Collage Theorem provides a bound for the distance between an element of a given space and a fixed point of a self-map on that space, in terms of the distance between the point and its image. We give in this paper some results of Collage type for Reich mutual contractions in b-metric and strong b-metric spaces. We give upper and lower bounds for this distance, in terms of the constants of the inequality involved in the definition of the contractivity. Reich maps contain the classical Banach contractions as particular cases, as well as the maps of Kannan type, and the results obtained are very general. The middle part of the article is devoted to the invertibility of linear operators. In particular we provide criteria for invertibility of operators acting on quasi-normed spaces. Our aim is the extension of the Casazza-Christensen type conditions for the existence of inverse of a linear map defined on a quasi-Banach space, using different procedures. The results involve either a single map or two operators. The latter case provides a link between the properties of both mappings. The last part of the article is devoted to study the construction of fractal curves in Bochner spaces, initiated by the first author in a previous paper. The objective is the definition of fractal curves valued on Banach spaces and Banach algebras. We provide further results on the fractal convolution of operators, defined in the same reference, considering in this case the nonlinear operators. We prove that some properties of the initial maps are inherited by their convolutions, if some conditions on the elements of the associated iterated function system are satisfied. In the last section of the paper we use the invertibility criteria given before in order to obtain perturbed fractal spanning systems for quasi-normed Bochner spaces composed of Banach-valued integrable maps. These results can be applied to Lebesgue spaces of real valued functions as a particular case.

拼贴定理提供了给定空间的元素与该空间上自映射的固定点之间的距离的界限，以该点与其图像之间的距离来表示。我们在本文中给出了 b 度量和强 b 度量空间中 Reich 相互收缩的 Collage 类型的一些结果。我们根据收缩性定义中涉及的不等式常数给出了该距离的上限和下限。 Reich图包含了作为特例的经典Banach收缩，以及Kannan型图，所得到的结果非常一般。文章的中间部分专门讨论线性算子的可逆性。特别是，我们提供了作用于拟范数空间的算子的可逆性标准。我们的目标是使用不同的过程扩展 Casazza-Christensen 类型条件，以证明准巴纳赫空间上定义的线性映射的逆存在。结果涉及单个地图或两个操作员。后一种情况提供了两个映射的属性之间的链接。本文的最后一部分致力于研究博赫纳空间中分形曲线的构造，该研究由第一作者在上一篇论文中发起。目标是定义巴拿赫空间和巴拿赫代数上的分形曲线。我们提供了在同一参考文献中定义的算子分形卷积的进一步结果，在这种情况下考虑了非线性算子。我们证明，如果相关迭代函数系统的元素的某些条件得到满足，则初始映射的某些属性将由它们的卷积继承。在本文的最后一节中，我们使用之前给出的可逆性准则来获得由 Banach 值可积映射组成的拟范数 Bochner 空间的扰动分形跨越系统。作为特殊情况，这些结果可以应用于实值函数的勒贝格空间。