This article introduces a measure of noncompactness in the generalized Morrey space. We study the applications of our new definition in investigating the conditions for the existence of solutions for systems of nonlinear integral equations. We can extend many useful theorems in \(L^{p}(\mathbb {R}^{N})\) for functions belonging to the generalized Morrey spaces. Compared to the \(L^{p}(\mathbb {R}^{N})\) spaces, the advantage of studying in the Morrey spaces is that we can research no compact support functions in our problems. Finally, significant examples are presented to show the efficiency of the main results.

中文翻译：

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一种新的非紧性度量在广义 Morrey 空间中非线性和分数积分方程组可解性中的应用

本文介绍了广义莫雷空间中的非紧性度量。我们研究了新定义在研究非线性积分方程组解的存在条件中的应用。对于属于广义莫雷空间的函数，我们可以在\(L^{p}(\mathbb {R}^{N})\)中扩展许多有用的定理。与\(L^{p}(\mathbb {R}^{N})\)空间相比，在 Morrey 空间中研究的优点是我们可以在问题中研究无紧支持函数。最后，给出了重要的例子来展示主要结果的有效性。