Mathematical Sciences: Asymptotic Theory of Difference Equations

Elaydi抽象Poincar-Perron定理标志着标量非自治差分方程渐近理论的开始。然而，由于两个原因，这个定理的使用有限。第一是要求极限方程的所有特征根都有不同的模数。第二个原因是该定理没有给出差分方程解的显式渐近表示。莱文森定理，或者更确切地说，它的离散类比，通过提供差分方程解的理想的渐近表示来解决第二个缺陷。然而，莱文森定理对所讨论的差分方程的系数施加了严格的限制，因为它要求它们是可求和的。后一条件使得该定理的适用性相当有限，特别是在正交多项式、组合学、特殊函数和连分式中。所提出的研究旨在解决上述两个重要定理中的不足。它将试图通过二分法理论，使用统一的方法来完善和统一渐近理论。差分方程组是计算的基本数学工具之一。几乎没有一项计算任务不是在某个时刻不依赖递归技术的。差分方程组的广泛使用可以归因于其内在的构造性，以及它们非常容易被机械化。另一方面，像大多数递归过程一样，差分方程容易出现误差增长。如果条件不利，由此产生的误差传播可能或将是灾难性的。因此，需要研究差分方程解的渐近和定性行为。这一点至关重要，因为差分方程在应用数学、经济学、人口动力学、混沌理论、数学物理等许多分支中都有突出的出现。渐近性在计算机科学中更加重要，在计算机科学中，它被用来对算法进行评级，以及它们的速度和精度。

Elaydi Abstract Poincar -Perron's Theorem marks the beginning of the asymptotic theory of scalar nonautonomous difference equations. However, this theorem is of limited use due to two reasons. The first is the requirement that all characteristic roots of the limiting equation have distinct moduli. The second reason is that the theorem does not provide explicit asymptotic representations of solutions of difference equations. Levinson's Theorem or rather its discrete analogue addressed the second deficiency by providing the desirable asymptotic representation of solutions of difference equations. However, Levinson's Theorem puts a severe restriction on the coefficients of the difference equation in question since it requires that they are summable. The latter condition makes the applicability of the the theorem rather limited, particulaply in orthogonal polynomials, combinatorics, special functions, and continued fractions. The proposed research is directed toward addressing the deficiencies in the above two important theorems. It will attempt to complete and unify asymptotic theory using a unified approach via the theory of dichotomy. Difference equations are one of the basic mathematical tools of computation. There is hardly a computational task which does not rely on recursive techniques, at one time or another. The widespread use of difference equations can be ascribed to their intrinsic constructive quality, and the great ease with which they are amenable to mechanization. On the other hand, like most recursive processes, difference equations are susceptible to error growth. If conditions are unfavorable, the resulting propagation of error may or will be disastrous. Hence the need to investigate the asymptotic and qualitative behavior of solutions of difference equations. This is of paramount importance since difference equations occur prominently in many branches of applied mathematics, Economics, population dynamics, Chaos Theory, mathematical physics, etc. Asymptotics is even more important in Computer Science where it is used to rate algorithms and how fast and accurate they are.