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 present some closed-form formulas for the general solution to the family of difference equations

   \begin{document}$ x_{n+1} = \Phi^{-1}\left(\Phi(x_{n-1})\frac{{\alpha} \Phi(x_{n-2})+{\beta} \Phi(x_{n-4})}{{\gamma} \Phi(x_{n-2})+{\delta} \Phi(x_{n-4})}\right), $\end{document} 

for $ n\in{\mathbb N}_0 $ where the initial values $ x_{-j} $, $ j = \overline{0, 4} $ and the parameters $ {\alpha}, {\beta}, {\gamma} $ and $ {\delta} $ are real numbers satisfying the conditions $ {\alpha}^2+{\beta}^2\ne 0, $ $ {\gamma}^2+{\delta}^2\ne 0 $ and $ \Phi $ is a function which is a homeomorphism of the real line such that $ \Phi(0) = 0, $ generalizing in a natural way some closed-form formulas to the general solutions to some very special cases of the family of difference equations which have been presented recently in the literature. Besides this, we consider in detail some of the recently formulated statements in the literature on the local and global stability of the equilibria as well as on the boundedness character of positive solutions to the special cases of the difference equation and give some comments and results related to the statements.

On a family of nonlinear difference equations of the fifth order solvable in closed form

Abstract We present generalizations of some results on the integer parts of the reciprocal remainders of the zeta function $\zeta (s)$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>ζ</mml:mi> <mml:mo>(</mml:mo> <mml:mi>s</mml:mi> <mml:mo>)</mml:mo> </mml:math> with $s=2$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>s</mml:mi> <mml:mo>=</mml:mo> <mml:mn>2</mml:mn> </mml:math> and $s=3$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>s</mml:mi> <mml:mo>=</mml:mo> <mml:mn>3</mml:mn> </mml:math> , and a very short and elegant proof of a recent result on the integer parts of the reciprocal remainders of the series $\zeta (3)$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>ζ</mml:mi> <mml:mo>(</mml:mo> <mml:mn>3</mml:mn> <mml:mo>)</mml:mo> </mml:math> . We also give some historical and theoretical remarks to problems of this type, conduct some analyses, and make some connections with the theory of linear difference equations with constant coefficients. 

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On integer parts of the reciprocal remainders of some sums

Let $\phi$ be an analytic self-map and $u$ be a fixed analytic function on the open unit disk $D$ in the complex plane $\CC.$ The weighted composition operator is defined\break by \begin{equation*} uC_\phi f =u \cdot (f\circ \phi), f \in H(D). \end{equation*} Weighted composition operators from Bergman-type spaces into Bloch spaces and little Bloch spaces are characterized by function theoretic properties of their inducing maps.

Weighted composition operators from Bergman-type spaces into Bloch spaces

Asymptotic behavior of solutions of a system of q-functional-difference equations

/pdf/on-a-new-product-type-operator-on-the-unit-ball-2ve21s4u.pdf

Stevo Stević

Papers

On a family of nonlinear difference equations of the fifth order solvable in closed form

On integer parts of the reciprocal remainders of some sums

Weighted composition operators from Bergman-type spaces into Bloch spaces

Asymptotic behavior of solutions of a system of q-functional-difference equations

On a new product-type operator on the unit ball