控制不动点原理及其在非线性积分方程中的应用
Majorization fixed point principle and application to nonlinear integral equations
摘要：
逐次逼近方法使我们能够解决有关问题的存在性和唯一性的不动点的广泛类的运营商。这一领域的经典结果，如Banach - Caccioppoli原理及其某些修正和推广，适用于满足小系数利普希茨条件的算子，即具有压缩性质的算子。然而，逐次逼近方法对于不是压缩的其他类别的算子也很有效。特别是，著名的Kantorovich不动点原则的可微算子处理的运营商，在一般情况下，不压缩;此外，这一原则涵盖了一些情况下，当Banach - Caccioppoli原则是不适用的。本文对Kantorovich不动点原理作了一些修改，使其涵盖了不可微算子。我们描述的精确（不可改进）的估计的内部和外部半径的域存在的一个唯一的不动点的运营商正在考虑中，此外，我们提出了新的先验和后验误差估计逐次逼近相应的不动点。还给出了新的不动点原理在不同类型的非线性积分算子中的一些应用。
英文摘要：
The successive approximations method allows us to solve problems concerning existence and uniqueness of fixed points of wide classes of operators. The classical result in this field, such as Banach -- Caccioppoli principle together with some its modification and generalisations, is applicable to operators satisfying Lipschitz condition with a small coefficient or, in other words, to operators with the compression property. However, the successive approximations method works well for other classes of operators that are not compressions. In particular, the well known Kantorovich fixed point principle for differentiable operators deals with operators that, in general, are not compression; moreover, this principle covers some cases when Banach -- Caccioppoli principle is nonapplicable. In the article it is presented some modification of Kantorovich fixed point principle that covers nondifferentiable operators. We describe the exact (unimprovable) estimates of the internal and external radius of the domain of existence of a unique fixed point of the operator under consideration and, in addition, we present new apriori and aposteriori error estimates for successive approximations to the corresponding fixed point. Some applications of the new fixed point principle to nonlinear integral operators of different types are given as well.
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