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

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The global attractivity of the rational difference equation _{}=1+\frac{_{-}}_{-}

Cesàro-type operators on some spaces of analytic functions on the unit ball

In this paper we consider positive solutions of the following difference equation

$$x_{n + 1} = \min \left\{ {\frac{A}{{x_n }},\frac{B}{{x_{n - 2} }}} \right\}, A, B > 0.$$

We prove that every positive solution is eventually periodic. Also, we present here some results concerning positive solutions of the difference equation

$$x_{n + 1} = \min \left\{ {\frac{A}{{x_n x_{n - 1} ...x_{n - k} }},\frac{B}{{x_{n - (k + 2)} ...x_{n - (2k + 2)} }}} \right\}, A, B > 0.$$

On positive solutions of a reciprocal difference equation with minimum

We study the difference equation x n + 1 = α − x n / x n − 1 , n ∈ ℕ 0 , where α ∈ ℝ and where x − 1 and x 0 are so chosen that the corresponding solution ( x n ) of the equation is defined for every n ∈ ℕ . We prove that when α = 3 the equilibrium x ¯ = 2 of the equation is not stable, which corrects a result due to X. X. Yan, W. T. Li, and Z. Zhao. For the case α = 1 , we show that there is a strictly monotone solution of the equation, and we also find its asymptotics. An explicit formula for the solutions of the equation are given for the case α = 0 .

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Stevo Stević

Papers

The global attractivity of the rational difference equation _{}=1+\frac{_{-}}_{-}

Cesàro-type operators on some spaces of analytic functions on the unit ball

On positive solutions of a reciprocal difference equation with minimum

On the Behaviour of the Solutions of a Second-Order Difference Equation

Norm of weighted composition operators from α-Bloch spaces to weighted-type spaces