Solvability of a higher-order multi-point boundary value problem at resonance

Based on the coincidence degree theory of Mawhin, we get a new general existence result for the following higher-order multi-point boundary value problem at resonance

Existence of solutions for singular boundary problems for higher order differential equations

We offer sufficient conditions for the existence of solutions for the boundary value problem $\begin{gathered} y(n) + f(t,y,y^1 ,...,y^{(n - 2)} = 0, 0< t< 1, n \geqslant 2 \\ y^{(i)} (0) = 0, 0 \leqslant i \leqslant n - 3 \\ y^{(n - 2)} (0) - \beta y^{(n - 1)} (0) = 0 \\ \gamma y^{(n - 2)} (1) + \delta y^{(n - 1)} (1) = 0 \\ \end{gathered} $ where α, β, γ and δ are constants satisfying αγ+αδ+βγ>0, β, δ≥0, β+α>0 and δ+γ>0.     

On some boundary value problems with integral conditions for functional differential equations

For the functional differential equationu ⁽ⁿ⁾ (t)=f(u)(t) we have established the sufficient conditions for solvability and unique solvability of the boundary value problems $$u^{(i)} (0) = c_i (i = 0,...,m - 1), \smallint _0^{ + \infty } |u^{(m)} (t)|^2 dt< + \infty $$ and $$\begin{gathered} u^{(i)} (0) = c_i (i = 0),...,m - 1, \hfill \\ \smallint _0^{ + \infty } t^{2j} |u^{(j)} (t)|^2 dt< + \infty (j = 0,...,m), \hfill \\ \end{gathered} $$ wheren≥2,m is the integer part of $\tfrac{n}{2}$ ,c _i ∈R, andf is the continuous operator acting from the space of (n?1)-times continuously differentiable functions given on an interval [0,+∞] into the space of locally Lebesgue integrable functions.     

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≤∞     

Oscillation of even order nonlinear functional differential equations with damping

This paper is concerned with a class of even order nonlinear damped differential equations

Criterion of polynomial increase of a function and its derivatives

ДОкАжАНО, ЧтО Дль тОгО, ЧтОБы Дльr РАж ДИФФЕРЕНцИРУЕМОИ НА пРОМЕжУткЕ [А, + ∞) ФУНкцИИf сУЩЕстВОВА л тАкОИ МНОгОЧлЕН (1) $$P(x) = \mathop \Sigma \limits_{\kappa = 0}^{r - 1} a_k x^k ,$$ , ЧтО (2) $$\mathop {\lim }\limits_{x \to + \infty } (f(x) - P(x))^{(k)} = 0,k = 0,1,...,r - 1,$$ , НЕОБхОДИМО И ДОстАтО ЧНО, ЧтОБы схОДИлсь ИН тЕгРАл (3) $$\int\limits_a^{ + \infty } {dt_1 } \int\limits_{t_1 }^{ + \infty } {dt_2 ...} \int\limits_{t_{r - 1} }^{ + \infty } {f^{(r)} (t)dt.}$$ ЕслИ ЁтОт ИНтЕгРАл сх ОДИтсь, тО Дль кОЁФФИц ИЕНтОВ МНОгОЧлЕНА (1) ИМЕУт МЕс тО ФОРМУлы $$\begin{gathered} a_{r - m} = \frac{1}{{(r - m)!}}\left( {\mathop \Sigma \limits_{j = 1}^m \frac{{( - 1)^{m - j} f^{(r - j)} (x_0 )}}{{(m - j)!}}} \right.x_0^{m - j} + \hfill \\ + ( - 1)^{m - 1} \left. {\mathop \Sigma \limits_{l = 0}^{m - 1} \frac{{x_0^l }}{{l!}}\int\limits_a^{ + \infty } {dt_1 } \int\limits_{t_1 }^{ + \infty } {dt_2 ...} \int\limits_{t_{m - l - 1} }^{ + \infty } {f^{(r)} (t_{m - 1} )dt_{m - 1} } } \right),m = 1,2,...,r. \hfill \\ \end{gathered}$$ ДОстАтОЧНыМ, НО НЕ НЕОБхОДИМыМ Усл ОВИЕМ схОДИМОстИ кРА тНОгО ИНтЕгРАлА (3) ьВльЕтсь схОДИМОсть ИНтЕгРАл А \(\int\limits_a^{ + \infty } {x^{r - 1} f^{(r)} (x)dx}\)      

Solvability for Some Higher Order Multi-Point Boundary Value Problems at Resonance

In this paper, by using the Mawhin’s continuation theorem, we obtain an existence theorem for some higher order multi-point boundary value problems at resonance in the following form: $$\begin{array}{lll}x^{(n)}(t) = f(t,x(t),x'(t),\ldots,x^{(n-1)}(t))+e(t),\ t\in(0,1),\\x^{(i)}(0) = 0, i=0,1,\ldots,n-1,\ i\neq p, \\x^{(k)}(1) = \sum\limits_{j=1}^{m-2}{\beta_j}x^{(k)}(\eta_j),\end{array}$$ where ${f:[0,1]\times \mathbb{R}^n \to \mathbb{R}=(-\infty,+\infty)}$ is a continuous function, ${e(t)\in L^1[0,1], p, k\in\{0,1,\ldots,n-1\}}$ are fixed, m ≥ 3 for p ≤ k (m ≥ 4 for p > k), ${\beta_j \in \mathbb{R}, j=1,2,\ldots,m-2, 0 < \eta_1 < \eta_2 < \cdots < \eta_{m-2} <1 }$ . We give an example to demonstrate our results.     

Boundedness of solutions for polynomial potentials withC 2 time dependent coefficients

In this paper we prove the existence of invariant curves and thus stability for all time for a class of Hamiltonian systems with time dependent potentials: $$\frac{{d^2 x}}{{dt^2 }} + Vx(x,t) = 0,x \in R^1 $$ where $\begin{gathered} V(x,t) = \tfrac{1}{{2n + 2}}x^{2n + 2} + \Sigma _{j = 0}^{2n} \tfrac{{Pj(t)}}{{j + 1}}x^{j + 1} ,p_j (t + 1) = p_j (t),p_j \in C^2 ,2n \geqslant j \geqslant n + 1;p_j \in \\ C^1 ,n \geqslant j \geqslant 0,n \geqslant 1. \\ \end{gathered} $      

Monotone empirical Bayes tests for some discrete nonexponential families

This paper deals with the empirical Bayes two-action problem of testingH ₀ : θ ≤ θ_o versusH ₁ : θ > θ_o using a linear error loss for some discrete nonexponential families having probability function either $\begin{gathered} f_1 (x|\theta ) = (x\alpha + 1 - \theta )\theta ^x /\prod\limits_{j = 0}^x {(j\alpha + 1)} \\ or \\ f_1 (x|\theta ) = \left[ {\theta \prod\limits_{j = 0}^{x - 1} {(j\alpha + 1 - \theta )} } \right]/\left[ {\prod\limits_{j = 0}^x {(j\alpha + 1)} } \right] \\ \end{gathered} $ Two empirical Bayes tests δ_n* and δ_n** are constructed. We have shown that both δ_n* and δ_n** are asymptotically optimal, and their regrets converge to zero at an exponential decay rate O(exp( -cn)) for some c > 0, wheren is the number of historical data available when the present decision problem is considered.     

BOUNDARY VALUE PROBLEMS OF SINGULARLY PERTURBED INTEGRO－DIFFERENTIAL EQUATIONS

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Estimating the diameter of one class of functions in L2

Let

Global structure of positive solutions for superlinear 2mth-boundary value problems

We consider boundary value problems for nonlinear 2mth-order eigenvalue problem $$ \begin{gathered} ( - 1)^m u^{(2m)} (t) = \lambda a(t)f(u(t)),0 < t < 1, \hfill \\ u^{(2i)} (0) = u^{(2i)} (1) = 0,i = 0,1,2,...,m - 1. \hfill \\ \end{gathered} $$ . where a ∈ C([0, 1], [0, ∞)) and a(t ₀) > 0 for some t ₀ ∈ [0, 1], f ∈ C([0, ∞), [0, ∞)) and f(s) > 0 for s > 0, and f ₀ = ∞, where $ \mathop {\lim }\limits_{s \to 0^ + } f(s)/s $ . We investigate the global structure of positive solutions by using Rabinowitz’s global bifurcation theorem.     

Eine Verschärfung der Abschätzung des Restes Taylorscher Näherungspolynome

I begin with a new short proof of: (I) LetP(t) inR ^d be a function oft havingn continuous derivatives fora≤t≤x. ThenP(x)∈ convK, where $$K = \left\{ {\sum\limits_{j = 0}^{n - 1} {\frac{{(x - a)^j }}{{j!}}} P^{(j)} (a) + \frac{{(x - a)^n }}{{n!}}P^{(n)} (t),a \leqslant t \leqslant x} \right\}.$$ for applying (I) let bef(t) a real function such that the point ((t?a) ⁿ⁺¹,f(t)) fulfills the conditions of (I). Then (I) gives a sharper estimate of then ^th remainder term off(x) than the Lagrange remainder formula. Iff( ⁿ )(t) is also convex ina≤t≤x, thenf(x)∈[c,d], where $$\begin{gathered} c = \sum\limits_{j = 0}^{n - 1} {\frac{{(x - a)^j }}{{j!}}f^{(j)} (a) + \frac{{(x - a)^n }}{{n!}}f^{(n)} \left( {\frac{{na + x}}{{n + 1}}} \right)} , \hfill \\ d = \sum\limits_{j = 0}^{n - 1} {\frac{{(x - a)^j }}{{j!}}f^{(j)} (a) + \frac{{(x - a)^n }}{{n!}}} \frac{{nf^{(n)} (a) + f^{(n)} (x)}}{{n + 1}}. \hfill \\ \end{gathered} $$      

Estimation of the remainder of a cubature formula on a Chebyshev grid

Let C(Q) denote the space of continuous functions f(x, y) in the square Q = [?1, 1] × [?1, 1] with the norm $\begin{gathered} \left\| f \right\| = \max \left| {f(x,y)} \right|, \hfill \\ (x,y) \in Q. \hfill \\ \end{gathered} $ On a Chebyshev grid, a cubature formula of the form $\int\limits_{ - 1}^1 {\int\limits_{ - 1}^1 {\frac{1} {{\sqrt {(1 - x^2 )(1 - y^2 )} }}f(x,y)dxdy = \frac{{\pi ^2 }} {{mn}}\sum\limits_{i = 1}^n {\sum\limits_{j = 1}^m {f\left( {\cos \frac{{2i - 1}} {{2n}}\pi ,\cos \frac{{2j - 1}} {{2m}}\pi } \right)} + R_{m,n} (f)} } } $ is considered in some class H(r ₁, r ₂) of functions f ?? C(Q) defined by a generalized shift operator. The remainder R _{m, n} (f) is proved to satisfy the estimate $\mathop {\sup }\limits_{f \in H(r_1 ,r_2 )} \left| {R_{m,n} (f)} \right| = O(n^{ - r_1 + 1} + m^{ - r_2 + 1} ), $ where r ₁, r ₂ > 1; ??^?1 ?? n/m ?? ?? with ?? > 0; and the constant in O(1) depends on ??.     

Boundary parametrization of planar self-affine tiles with collinear digit set

We consider a class of planar self-affine tiles T = M-1 a∈D(T + a) generated by an expanding integral matrix M and a collinear digit set D as follows:M =(0-B 1-A),D = {(00),...,(|B|0-1)}.We give a parametrization S1 →T of the boundary of T with the following standard properties.It is H¨older continuous and associated with a sequence of simple closed polygonal approximations whose vertices lie on T and have algebraic preimages.We derive a new proof that T is homeomorphic to a disk if and only if 2|A| |B + 2|.     

Solvability of a class of elastic beam equations with strong Carathéodory nonlinearity

We study the existence of a solution to the nonlinear fourth-order elastic beam equation with nonhomogeneous boundary conditions

Certain linear differential operators and generalized differences

The solution of the problem of finding the quantity 1 $$|\vartriangle \mathop n\nolimits_{v_k }^{\sup } | \leqslant 1 1\begin{array}{*{20}c} {1nf} \\ {(k) = 1/_k } \\ {(k = 0, \pm 1. \pm 2, ...)} \\ \end{array} || /^{(n)} (x)||_C ( - \infty ,\infty )'$$ obtained by Subbotin, is extended to the case of formally self-adjoint differential operators with constant coefficients and corresponding generalized differences.     

Best quadrature formula on the class W * r L2 of periodic functions

For the classes of periodic functions with r-th derivative integrable in the mean,we obtain a best quadrature formula of the form $$\begin{gathered} \int_0^1 {f(x)dx = \sum\nolimits_{k = 0}^{m - 1} {\sum\nolimits_{l = 0}^\rho {p_{k,l} } } } f^{(l)} (x_k ) + R(f),0 \leqslant \rho \leqslant r - 1, \hfill \\ 0 \leqslant x_0< x_1< ...< x_{m - 1} \leqslant 1, \hfill \\ \end{gathered}$$ where ρ=r?2 and r?3, r=3, 5, 7, ..., and we determine an exact bound for the error of this formula.     

The number of strictly increasing mappings of fences

Let X and Y be fences of size n and m, respectively and n, m be either both even or both odd integers (i.e., |m-n| is an even integer). Let \(r = \left\lfloor {{{(n - 1)} \mathord{\left/ {\vphantom {{(n - 1)} 2}} \right. \kern-0em} 2}} \right\rfloor\) . If 1<n<-m then there are \(a_{n,m} = (m + 1)2^{n - 2} - 2(n - 1)(\begin{array}{*{20}c} {n - 2} \\ r \\ \end{array} )\) of strictly increasing mappings of X to Y. If 1<-m<-n<-2m and s=1/2(n?m) then there are a _n,m+b _n,m+c _n of such mappings, where $$\begin{gathered} b_{n,m} = 8\sum\limits_{i = 0}^{s - 2} {\left( {\begin{array}{*{20}c} {m + 2i + 1} \\ l \\ \end{array} } \right)4^{s - 2 - 1} } \hfill \\ {\text{ }}c_n = \left\{ \begin{gathered} \left( {\begin{array}{*{20}c} {n - 1} \\ {s - 1} \\ \end{array} } \right){\text{ if both }}n,m{\text{ are even;}} \hfill \\ {\text{ 0 if both }}n,m{\text{ are odd}}{\text{.}} \hfill \\ \end{gathered} \right. \hfill \\ \end{gathered} $$