In this article, we first establish a series of user-friendly versions of fixed point theorems in cones for sum of two operators with one being either contractive or expansive and the other being compact, one of which covers the classical Krasnoselskii’s fixed point theorem concerning cone expansion and compression of norm type. Along the way, we also offer some sufficient conditions which slightly relax the compactness requirement on the one summand operator. Second, as applications to some of our main results, we consider the eigenvalue problems of Krasnoselskii type in critical case and the existence of one positive solution to one parameter operator equations. Finally, to illustrate the usefulness and the applicability of our fixed point results, we study the existence of one nontrivial positive solution to certain integral equations of Hammerstein type and of perturbed Volterra type.

中文翻译：

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两个算子之和的锥展开和锥压缩不动点定理及其应用

在本文中，我们首先在锥中建立一系列易于使用的定点定理版本，以求两个算子之和，一个算子是收缩的或膨胀的，另一个算子是紧的，其中一个涉及经典的Krasnoselskii关于锥的不动点定理规范类型的扩展和压缩。在此过程中，我们还提供了一些充分的条件，这些条件可以稍微放松对一个求和运算符的紧凑性要求。第二，作为对我们一些主要结果的应用，我们考虑了临界情况下的Krasnoselskii型特征值问题和一个参数算子方程的一个正解的存在。最后，为了说明定点结果的有用性和适用性，