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

In this paper, we investigate the periodicity character of some classes of difference equations. Among other results, we give a simpler proof of an Open Problem addressed in Csornyei and Laczkovich, 2001, “Some periodic and non-periodic recursions”, Monath. Math., 132, 215–236. We also consider Open Problem 2.8 in Grove and Ladas, 2005, Periodicities in Nonlinear Difference Equations, (Chapman and Hall/CRC), and the case when a difference equation has periodic solutions depending on arbitrary parameters.

Periodicity of some classes of holomorphic difference equations

On operator P g φ from the logarithmic Bloch-type space to the mixed-norm space on the unit ball

On some product-type operators from Hardy–Orlicz and Bergman–Orlicz spaces to weighted-type spaces

We prove that, for every k ∈ ℕ, the following generalization of the Putnam difference equation $$
x_{n + 1} = \frac{{x_n + x_{n - 1} + \cdots + x_{n - (k - 1)} + x_{n - k} x_{n - (k + 1)} }}
{{x_n x_{n - 1} + x_{n - 2} + \cdots + x_{n - (k + 1)} }}, n \in \mathbb{N}_0 ,
$$
 has a positive solution with the following asymptotics $$
x_n = 1 + (k + 1)e^{ - \lambda ^n } + (k + 1)e^{ - c\lambda ^n } + o(e^{ - c\lambda ^n } )
$$
 for some c > 1 depending on k, and where λ is the root of the polynomial P(λ) = λ
 k+2 − λ − 1 belonging to the interval (1, 2). Using this result, we prove that the equation has a positive solution which is not eventually equal to 1. Also, for the case k = 1, we find all positive eventually equal to unity solutions to the equation.

Nontrivial solutions of a higher-order rational difference equation

We describe a method for finding monotone solutions of some
classes of difference equations converging to the corresponding
equilibria. The method enables us to confirm three conjectures
posed by the present author in a talk, which are extensions of
three conjectures by M. R. S. Kulenovic and G. Ladas,
Dynamics of Second Order Rational Difference Equations.
With Open Problems and Conjectures. Chapman and Hall/CRC, 2002.
It is interesting that the method, in some cases, can be applied
also when the parameters are variable.

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Stevo Stević

Papers

Periodicity of some classes of holomorphic difference equations

On operator P g φ from the logarithmic Bloch-type space to the mixed-norm space on the unit ball

On some product-type operators from Hardy–Orlicz and Bergman–Orlicz spaces to weighted-type spaces

Nontrivial solutions of a higher-order rational difference equation

On monotone solutions of some classes of difference equations