Conditionally oscillatory half-linear differential equations

We consider a nonoscillatory half-linear second order differential equation (*) $$ (r(t)\Phi (x'))' + c(t)\Phi (x) = 0,\Phi (x) = \left| x \right|^{p - 2} x,p > 1, $$ and suppose that we know its solution h. Using this solution we construct a function d such that the equation (**) $$ (r(t)\Phi (x'))' + [c(t) + \lambda d(t)]\Phi (x) = 0 $$ is conditionally oscillatory. Then we study oscillations of the perturbed equation (**). The obtained (non)oscillation criteria extend existing results for perturbed half-linear Euler and Euler-Weber equations.     

Oscillation criteria for a class of nonlinear fourth order neutral differential equations

In this paper, sufficient conditions are obtained for oscillation of a class of nonlinear fourth order mixed neutral differential equations of the form (E) $$\left( {\frac{1} {{a\left( t \right)}}\left( {\left( {y\left( t \right) + p\left( t \right)y\left( {t - \tau } \right)} \right)^{\prime \prime } } \right)^\alpha } \right)^{\prime \prime } = q\left( t \right)f\left( {y\left( {t - \sigma _1 } \right)} \right) + r\left( t \right)g\left( {y\left( {t + \sigma _2 } \right)} \right)$$ under the assumption $$\int\limits_0^\infty {\left( {a\left( t \right)} \right)^{\tfrac{1} {\alpha }} dt} = \infty .$$ where α is a ratio of odd positive integers. (E) is studied for various ranges of p(t).     

Functional Equations and Weierstrass Transforms

The purpose of this article is to illustrate the utility of the Weierstrass transform in the study of functional equations (and systems) of the form 1 $${\mathop \sum^N\limits_{k=0}}\alpha_{k}f(x+r_{k})=f_{0}(x)\ \ \ \, x\in\ {\rm R}.$$ One may think of α₀, α₁,…, α_N as given complex numbers, r₀, r₁,…, r_N as given real numbers, ?₀: ? → C as a given function and ? as the unknown.     

On an equation involving weighted quasi-arithmetic means

Let I ? ? be an interval and κ, λ ∈ ? / {0, 1}, µ, ν ∈ (0, 1). We find all pairs (φ, ψ) of continuous and strictly monotonic functions mapping I into ? and satisfying the functional equation $$ \kappa x + (1 - \kappa )y = \lambda \phi ^{ - 1} (\mu \phi (x) + (1 - \mu )\phi (y)) + (1 - \lambda )\psi ^{ - 1} (\nu \psi (x) + (1 - \nu )\psi (y)) $$ which generalizes the Matkowski-Sutô equation. The paper completes a research stemming in the theory of invariant means.     

On a Functional Equation Associated with Simpson’s Rule

In this paper, we determine the general solution of the functional equation $$f(x)-g(y)=(x-y)\lbrack h(x+y)+\psi (x)+\phi (y)\rbrack$$ for all real numbers x and y. This equation arises in connection with Simpson’s Rule for the numerical evaluation of definite integrals. The solution of this functional equation is achieved through the functional equation $$g(x)-g(y)=(x-y)f(x+y)+(x+y)f(x-y).$$      

On The Generalized Cocycle Equation of Ebanks and Ng

I show that in order to solve the functional equation $$F_{1}(x+y,z)+F_{2}(y+z,x)F_{3}(z+x,\ y)+F_{4}(x,y)+F_{5}(y,z)+F_{6}(z,x)=0$$ for six unknown functions (x,y,z are elements of an abelian monoid, and the codomain of each F _j is the same divisible abelian group) it is necessary and sufficient to solve each of the following equations in a single unknown function $$\matrix{\quad\quad\quad\quad\quad\quad\quad \quad\quad\quad\quad\quad\quad G(x+y,\ z)- G(x,z)- G(y,z)=G(y+z,x)- G(y,x)- G(z,x)\cr \quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad H(x+y,\ z)- H(x,z)- H(y,x)+H(y+z,\ x)- H(y,x)- H(z,x)\cr +H(z+x,\ y)- H(z,y)- H(x,y)=0.}$$      

Additivity and exponentiality are alien to each other

Let (S, +) be a (semi)group and let (R,+, ·) be an integral domain. We study the solutions of a Pexider type functional equation $$f(x+y) + g(x+y) = f(x) + f(y) + g(x)g(y)$$ for functions f and g mapping S into R. Our chief concern is to examine whether or not this functional equation is equivalent to the system of two Cauchy equations $$\left\{\begin{array}{@{}ll} f(x+y) = f(x) + f(y)\\ g(x+y) = g(x)g(y)\end{array}\right.$$ for every ${x,y \in S}$ .     

Functions on adjacent vertex degrees of trees with given degree sequence

In this note we consider a discrete symmetric function f(x, y) where $$f(x,a) + f(y,b) \geqslant f(y,a) + f(x,b) for any x \geqslant y and a \geqslant b,$$ associated with the degrees of adjacent vertices in a tree. The extremal trees with respect to the corresponding graph invariant, defined as $$\sum\limits_{uv \in E(T)} {f(deg(u),deg(v))} ,$$ are characterized by the “greedy tree” and “alternating greedy tree”. This is achieved through simple generalizations of previously used ideas on similar questions. As special cases, the already known extremal structures of the Randic index follow as corollaries. The extremal structures for the relatively new sum-connectivity index and harmonic index also follow immediately, some of these extremal structures have not been identified in previous studies.     

Nonlinear wavelet methods for high-dimensional backward heat equation

The backward heat equation is a typical ill-posed problem. In this paper, we shall apply a dual least squares method connecting Shannon wavelet to the following equation ut (x, y, t) = u xx (x, y, t) + uyy (x, y, t), x ∈ R, y ∈ R, 0 ≤ t 1, u(x, y, 1) = (x, y), x ∈ R, y ∈ R. Motivated by Regińska's work, we shall give two nonlinear approximate methods to regularize the approximate solutions for high-dimensional backward heat equation, and prove that our methods are convergent.     

On the convergence to infinity of Fourier series along dense subsequences of numbers

В статье доказываетс я Теорема.Какова бы ни была возрастающая последовательность натуральных чисел {H _k } _{k = 1} ^∞ c $$\mathop {\lim }\limits_{k \to \infty } \frac{{H_k }}{k} = + \infty$$ , существует функцияf∈L(0, 2π) такая, что для почт и всех x∈(0, 2π) можно найти возраст ающую последовательность номеров {n_k(x)} _k=1 ^∞ ,удовлетворяющую усл овиям 1) $$n_k (x) \leqq H_k , k = 1,2, ...,$$ 2) $$\mathop {\lim }\limits_{t \to \infty } S_{n_{2t} (x)} (x,f) = + \infty ,$$ 3) $$\mathop {\lim }\limits_{t \to \infty } S_{n_{2t - 1} (x)} (x,f) = - \infty$$ .     

On a result of Hardy and Littlewood concerning Fourier series

Let f ∈ L ¹( $ \mathbb{T} $ ) and assume that $$ f\left( t \right) \sim \frac{{a_0 }} {2} + \sum\limits_{k = 1}^\infty {\left( {a_k \cos kt + b_k \sin kt} \right)} $$ Hardy and Littlewood [1] proved that the series $ \sum\limits_{k = 1}^\infty {\frac{{a_k }} {k}} $ converges if and only if the improper Riemann integral $$ \mathop {\lim }\limits_{\delta \to 0^ + } \int_\delta ^\pi {\frac{1} {x}} \left\{ {\int_{ - x}^x {f(t)dt} } \right\}dx $$ exists. In this paper we prove a refinement of this result.     

Nonlocal problem for a parabolic-hyperbolic equation in a rectangular domain

For an equation of mixed type, namely, $$ \left( {1 - \operatorname{sgn} t} \right)u_{tt} + \left( {1 - \operatorname{sgn} t} \right)u_t - 2u_{xx} = 0 $$ in the domain {(x, t) | 0 < x < 1, ?α < t < β}, where α, β are given positive real numbers, we study the problem with boundary conditions $$ u\left( {0,t} \right) = u\left( {1,t} \right) = 0, - \alpha \leqslant t \leqslant \beta , u\left( {x, - \alpha } \right) - u\left( {x,\beta } \right) = \phi \left( x \right), 0 \leqslant x \leqslant 1. $$ . We establish a uniqueness criterion for the solution constructed as the sum of Fourier series. We establish the stability of the solution with respect to its nonlocal condition φ(x).     

Diophantine properties of IETs and general systems: quantitative proximality and connectivity

Three properties of dynamical systems (recurrence, connectivity and proximality) are quantified by introducing and studying the gauges (measurable functions) corresponding to each of these properties. The properties of the proximality gauge are related to the results in the active field of shrinking targets. The emphasis in the present paper is on the IETs (interval exchange transformations) $( \mathcal {I},T)$ , $\mathcal {I}=[0,1)$ . In particular, we prove that if an IET T is ergodic (relative to the Lebesgue measure λ), then the equality A1 $$ \liminf_{n\to\infty} \, n\, \bigl|T^n(x)-y \bigr|=0 $$ holds for λ×λ-a.a. $(x,y)\in \mathcal {I}^{2}$ . The ergodicity assumption is essential: the result does not extend to all minimal IETs. Also, the factor? n? in (A1) is optimal (e.g., it cannot be replaced by n?ln(ln(lnn))). On the other hand, for Lebesgue almost all 3-IETs $( \mathcal {I},T)$ we prove that for all ?>0 A2 $$ \liminf_{n\to\infty} \, n^ \epsilon \bigl |T^n(x)-T^n(y)\bigr| = \infty,\quad\text{for Lebesgue a.a.} \ (x,y)\in \mathcal {I}^2. $$ This should be contrasted with the equality lim?inf _n→∞?|T ⁿ (x)?T ⁿ (y)|=0, for a.a. $(x,y)\in \mathcal {I}^{2}$ , which holds since $( \mathcal {I}^{2}, T\times T)$ is ergodic (because generic 3-IETs $( \mathcal {I},T)$ are weakly mixing). We introduce the notion of τ-entropy of an IET which is related to obtaining estimates of type (A2). We also prove that no 3-IET is strongly topologically mixing.     

Reelle Abstandsräume und hyperbolische Geometrie

Ein reeller Abstandsraum ist eine Menge S ≠ ø zusammen mit einer Abbildung d: S × S → ?. Für x,y ∈ S hei\t d(x,y) der Abstand von x und y. Für beliebige reelle Abstandsräume definieren wir Begriffe wie Gerade, sphärischer Teilraum, konvexe Teilmenge, Winkelma\e usf. derart, da\ diese Begriffe im Falle $$={\rm R}^{n}\ {\rm und}\ d(x,y)=\sqrt{x^{2}+y^{2}- 2xy}$$ klassische Objekte gleichen Namens ergeben. Sodann wenden wir unseren Begriffsapparat auf $$ S={\rm R}^{n}\ {\rm und}\ d(x,y)=\sqrt{1+x^{2}}\sqrt{1+y^{2}}- xy $$ an. Dabei entsteht dann die n-dimensionale hyperbolische Geometrie mit dem ?ⁿ als Punktmenge und geeigneten Geraden, Strecken usf. als dortige Grundobjekte.     

On oscillatory integral Hilbert transformation with trigonometric polynomial phase

Let $ \mathcal{P}_n $ denote the set of algebraic polynomials of degree n with the real coefficients. Stein and Wpainger [1] proved that $$ \mathop {\sup }\limits_{p( \cdot ) \in \mathcal{P}_n } \left| {p.v.\int_\mathbb{R} {\frac{{e^{ip(x)} }} {x}dx} } \right| \leqslant C_n , $$ where C _n depends only on n. Later A. Carbery, S. Wainger and J. Wright (according to a communication obtained from I. R. Parissis), and Parissis [3] obtained the following sharp order estimate $$ \mathop {\sup }\limits_{p( \cdot ) \in \mathcal{P}_n } \left| {p.v.\int_\mathbb{R} {\frac{{e^{ip(x)} }} {x}dx} } \right| \sim \ln n. $$ . Now let $ \mathcal{T}_n $ denote the set of trigonometric polynomials $$ t(x) = \frac{{a_0 }} {2} + \sum\limits_{k = 1}^n {(a_k coskx + b_k sinkx)} $$ with real coefficients a _k , b _k . The main result of the paper is that $$ \mathop {\sup }\limits_{t( \cdot ) \in \mathcal{T}_n } \left| {p.v.\int_\mathbb{R} {\frac{{e^{it(x)} }} {x}dx} } \right| \leqslant C_n , $$ with an effective bound on C _n . Besides, an analog of a lemma, due to I. M. Vinogradov, is established, concerning the estimate of the measure of the set, where a polynomial is small, via the coefficients of the polynomial.     

General Solution of Two Functional Equations Concerning Measures of Information

In the characterization of multidimensional sum form information measures the two functional equations $$f(pq) + f(p(1 - q)) = f(p)\lbrace f(q) + f(1 - q)\rbrace \ \ \ p,q,\in I,$$ $$f(pq) + f(p(1 - q)) = f(p)\lbrace M(q) + M(1 - q)\rbrace \ \ \ p,q,\in I,$$ arise. For the one-dimensional case, these equations were studied by Maksa [2] and Kannappan and Sahoo [1], respectively. This paper extends their results to the n-dimensional case.     

A sequential implicit function theorem for the chords iteration

In this paper we study the local convergence of the method $$0 \in f\left( {p,x_k } \right) + A\left( {x_{k + 1} - x_k } \right) + F\left( {x_{k + 1} } \right),$$ in order to find the solution of the generalized equation $$find x \in X such that 0 \in f\left( {p,x} \right) + F\left( x \right).$$ We first show that under the strong metric regularity of the linearization of the associated mapping and some additional assumptions regarding dependence on the parameter and the relation between the operator A and the Jacobian $\nabla _x f\left( {\bar p,\bar x} \right)$ , we prove linear convergence of the method which is uniform in the parameter p. Then we go a step further and obtain a sequential implicit function theorem describing the dependence of the set of sequences of iterates of the parameter.     

An estimate for the sum of Legendre symbols

For the sum S of the Legendre symbols of a polynomial of odd degree n ≥ 3 modulo primes p ≥ 3, Weil’s estimate |S| ≤ (n ? 1) $ \sqrt p $ and Korobov’s estimate $$ \left| S \right| \leqslant (n - 1)\sqrt {p - \frac{{(n - 3)(n - 4)}} {4}} forp \geqslant \frac{{n^2 + 9}} {2} $$ are well known. In this paper, we prove a stronger estimate, namely, $$ \left| S \right| < (n - 1)\sqrt {p - \frac{{(n - 3)(n + 1)}} {4}} $$ .     

The problem of differentiating an asymptotic expansion in real powers. Part II: Factorizational theory

In Part II of our work we approach the problem discussed in Part I from the new viewpoint of canonical factorizations of a certain nth order differential operator L. The main results include:

1. characterizations of the set of relations $$ f^{(k)} (x) = P^{(k)} (x) + o^{(k)} (x^{\alpha _n - k} ),x \to + \infty ,0 \leqslant k \leqslant n - 1, $$ where $$ P(x) = a_1 x^{\alpha _1 } + \cdots + a_n x^{\alpha _n } and \alpha _1 > \alpha _2 > \cdots > \alpha _n , $$ by means of suitable integral conditions
2. formal differentiation of a real-power asymptotic expansion under a Tauberian condition involving the order of growth of L
3. remarkable properties of asymptotic expansions of generalized convex functions.

     

Stability of mixed additive-quadratic Jensen type functional equation in non-Archimedean \ell -fuzzy normed spaces

In this papers we prove the generalized Hyers–Ulam–Rassias stability of the following mixed additive-quadratic Jensen functional equation $$\begin{aligned} 2f\left( \frac{x+y}{2}\right) +f\left( \frac{x-y}{2}\right) +f\left( \frac{y-x}{2}\right) =f(x)+f(y) \end{aligned}$$ in non- Archimedean \(\ell \) -fuzzy normed spaces.