Asymptotic Behavior of Solutions of Some Linear Difference Equations with Oscillatory Decreasing Coefficients

We investigate the asymptotic behavior of solutions of the system x ( n +1)=[ A + B ( n ) V ( n )+ R ( n )] x ( n ), n S n 0 , where A is an invertible m 2 m matrix with real eigenvalues, B ( n )= ~ j =1 r B j e i u j n , u j are real and u j p ~ (1+2 M ) for any M ] Z , B j are constant m 2 m matrices, the matrix V ( n ) satisfies V ( n ) M 0 as n M X , ~ n =0 X Á V ( n +1) m V ( n ) Á < X , ~ n =0 X Á V ( n ) Á 2 < X , and ~ n =0 X Á R ( n ) Á < X . If AV ( n )= V ( n ) A , then we show that the original system is asymptotically equivalent to a system x ( n +1)=[ A + B 0 V ( n )+ R 1 ( n )] x ( n ), where B 0 is a constant matrix and ~ n =0 X Á R 1 ( n ) Á < X . From this, it is possible to deduce the asymptotic behavior of solutions as n M X . We illustrate our method by investigating the asymptotic behavior of solutions of x 1 ( n +2) m 2(cos f 1 ) x 1 ( n +1)+ x 1 ( n )+ a sin n f n g x 2 ( n )=0 x 2 ( n +2) m 2(cos f 2 ) x 2 ( n +1)+ x 2 ( n )+ b sin n f n g x 1 ( n )=0 , where 0< f 1 , f 2 < ~ , 1/2< g h 1, f 1 p f 2 , and 0< f <2 ~ .     

URS' For Weierstrass Products without Exponential Factors

Let $ \cal W $ be the set of entire functions equal to a Weierstrass product of the form $ {f(x)= Ax^q\lim_{r \to \infty} \prod_{|a_j|\leq r}{(1- \fraca {x} {a_j})}} $ where the convergence is uniform in all bounded subsets of $ {\shadC} $ , let $ \cal V $ be the set of $ f\in {\cal W} $ such that $ {\shadC} [\,f]\subset {\cal W} $ , and let $ {\cal H} $ be the $ {\shadC} $ -algebra of entire functions satisfying $ { {\lim_{r\to \infty } } ({\ln M(r,f) / r})=0} $ . Then $ \cal H $ is included in $ {\cal V} $ and strictly contains the set of entire functions of genus zero, (which, itself, strictly contains the $ {\shadC} $ -algebra of entire functions of order 𝜌 < 1). Let $ n, m\in {\shadN} ^* $ satisfy n > m S 3. Let $ a\in {\shadC}^* $ satisfies $ {a^n\not = \fraca{n^n}{(m^m(n-m)^{n-m}})} $ and assume that for every ( n m m )-th root ξ of 1 different from m 1, a satisfies further $ {a^{n}\neq (1+\xi )^{n-m} (\fraca{n^n}{((n-m)^{n-m}m^m}))} $ . Let P ( X ) = X n m aX m + 1 and let T n,m ( a ) be the set of its zeros. Then T n,m ( a ) has n distinct points and is a urs for $ {\cal V} $ . In particular this applies to functions such as sin x and cos x .     

Asymptotic Behavior of a Class of Nonlinear Delay Difference Equations

In this paper, we shall study the asymptotic behavior of solutions of difference equations of the form x n +1 = x n p f ( x n m k 1 , x n m k 2 ,…, x n m k r ), n =0,1,…, where p is a positive constant and k 1 ,…, k r are (fixed) nonnegative integers. In particular, permanence and global attractivity will be discussed.     

Minimal Cantor Systems and Unimodal Maps

We consider the restriction of unimodal maps f to the omega-limit set y ( c ) of the critical point for certain cases where y ( c ) is a Minimal Cantor set. We investigate the relation of these minimal systems to enumeration scales (generalized adding machines), to Vershik adic transformations on ordered Bratelli diagrams and to substitution shifts. Sufficient conditions are given for ( y ( c ), f ) to be uniquely ergodic.     

Isoliertheit und Stabilität von Flächen konstanter mittlerer Krümmung

In the Sobolev space H^m(B,?³), B the open unit disc in ?², we consider the set M_n of all conformally parametrized surfaces of constant mean curvature H with exactly n simple interior branch points (and no others). We denote by M*_n the set of all xεM_n with the following properties:

1. in every branch point the geometrical condition K_G¦x_Z¦≡O holds (K_G is the Gauss curvature and x_z is the complex gradient of the surface x).
2. the corresponding boundary value problem Δh+×_z{²(2H²-K_G)h=O,^hδB=O, is uniquely solvable.

We prove then, that the manifold M*=UM*_n is open and dense in the set of all surfaces of constant mean curvature H and that all x εM*_n are isolated and stable solutions of the Plateau problem corresponding to their boundary curves. In addition, the submanifold M*_n contains exactly all surfaces x for which the space of Jacobi fields is transversal (with exception of the 3-dimensional space of conformai directions) to the tangent space T_xM_n.     

On Globally Periodic Solutions of the Difference Equation x n + 1 = f ( x n )/ x n − 1

We study the second-order difference equation x n +1 = f ( x n ) x n m 1 where f ] C 1 ([0, X ),[0, X )) and x n ] (0, X ) for all n ] Z . For the cases p h 5, we find necessary and sufficient conditions on f for all solutions to be periodic with period p . We answer some questions and conjectures of Kulenovi ' and Ladas.     

A Note on the Difference Equation x n +1 = ∑ i =0 k α i x n − i p i

In this note we improve Theorem 2 in Ref. [3] , about the difference equation x n +1 = ~ i =0 k f i x n m i p i , n =0,1,2,..., where k is a positive integer, f i , p i ] (0, X ) for i =0,..., k , and the initial conditions x m k , x m k +1 ,..., x 0 are arbitrary positive numbers.     

Steiner minimal trees for regular polygons

Fifty years ago Jarnik and Kössler showed that a Steiner minimal tree for the vertices of a regularn-gon contains Steiner points for 3 n5 and contains no Steiner point forn=6 andn13. We complete the story by showing that the case for 7n12 is the same asn13. We also show that the set ofn equally spaced points yields the longest Steiner minimal tree among all sets ofn cocircular points on a given circle.     

Common periodic trajectories of interval maps

We prove that for any continuous piecewise monotone or smooth interval map f and any subset of the set of periods of periodic trajectories of f, there is another map such that the set of periods of periodic trajectories common for f and , which is denoted by , coincides with . At the same time, for each integer , there exists a continuous map f such that for any map if is an infinite set. Dedicated to Vladimir Igorevich Arnold     

On the Global Character of the Difference Equation X n + 1 =

We investigate the global stability, the periodic character, and the boundedness nature of solutions of the difference equation x n +1 = f + n x n m (2 k +1) + i x n m 2 l A + x n m 2 l , n =0,1,… where k and l are non-negative integers, the parameters f , n , i , A are non-negative real numbers with f + n + i >0, and the initial conditions are non-negative real numbers. We show that the solutions exhibit a trichotomy character depending upon the parameters n , i and A .     

On Ω-limit Sets of Discrete-time Dynamical Systems

The paper investigates z -limit sets for discrete-time dynamical systems of the form x n +1 = f n +1 ( x n ), n S 0, with each f n mapping an interval I of R into itself. For autonomous systems, i.e. f n = f for all n , and f continuous on I =[ a , b ], the case that all z -limit sets consist of one point only is characterized by several equivalent conditions, one being that f has no 2-periodic points. The non-autonomous case assumes that the functions f n converge uniformly to a continuous function f X that has no 2-periodic points. It is shown that the z -limit sets are closed intervals consisting of fixed points of f X only. Under certain conditions these closed intervals contain exactly one point each. This allows a treatment of certain discrete-time dynamical systems in R n .     

Maximal inequalities for a continuous semimartingale

Let X =( X t ) t S 0 be a continuous semimartingale given by d X t = f ( t ) w ( X t )d d M ¢ t + f ( t ) σ ( X t )d M t , X 0 =0, where M =( M t , F t ) t S 0 is a continuous local martingale starting at zero with quadratic variation d M ¢ and f ( t ) is a positive, bounded continuous function on [0, X ), and w , σ both are continuous on R and σ ( x )>0 if x p 0. Denote X 𝜏 * =sup 0 h t h 𝜏 | X t | and J t = Z 0 t f ( s ) } ( X s )d d M ¢ s ( t S 0) for a nonnegative continuous function } . If w ( x ) h 0 ( x S 0) and K 1 | x | n σ 2 ( x ) h | w ( x )| h K 2 | x | n σ 2 ( x ) ( x ] R , n >0) with two fixed constants K 2 S K 1 >0, then under suitable conditions for } we show that the maximal inequalities c p , n log 1 n +1 (1+ J 𝜏 ) p h Á X 𝜏 * Á p h C p , n log 1 n +1 (1+ J 𝜏 ) p (0< p < n +1) hold for all stopping times 𝜏 .     

Attractive Periodic Orbits in Nonlinear Discrete-time Neural Networks with Delayed Feedback

We consider the discrete-time system x ( n )= g x ( n m 1)+ f ( y ( n m k )), y ( n )= g y ( n m 1)+ f ( x ( n m k )), n ] N describing the dynamic interaction of two identical neurons, where g ] (0,1) is the internal decay rate, f is the signal transmission function and k is the signal transmission delay. We construct explicitly an attractive 2 k -periodic orbit in the case where f is a step function (McCulloch-Pitts Model). For the general nonlinear signal transmission functions, we use a perturbation argument and sharp estimates and apply the contractive map principle to obtain the existence and attractivity of a 2 k -periodic orbit. This is contrast to the continuous case (a delay differential system) where no stable periodic orbit can occur due to the monotonicity of the associated semiflow.     

A generalization of addition formulae

Let R denote the set of reals, J a real interval and X a real linear space. We determine the functions g : X J, M : J R and H : J ² R satisfying the equationg(x+M(g(x))y)=H(g(x),g(y)),under the assumptions that g is continuous on rays, M is continuous, and H is symmetric. As a consequence we obtain characterizations of some groups and semigroups.     

Fundamental equation of information revisited

Summary This paper presents a new, shorter and more direct proof of the following result of J. Aczél and C. T. Ng: IfM: J R (J =]0, 1[ ^k ) is both multiplicative and additive, then the general solution: J R of(x) + M(1 – x)(y/1 – x) = (y) + M(1 – y)(x/1 – y) (x, y, x + y J) is given by(x) = ifM = 0,(x) = M(x)[L(x) + ] + M(1 – x)L(1 – x) ifM 0,where is an arbitrary constant andL: J R is an arbitrary solution of the logarithmic functional equationL(xy) = L(x) + L(y) (x, y J). Also, some extensions of this result to fields more general than the reals are given.     

Asymptotically Constant Solutions of Functional Difference Systems

We consider the functional difference system ( A ) j x i ( n )= f i ( n ; X ), 1 h i h k , where X =( x 1 ,…, x k ) and f 1 (·; X ),…, f k (·; X ) are real-valued functionals of X , which may depend quite arbitrarily on values of X ( l ) for multiple values of l ] Z . We give sufficient conditions for ( A ) to have solutions that approach specified constant vectors as n M X . Some of the results guarantee only that the solutions are defined for n sufficiently large, while others are global. The proof of the main theorem is based on the Schauder-Tychonoff theorem. Applications to specific quasi-linear systems are included.     

A General Comparison Result for Higher Order Nonlinear Difference Equations With Deviating Arguments

The authors consider m -th order nonlinear difference equations of the form D m p x n + i h j ( n , x s j ( n ) )=0, j =1,2,( E j ) where m S 1, n ] N 0 ={0,1,2,…}, D 0 p x n = x n , D i p x n = p n i j ( D i m 1 p x n ), i =1,2,…, m , j x n = x n +1 m x n , { p n 1 },…,{ p n m } are real sequences, p n i >0, and p n m L 1. In Eq. ( E 1 ) , p = a and p n i = a n i , and in Eq. ( E 2 ) , p = A and p n i = A n i , i =1,2,…, m . Here, { s j ( n )} are sequences of nonnegative integers with s j ( n ) M X as n M X , and h j : N 0 2 R M R is continuous with uh j ( n , u )>0 for u p 0. They prove a comparison result on the oscillation of solutions and the asymptotic behavior of nonoscillatory solutions of Eq. ( E j ) for j =1,2. Examples illustrating the results are also included.     

On the Periodic Structure of Delayed Difference Equations of the Form x n = f ( x n − k ) on I and S 1

The aim of this paper is to give an account of some results recently obtained in Combinatorial Dynamics and apply them to get for k S 2 the periodic structure of delayed difference equations of the form x n = f ( x n m k ) on I and S 1 .     

On the Center of Continuous Maps of Dendrites

The structure and the depth of the center of a continuous map of a dendrite with a closed countable set of branch points of a finite order are studied. It is proved that the center of that map coincides with the closure of the set of periodic points. It is shown also that for an arbitrary natural number n S 2 there are the dendrite X n with a closed countable set of branch points of a finite order and the continuous map f n : X n M X n with n as the depth of the center.     

BMO ESTIMATES FOR MULTILINEAR FRACTIONAL INTEGRALS

In this paper,the authors prove that the multilinear fractional integral operator T A 1,A 2 ,α and the relevant maximal operator M A 1,A 2 ,α with rough kernel are both bounded from L p (1 p ∞) to L q and from L p to L n/(n α),∞ with power weight,respectively,where T A 1,A 2 ,α (f)(x)=R n R m 1 (A 1 ;x,y)R m 2 (A 2 ;x,y) | x y | n α +m 1 +m 2 2 (x y) f (y)dy and M A 1,A 2 ,α (f)(x)=sup r0 1 r n α +m 1 +m 2 2 | x y | r 2 ∏ i=1 R m i (A i ;x,y)(x y) f (y) | dy,and 0 α n, ∈ L s (S n 1) (s ≥ 1) is a homogeneous function of degree zero in R n,A i is a function defined on R n and R m i (A i ;x,y) denotes the m i t h remainder of Taylor series of A i at x about y.More precisely,R m i (A i ;x,y)=A i (x) ∑ | γ | m i 1 γ ! D γ A i (y)(x y) r,where D γ (A i) ∈ BMO(R n) for | γ |=m i 1(m i 1),i=1,2.