On some algebraic difference equations in , related to families of conics or cubics: generalization of the Lyness' sequences

In this paper and in a forthcoming one, we study difference equations in of the types (2)(4)(6) which are linked to families of conics, cubics and quartics, respectively. These equations generalize Lyness' one u_n+2u_n=a+u_n+1 studied in several papers, whose behavior was completely elucidated in [G. Bastien, M. Rogalski, in press] through methods which are transposed in the present paper for the study of (1) and (2), and in the forthcoming one for (3). In particular we prove in the present paper a form of chaotic behavior for solutions of difference equations (1) and (2), and find all the possible periods for these solutions.     

On the Recursive Sequence x_{n + 1} = {{ \mgreek{g} x_{n-1} + \mgreek{d} x_{n-2}} \over {Cx_{n-1} + Dx_{n-2}}}

We investigate the global character of solutions of the equation in the title with positive parameters and positive initial conditions. We obtain results about the global attractivity of the equilibrium, the existence and attractivity of the period-two solution and the semicycles.     

On the Dynamics of x_{\bf{n} + \bf{1}} = {{ \mgreek{a} + \mgreek{g} x_{n-1} + \mgreek{d} x_{n-2}} \over {A + x_{n-2}}}

We investigate the global stability, the periodic character and the boundedness nature of solutions of the equation in the title for all admissible nonnegative values of the parameters and the initial conditions. We show that the solutions exhibit a trichotomy character depending on how the parameter γ compares to the sum of the parameters δ and A.     

Asymptotic Behavior of Solutions to the Difference Equation Using the Second Differences of a Liapunov Function

We introduce the concept of the second Liapunov differences in difference equations and show that, in the autonomous case, the second variations can guarantee the existence of solutions tending to zero while others starting arbitrarily near 0 go to ∞. These are based on Yorke's results for ordinary differential equations.     

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 .     

Multiplicity one theorem for ({\rm GL}_{n+1}({\mathbb{R}}), {\rm GL} _ {n} ({ \mathbb{R}}))

Let F be either or . Consider the standard embedding and the action of GL_n(F) on GL_n+1(F) by conjugation. We show that any GL_n(F)-invariant distribution on GL_n+1(F) is invariant with respect to transposition. We prove that this implies that for any irreducible admissible smooth Fréchet representations π of GL_n+1(F) and of GL_n(F),

On lattices of convex sets in {\mathbb{R}}^{n}

Properties of several sorts of lattices of convex subsets of are examined. The lattice of convex sets containing the origin turns out, for n > 1, to satisfy a set of identities strictly between those of the lattice of all convex subsets of and the lattice of all convex subsets of The lattices of arbitrary, of open bounded, and of compact convex sets in all satisfy the same identities, but the last of these is join-semidistributive, while for n > 1 the first two are not. The lattice of relatively convex subsets of a fixed set satisfies some, but in general not all of the identities of the lattice of “genuine” convex subsets of To the memory of Ivan RivalReceived April 22, 2003; accepted in final form February 16, 2005.This revised version was published online in August 2005 with a corrected cover date.     

Atomic Decomposition of a Weighted Inductive Limit in \mathbb{C}^{n}

We construct an atomic decomposition of the space H_V^∞ consisting of analytic functions on the open unit disc. The research of the second named author was supported by the Academy of Finland project “Functional Analysis and Applications”.     

On the Number of Solutions of the Equation a_{1}x_{1}^{m_{1}}+\cdots +a_{n}x_{n}^{m_{n}}=bx_{1}\cdots x_{n} in a Finite Field

We obtain an explicit formula for the number of solutions of a special equation in a finite field under a certain restriction on the exponents. Mathematics Subject Classifications (2000) primary: 11G25, secondary: 11T24.     

On the functional equationf(x) g(y) = p(x + y) q(x/y)

Aequationes mathematicae -     

On the Behavior of the Solutions for Certain Linear Autonomous Difference Equations

Certain linear autonomous delay as well as neutral delay difference equations are considered. A class of linear autonomous delay difference equations with continuous variable is also considered. Some results on the behavior of the solutions are established via two distinct positive roots of the corresponding characteristic equation.