In this paper, we introduce the concept of fuzzy double controlled metric space that can be regarded as the generalization of fuzzy b-metric space, extended fuzzy b-metric space and controlled fuzzy metric space. We use two non-comparable functions α and β in the triangular inequality as: M q ( x , z , t α ( x , y ) + s β ( y , z ) ) ≥ M q ( x , y , t ) ∗ M q ( y , z , s ) . We prove Banach contraction principle in fuzzy double controlled metric space and generalize the Banach contraction principle in aforementioned spaces. We give some examples to support our main results. An application to existence and uniqueness of solution for an integral equation is also presented in this work.

In this study, we introduce fuzzy triple controlled metric space that generalizes certain fuzzy metric spaces, like fuzzy rectangular metric space, fuzzy rectangular b -metric space, fuzzy b -metric space, and extended fuzzy b -metric space. We use f , g , h , three noncomparable functions as follows: m q μ , η , t + s + w ≥ m q μ , ν , t / f μ , ν ∗ m q ν , ξ , s / g ν , ξ ∗ m q ξ , η , w / h ξ , η . We prove Banach fixed point theorem in the settings of fuzzy triple controlled metric space that generalizes Banach fixed point theorem for aforementioned spaces. An example is presented to support our main results. We also apply our technique to the uniqueness for the solution of an integral equation.

In this article, we introduce the concept of a fuzzy triple controlled metric like space in the sense that the self distance may not be equal to one. We have used three functions in our space that generalize fuzzy controlled rectangular, extended fuzzy rectangular, fuzzy $b–$rectangular and fuzzy rectangular metric like spaces. Various examples are given to justify our definitions and results. As for the topological aspect, we prove a fuzzy triple controlled metric like space is not Hausdorff. We also apply our main result to solve the uniqueness of the solution of a fractional differential equation.

As it is not always true that the distance between the points in fuzzy rectangular metric spaces is one, so we introduce the notions of rectangular and b-rectangular metric-like spaces in fuzzy set theory that generalize many existing results, which can be regarded as the main advantage of this paper. In these spaces, the symmetry property is preserved, but the self distance may not be equal to one. We discuss topological properties and demonstrate that neither of these spaces is Hausdorff. Using $\alpha -\psi$-contraction and Geraghty contractions, respectively, in our main results, we establish fixed point results in these spaces. We present examples that justify our definitions and results. We use our main results to demonstrate that the solution of a nonlinear fractional differential equation for HIV is unique.

The paper is well-written, and the results are interesting. The authors should address the following comment. 1. The authors should add an example to support Theorem 3.3. 2. The authors should add an example to support Theorem 3.4. 3. There should...