Periodic solutions to a difference equation with maximum

An open problem posed by G. Ladas is to investigate the difference equation

where are any nonnegative real numbers with 0$">. We prove that there exists a positive integer such that every positive solution of this equation is eventually periodic of period .

     

On the recursive sequence

We show that every positive solution of the equation

where , converges to a period two solution.

     

Singular integral operators associated to curves with rational components

We prove , bounds for

and

where , are rational functions. Our bounds depend only on the degrees of the polynomials and, in particular, they do not depend on the coefficients of these polynomials.

     

On positive periodic solutions of Lotka-Volterra competition systems with deviating arguments

By using Krasnoselskii's fixed point theorem, we prove that the following periodic species Lotka-Volterra competition system with multiple deviating arguments

has at least one positive periodic solution provided that the corresponding system of linear equations

has a positive solution, where and are periodic functions with

Furthermore, when and , , are constants but , remain -periodic, we show that the condition on is also necessary for to have at least one positive periodic solution.

     

On the irrationality of a certain multivariate series

We prove that for integers 1,m\geq 1$"> and positive rationals the series

is irrational. Furthermore, if all the positive rationals are less than then the series

is also irrational.

     

A problem of prescribing Gaussian curvature on

A class of functions and the corresponding solutions of

are obtained as a special case of the solutions of

where is defined as .

     

Forced oscillation of second order linear and half-linear difference equations

Oscillation properties of solutions of the forced second order linear difference equation

are investigated. The authors show that if the forcing term does not oscillate, in some sense, too rapidly, then the oscillation of the unforced equation implies oscillation of the forced equation. Some results illustrating this statement and extensions to the more general half-linear equation

1, \end{displaymath}">

are also given.

     

Explicit merit factor formulae for Fekete and Turyn polynomials

We give explicit formulas for the norm (or equivalently for the merit factors) of various sequences of polynomials related to the Fekete polynomials

where is the Legendre symbol. For example for an odd prime,

where is the class number of . Similar explicit formulas are given for various polynomials including an example of Turyn's that is constructed by cyclically permuting the first quarter of the coefficients of . This is the sequence that has the largest known asymptotic merit factor. Explicitly,

where denotes the nearest integer, satisfies

where

Indeed we derive a closed form for the norm of all shifted Fekete polynomials

Namely

and if .

     

Multiple symmetric positive solutions of a class of boundary value problems for higher order ordinary differential equations

In this paper, the authors consider the boundary value problem

and give sufficient conditions for the existence of any number of symmetric positive solutions of (E)-(B). The relationships between the results in this paper and some recent work by Henderson and Thompson (Proc. Amer. Math. Soc. 128 (2000), 2373-2379) are discussed.

     

Qualitative analysis of a fourth order difference equation

In this paper, we will investigate some qualitative behavior of solutions of the following fourth order difference equation $x_{n+1}=ax_{n-1}+\frac{bx_{n-1}}{cx_{n-1}-dx_{n-3}},$ \ $n=0,1,...,$ where the initial conditions $x_{-3,}x_{-2},\ x_{-1}$\ and\ $x_{0}\ $are arbitrary real numbers and the values $a,\ b,\ c\ $and$\;d$ are\ defined as positive real numbers.     

The full Markov-Newman inequality for Müntz polynomials on positive intervals

For a function defined on an interval let

The principal result of this paper is the following Markov-type inequality for Müntz polynomials. Theorem. Let be an integer. Let be distinct real numbers. Let . Then

where the supremum is taken for all (the span is the linear span over ).

     

Irrational rotation numbers and unboundedness of solutions of the second order differential equations with asymmetric nonlinearities

In this paper, we study the dynamics of the mappings

where is a irrational rotation number. We prove the existence of orbits that go to infinity in the future or in the past by using the well-known Birkhoff Ergodic Theorem. Applying this conclusion, we deal with the unboundedness of solutions of Liénard equations with asymmetric nonlinearities.

     

Inequalities for sine sums with more variables

A result of Vietoris states that if the real numbers \(a_1,\ldots ,a_n\) satisfy

$$\begin{aligned} \text{(*) } \qquad a_1\ge \frac{a_2}{2} \ge \cdots \ge \frac{a_n}{n}>0 \quad \text{ and } \quad a_{2k-1}\ge a_{2k} \quad (1\le k\le n/2), \end{aligned}$$

then, for \(x_1,\ldots ,x_m>0\) with \(x_1+\cdots +x_m <\pi \),

$$\begin{aligned} \begin{aligned} \text{(**) } \qquad \sum _{k=1}^n a_k \frac{\sin (k x_1) \cdots \sin (k x_m)}{k^m}>0. \end{aligned} \end{aligned}$$

We prove that \((**)\) (with “\(\ge \)” instead of “>”) holds under weaker conditions. It suffices to assume, instead of \((*)\), that

$$\begin{aligned} \sum _{k=1}^N a_k \frac{\sin (kt)}{k}>0 \quad (N=1,\ldots ,n; \, 0<t<\pi ), \end{aligned}$$

and, moreover, \((**)\) is valid for a larger region, namely, \(x_1,\ldots ,x_m\in (0,\pi )\).

     

A simple proof of a curious congruence by Sun

In this note, we give a simple and elementary proof of the following curious congruence which was established by Zhi-Wei Sun:

     

The linear heat equation with highly oscillating potential

In this paper we consider the following initial value problem:

where and . Nonexistence of positive solutions is analyzed.

     

Twisted higher moments of Kloosterman sums

Let be a nontrivial Dirichlet character modulo an odd prime . Write

We shall prove

and, for complex ,

0, \end{displaymath}">

where is a constant depending only on .

     

On Littlewood's boundedness problem for sublinear Duffing equations

In this paper, we are concerned with the boundedness of all the solutions and the existence of quasi-periodic solutions for second order differential equations

where the 1-periodic function is a smooth function and satisfies sublinearity:

     

Global Behavior of Rational Sequences Involving Piecewise Power Function

In this paper, we are concerned with the global behavior ofthe solutions of the difference equation

On boundedness of solutions of the difference equation x_{n+1}=p+\frac{x_{n-1}}{x_{n}} for p<1

In this paper, we study the difference equation $$x_{n+1}=p+\frac{x_{n-1}}{x_n}, \quad n=0,1,\ldots, $$ where initial values x _?1,x ₀∈(0,+∞) and 0<p<1, and obtain the set of all initial values (x _?1,x ₀)∈(0,+∞)×(0,+∞) such that the positive solutions $\{x_{n}\}_{n=-1}^{\infty}$ are bounded. This answers the Open problem 4.8.11 proposed by Kulenovic and Ladas (Dynamics of Second Order Rational Difference Equations, with Open Problems and Conjectures, 2002).     

Values of the Legendre chi and Hurwitz zeta functions at rational arguments

We show that the Hurwitz zeta function, , and the Legendre chi function, , defined by

and

respectively, form a discrete Fourier transform pair. Many formulae involving the values of these functions at rational arguments, most of them unknown, are obtained as a corollary to this result. Among them is the further simplification of the summation formulae from our earlier work on closed form summation of some trigonometric series for rational arguments. Also, these transform relations make it likely that other results can be easily recovered and unified in a more general context.