Nontrivial solutions of a higher-order rational difference equation

We prove that, for every k ∈ ?, the following generalization of the Putnam difference equation $$ x_{n + 1} = \frac{{x_n + x_{n - 1} + \cdots + x_{n - (k - 1)} + x_{n - k} x_{n - (k + 1)} }} {{x_n x_{n - 1} + x_{n - 2} + \cdots + x_{n - (k + 1)} }}, n \in \mathbb{N}_0 , $$ has a positive solution with the following asymptotics $$ x_n = 1 + (k + 1)e^{ - \lambda ^n } + (k + 1)e^{ - c\lambda ^n } + o(e^{ - c\lambda ^n } ) $$ for some c > 1 depending on k, and where λ is the root of the polynomial P(λ) = λ ^k+2 ? λ ? 1 belonging to the interval (1, 2). Using this result, we prove that the equation has a positive solution which is not eventually equal to 1. Also, for the case k = 1, we find all positive eventually equal to unity solutions to the equation.     

A problem of Carlitz and its generalizations

Let ${\mathbb{F}_q}$ be the finite field of characteristic p > 2 with q elements. Carlitz proposed the problem of finding an explicit formula for the number of solutions to the equation $$(x_1+ x_2+\cdots+x_n)^2=a\, x_1x_2\cdots x_n,$$ where ${a\in \mathbb{F}_q^*}$ and n ≥ 3. By using the augmented degree matrix and Gauss sums, we consider the generalizations of the above equation and partially solve Carlitz’s problem. Moreover, the technique developed in this paper may be applied to other equations of the form ${h_1^\lambda=h_2}$ with ${h_1, h_2 \in \mathbb{F}_q[x_1,\ldots,x_n]}$ and ${\lambda \in \mathbb{N}}$ .     

On positive solutions of a reciprocal difference equation with minimum

In this paper we consider positive solutions of the following difference equation $$x_{n + 1} = \min \left\{ {\frac{A}{{x_n }},\frac{B}{{x_{n - 2} }}} \right\}, A, B > 0.$$ We prove that every positive solution is eventually periodic. Also, we present here some results concerning positive solutions of the difference equation $$x_{n + 1} = \min \left\{ {\frac{A}{{x_n x_{n - 1} ...x_{n - k} }},\frac{B}{{x_{n - (k + 2)} ...x_{n - (2k + 2)} }}} \right\}, A, B > 0.$$      

Subalgebras of free products of algebras of the variety\mathfrak{u}_{m,n}

The variety \(\mathfrak{u}_{m,n} \) is defined by the system of n-ary operations ω_i,..., ω_m, the system of m-ary operations ?_i,..., ?_n, 1≤ m ≤ n, and the system of identities $$\begin{gathered} x_1 ...x_n \omega _1 ...x_1 ...x_n \omega _m \varphi _i = x_i (i = 1,...,n), \hfill \\ x_1 ...x_m \varphi _1 ...x_1 ...x_m \varphi _n \omega _j = x_j (i = 1,...,m), \hfill \\ \end{gathered} $$ It is proved in this paper that the subalgebra U of the free product \(\Pi _{i \in I}^* A_i \) of the algebras A_i (i ε I) can be expanded as the free product of nonempty intersections U ∩ A_i (i ε I) and a free algebra.     

Asymptotic formula for the mean value of a multiple trigonometric sum

When k≥k₀=10 Mr²n log (rn) we have for the trigonometric integral $$J_n (k,P) = \int_E {|S(A)|^{2k} dA,} $$ where $$\begin{gathered} S(A) = \sum _{x_1 = 1}^P \cdots \sum _{x_r = 1}^P \exp (2\pi if_A (x_1 , \ldots ,x_r )), \hfill \\ f_A (x_1 , \ldots ,x_r ) = \sum _{t_1 = 0}^n \cdots \sum _{t_r = 0}^n \alpha _{t_1 \cdots l_r } x_1^{t_1 } \cdots x_{r^r }^t \hfill \\ \end{gathered} $$ and E is the M-dimensional unit cube, the asymptotic formula $$J_n (k,P) = \sigma \theta P^{2kr - rnM/2} + O(P^{2kr - rnM/2 - 1/(2M)} ) + O(P^{2kr - rnM/2 - 1/(500r^2 \log (rn))} ),$$ where σ is a singular series and θ is a singular integral.     

Asymptotic behaviour and existence of nonoscillatory solutions of second-order neutral delay difference equations

In this paper, we give a classification of nonoscillatory solution of a second-order neutral delay difference equation of the form $$\Delta ^2 (x_n - c_n x_{n - \tau } ) = f(n, x_{g_1 (n)} ,..., x_{g_m (n)} ).$$ Some existence results for each kind of nonoscillatory solutions are also established.     

A functional equation related to the product in a quadratic number field

The functional equation $$f(x_{1},y_{1})f(x_{2},y_{2})=f(x_{1}x_{2}+\alpha y_{1}y_{2},x_{1}y_{2}+x_{2}y_{1}),\ (x_{1},y_{1}),\,(x_{2},y_{2})\in \mathbb{ R}^{2}$$ arises from the formula for the product of two numbers in the quadratic field ${\mathbb{Q}(\sqrt{\alpha})}$ . The general solution ${f:\mathbb{R}\rightarrow \mathbb{R}}$ to this equation is determined. Moreover, it is shown that no more general equations arise from a change of basis in the field.     

Stability of a mixed additive and cubic functional equation in several variables in non-Archimedean spaces

Consider the functional equation ${\Im_1(f ) = \Im_2(f )\,\,(\Im)}$ in a certain general setting. A function g is an approximate solution of ${(\Im)}$ if ${\Im_1(g)}$ and ${\Im_2(g)}$ are close in some sense. The Ulam stability problem asks whether or not there is a true solution of ${(\Im)}$ near g. In this paper, we achieve the general solution and the stability of the following functional equation $$\begin{array}{ll}f\left(\sum\limits^{n}_{i=1}x_{i} \right)+f\left(\sum\limits^{n-1}_{i=1} x_{i}-x_{n} \right)\\\quad=2f\left(\sum\limits^{n-1}_{i=1}x_{i} \right)+\sum\limits^{n-1}_{i=1}(f(x_{i}+x_{n}) +f(x_{i}-x_{n})-2f(x_{i}))\end{array}$$ for all x _i (i =? 1,2, . . . , n), in non-Archimedean spaces.     

Oscillatory behavior of the second-order nonlinear neutral difference equations

In this paper, we consider the oscillation of the second-order neutral difference equation $$\Delta ^2 \left( {x_n - px_{n - \tau } } \right) + q_n f\left( {x_{n - \sigma _n } } \right) = 0$$ as well as the oscillatory behavior of the corresponding ordinary difference equation $$\Delta ^2 z_n + q_n f\left( {R\left( {n,\lambda } \right)z_n } \right) = 0$$ .     

Global integrability for minimizers of anisotropic functionals

We consider integral functionals in which the density has growth p _i with respect to ${\frac{\partial u}{\partial x_i}}$ , like in $$\int\limits_{\Omega}\left( \left| \frac{\partial u}{\partial x_1}(x) \right|^{p_1} + \left|\frac{\partial u}{\partial x_2}(x)\right|^{p_2} + \cdots + \left|\frac{\partial u}{\partial x_n}(x) \right|^{p_n} \right) dx.$$ We show that higher integrability of the boundary datum forces minimizer to be more integrable.     

Estimate of a multidimensional sum with the Legendre symbol for a polynomial of odd degree

The estimate $\left| {\sum\nolimits_{x_1 ,...,x_n \in F_q } {x(f(x_1 ,...,x_n ))} } \right| \leqslant (d - 1)^n q^{n/2} $ is derived for the quadratic character Λ of a field F_q of q elements and a polynomial f of odd degree d over F_q under certain natural conditions.     

Oscillation theorems for second-order nonlinear delay difference equations

By means of Riccati transformation technique, we establish some new oscillation criteria for second-order nonlinear delay difference equation $$\Delta (p_n (\Delta x_n )^\gamma ) + q_n f(x_{n - \sigma } ) = 0,\;\;\;\;n = 0,1,2,...,$$ when $\sum\limits_{n = 0}^\infty {\left( {\frac{1}{{Pn}}} \right)^{\frac{1}{\gamma }} = \infty }$ . When $\sum\limits_{n = 0}^\infty {\left( {\frac{1}{{Pn}}} \right)^{\frac{1}{\gamma }} < \infty }$ we present some sufficient conditions which guarantee that, every solution oscillates or converges to zero. When $\sum\limits_{n = 0}^\infty {\left( {\frac{1}{{Pn}}} \right)^{\frac{1}{\gamma }} = \infty }$ holds, our results do not require the nonlinearity to be nondecreasing and are thus applicable to new classes of equations to which most previously known results are not.     

A two-dimensional additive problem with an increasing number of terms

In this paper there is established an asymptotic formula for the number of simultaneous representations of two numbers as sums of an increasing number of terms involving a power function, i.e., an asymptotic (as n→∞) formula is found for the number of solutions in integers x_i, 0 ≤ x_i ≤ p, of the following system of diophantine equations: $$\left\{ {\begin{array}{*{20}c} {x_1 + x_2 + \ldots x_n = N_{1,} } \\ {x_1^2 + x_{_2 }^{_2 } + \ldots x_n = N_{2.} } \\ \end{array} } \right.$$ . The analysis is carried out as in the proof of a local limit theorem of probability theory and involves estimates of Weyl sums.     

On asymptotic behaviour of the difference equationX_{N + 1} = \alpha + \frac{{X_{N - 1} ^P }}{{X_N ^P }}

In this paper, we investigate local stability, oscillation and boundeness character of positive solutions of the difference equation $$x_{n + 1} = \alpha + \frac{{x_{n - 1} ^p }}{{x_n ^p }},n = 0,1,...$$ under specified conditions.     

Transformations z(t) = L(t)y(ϕ(t)) of ordinary differential equations

The paper describes the general form of an ordinary differential equation of an order n + 1 (n ≥ 1) which allows a nontrivial global transformation consisting of the change of the independent variable and of a nonvanishing factor. A result given by J. Aczél is generalized. A functional equation of the form $f\left( {s,w_{00} \upsilon _0 ,...,\sum\limits_{j = 0}^n {w_{nj\upsilon _j } } } \right) = \sum\limits_{j = 0}^n {w_{n + 1j\upsilon j} + w_{n + 1n + 1} f\left( {x,\upsilon ,\upsilon _1 ,...,\upsilon _n } \right),}$ where $w_{n + 10} = h\left( {s,x,x_1 ,u,u_1 ,...,u_n } \right),w_{n + 11} = g\left( {s,x,x_1 ,...,x_n ,u,u_1 ,...,u_n } \right){\text{ and }}w_{ij} = a_{ij} \left( {x_i ,...,x_{i - j + 1} ,u,u_1 ,...,u_{i - j} } \right)$ for the given functions a _ij is solved on $\mathbb{R},u \ne {\text{0}}$ .     

Global asymptotic stability of a second-order system of difference equations

In this paper a sufficient condition is obtained for the global asymptotic stability of the following system of difference equations $$x_{n + 1} = \frac{{x_n y_{n - 1}^b + 1}} {{x_n + y_{n - 1}^b }}, y_{n + 1} = \frac{{y_n x_{n - 1}^b + 1}} {{y_n + x_{n - 1}^b }}n = 0,1,2 \ldots$$ where the parameter b ∈ [0, ∞) and the initial values (x _k , y _k ) ∈ (0, ∞) (for k = ?1, 0).     

Mean Value Properties for the Weinstein Equation Using the Hyperbolic Metric

In this paper we consider solutions of the Weinstein equation $$\begin{aligned} \Delta u-\frac{k}{x_{n}}\frac{\partial u}{\partial x_{n}}+\frac{\ell }{ x_{n}^{2}}u=0, \end{aligned}$$ on some open subset $\Omega \subset \mathbb R ^{n}\cap \{x_{n}>0\}$ subject to the conditions $4\ell \le (k+1)^{2}$ . If $l=0$ , the operator $x_{n}^{2k/n-2}\left( \Delta u-\frac{k}{x_{n}}\frac{\partial u}{\partial x_{n}}\right) $ is the Laplace–Beltrami operator with respect to the Riemannian metric $ds^{2}=x_{n}^{-2k/n-2}\left( \sum _{i=1}^{n}dx_{i} ^{2}\right) $ . In case $k=n-2$ the Riemannian metric is the hyperbolic distance of Poincaré upper half space. The Weinstein equation is connected to the axially symmetric potentials. The solutions of of the Weinstein equation form a so-called Brelot harmonic space and therefore it is known they satisfy the mean value properties with respect to the harmonic measure. We present the explicit mean value properties which give a formula for a harmonic measure evaluated in the center point of the hyperbolic ball. The key idea is to transform the solutions to the eigenfunctions of the Laplace–Beltrami operator in the Poincaré upper half-space model.     

Abelian covers of alternating groups

Let \({C={\rm inf} (k/n)\sum_{i=1}^n x_i(x_{i+1}+\cdots+x_{i+k})^{-1}}\), where the infimum is taken over all pairs of integers \({n\geq k\geq 1}\) and all positive \({x_1,\ldots,x_n}\), \({x_{n+i}=x_i}\). We prove that \({\ln 2 \leq C < 0.9305}\). In the definition of the constant C, the operation \({{\rm inf}_{k}\, {\rm inf}_{n}\, {\rm inf}_{x}}\) can be replaced by \({{\rm lim}_{k \to \infty}\, {\rm lim}_{n \to \infty} {\rm inf}_{x}}\).     

Lower and upper bounds of Dirichlet eigenvalues for totally characteristic degenerate elliptic operators

Let λkbe the k-th Dirichlet eigenvalue of totally characteristic degenerate elliptic operator-ΔB defined on a stretched cone B0 ■ [0,1) × X with boundary on {x1 = 0}. More precisely,ΔB=(x1αx1)2+ α2x2+ + α2xnis also called the cone Laplacian. In this paper,by using Mellin-Fourier transform,we prove thatλk Cnk2 n for any k 1,where Cn=(nn+2)(2π)2(|B0|Bn)-2n,which gives the lower bounds of the Dirchlet eigenvalues of-ΔB. On the other hand,by using the Rayleigh-Ritz inequality,we deduce the upper bounds ofλk,i.e.,λk+1 1 +4n k2/nλ1. Combining the lower and upper bounds of λk,we can easily obtain the lower bound for the first Dirichlet eigenvalue λ1 Cn(1 +4n)-12n2.     

Classification and existence of nonoscillatory solutions of higher order nonlinear neutral difference equations

In this paper, we consider the higher order nonlinear neutral delay difference equation of the form $$\Delta ^r (x_n + px_{n - \tau } ) + f(n,x_{n - \sigma _1 (n)} ,x_{n - \sigma _2 (n)} ,...,x_{n - \sigma _m (n)} ) = 0.$$ We give an integrated classification of nonoscillatory solutions of the above equation according to their asymptotic behaviours. Necessary and sufficient conditions for the existence of nonoscillatory solutions with designated asymptotic properties are also established.