Abstract
In this paper, we introduce the notion of controlled rectangular metric spaces as a generalization of rectangular metric spaces and rectangularb-metric spaces. Further, we establish some related fixed point results. Our main results extend many existing ones in the literature. The obtained results are also illustrated with the help of an example. In the last section, we apply our results to a common real-life problem in a general form by getting a solution for the Fredholm integral equation in the setting of controlled rectangular metric spaces.
Highlights
The fixed point theory is a growing and exciting field of mathematics with a variety of variant applications in mathematical sciences, proposing newer applications in discrete dynamics and super fractals
Fixed point techniques have been applied in such diverse fields; see [4, 5]
There are particular real-life problems, whose statements are fairly easy to understand, which can be argued using some versions of fixed point theorems; see [6, 7]
Summary
The fixed point theory is a growing and exciting field of mathematics with a variety of variant applications in mathematical sciences, proposing newer applications in discrete dynamics and super fractals. The notion of extended b-metric spaces was introduced by Kamran et al [9] as a generalization of metric spaces and b-metric spaces [10, 11]. Our goal is to introduce the notion of controlled rectangular metric spaces, which is different from controlled rectangular b-metric spaces, and generalize rectangular metric spaces as well as rectangular b -metric spaces. As an application, we give an existence theorem for the Fredholm integral equation in the setting of controlled rectangular metric spaces. The topological structure of rectangular metric spaces is not compatible with the topology of classic metric spaces; see Example 7 in the paper of Suzuki [26]. In the same direction, extended rectangular b-metric spaces cannot be Hausdorff
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