The method of successive approximation for functional equations
The method of successive approximation for functional equations
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DOI：
10.1007/bf02547750
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发表时间：
1939-12
期刊：
影响因子：
3.7
通讯作者：
L. Kantorovitch
中科院分区：
文献类型：
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作者：
L. Kantorovitch
在泛函分析中，抽象线性空间被考虑为其元素可能具有各种性质的数学对象：数、数列、函数等。因此，为这种抽象空间建立的定理通常可以应用于非常不同的数学分析分支。因此，泛函方程的一般理论，即未知量是线性空间的元素的这种方程的一般理论，包括分析中考虑的微分方程组、积分方程组和其他一些方程的理论，以及有限和无限代数方程组的理论。证明解的存在性和研究这些解的最重要的方法之一是逐次逼近法。在这里，我们将给出线性和非线性泛函方程在非常广泛的一类空间中的这种方法的几何理论。用半序空间的元素赋范的空间。这类空间包括区间Banach空间和半序空间。这种方法的理论将以优势原则为基础。我们还将给出一般理论在代数方程组、微分方程组和积分方程组中的一些应用。
In the functional analysis abstract linear spaces are considered which may have for their elements mathematical objects of a various nature: numbers, sequences of numbers, functions etc. Therefore theorems established for such abstract spaces usually can be applied to very different branches of mathematical analysis. Thus the general theory of functional equations, ie of-such equations where the unknown quantities are elements of a linear space, comprises the theories of differential, integral and some other equations considered in analysis as well as the theory of finite and infinite systems of algebraic equations. One of the most important methods for establishing the existence of solutions and for the investigations of these solutions is the method of successive approximations. We shall give here the geaera] theory of this method for linear and non-linear functional equations in a very wide class of spaces viz. the spaces normed with the elements of a semi-ordered space. This class comprises inter,~ lia Banach's spaces and semi-ordered spaces. The theory of this method will be based on the principle of majorants. We shall give also some applications of the general theory to the systems of alo'ebraic equations and to the differential and integral equations.