Some boundedness results for systems of two rational difference equations

We study k ^th order systems of two rational difference equations

Refined functional equations stemming from cubic,quadratic and additive mappings

Let n ≥ 2 be an integer. In this paper, we investigate the generalized Hyers-Ulam stability problem for the following functional equation f（n-1∑j=1 xj＋2xn）＋f（n-1∑j=1 xj-2xn）＋8 n-1∑j=1f（xj）=2f（n-1∑j=1 xj） ＋4 n-1∑j=1[f（xj＋xn）＋f（xj-xn）] which contains as solutions cubic, quadratic or additive mappings.     

The generalized Roper-Suffridge extension operator in Banach spaces (III)

Suppose that X is a complex Banach space with the norm ‖·‖ and n is a positive integer with dim X ⩾ n ⩾ 2. In this paper, we consider the generalized Roper-Suffridge extension operator $ \Phi _{n,\beta _2 ,\gamma _2 , \ldots ,\beta _{n + 1} ,\gamma _{n + 1} } (f) $ \Phi _{n,\beta _2 ,\gamma _2 , \ldots ,\beta _{n + 1} ,\gamma _{n + 1} } (f) on the domain $ \Omega _{p_1 ,p_2 , \ldots ,p_{n + 1} } $ \Omega _{p_1 ,p_2 , \ldots ,p_{n + 1} } defined by

Real moments of the restrictive factor

Let λ be a real number such that 0 < λ < 1. We establish asymptotic formulas for the weighted real moments Σ _n≤x R ^λ (n)(1 − n/x), where R(n) =$ \prod\nolimits_{\nu = 1}^k {p_\nu ^{\alpha _\nu - 1} } $ \prod\nolimits_{\nu = 1}^k {p_\nu ^{\alpha _\nu - 1} } is the Atanassov strong restrictive factor function and n =$ \prod\nolimits_{\nu = 1}^k {p_\nu ^{\alpha _\nu } } $ \prod\nolimits_{\nu = 1}^k {p_\nu ^{\alpha _\nu } } is the prime factorization of n.     

On the rational (K + 1,K + 1)-type difference equation

In this paper we investigate the boundedness character of the positive solutions of the rational difference equation of the form $$x_{n + 1} = \frac{{a_0 + \sum\nolimits_{j = 1}^k {a_j x_{n - j + 1} } }}{{b_0 + \sum\nolimits_{j = 1}^k {b_j x_{n - j + 1} } }}, n = 0,1,...$$ where k ε N, andaj,bj, j = 0,1,…, k, are nonnegative numbers such thatb ₀+∑ _j=1 ^k b _j x _n-j+1>0 for everyn ∈N ∪{0}. In passing we confirm several conjectures recently posed in the paper: E. Camouzis, G. Ladas and E. P. Quinn, On third order rational difference equations (part 6).     

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).     

Some properties for a class of symmetric functions with applications

For x = (x ₁, x ₂, ..., x _n ) ∈ ℝ₊ ⁿ , the symmetric function ψ _n (x, r) is defined by $\psi _n (x,r) = \psi _n \left( {x_1 ,x_2 , \cdots ,x_n ;r} \right) = \sum\limits_{1 \leqslant i_1 < i_2 \cdots < i_r \leqslant n} {\prod\limits_{j = 1}^r {\frac{{1 + x_{i_j } }} {{x_{i_j } }}} } ,$\psi _n (x,r) = \psi _n \left( {x_1 ,x_2 , \cdots ,x_n ;r} \right) = \sum\limits_{1 \leqslant i_1 < i_2 \cdots < i_r \leqslant n} {\prod\limits_{j = 1}^r {\frac{{1 + x_{i_j } }} {{x_{i_j } }}} } ,     

Determination of Riesz bounds for the spline basis with the help of trigonometric polynomials

The problem of determining the upper and lower Riesz bounds for the mth order B-spline basis is reduced to analyzing the series $ \sum\nolimits_{j = - \infty }^\infty {\frac{1} {{(x - j)^{2m} }}} $ \sum\nolimits_{j = - \infty }^\infty {\frac{1} {{(x - j)^{2m} }}} . The sum of the series is shown to be a ratio of trigonometric polynomials of a particular shape. Some properties of these polynomials that help to determine the Riesz bounds are established. The results are applied in the theory of series to find the sums of some power series.     

Nearly Quartic Mappings in β-Homogeneous F-Spaces

In this paper, we investigate the Hyers–Ulam stability of the following quartic equation $$\begin{array}{ll} {\sum\limits^{n}_{k=2}}\left({\sum\limits^{k}_{i_{1}=2}}{\sum\limits^{k+1}_{i_{2}=i_{1}+1}} \ldots {\sum\limits^{n}_{i_{n-k+1}=i_{n-k}+1}}\right)\\ \quad\times f \left({\sum\limits^{n}_{i=1,i \neq i_{1},\ldots,i_{n-k+1}}} x_{i}-{\sum\limits^{n-k+1}_{r=1}}x_{i_{r}}\right) + f \left({\sum\limits^{n}_{i=1}}x_{i}\right)\\ \quad-2^{n-2}{\sum\limits^{}_{1 \leq{i} \leq{j} \leq{n}}}(f(x_{i} + x_{j}){+f(x_{i} - x_{j})){+2^{n-5}(n - 2){\sum\limits^{n}_{i=1}}f(2x_{i})}} = \theta \end{array} $$ $({n \in \mathbb{N}, n \geq 3})$ in β-homogeneous F-spaces.     

On the strong law of large numbers and additive functions

Let f(n) be a strongly additive complex-valued arithmetic function. Under mild conditions on f, we prove the following weighted strong law of large numbers: if X,X ₁,X ₂, … is any sequence of integrable i.i.d. random variables, then

Complexity of self-correcting circuits for some sequence of Boolean functions

k-Self-correcting circuits of functional elements in the basis {x ₁&x ₂, $ \bar x $ \bar x } are considered. It is assumed that constant faults on outputs of functional elements are of the same type. Inverters are supposed to be reliable in service. The weight of each inverter is equal to 1. Conjunctors can be reliable in service, or not reliable. Each reliable conjunctor implements a conjunction of two variables and has a weight p > k + 2. Each unreliable conjunctor implements a conjunction in its correct state and implements a Boolean constant δ (δ ∈ {0, 1}) otherwise. Each unreliable conjunctor has the weight 1. It is stated that the complexity of realization of monotone threshold symmetric functions $ f_2^n \left( {x_1 ,...,x_n } \right) = \mathop \vee \limits_{1 \leqslant i < j \leqslant n} x_1 x_j ,n = 3,4 $ f_2^n \left( {x_1 ,...,x_n } \right) = \mathop \vee \limits_{1 \leqslant i < j \leqslant n} x_1 x_j ,n = 3,4 , ... by such circuits asymptotically equals (k + 3)n.     

Boundedness and compactness of the embedding between spaces with multiweighted derivatives when 1 ⩽ <Emphasis Type="Italic">q</Emphasis> < <Emphasis Type="Italic">p</Emphasis> < ∞

We consider a new Sobolev type function space called the space with multiweighted derivatives $ W_{p,\bar \alpha }^n $ W_{p,\bar \alpha }^n , where $ \bar \alpha $ \bar \alpha = (α ₀, α ₁,…, α _n ), α _i ∈ ℝ, i = 0, 1,…, n, and $ \left\| f \right\|W_{p,\bar \alpha }^n = \left\| {D_{\bar \alpha }^n f} \right\|_p + \sum\limits_{i = 0}^{n - 1} {\left| {D_{\bar \alpha }^i f(1)} \right|} $ \left\| f \right\|W_{p,\bar \alpha }^n = \left\| {D_{\bar \alpha }^n f} \right\|_p + \sum\limits_{i = 0}^{n - 1} {\left| {D_{\bar \alpha }^i f(1)} \right|} ,

The Law of Iterated Logarithm of Rescaled Range Statistics for AR（1） Model

Let {Xn,n ≥ 0} be an AR（1） process. Let Q（n） be the rescaled range statistic, or the R/S statistic for {Xn} which is given by （max1≤k≤n（∑j=1^k（Xj - ^-Xn）） - min 1≤k≤n（∑j=1^k（ Xj - ^Xn ））） /（n ^-1∑j=1^n（Xj -^-Xn）^2）^1/2 where ^-Xn = n^-1 ∑j=1^nXj. In this paper we show a law of iterated logarithm for rescaled range statistics Q（n） for AR（1） model.     

The Stable Periodic Solutions of a Discrete Neural Network System

In this paper, a nonlinear difference system {xn=βxn-1＋f（yn-κ）,yn=βyn-1＋f（xn-κ）,n∈N is considered a,nd sufficient conditions for the existe~lce of the stable 2κ ＋ 1 periodic solution are obtained.     

Stability problem of Hyers-Ulam-Rassias for generalized forms of cubic functional equation

Let n≥2 be an integer number. In this paper, we investigate the generalized Hyers Ulam- Rassias stability in Banach spaces and also Banach modules over a Banach algebra and a C＊-algebra and the stability using the alternative fixed point of an n-dimensional cubic functional equation in Banach spaces：f（2∑j=1^n-1 xj＋xn）＋f（2∑j=1^n-1 xj-xn）＋4∑j=1^n-1f（xj）=16f（∑j=1^n-1 xj）＋2∑j=1^n-1（f（xj＋xn）＋f（xj-xn）     

Studies on a multi-point boundary value problem for second order differential equation with <Emphasis Type="Italic">p</Emphasis>-Laplacian

The existence of at least one solution of the following multi-point boundary value problem

On boundary values of solutions of the dirichlet problem for second order elliptic equation

The paper suggests some conditions on the lower order terms, which provide that the solution of the Dirichlet problem for the general elliptic equation of the second order

Fundamental Solution of the Anisotropic Porous Medium Equation

We establish the existence of fundamental solutions for the anisotropic porous medium equation, ut = ∑n i=1（u^mi）xixi in R^n × （O,∞）, where m1,m2,..., and mn, are positive constants satisfying min1≤i≤n{mi}≤ 1, ∑i^n=1 mi 〉 n - 2, and max1≤i≤n{mi} ≤1/n（2 ＋ ∑i^n=1 mi）.     

Strong convergence theorems of <Emphasis Type="Italic">k</Emphasis>-strict pseudo-contractions in Hilbert spaces

Let K be a nonempty closed convex subset of a real Hilbert space H such that K ± K ⊂ K, T: K → H a k-strict pseudo-contraction for some 0 ⩽ k < 1 such that F(T) = {x ∈ K: x = Tx} ≠ $ \not 0 $ \not 0 . Consider the following iterative algorithm given by

Analogs of the Luzin-Danzhu and Cantor-Lebesgue theorems for double trigonometric series

Let ∥ · ∥ be some norm in R², Γ be the unit sphere induced in R² by this norm, and {Aj} a sequence of disjoint subsets of R₊ such that if ν ε A_j, then ν · Γ ∩ Z^N ≠ Ø. For series of the form $$\sum\nolimits_{j = 1}^\infty {} \sum\nolimits_{\parallel n\parallel \in A_j } {c_n e^{2\pi _i (n_1 x_1 + n_2 x_2 )} } $$ analogs of the Luzin-Danzhu and Cantor-Lebesgue theorems are established.