Mathematical Sciences: Asymptotic Theory of Difference Equations

埃莱迪 摘要 庞加莱-佩龙定理标志着渐近 标量非自治差分方程理论 然而，在这方面， 由于两个原因，该定理的使用受到限制。 一是 要求极限的所有特征根 方程具有不同的模量。 第二个原因是 定理没有提供显式的渐近表示 差分方程的解 莱文森定理或者更确切地说 它的离散类似物解决了第二个缺陷， 差分解理想渐近表示 方程 然而，Levinson定理提出了严格的限制 在差分方程的系数问题，因为 它要求它们是可求和的。 后一种情况使 该定理的适用性相当有限，特别是 在正交多项式、组合数学、特殊函数和 连续分数该研究旨在 解决了上述两个重要定理的不足。 它将试图用一个 通过二分法理论的统一方法。 差分方程是求解微分方程的基本数学工具之一。 计算几乎没有一个计算任务不 依赖于递归技术。 广泛 差分方程的使用可以归因于其内在的 建设性的质量，以及他们所拥有的巨大便利， 适合机械化。 另一方面，像大多数递归 过程中，差分方程易受误差增长的影响。 如果条件不利， 可能或将是灾难性的。 因此，有必要调查 差分解的渐近性和定性性 方程 这一点至关重要，因为差异 方程在应用数学的许多分支中都有突出的表现， 经济学，人口动力学，混沌理论，数学物理， 渐近在计算机科学中甚至更重要， 它用于评估算法以及它们的速度和准确性。

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.