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

The following discrete initial value problem $$ x_{n+1}=x_{n}\bigl(x_{n-1}^{2}-2 \bigr)-x_{1},\quad n\in {\mathbb{N}}, $$
 $x_{0}=2$
 and $x_{1}=5/2$
 , appeared at an international competition It is known that the problem can be solved in closed form Here we discuss the solvability of a more general initial value problem which includes the former one We show that, in a sense, there are not so many solvable discrete initial value problems related to this one, showing its specificity, which is a bit surprising result

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Note on a discrete initial value problem from a competition

We calculate the norms of several concrete operators, mostly of some integral-type ones between weighted-type spaces of continuous functions on several domains. We also calculate the norm of an integral-type operator on some subspaces of the weighted Lebesgue spaces.

Norms of some operators between weighted-type spaces and weighted Lebesgue spaces

Abstract By using a comparison method and some difference inequalities we show that the following higher order difference equation $$ x_{n+k}=\frac{1}{f(x_{n+k-1},\ldots ,x_{n})},\quad n\in{\mathbb{N}},$$ <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:mi>k</mml:mi> </mml:mrow> </mml:msub> <mml:mo>=</mml:mo> <mml:mfrac> <mml:mn>1</mml:mn> <mml:mrow> <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>k</mml:mi> <mml:mo>−</mml:mo> <mml:mn>1</mml:mn> </mml:mrow> </mml:msub> <mml:mo>,</mml:mo> <mml:mo>…</mml:mo> <mml:mo>,</mml:mo> <mml:msub> <mml:mi>x</mml:mi> <mml:mi>n</mml:mi> </mml:msub> <mml:mo>)</mml:mo> </mml:mrow> </mml:mfrac> <mml:mo>,</mml:mo> <mml:mspace /> <mml:mi>n</mml:mi> <mml:mo>∈</mml:mo> <mml:mi>N</mml:mi> <mml:mo>,</mml:mo> </mml:math> where $k\in{\mathbb{N}}$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>k</mml:mi> <mml:mo>∈</mml:mo> <mml:mi>N</mml:mi> </mml:math> , $f:[0,+\infty )^{k}\to [0,+\infty )$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>f</mml:mi> <mml:mo>:</mml:mo> <mml:mo>[</mml:mo> <mml:mn>0</mml:mn> <mml:mo>,</mml:mo> <mml:mo>+</mml:mo> <mml:mi>∞</mml:mi> <mml:msup> <mml:mo>)</mml:mo> <mml:mi>k</mml:mi> </mml:msup> <mml:mo>→</mml:mo> <mml:mo>[</mml:mo> <mml:mn>0</mml:mn> <mml:mo>,</mml:mo> <mml:mo>+</mml:mo> <mml:mi>∞</mml:mi> <mml:mo>)</mml:mo> </mml:math> is a homogeneous function of order strictly bigger than one, which is nondecreasing in each variable and satisfies some additional conditions, has unbounded solutions, presenting a large class of such equations. The class can be used as a useful counterexample in dealing with the boundedness character of solutions to some difference equations. Some analyses related to such equations and a global convergence result are also given. 

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Higher order difference equations with homogeneous governing functions nonincreasing in each variable with unbounded solutions

More on a rational second-order difference equation

We give detailed theoretical explanations for getting the closed-form formulas and representations for the general solutions to four special cases of a class of nonlinear difference equations of second order considered in the literature, present an extension of the class of difference equations which is solvable in closed form, analyze some results on the long-term behavior of the solutions to the class of equations, and give some results on convergence.

Stevo Stević

Papers

Note on a discrete initial value problem from a competition

Norms of some operators between weighted-type spaces and weighted Lebesgue spaces

Higher order difference equations with homogeneous governing functions nonincreasing in each variable with unbounded solutions

More on a rational second-order difference equation

Solvability and representations of the general solutions to some nonlinear difference equations of second order