A New Extension to the Intuitionistic Fuzzy Metric ‐ like Spaces

: In this manuscript, we introduce the concept of intuitionistic fuzzy controlled metric ‐ like spaces via continuous t ‐ norms and continuous t ‐ conorms. This new metric space is an extension to intuitionistic fuzzy controlled metric ‐ like spaces, controlled metric ‐ like spaces and controlled fuzzy metric spaces, and intuitionistic fuzzy metric spaces. We prove some fixed ‐ point theorems and we present non ‐ trivial examples to illustrate our results. We used different techniques based on the properties of the considered spaces notably the symmetry of the metric. Moreover, we present an application to non ‐ linear fractional differential equations.


Introduction
In todayʹs multifaceted environment, uncertainty and fuzziness are widespread in many applications. Zadeh [1] pioneered the concept of fuzzy sets (FSs) to capture the ambiguity and fuzziness of information. Since its origin, many extensions of FSs have been proposed to better represent sophisticated information, including intuitionistic fuzzy sets (IFSs), picture FSs, q-rung orthopair FSs, and neutrosophic sets.
generalized the concept of controlled type metric spaces and introduced the concept of Controlled fuzzy metric spaces (CFMS). In this sequel, Shukla and Abbas [15] generalized the concept of metric-like spaces and introduced fuzzy metric-like spaces (FMLSs). Javed et al. [16] proposed fuzzy b-metric-like spaces. Shukla et al. [17] proposed an amazing notion of 1-M complete FMSs and proved various theorems.
The approach of intuitionistic fuzzy metric spaces (IFMSs) via continuous t-norms and continuous t-conorms was presented by Park in [18]. Rafi and Noorani [19] proved several fixed-point results in the context of IFMSs. Sintunavarat and Kumam [20] proved fixed point theorems for a generalized intuitionistic fuzzy contraction in IFMSs. Later, Konwar [21] presented intuitionistic fuzzy b-metric space (IFBMS). Alaca et al. [22] and Mohamad [23] proved several fixed-point results. Saadati and Park [24] did amazing work on intuitionistic fuzzy topological spaces. In addition, Sezen in [14], introduced the concept of controlled fuzzy metric spaces.
The goal of this manuscript is to introduce intuitionistic fuzzy controlled metric-like spaces (IFCMLSs) by using the approach in [5], also to extend various fixed point (FP) results for contraction mappings, which is an improvement of the present literatureʹs methodology using different techniques based on the properties of contractions and the considered metric such as the triangle inequality and the symmetry. In closing, and inspired by work carried out in [25][26][27][28][29], we present an application of our results to fractional differential equations.

Preliminaries
Now, we start this section by listing various helpful definitions for readers and 0, 1 be used in this study.

Main Results
In this section, we present the concept of an IFCMLS and prove several FP results.
In addition, Then K, ℵ , ℜ , * ,∘ is an IFCMLS with CTN a * b ab and CTCN a ∘ b max a, b .

Application to Nonlinear Fractional Differential Equations
In present section, we aim to apply Theorem 3 to obtain the existence and uniqueness of a solution to a nonlinear fractional differential equation (NFDE), with the boundary conditions, where 1 2 is a number, is the Caputo fractional derivative and ∶ 0,1 0, ∞ → 0, ∞ is a continuous function. Let K 0,1 , ℝ denote the space of all continuous functions defined on 0, 1 equipped with the CTN * . and CTCN ∘ max , for all , 0,1 and define an IFCMLS on K as follows: Observe that Þ ∈ K solves (4A) whenever Þ ∈ K solves the below integral equation: Theorem 3. The integral operator : K → K is given by, for some , 0. Observe that the conditions of the Theorem1 are fulfilled. Resultantly, has a unique fixed point; accordingly, the specified NFDE has a unique solution. □

Conclusions
We present intuitionistic fuzzy controlled metric-like spaces in this paper and established several new types of fixed-point theorems in this new context. Moreover, we provided non-trivial examples and an application to non-linear fractional differential equations is given to demonstrate the viability of the proposed method. Our findings and concepts expand and generalize the existing literature. The structures of intuitionistic fuzzy double controlled metric-like spaces, intuitionistic pentagonal fuzzy controlled metriclike spaces, and neutrosophic controlled metric-like spaces etc. can all be extended using this study.