ASYMPTOTIC BEHAVIOR OF SOLUTIONS FOR A CLASS OF DELAY DIFFERENCE EQUATION

We propose a class of delay difference equation with piecewise constant nonlinearity. Such a delay difference equation can be regarded as the discrete analog of a differential equation. The convergence of solutions and the existence of asymptotically stable periodic solutions are investigated for such a class of difference equation.     

EXISTENCE OF PERIODIC SOLUTIONS OF SYSTEM OF DIFFERENCE EQUATIONS

This paper studies the nonautonomous nonlinear system of difference equationsΔx(n)=A(n)x(n)+f(n,x(n)),n∈Z,(*) where x(n)∈R~N,A(n)=(a_(ij)(n))N×N is an N×N matrix,with a-(ij)∈C(R,R) for i,j= 1,2,3,...,N,and f=(f_1,f_2,...,f_N)~T∈C(R×R~N,R~N),satisfying A(t+ω)=A(t),f(t+ω,z)=f(t,z) for any t∈R,(t,z)∈R×R~N andωis a positive integer.Sufficient conditions for the existence ofω-periodic solutions to equations (*) are obtained.     

OSCILLATION CRITERIA OF SOLUTIONS OF NONLINEAR DIFFERENCE EQUATIONS OF SECOND ORDER

Oscillatory behavior of solutions of second order nonlinear difference equation is studied. Oscillation criteria for its solutions are given. Examples are given in the text to illustrate the results.     

GLOBAL ATTRACTIVITY IN A CLASS OF HIGHER-ORDER NONLINEAR DIFFERENCE EQUATION

In this paper the global attractivity of the nonlinear difference equation xn 1 = a bxn / A xn-k, n =0, 1, …，is investigated, where a, b, A ∈ (0, ∞), k is an positive integer and the initial conditions x- k, …，x- 1 and x0 are arbitrary positive numbers. It is shown that the unique positive equilibrium of the equation is global attractive. As a corollary, the result gives a positive confirmation on the conjecture presented by Kocic and Ladas [1,p154].     

OSCILLATORY AND ASYMPTOTIC BEHAVIOR OF A CLASS OF FIRST ORDER NEUTRAL DIFFERENTIAL EQUATION WITH PIECEWISE CONSTANT DELAY

The oscillatory and asymptotic behavior of a class of first order nonlinear neutral differential equation with piecewise constant delay and with diverse deviating arguments are considered. We prove that all solutions of the equation are nonoscillatory and give sufficient criteria for asymptotic behavior of nonoscillatory solutions of equation.     

EXISTENCE OF SOLUTIONS OF A FAMILY OF NONLINEAR BOUNDARY VALUE PROBLEMS IN L^2-SPACES

By using the perturbation results of sums of ranges of accretive mappings of Calvert and Gupta (1978),the abstract results on the existence of solutions of a family of nonlinear boundary value problems in L^2 (Ω) are studied. The equation discussed in this paper and the methods used here are extension and complement to the corresponding results of Wei Li and He Zhen‘s previous papers. Especially,some new techniques are used in this paper.     

ASYMPTOTIC BEHAVIOR OF SOLUTIONS OF SOME NONLINEAR DIFFERENCE EQUATIONS

In this paper, we investigate the asymptotic behavior of the extremal solutions of a difference equation and their character and prove the existence of the non-extremal solutions.     

EXPLICIT DESCRIPTION OF A CLASS OF INDECOMPOSABLE INJECTIVE MODULES

Let R be a commutative Noetherian ring and p be a prime ideal of R such that the ideal pRp is principal and ht(p)≠ 0. In this note, the anthors describe the explicit structure of the injective envelope of the R-module R/p.     

ON THE DIVIDED DIFFERENCE FORM OF FAA DI BRUNO＇S FORMULA

The n-divided difference of the composite function h ：= f o g of functions f, g at a group of nodes t0,t1,…,tn is shown by the combinations of divided differences of f at the group of nodes g（t0）,g（t1）,…,g（tm） and divided differences of g at several partial group of nodes t0,t1,…,tn, where m = 1,2,…,n. Especially, when the given group of nodes are equal to each other completely, it will lead to Faà di Bruno＇s formula of higher derivatives of function h.     

LARGE-TIME BEHAVIOR OF SOLUTIONS OF QUANTUM HYDRODYNAMIC MODEL FOR SEMICONDUCTORS

A one-dimensional quantum hydrodynamic model (or quantum Euler-Poisson system) for semiconductors with initial boundary conditions is considered for general pressure-density function. The existence and uniqueness of the classical solution of the corresponding steady-state quantum hydrodynamic equations is proved. Furthermore, the global existence of classical solution, when the initial datum is a perturbation of the steady-state solution, is obtained. This solution tends to the corresponding steady-state solution exponentially fast as the time tends to infinity.     

A critical case of stability of one quasilinear difference equation of the second order

We obtain sufficient conditions for the Perron stability of the trivial solution of a real difference equation of the form

Asymptotic approximation of solutions of nonlinear third order difference equations

In this paper, we obtain asymptotic bounds, under appropriate conditions, of solutions of third order difference equations of the form

Modified mann iterations for nonexpansive semigroups in Banach space

Let E be a real reflexive Banach space which admits a weakly sequentially continuous duality mapping from E to E^＊, and C be a nonempty closed convex subset of E. Let {T（t） ： t ≥ 0} be a nonexpansive semigroup on C such that F ：=∩t≥0 Fix（T（t）） ≠ 0, and f ： C → C be a fixed contractive mapping. If {αn}, {βn}, {an}, {bn}, {tn} satisfy certain appropriate conditions, then we suggest and analyze the two modified iterative processes as：{yn=αnxn＋（1-αn）T（tn）xn,xn=βnf（xn）＋（1-βn）yn
{u0∈C,vn=anun＋（1-an）T（tn）un,un＋1=bnf（un）＋（1-bn）vn
We prove that the approximate solutions obtained from these methods converge strongly to q ∈∩t≥0 Fix（T（t））, which is a unique solution in F to the following variational inequality：
〈（I-f）q,j（q-u）〉≤0 u∈F Our results extend and improve the corresponding ones of Suzuki [Proc. Amer. Math. Soc., 131, 2133-2136 （2002）], and Kim and XU [Nonlear Analysis, 61, 51-60 （2005）] and Chen and He [Appl. Math. Lett., 20, 751-757 （2007）].     

Some boundedness results for systems of two rational difference equations

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

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

Inequalities of Paley type for noncommutative martingales

Summary The purpose of this paper is to study the validity of the Paley inequality on square function, for noncommutative martingales. Let be a regular gage space, and a sequence of von-Neumann algebras such that we prove that for every , where ɛ_n(F) is the conditional expectation of F with respect to the subalgebra : We also consider the case of a martingale arising in the context of harmonic analysis on noncommutative discrete groups, in analogy to the theorem of R.E.A.C. Paley on Fourier-Walsh series. Entrata in Redazione il 26 gennaio 1977. Partially sponsored by C.N.R.     

On analytic continuation of a hypergeometric series using the Euler-Knopp transformation

The Euler-Knopp transformation is considered in terms of the problems of regularity and acceleration of the rate of convergence. The object of study is the hypergeometric series

Approximation of fixed points of strongly pseudocontractive maps without Lipschitz assumption

In the present paper, the following result is shown: Let be a real Banach space with a uniformly convex dual , and let be a nonempty closed convex and bounded subset of . Assume that is a continuous strong pseudocontraction. Let and be two real sequences satisfying (i) for all ; (ii) ; and (iii) as Then the Ishikawa iterative sequence generated by

converges strongly to the unique fixed point of .

     

On the Stability of the Maximum Term of the Entire Dirichlet Series

We establish necessary and sufficient conditions for the logarithms of the maximum terms of the entire Dirichlet series and to be asymptotically equivalent as Re z → +∞ outside a certain set of finite measure. __________ Translated from Ukrains'kyi Matematychnyi Zhurnal, Vol. 57, No. 4, pp. 571–576, April, 2005.