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

This paper studies the boundedness, global attractivity, and periodicity of the positive solutions of the difference equation x n + 1 = A + x n p / x n − 1 p , n ∈ ℕ 0 , with p , A ∈ ( 0 , ∞ ) . The main results give a complete picture regarding the boundedness character of the positive solutions of the equation.

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On the Recursive Sequence xn+1=A+xnp/xn−1p

We show that every well-defined solution of the difference equation x n + 1 = max { A x n , x n − 2 } , n ∈ N 0 , where A ∈ R is eventually periodic with period three.

On the max-type equation xn+1=max{Axn,xn−2}

Note on the bilinear difference equation with a delay

and Applied Analysis 3 2. Auxiliary Results Here we quote some auxiliary results which will be used in the proofs of the main results. The first lemma can be proved in a standard way see, e.g., in 13, Proposition 3.11 or in 15, Lemma 3 . Lemma 2.1. Assume that m ∈ N0, n ∈ N, p, q > 0, γ > −1, φ is an analytic self-map of D and u ∈ H D . Then the operator D φ,u : Hp,q,γ → W n μ is compact if and only if D φ,u : Hp,q,γ → W n μ is bounded and for any bounded sequence fk k∈N in Hp,q,γ which converges to zero uniformly on compact subsets of D, D φ,ufk → 0 inW n μ as k → ∞. The next lemma is known, but we give a proof of it for the benefit of the reader. Lemma 2.2. Assume that n ∈ N0, 0 −1 and m > 1 β one has ∫1 0 1 − r β ( 1 − ρrm ≤ C ( 1 − ρ) β−m, 0 0 and Dn a ∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣ 1 1 · · · 1 a a 1 · · · a n − 1 a a 1 a 1 a 2 · · · a n − 1 a n · · · n−2 ∏

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Stevo Stević

Papers

On the Recursive Sequence xn+1=A+xnp/xn−1p

On the max-type equation xn+1=max{Axn,xn−2}

Note on the bilinear difference equation with a delay

Weighted Differentiation Composition Operators from the Mixed-Norm Space to the th Weigthed-Type Space on the Unit Disk

On an integral-type operator from logarithmic Bloch-type spaces to mixed-norm spaces on the unit ball