Abstract

We study a class of nonlinear integral equations with a noncompact Hammerstein– Nemytskii operator on the entire line. Some special cases of such equations have specific applications in various fields of natural science. The combination of a method for constructing invariant cone segments for the corresponding nonlinear monotone operator with methods of the theory of functions of a real variable allows one to prove a constructive theorem on the existence of bounded positive solutions of equations of the class under consideration. The asymptotic behavior of the solution at \( \pm \infty \) is studied as well. In particular, we prove that the solution constructed in the paper is an integrable function on the negative half-line and that the difference between the limit at \(+\infty \) and the solution is integrable on the positive half-line. In one special case, we show that our solution generates a one-parameter family of bounded positive solutions. At the end of the paper, we give specific applied examples of nonlinearities to illustrate the results.

中文翻译：

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一类具有Hammerstein-Nemytskii算子的非线性积分方程的正有界解

摘要

我们研究了一类非线性积分方程，在整条线上具有非紧的 Hammerstein-Nemytskii 算子。这种方程的一些特殊情况在自然科学的各个领域都有特定的应用。为相应的非线性单调算子构造不变锥段的方法与实变量的函数理论方法的组合允许证明关于所考虑的类的方程的有界正解的存在性的构造定理。还研究了解在\( \pm \infty \)处的渐近行为。特别地，我们证明了论文中构造的解是负半线上的可积函数，并且在\(+\infty \) 并且解在正半线上是可积的。在一个特殊情况下，我们展示了我们的解决方案生成了一个单参数的有界正解决方案族。在论文的最后，我们给出了非线性的具体应用示例来说明结果。