关于算子系统压缩-扩展不动点定理的统一方法及其应用
A unified approach to compression-expansion fixed point theorems for operators systems and applications
AI总结：
本文提出了一种统一方法，用于压缩-扩展不动点定理的算子系统，应用于积分方程组和非线性系统。
摘要：
在本文中，我们提出了若干关于算子系统的不动点定理，沿着Krasnosel'skii定理在锥中的思路。锥压缩和锥扩展类型的条件以分量方式施加。与文献中的相关结果不同，算子允许在由非凸集界定的锥形区域的笛卡尔积中定义。我们的方法基于不动点指数，确保存在一个共存不动点——即具有非平凡分量的不动点。作为第一个应用，我们建立了在严格星形集之间由泛函定义的积分方程组的若干局部化结果。这些结果不能仅通过之前研究中处理环形区域的算子得出。第二个应用涉及非线性系统，其中包括Φ-拉普拉斯算子。
英文摘要：
In this paper, we present some fixed point theorems for operator systems in the line of Krasnosel'skii's theorem in cones. The cone-compression and cone-expansion type conditions are imposed in a component-wise manner. Unlike related results in the literature, the operators are allowed to be defined in the Cartesian product of conical regions delimited by nonconvex sets. Our approach, based on the fixed point index, ensures the existence of a coexistence fixed point--that is, one with nontrivial components. As a first application, we establish several localization results for systems of integral equations between strictly star-shaped sets defined by functionals. These results cannot be derived solely from previous studies dealing with operators in annular regions. A second application concerns nonlinear systems involving the Φ-Laplacian.
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