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

/pdf/aufgaben-und-lehrsatze-aus-der-analysis-mva9fm8nx4.pdf

Aufgaben und Lehrsätze aus der Analysis

We show that a holomorphic function in the unit polydisc is the image of a bounded holomorphic function by the weighted Bergman projection if and only if some weighted derivations of the function are bounded. This result improves Theorem 4 in the paper K. Zhu, Weighted Bergman projections on the polydisc

A note on a theorem of Zhu on weighted Bergman projections on the polydisc

Abstract We present an example of a difference equation of arbitrary order, possessing the right-hand side function that is homogeneous to a certain degree and nonincreasing in each variable, which has a unique positive equilibrium, as well as solutions that do not converge to the equilibrium. The example shows that the main result in the paper: O. Moaaz, Dynamics of difference equation $x_{n+1}=f(x_{n-l}, x_{n-k})$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mi>x</mml:mi> <mml:mrow> <mml:mi>n</mml:mi> <mml:mo>+</mml:mo> <mml:mn>1</mml:mn> </mml:mrow> </mml:msub> <mml:mo>=</mml:mo> <mml:mi>f</mml:mi> <mml:mo>(</mml:mo> <mml:msub> <mml:mi>x</mml:mi> <mml:mrow> <mml:mi>n</mml:mi> <mml:mo>−</mml:mo> <mml:mi>l</mml:mi> </mml:mrow> </mml:msub> <mml:mo>,</mml:mo> <mml:msub> <mml:mi>x</mml:mi> <mml:mrow> <mml:mi>n</mml:mi> <mml:mo>−</mml:mo> <mml:mi>k</mml:mi> </mml:mrow> </mml:msub> <mml:mo>)</mml:mo> </mml:math> (Adv. Differ. Equ. 2018:447, 2018), is incorrect. 

/pdf/note-on-difference-equations-with-the-right-hand-side-1f7w6zz1.pdf

Note on difference equations with the right-hand side function nonincreasing in each variable

We present a closed-form formula for the general solution to the difference equation $$x_{n+k}-q_{n}x_{n}=f_{n},\quad n\in \mathbb {N}_{0}, $$
 where $k\in \mathbb {N}$
 , $(q_{n})_{n\in \mathbb {N}_{0}}$
 , $(f_{n})_{n\in \mathbb {N}_{0}}\subset \mathbb {C}$
 , in the case $q_{n}=q$
 , $n\in \mathbb {N}_{0}$
 , $q\in \mathbb {C}\setminus\{0\}$
 . Using the formula, we show the existence of a unique bounded solution to the equation when $|q|>1$
 and $\sup_{n\in \mathbb {N}_{0}}|f_{n}|<\infty$
 by finding a solution in closed form. By using the formula for the bounded solution we introduce an operator that, together with the contraction mapping principle, helps in showing the existence of a unique bounded solution to the equation in the case where the sequence $(q_{n})_{n\in \mathbb {N}_{0}}$
 is real and nonconstant, which shows that, in this case, there is an elegant method of proving the result in a unified way. We also obtain some interesting formulas.

/pdf/general-solution-to-a-higher-order-linear-difference-3s5hy5niym.pdf

General solution to a higher-order linear difference equation and existence of bounded solutions

Generalized Hilbert operator and Fejér-Riesz type inequalities on the polydisc

This paper studies the boundedness character and the global attractivity of positive solutions of the difference equation xn

/pdf/on-the-difference-equation-xn-1-j-0kajfj-xn-j-2lka7e51r5.pdf

Stevo Stević

Papers

A note on a theorem of Zhu on weighted Bergman projections on the polydisc

Note on difference equations with the right-hand side function nonincreasing in each variable

General solution to a higher-order linear difference equation and existence of bounded solutions

Generalized Hilbert operator and Fejér-Riesz type inequalities on the polydisc

On the Difference Equation xn+1=∑j=0kajfj(xn−j)