Differential-Difference Equations. By Richard Bellman and Kenneth L. Cooke. Pp. xvi, 465. 1963. (Academic Press)

Real and Complex Analysis

The work is giving estimations of the discrepancy between solutions of the initial and the homogenized problems for a one{dimensional second order elliptic operators with random coeecients satisfying strong or uniform mixing conditions. We obtain several sharp estimates in terms of the corresponding mixing coeecient. Abstract. In the theory of homogenisation it is of particular interest to determine the classes of problems which are stable on taking the homogenisation limits. A notable situation where the limit enlarges the class of original problems is known as memory (nonlocal) eeects. A number of results in that direction has been obtained for linear problems. Tartar (1990) innitiated the study of the eeective equation corresponding to nonlinear equation: @ t u n + a n u 2 n = f: Signiicant progress has been hampered by the complexity of required computations needed in order to obtain the terms in power{series expansion. We propose a method which overcomes that diiculty by introducing graphs representing the domain of integration of the integrals in each term. The graphs are relatively simple, it is easy to calculate with them and they give us a clear image of the form of each term. The method allows us to discuss the form of the eeective equation and the convergence of power{series expansions. The feasibility of our method for other types of nonlinearities will be discussed as well.

Applied Mathematics and Computation

Generalized composition operators on zygmund spaces and bloch type spaces

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Aufgaben und Lehrsätze aus der Analysis

We give a complete picture regarding the asymptotic periodicity of positive
solutions of the following difference equation:
 x n = f ( x n − p 1 , … , x n − p k , x n − q 1 , … , x n − q m ) , n ∈ ℕ 0 , where p i , i ∈ { 1 , … , k } , 
and q j , j ∈ { 1 , … , m } , 
are natural numbers such that p 1 p 2 ⋯ p k , q 1 q 2 ⋯ q m and gcd ( p 1 , … , p k , q 1 , … , q m ) = 1 , the function f ∈ C [ ( 0 , ∞ ) k + m ,
 ( α , ∞ ) ] , α > 0 , is
increasing in the first k arguments and decreasing in other m 
arguments, there is a decreasing function g ∈ C [ ( α , ∞ ) , ( α , ∞ ) ] such that g ( g ( x ) ) = x , x ∈ ( α , ∞ ) ,
 x = f ( x , … , x ︸ k , g ( x ) , … , g ( x ) ︸ m ) , x ∈ ( α , ∞ ) , lim x → α + g ( x ) = + ∞ , and lim x → + ∞ g ( x ) = α . It is proved that if all p i , i ∈ { 1 , … , k } , 
are even and all q j , j ∈ { 1 , … , m } 
are odd, every positive solution of the equation converges to 
(not necessarily prime) a periodic solution of period two, 
otherwise, every positive solution of the equation converges to a 
unique positive equilibrium.

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Asymptotic Periodicity of a Higher-Order Difference Equation

Iterated differentiation followed by composition from Bloch-type spaces to weighted BMOA spaces

On a system of difference equations which can be solved in closed form

In this note, we prove that every positive solution of the difference equation where k \in \bf{N}, p,q \in [0, \infty ) and the initial conditions x_{-k}, \ldots ,x_{-1},x_{0} are positive real numbers, converges to a periodic solution of period 2k. This result confirms conjectures 11.4.10, 11.4.16 (a) and 11.4.17 in Kulenovic and Ladas (2001) Dynamics of Second Order Rational Difference Equations (Chapman and Hall/CRC).

Stevo Stević

Papers

Asymptotic Periodicity of a Higher-Order Difference Equation

Iterated differentiation followed by composition from Bloch-type spaces to weighted BMOA spaces

On a system of difference equations which can be solved in closed form

Periodic Character of a Class of Difference Equation

Norms of some operators on bounded symmetric domains