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 show that the difference equation where , has a positive solution converging to zero, by finding a finite asymptotic expansion of the solution. We also show that if and are arbitrarily given positive numbers, then there exists a solution of the equation such that the subsequences , have partial sums of exponential power series as finite asymptotic expansions. Finally, we sketch the case when q of the limits ( ) are vanishing, where we determine only heuristically the next term of the asymptotic expansion.

On the asymptotics of the difference equation y n (1 + y n − 1 … y n − k + 1) = y n − k

Products of composition and differentiation operators from Zygmund spaces to Bloch spaces and Bers spaces

The boundedness, global attractivity, oscillatory and asymptotic periodicity of the positive solutions of the difference equation of the form
 
$$x_{n + 1} = \alpha + \frac{{x_{n - 1}^p }}{{x_n^p }}, n = 0,1,...$$

 is investigated, where all the coefficients are nonnegative real numbers.

On the recursive sequence $$x_{n + 1} = \alpha + \frac{{x_{n - 1}^p }}{{x_n^p }}$$

Let $g:B\to \mathbb C^1$ be a holomorphic map of the unit ball $B$. We study the integral operators $$ T_gf(z)=\int_0^1f(tz)\Re g(tz)\frac{dt}{t}; \ \ L_gf(z)= \int_0^1 \Re f(tz) g(tz)\frac{dt}{t},\qquad z\in B. $$ The boundedness and compactness of the operators $T_g$ and $L_g$ on the Hardy space $H^2$ in the unit ball are discussed in this paper.

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Riemann-Stieltjes operators on Hardy spaces in the unit ball of $\mathbb C^n$

This paper studies the boundedness and compactness of the following integral-type operator, recently introduced by this author, P φ g f ( z ) = ∫ 0 1 f ( φ ( t z ) ) g ( t z ) d t t , z ∈ B , where φ is a holomorphic self-map of the unit ball B in C n and g is a holomorphic function on B such that g ( 0 ) = 0 , from logarithmic Bloch-type and mixed-norm spaces to Bloch-type spaces.

Stevo Stević

Papers

On the asymptotics of the difference equation y n (1 + y n − 1 … y n − k + 1) = y n − k

Products of composition and differentiation operators from Zygmund spaces to Bloch spaces and Bers spaces

On the recursive sequence $$x_{n + 1} = \alpha + \frac{{x_{n - 1}^p }}{{x_n^p }}$$

Riemann-Stieltjes operators on Hardy spaces in the unit ball of $\mathbb C^n$

On an integral-type operator from logarithmic Bloch-type and mixed-norm spaces to Bloch-type spaces