Linear Differential Equations with Variable Coefficients in Regular and Some Singular Cases

In this paper the asymptotic properties as t → + ∞ for a single linear differential equation of the form x⁽ⁿ⁾ + a₁ (t)x^(n?1)+…. + a_n(t)x = 0, where the coefficients aj (z) are supposed to be of the power order of growth, are considered. The results obtained in the previous publications of the author were related to the so called regular case when a complete set of roots {λ,(t)}, j = 1, 2, …, n of the characteristic polynomial yⁿ + a₁ (t)y^n?1 + … + a_n(t) possesses the property of asymptotic separability. One of the main restrictions of the regular case consists of the demand that the roots of the set {λ,(t)} have not to be equivalent in pairs for t → + ∞. In this paper we consider the some more general case when the set of characteristic roots possesses the property of asymptotic independence which includes the case when the roots may be equivdent in pairs. But some restrictions on the asymptotic behaviour of their differences λ_i(t)→ λ_j(t) are preserved. This case demands more complicated technique of investigation. For this purpose the so called asymptotic spaces were introduced. The theory of asymptotic spaces is used for formal solution of an operator equation of the form x = A(x) and has the analogous meaning as the classical theory of solving this equation in Band spaces. For the considered differential equation, the main asymptotic terms of a fundamental system of solution is given in a simple explicit form and the asymptotic fundamental system is represented in the form of asymptotic Emits for several iterate sequences.     

More on Poincaré’s and Perron’s Theorems for Difference Equations∗

Consider the scalar kth order linear difference equation: x(n + k) + pi(n)x(n + k - 1) + … + pk(n)x(n) = 0 where the limits qi=lim_n→∞Pi(n) (i=1,…,k) are finite. In this paper, we confirm the conjecture formulated recently by Elaydi. Namely, every nonzero solution x of (?) satisfies the same asymptotic relation as the fundamental solutions described earlier by Perron, ie., ?= lim sup_n→∞ |x(n)| is equal to the modulus of one of the roots of the characteristics equation χ ^k + q ₁χ ^k?1+…+qk=0. This result is a consequence of a more general theorem concerning the Poincaré difference system x(n+1)=[A+B(n]x(n), where A and B(n) (n=0,1,…) are square matrices such that ‖B(n)‖ →0 as n → ∞. As another corollary, we obtain a new limit relation for the solutions of (?).     

Preservation of Absolutely Continuous Spectrum of?Periodic Jacobi Operators Under Perturbations of?Square-Summable Variation

We study self-adjoint bounded Jacobi operators of the form:

Cantor-Lebesgue theorem for double trignometric series

Let ∥·∥ be a norm in R² and let γ be the unit sphere induced by this norm. We call a segment joining points x,y ε R² rational if (x₁ ? y₁)/(x₂ ? y₂) or (x₂ ? y₂)/(x₁ ? y₁) is a rational number. Let γ be a convex curve containing no rational segments. Satisfaction of the condition $$T_\nu (x) = \sum\nolimits_{\parallel n\parallel = \nu } {c_n e^{2\pi i(n_1 x_1 + n_2 x_2 )} } \to 0(\nu \to \infty )$$ in measure on the set e? [- 1/2,1/2)×[- 1/2, 1/2) =T² of positive planar measure implies ∥T _v ∥L₄ (T²) → 0(v → ∞). if, however, γ contains a rational segment, then there exist a sequence of polynomials {T _v } and a set E ? T², ¦E¦ > 0, such that T _v (x) → 0(v → ∞) on E; however, ¦c_n¦ ? 0 for ∥n∥ → ∞.     

On additive maps of prime rings satisfying the engel condition

Let A be a prime ring with nonzero right ideal R and f : R → A an additive map. Next, let k,n₁, n₂,…,n_k be natural numbers. Suppose that […[[(x), xn¹], xn²],…, xn^k]=0 for all x ∈ R. Then it is proved in Theorem 1.1 that [f(x),x]=0 provided that either char(A)=0 or char (A)> n₁+n₂+ …+n_k Theorem 1.1 is a simultaneous generalization of a number of results proved earlier.     

Solution of the Szili type conjugate problem on the roots of the laguerre polynomials

Let ?1<α≤0 and let $$L_n^{(\alpha )} (x) = \frac{1}{{n!}}x^{ - \alpha } e^x \frac{{d^n }}{{dx^n }}(x^{\alpha + n} e^{ - x} )$$ be the generalizednth Laguerre polynomial,n=1,2,… Letx ₁,x ₂,…,x _n andx*₁,x*₂,…,x* _n?1 denote the roots ofL _n ^(α) (x) andL _n ^(α)′ (x) respectively and putx*₀=0. In this paper we prove the following theorem: Ify ₀,y ₁,…,y _n ?1 andy ₁ ^′ ,…,y _n ^′ are two systems of arbitrary real numbers, then there exists a unique polynomialP(x) of degree 2n?1 satisfying the conditions $$\begin{gathered} P\left( {x_k^* } \right) = y_k (k = 0,...,n - 1) \hfill \\ P'\left( {x_k } \right) = y_k^\prime (k = 1,...,n). \hfill \\ \end{gathered} $$ .     

Nontrivial solution of a quasilinear elliptic equation with critical growth in ℝn

Suppose Δ_n u = div (¦ ?u ¦^n-2?u) denotes then-Laplacian. We prove the existence of a nontrivial solution for the problem $$\left\{ \begin{gathered} - \Delta _n u + \left| u \right|^{n - 2} u = \int {(x,u)u^{n - 2} in \mathbb{R}^n } \hfill \\ u \in W^{1,n} (\mathbb{R}^n ) \hfill \\ \end{gathered} \right.$$ wheref(x, t) =o(t) ast → 0 and ¦f(x, t)¦ ≤C exp(α_n¦t¦^n/(n-1)) for some constantC > 0 and for allx∈?;t∈? with α_n =nω _n ^1/(n-1) , ω_n = surface measure ofS ^n-1.     

GlobalL 2-bifurcation of nonlinear Sturm-Liouville problems

We study the nonlinear Sturm-Liouville problem $$ - u ''(x) = f(u(x)) - \mu u(x), 0< x< 1, u(0) = u(1) = 0.$$ Let (u _n,(μ,x), μ), (n ωN) be a solution pair and α₂ (μ)=∥μ_μ∥2. The purpose of this paper is to study the globalL ²-bifurcation, that is, to establish an asymptotic formula of α_n(υ) as μ → ∞. Furthermore, we give an asymptotic formula ofu _n(μ,x) as μ → ∞.     

A Functional Equation Of J.A. Baker Involving Integral Means

The functional equation $$f(x)={1\over 2}\int^{x+1}_{x-1}f(t)\ dt\ \ \ {\rm for}\ \ \ x\ \in\ {\rm R}$$ has the linear functions ?(x) = a + bx (a, b ∈ ?) as trivial solutions. It is shown that there are two kinds of nontrivial solutions, (i) ?(x) = eλ_i ^x (i = 1, 2, …), where the λ_i∈ ? are the fixed points of the map z ? sinh z, and (ii) C^∞-solutions ? for which the values in the interval [?1,1] can be prescribed arbitrarily, but with the provision that ?^(j)(? 1) = ?^(j)(0) = ?^(j)(1) = 0 for all j = 0, 1, 2 …     

Distribution of the spectrum of a singular positive Sturm–Liouville operator perturbed by the Dirac delta function

We consider the Sturm–Liouville operator generated in the space L ²[0,+∞) by the expression l _a,b:= ?d ²/dx ² +x+aδ(x?b) and the boundary condition y(0) = 0. We prove that the eigenvalues λ _n of this operator satisfy the inequalities λ₁ ⁰ < λ₁ < λ₂ ⁰ and λ_n ⁰ ≤ λ_n < λ_n+1 ⁰, n = 2, 3,..., where {?λ_n ⁰} is the sequence of zeros of the Airy function Ai (λ). We find the asymptotics of λ_n as n → +∞ depending on the parameters a and b.     

On oscillation of solutions of a nonautonomous quasilinear second-order equation

Sufficient conditions are obtained for the initial values of nontrivial oscillating (for t=ω) solutions of the nonautonomous quasilinear equation $$y'' \pm \lambda (t)y = F(t,y,y'),$$ wheret ∈ Δ=[a, ω[,-∞ <a < ω ≤+ ∞, λ(t) > 0, λ(t) ∈ C _Δ ⁽¹⁾ , |F((t,x,y))|≤L(t)(|x|+|y|)^1+α, L(t) ≥-0, α ∈ [0,+∞[, F: Δ × R² →R,F ∈C _Δ×R ²,R is the set of real numbers, and R² is the two-dimensional real Euclidean space.     

高阶常系数非齐次线性微分方程的新解法

提出了高阶常系数非齐次线性微分方程y(n)+P1y(n-1)+…+Pny=f(x)(P1,P2,…,Pn是实数)的一种新解法.首先将该方程降为n个一阶非齐次线性微分方程组:y1′-w1y1=f(x),y2′-w2y2=y1,…………………yn′-wnyn=yn-1,其中w1,w2,…,wn是对应的齐次方程的特征方程tn+P1tn-1+…+Pn=0的n个根.然后求出它的通解y=yn,最后得出了求原方程一个特解的迭代公式.     

A Tauberian theorem related to Weyl's criterion

Let {x_n} be a sequence of real numbers and let a(n) be a sequence of positive real numbers, with A(N) = Σ_n=1^Na(n). Tsuji has defined a notion of a(n)-uniform distribution mod 1 which is related to the problem of determining those real numbers t₀ for which A(N)^?1 Σ_n=1^Na(n)e^?it₀x_n → 0 as N → ∞. In case f(s) = Σ_n=1^∞a(n)e^?sx_n, s = σ + it, is analytic in the right half-plane 0 < σ, and satisfies a certain smoothness condition as σ → 0 +, we show that f(σ)^?1f(σ + it₀) → 0 as σ → 0 + if and only if A(N)^?1 Σ_n=1^Na(n)e^?it₀x_n → 0 as N → ∞.     

A note on a problem by H. S. Shapiro

В РАБОтЕ ДОкАжАНО, ЧтО limk _a *f(x)=f(x) пОЧтИ ВсУДУ, гДЕk _a(t)=a^?n k(a^?1t), t?Rⁿ, Для Дль ДОВОльНО шИРОкОг О клАссА ФУНкцИИk(t). ДАНы УслОВИь, пРИ кОтО Рых пОлУЧЕННыИ РЕжУл ьтАт РАспРОстРАНьЕтсь НА ФУНкцИУ $$k(x,y) = \gamma \frac{1}{{1 + |x|^\alpha }} \cdot \frac{1}{{1 + |y|^\beta }},$$ гДЕ α, β>1, А γ — НОРМИРУУЩ ИИ МНОжИтЕль тАкОИ, Чт О∫∫k(x, y) dx dy=1.     

关于一类微分方程周期解的存在性的注记

本文主要探讨下列周期系数微分方程dy/dt=(A₁(t)y+A₂(t)y²+A₃(t)y³)/(a₀(t)+a₁(t)y+a₂(t)y²)(**)的周期解个数问题，利用方程（**）解的差率法得到了方程（**）周期解的个数定理．本文仅在Ａ_i（ｔ），ａ_j（ｔ）（ｉ＝１，２，３，ｊ＝０，１，２）是连续周期函数的条件下得到这一结论，从而减弱了文［２］中相应定理的条件，即Ａ_i（ｔ），ａ_j（ｔ）均是连续可微的周期函数．     

An extremal result for sum-free sequences

Let a₁ < a₂ < … be a sequence of positive integers such that no a_k is a sum of distinct other terms. Erdös conjectured that if a₁ ≥ n, then $\text{Σ1}\text{a}{}_{\mathrm{k}}\text{<}\text{log}\text{2 + ?}{}_{\mathrm{n}}$, where, ?_n → 0 as n → ∞. This result, which is the best possible, is established in this paper.     

Existence of solution for a class of nth-order multi-point boundary value problem

In this paper, we are concerned with the following nth-order ordinary differential equation $$x^{(n)}(t)+f(t,x(t),x'(t),\ldots,x^{(n-1)}(t))=0,\quad t\in (0,1),$$ with the nonlinear boundary conditions $$\begin{array}{l}x^{(i)}(0)=0,\quad i=0,1,\ldots,n-3,\\[3pt]g(x^{(n-2)}(0),x^{(n-1)}(0),x(\xi_1),\ldots,x(\xi_{m-2}))=A,\\[3pt]h(x^{(n-2)}(1),x^{(n-1)}(1),x(\eta_1),\ldots,x(\eta_{l-2}))=B,\end{array}$$ here A,B∈R, f:[0,1]×R ⁿ →R is continuous, g:[0,1]×R ^m →R is continuous, h:[0,1]×R ^l →R is continuous, ξ _i ∈(0,1), i=1,…,m?2, and η _j ∈(0,1), j=1,…,l?2. The existence result is given by using a priori estimate, Nagumo condition, the method of upper and lower solutions and Leray-Schauder degree. We also give an example to demonstrate our result.     

A class of functions of a real variable

An investigation of measurable almost-everywhere finite functions ξ(t), -∞ $$\varphi _T^\xi (\tau _{(n)} , \lambda _{(n)} ) = \frac{1}{{2T}}\int_{ - T}^T {\exp i} \sum\nolimits_{k - 1}^n {\lambda _k \xi (t - \tau _k )dt} $$ tends to an asymptotic characteristic function? _∞ ^ξ (τ _(n), λ_(n)) when T → ∞. Here n is any positive integer and T_(n)=(τ₁; τ₂, ..., τ_n) is arbitrary. It is proved that the class of such functions ξ(t) is larger than the class of Besicovich almost-periodic functions.     

Ein Null-Zwei-Gesetz auf lokalkompakten abelschen Halbgruppen

Let λ, μ be regular probability measures on a locally compact abelian semigroup S, λ * μ the convolution of λ and μ, λⁿ the n^th iterated convolution of λ, δ_x the point measure of x?S. We study the totalvariation of λⁿ–δ_x * λⁿ for n → ∞. We shall see that for a certain class of semigroups the limit of this sequence is either 0 or 2.     

On a conjecture of Chowla and Chowla

The equation y² ≡ x(x + a₁)(x + a₂) … (x + a_r) (mod p), where a₁, a₂, …, a_r are integers is shown to have a solution in integers x, y with 1 ≦ x ≦C, where C is a constant depending only on a₁, a₂, …, a_r.