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

Weighted iterated radial composition operators from weighted Bergman–Orlicz spaces to weighted‐type spaces on the unit ball

We consider a nonoscillatory second-order linear dynamic equation on a time scale together with a linear perturbation of this equation and give conditions on the perturbation that guarantee that the perturbed equation is also nonoscillatory and has solutions that behave asymptotically like a recessive and dominant solutions of the unperturbed equation. As the theory of time scales unifies continuous and discrete analysis, our results contain as special cases results for corresponding differential and difference equations by William F. Trench.

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Trench's Perturbation Theorem for Dynamic Equations

The boundedness, global attractivity, oscillatory and asymptotic periodicity of the nonnegative solutions of the difference equation
 
$$x_{n + 1} = \frac{{ax_{n - 2m + 1}^p }}{{b + cx_{n - 2k}^{p - 1} }}, n = 0, 1,...$$

 wherem, k ∈ N, 2k > 2m−1,a, b, c are nonnegative real numbers andp < 1, are investigated.

On the recursive sequence$x_{n + 1} = \frac{{ax_{n - 2m + 1}^p }}{{b + cx_{n - 2k}^{p - 1} }}$

The following recurrent relation/partial difference equation wn,k = awn−1,k−1 + bwn−1,k, where k, n, a, b ∈ N, appears in a problem in combinatorics. Here we show that an extension of the recurrent relation is solvable on, the, so called, combinatorial domain C = { (n, k) ∈N0 : 0 ≤ k ≤ n } \\ {(0, 0)}, when its coefficients and the boundary values wj,0, wj,j, j ∈ N, are complex numbers, by presenting a representation of the general solution to the recurrent relation on the domain in terms of the boundary values. As a special case we obtain a solution to the problem in combinatorics in an elegant way. From the general solution along with an application of the linear first-order difference equation is also obtained the solution of the recurrent relation in the case wj,j = c ∈ C, j ∈N.

On an extension of a recurrent relation from combinatorics

This paper studies the boundedness, global asymptotic stability, and periodicity of positive solutions of the equation xn=f(xn−2)/g(xn−1), n∈ℕ0, where f,g∈C[(0,∞),(0,∞)]. It is shown that if f and g are nondecreasing, then for every solution of the equation the subsequences {x2n} and {x2n−1} are eventually monotone. For the case when f(x)=α

https://downloads.hindawi.com/journals/aaa/2008/653243.pdf

Stevo Stević

Papers

Weighted iterated radial composition operators from weighted Bergman–Orlicz spaces to weighted‐type spaces on the unit ball

Trench's Perturbation Theorem for Dynamic Equations

On the recursive sequence\(x_{n + 1} = \frac{{ax_{n - 2m + 1}^p }}{{b + cx_{n - 2k}^{p - 1} }}\)

On an extension of a recurrent relation from combinatorics

The Behavior of Positive Solutions of a Nonlinear Second-Order Difference Equation