Abstract

We consider a nonlinear integro-differential equation with zero operator in the differential part whose integral operator contains several rapidly varying kernels. This paper continues the research carried out earlier for equations with only one rapidly varying kernel. We prove that the conditions for the solvability of the corresponding iteration problems, as in the linear case, are not differential (as in problems with nonzero operator in the differential part) but rather integro-differential equations, and the structure of these equations is substantially influenced by the nonlinearity. In the nonlinear case, so-called resonances can arise, significantly complicating the development of the corresponding algorithm of the regularization method. The paper deals with the nonresonant case.

中文翻译：

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微分部分为零算子且有几个快速变化核的非线性积分-微分方程的正则化渐近解

摘要

我们考虑在微分部分具有零算子的非线性积分微分方程，其积分算子包含几个快速变化的内核。本文继续之前对只有一个快速变化内核的方程进行的研究。我们证明了对应迭代问题的可解性条件，如在线性情况下，不是微分的（如在微分部分具有非零算子的问题），而是积分微分方程，并且这些方程的结构实质上是受非线性影响。在非线性情况下，可能会出现所谓的共振，这使正则化方法的相应算法的开发变得非常复杂。该论文涉及非共振情况。