of Qualitative Theory of Differential Equations
of Qualitative Theory of Differential Equations
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DOI：
10.1515/9781400875955
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发表时间：
2016
期刊：
影响因子：
--
通讯作者：
T. Burton
中科院分区：
文献类型：
--
作者：
T. Burton
在本文中，我们延长了1998年开始的工作，作者和柯克的积分方程，我们结合Krasnoselskii的不动点定理的总和两个运营商与Schaefer的不动点定理。谢弗定理消除了克拉斯诺塞尔斯基定理中的一个困难假设，但需要解的先验界。在这里，我们简化了工作，通过一个变换，往往减少了一个平凡的先验约束。我们的工作集中在一个积分方程上，目标是证明在[0，∞）上存在唯一的连续正解.除了转换之外，还有两种技术我们要强调。介绍了一种技术产生的解决方案，使我们能够处理威胁非唯一性的问题的下限。这项技术为一个经典问题提供了一个解决方案，它似乎是全新的。我们表明，当方程定义的压缩和Lipschitz算子的总和，那么我们首先得到存在的任意区间[0，E]，然后引入一种技术，我们称之为渐进的压缩，这使我们能够证明唯一性，然后parlay解决方案[0，∞）。该技术非常适合积分方程。
In this paper we extend the work begun in 1998 by the author and Kirk for integral equations in which we combined Krasnoselskii’s fixed point theorem on the sum of two operators with Schaefer’s fixed point theorem. Schaefer’s theorem eliminates a difficult hypothesis in Krasnoselskii’s theorem, but requires an a priori bound on solutions. Here, we simplify the work by means of a transformation which often reduces the a priori bound to a triviality. Our work is focused on an integral equation in which the goal is to prove that there is a unique continuous positive solution on [0, ∞). In addition to the transformation, there are two techniques which we would emphasize. A technique is introduced yielding a lower bound on the solutions which enables us to deal with problems threatening non-uniqueness. The technique offers a solution to a classical problem and it seems entirely new. We show that when the equation defines the sum of a contraction and a Lipschitz operator, then we first get existence on arbitrary intervals [0, E] and then introduce a technique which we call a progressive contraction which allows us to prove uniqueness and then parlay the solution to [0, ∞). The technique is well suited to integral equations.