A non-constructive existence theory for certain operator equationsLu=Du,using the substitution u=B12ξ with B = L−1, is developed, where L is a linear operator (in a suitable Banach space) and D is a homogeneous nonlinear operator such that Dλu = λα D u for all λ ≥ 0 and some α∈R, α ≠ ~1. This theory is based on the positive-operator approach of Krasnosel’skii. The method has the advantage of being able to tackle the nonlinear right-hand side D in cases where conventional operator techniques fail. By placing the requirement that the operator B must have a positive square root, it is possible to avoid the usual regularity condition on either the mapping D or its Fréchet derivative. The technique can be applied in the case of elliptic PDE problems, and we show the existence of solitary waves for a generalization of Benjamin’s fluid dynamics problem.

中文翻译：

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使用正平方根算子解决非线性非局部问题

对于某些算子方程 Lu=Du 的非构造存在理论，使用替换 u=B12ξ 和 B = L−1，被开发，其中 L 是线性算子（在合适的 Banach 空间中），D 是齐次非线性算子，例如对于所有 λ ≥ 0 和一些 α∈R，α ≠ ~1，Dλu = λα D u。该理论基于 Krasnosel'skii 的正算子方法。该方法的优点是能够在传统算子技术失败的情况下处理非线性右侧 D。通过要求算子 B 必须具有正平方根，可以避免映射 D 或其 Fréchet 导数上的通常的正则性条件。该技术可以应用于椭圆 PDE 问题的情况，并且我们展示了孤立波的存在以推广 Benjamin 的流体动力学问题。