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1.
On the basis of the exact solution of the linear Dirichlet problem , we obtain conditions for the solvability of the corresponding Dirichlet problem for the quasilinear equation u ttu xx = f(x, t, u, u t).  相似文献   

2.
3.
A general algorithm is proposed for constructing interlineation operators , x=(x1, x2) with the properties
  相似文献   

4.
We prove the existence of continuously differentiable solutions such that
or
where
  相似文献   

5.
For functionsf which have an absolute continuous (n–1)th derivative on the interval [0, 1], it is proved that, in the case ofn>4, the inequality
  相似文献   

6.
We establish lower and upper bounds for the quantity
, where
2m,\quad x_l = \frac{{2\pi l}}{q},\quad l = 0,\;1,\;...\;,\;q - 1,$$ " align="middle" vspace="20%" border="0">
, and D m (t) is the Dirichlet kernel, for the class W r of 2π-periodic functions, whose rth derivative satisfies the condition |f r (x)| ≤ 1.__________Translated from Ukrains’kyi Matematychnyi Zhurnal, Vol. 56, No. 12, pp. 1691 – 1698, December, 2004.  相似文献   

7.
Let = (1,...,d) be a vector with positive components and let D be the corresponding mixed derivative (of order j with respect to the jth variable). In the case where d > 1 and 0 < k < r are arbitrary, we prove that
and
for all Moreover, if is the least possible value of the exponent in this inequality, then
Deceased.Translated from Ukrainskyi Matematychnyi Zhurnal, Vol. 56, No. 5, pp. 579–594, May, 2004.  相似文献   

8.
Estimates are obtained among moduli of continuity of functions in several variables that belong to various Lorentz spaces. The functions considered are periodic with period 1 in each variable. More exactly the following theorem is proved: If 0<,<(t) and (t) are so-called -functions such that ;>1>1, and
then for any 0 <1 we have
. The exactness of this estimate is also discussed.  相似文献   

9.
Consider the region Ω0 on the hyperboloid 1 = b2 + ac defined by the conditions
0 < L1 \leqslant a < L2 < 1,    0 < t1 \leqslant \fracba \leqslant t2 < 1.0 < L_1 \leqslant a < L_2 < 1,\quad 0 < t_1 \leqslant \frac{b}{a} \leqslant t_2 < 1.  相似文献   

10.
We prove the existence of continuously differentiable solutions with required asymptotic properties as t +0 and determine the number of solutions of the following Cauchy problem for a functional differential equation:
where : (0, ) (0, +), g: (0, ) (0, +), and h: (0, ) (0, +) are continuous functions, 0 < g(t) t, 0 < h(t) t, t (0, ), , and the function is continuous in a certain domain.  相似文献   

11.
Classical theorems on differential inequalities [1, 2, 3] are generalized for initial value problems of the kind and where is a singular Volterra operator, is continuous and positive on ]a, b], is a norm in R n, and [u]+ and [u] are respectively the positive and the negative part of the vector u R n.  相似文献   

12.
For arbitrary t [0, 1], s [1, ], and A 2, we determine the unimprovable constant B for the inequality
  相似文献   

13.
We obtain new unimprovable Kolmogorov-type inequalities for differentiable periodic functions. In particular, we prove that, for r = 2, k = 1 or r = 3, k = 1, 2 and arbitrary q, p [1, ], the following unimprovable inequality holds for functions :
where and r is the perfect Euler spline of order r.  相似文献   

14.
We obtain the new exact Kolmogorov-type inequality
for 2-periodic functions and any k, r N, k < r. We present applications of this inequality to problems of approximation of one class of functions by another class and estimates of K-functional type.  相似文献   

15.
Let be a nondecreasing sequence of positive numbers and let l 1,α be the space of real sequences for which . We associate every sequence ξ from l 1,α with a sequence , where ϕ(·) is a permutation of the natural series such that , j ∈ ℕ. If p is a bounded seminorm on l 1,α and , then
Using this equality, we obtain several known statements. __________ Translated from Ukrains'kyi Matematychnyi Zhurnal, Vol. 57, No. 7, pp. 1002–1006, July, 2005.  相似文献   

16.
To the sh-Gordon equation
  相似文献   

17.
Given a stochastic differential equation based on semimartingale with spatial parameter (1) $$\varphi _t = x_0 + \int_{t_0 }^t {F(\varphi _s ,ds) } on t \geqslant t_0 $$ and it perturbed system (2) $$\psi _t = x_0 + \int_{t_0 }^t {F\left( {\psi \alpha _s , ds} \right)} + \int_{t_0 }^t {G\left( {\psi _s , ds} \right)} on t \geqslant t_0 $$ In this paper we give some sufficient conditions under which the eventual uniform asymptotic stability of Eq. (1) is shared by Eq. (2).  相似文献   

18.
Let f C[0, 1], k = 5, 6, 7. We prove that if f(i/(k – 1)) = 0, i = 0, 1,..., k – 1, then   相似文献   

19.
In this paper, we are concerned with the global behavior ofthe solutions of the difference equation
where
and . Necessary and sufficient conditions for boundedness, persistence,and periodicity of all solutions will be established. The oscillatorybehavior will be investigated.  相似文献   

20.
Estimates for deviations are established for a large class of linear methods of approximation of periodic functions by linear combinations of moduli of continuity of different orders. These estimates are sharp in the sense of constants in the uniform and integral metrics. In particular, the following assertion concerning approximation by splines is proved: Suppose that is odd, . Then
moreover, for it is impossible to decrease the constants on . Here, are some explicitly constructed constants, is the modulus of continuity of order r for the function f, and are explicitly constructed linear operators with the values in the space of periodic splines of degree of minimal defect with 2n equidistant interpolation points. This assertion implies the sharp Jackson-type inequality
. Bibliography: 17 titles.  相似文献   

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