the energy functional associated to the Dirichlet problem $$\left\{ \begin{gathered} - \Delta {\kern 1pt} {\kern 1pt} u = f\left( {x,u} \right){\text{in}}\Omega \hfill \\ u_{\left| {\wp \Omega } \right.} = 0 \hfill \\ \end{gathered} \right.$$ Positive answers to these problems would produce innovative multiplicity results on problem (P_f).     

Interpolation by polyhedral functions

A polyhedral functionlp(Δ_n) (f). interpolating a function f, defined on a polygon Φ, is defined by a set of interpolating nodes Δ_n ?Φ and a partition P(Δ_n) of the polygon Φ into triangles with vertices at the points of Δ_n. In this article we will compute for convex moduli of continuity the quatities $$\begin{gathered} E (H_\Phi ^\omega ; P (\Delta _n )) = sup || f - l_{p(\Delta _n )} (f)||, \hfill \\ f \in H_\Phi ^\omega \hfill \\ \end{gathered} $$ and also give an asymptotic estimate of the quantities $$\begin{gathered} E_n (H_\Phi ^\omega ) = infinf E (H_\Phi ^\omega ; P (\Delta _n )). \hfill \\ \Delta _n P(\Delta _n ) \hfill \\ \end{gathered} $$      

Finite difference method of first boundary problem for quasilinear parabolic systems (IV)

More work is done to study the explicit, weak and strong implicit difference solution for the first boundary problem of quasilinear parabolic system: $$\begin{gathered} u_t = ( - 1)^{M + 1} A(x,t,u, \cdots ,u_x M - 1)u_x 2M + f(x,t,u, \cdots u_x 2M - 1), \hfill \\ (x,t) \in Q_T = \left| {0< x< l,0< t \leqslant T} \right|, \hfill \\ u_x ^k (0,t) = u_x ^k (l,t) = 0 (k = 0,1, \cdots ,M - 1),0< t \leqslant T, \hfill \\ u(x,0) = \varphi (x),0 \leqslant x \leqslant l, \hfill \\ \end{gathered} $$ whereu, ?, andf arem-dimensional vector valued functions, A is anm×m positively definite matrix, and $u_t = \frac{{\partial u}}{{\partial t}},u_x ^k = \frac{{\partial ^k u}}{{\partial x^k }}$ . For this problem, the convergence of iteration for the general difference schemes is proved.     

О последовательност ях сумм Фурье-Уолша ограниченных функци й

Let L^∞ denote the space of measurable 1-periodic essentially bounded functionsf(x) with ∥f∥=vrai sup ¦f(x)¦,S _k (f, x) thek-th partial sum of the Walsh-Fourier series off(x),L _k thek-th Lebesgue constant. The following theorem is proved. Theorem. Letλ={λ _K } be a sequence of nonnegative numbers, $$\left\| \lambda \right\|_1 = \mathop \sum \limits_{k = 1}^\infty \lambda _k< \infty ,\left\| \lambda \right\|_2 = (\mathop \sum \limits_{k = 1}^\infty \lambda _k^2 )^{1/2} ,m = log[(\left\| \lambda \right\|_1 /\left\| \lambda \right\|_2 )]$$ .Then for an arbitrary function f∈L ^∞ the following inequalities hold true $$\begin{gathered} \left\| {\mathop \sum \limits_{k = 1}^\infty \lambda _k \left| {S_k (f,x)} \right|} \right\| \leqq \mathop \sum \limits_{k = 1}^\infty \lambda _k (L_{[k2 - 2m]} + c)\left\| f \right\|, \hfill \\ \hfill \\ \mathop \sum \limits_{k = 1}^\infty \lambda _k \left\| {S_k (f)} \right\| \leqq \mathop \sum \limits_{k = 1}^\infty \lambda _k (L_{[k2 - m]} + c)\left\| f \right\| \hfill \\ \end{gathered} $$ , where[y] denotes integral part of a number y>0 and c is an absolute constant. A corollary of the above theorem is that for each functionfεL ^∞ the Lebesgue estimate can be refined for a certain sequence of indices, while the growth order of Lebesgue constants along that sequence can be arbitrarily close to the logarithmic one. “In the mean”, however, the Lebesgue estimate is exact. A further corollary deals with strong summability.     

Wave recursive interpolation

In this paper it is studied that the generated theory of wave recursive interpolation of uniform T-subdivi-ston scheme include wave parameter.The paper analyses the convergence of sequences of control polygons produced by wave recursive interpolation T-subdivision scheme of the formj=l,2,…,T-1;m=O,l,…,nTk;k=0,l,2,…,and differentiability of the limit curve.     

Weighted a priori estimates for solution of (−Δ)<Superscript><Emphasis Type="Italic">m</Emphasis></Superscript><Emphasis Type="Italic">u</Emphasis> = <Emphasis Type="Italic">f</Emphasis> with homogeneous dirichlet conditions

Let u be a weak solution of (-△)mu = f with Dirichlet boundary conditions in a smooth bounded domain Ω  Rn. Then, the main goal of this paper is to prove the following a priori estimate:‖u‖ Wω2 m,p(Ω) ≤ C ‖f‖ Lωp (Ω),where ω is a weight in the Muckenhoupt class Ap.     

An asymptotic Cauchy problem for the Laplace equation

The Cauchy problem for the Laplace operator $$\sum\limits_{k = 1}^\infty {\frac{{\left| {\hat f(n_k )} \right|}}{k}} \leqslant const\left\| f \right\|1$$ is modified by replacing the Laplace equation by an asymptotic estimate of the form $$\begin{gathered} \Delta u(x,y) = 0, \hfill \\ u(x,0) = f(x),\frac{{\partial u}}{{\partial y}}(x,0) = g(x) \hfill \\ \end{gathered} $$ with a given majoranth, satisfyingh(+0)=0. Thisasymptotic Cauchy problem only requires that the Laplacian decay to zero at the initial submanifold. It turns out that this problem has a solution for smooth enough Cauchy dataf, g, and this smoothness is strictly controlled byh. This gives a new approach to the study of smooth function spaces and harmonic functions with growth restrictions. As an application, a Levinson-type normality theorem for harmonic functions is proved.     

Mean value of weyl sums

We give a simple proof of a mean value theorem of I. M. Vinogradov in the following form. Suppose P, n, k, τ are integers, P≥1, n≥2, k≥n (τ+1), τ≥0. Put $$J_{k,n} (P) = \int_0^1 \cdots \int_0^1 {\left| {\sum\nolimits_{x = 1}^P {e^{2\pi i(a_1 x + \cdots + a_n x^n )} } } \right|^{2k} da_1 \ldots da_n .} $$ Then $$J_{k,n} \leqslant n!k^{2n\tau } n^{\sigma n^2 u} \cdot 2^{2n^2 \tau } P^{2k - \Delta } ,$$ where $$\begin{gathered} u = u_\tau = min(n + 1,\tau ), \hfill \\ \Delta = \Delta _\tau = n(n + 1)/2 - (1 - 1/n)^{\tau + 1} n^2 /2. \hfill \\ \end{gathered} $$      

Oscillation of solutions of nonlinear partial differential equations of neutral type

In this paper, we deal with the oscillatory behavior of solutions of the neutral partial differential equation of the form $$\begin{gathered} \frac{\partial }{{\partial t}}\left[ {p\left( t \right)\frac{\partial }{{\partial t}}(u\left( {x,t} \right) + \sum\limits_{i = 1}^t {p_i \left( t \right)u\left( {x,t - \tau _i } \right)} )} \right] + q\left( {x,t} \right)f_j (u(x,\sigma _j (t))) \hfill \\ = a\left( t \right)\Delta u\left( {x,t} \right) + \sum\limits_{k = 1}^n {a_k \left( t \right)} \Delta u\left( {x,\rho _k \left( t \right)} \right), \left( {x,t} \right) \in \Omega \times R_ + \equiv G \hfill \\ \end{gathered} $$ where Δ is the Laplacian in EuclideanN-spaceR ^N, R₊=(0, ∞) and Ω is a bounded domain inR ^N with a piecewise smooth boundary δΩ.     

On the normalizing multiplier of the generalized Jackson kernel

We consider the question of evaluating the normalizing multiplier $$\gamma _{n,k} = \frac{1}{\pi }\int_{ - \pi }^\pi {\left( {\frac{{sin\tfrac{{nt}}{2}}}{{sin\tfrac{t}{2}}}} \right)^{2k} dt} $$ for the generalized Jackson kernel J _n,k (t). We obtain the explicit formula $$\gamma _{n,k} = 2\sum\limits_{p = 0}^{\left[ {k - \tfrac{k}{n}} \right]} {( - 1)\left( {\begin{array}{*{20}c} {2k} \\ p \\ \end{array} } \right)\left( {\begin{array}{*{20}c} {k(n + 1) - np - 1} \\ {k(n - 1) - np} \\ \end{array} } \right)} $$ and the representation $$\gamma _{n,k} = \sqrt {\frac{{24}}{\pi }} \cdot \frac{{(n - 1)^{2k - 1} }}{{\sqrt {2k - 1} }}\left[ {1\frac{1}{8} \cdot \frac{1}{{2k - 1}} + \omega (n,k)} \right],$$ , where $$\left| {\omega (n,k)} \right| < \frac{4}{{(2k - 1)\sqrt {ln(2k - 1)} }} + \sqrt {12\pi } \cdot \frac{{k^{\tfrac{3}{2}} }}{{n - 1}}\left( {1 + \frac{1}{{n - 1}}} \right)^{2k - 2} .$$ .     

New Orlicz variants of Hardy type inequalities with power, power-logarithmic, and power-exponential weights

We obtain Hardy type inequalities $$\int_0^\infty {M\left( {\omega \left( r \right)\left| {u\left( r \right)} \right|} \right)\rho \left( r \right)dr} \leqslant C_1 \int_0^\infty {M\left( {\left| {u\left( r \right)} \right|} \right)\rho \left( r \right)dr + C_2 \int_0^\infty {M\left( {\left| {u'\left( r \right)} \right|} \right)\rho \left( r \right)dr,} }$$ and their Orlicz-norm counterparts $$\left\| {\omega u} \right\|_{L^M (\mathbb{R}_ + ,\rho )} \leqslant \tilde C_1 \left\| u \right\|_{L^M (\mathbb{R}_ + ,\rho )} + \tilde C_2 \left\| {u'} \right\|_{L^M (\mathbb{R}_ + ,\rho )} ,$$ with an N-function M, power, power-logarithmic and power-exponential weights ??, ??, holding on suitable dilation invariant supersets of C ₀ ^?? (?₊). Maximal sets of admissible functions u are described. This paper is based on authors?? earlier abstract results and applies them to particular classes of weights.