A Weak Tripled Contraction for Solving a Fuzzy Global Optimization Problem in Fuzzy Metric Spaces

: In the setting of fuzzy metric spaces (FMSs), a global optimization problem (GOP) obtaining the distance between two subsets of an FMS is solved by a tripled ﬁxed-point (FP) technique here. Also, fuzzy weak tripled contractions (WTCs) for that are given. This problem was known before in metric space (MS) as a proximity point problem (PPP). The result is correct for each continuous τ − norms related to the FMS. Furthermore, a non-trivial example to illustrate the main theorem is discussed.


Introduction and Preliminaries
During the last decade, the PPP has been discussed as it means determining the distance between two subsets of the MS. In optimization, this problem is mainly considered and it is treated by FP analysis by viewing the problem as that of finding an optimal approximate solution of an FP equation, this problem was known as a GOP. Research interests played a prominent role in finding a solution to such kinds of problems such as the papers [1][2][3][4][5][6][7][8].
The authors in [9] are the first ones to use the concept of non-self-coupled mappings to be used in these problems, they followed the important results of [10]. Separately, Choudhury and Maity [11], obtained coupled proximity points in general FMSs.
The goal of this work is to consider the global GOP of obtaining the distance between two subsets of an FMS and solve it by FP methodology through the determination of two different pairs of points each of which determines the fuzzy distance for which we use a tripled mapping from one set to the other.
In 1965, fuzzy mathematics was initiated by Zadeh [12]. After that, the fuzzy ideas appeared in many branches of mathematics through the work of the researchers over the years. In particular, FMSs [13] have been studied extensively by many authors in many mathematical disciplines because it has a naturally defined Hausdorff topology which is in a large way the cause for the successful improvement of metric FP theory in these spaces, some examples of these works were provided by [14][15][16][17].
The problem has a fixed-point approach which we adopt here. It seeks to find an approximate solution to the equation ϑ = ϑ in the optimal way where the optimal approximate solution ϑ verifies d( ϑ, ϑ) = d(Ξ, Θ). The solution of the FP equation ϑ = ϑ do not have an exact solution if Ξ and Θ are separately (Ξ ∩ Θ = ∅). This is actually our concern. This problem has been turned to the FMS by several researchers such as [18][19][20]. As usual in a MS, the problem is solved by applying different types of contractions like, for instance, Choudhury and Maity [3], Jleli and Samet [4], Samet et al. [5] and Raj et al. [7,8]. Here, we proposed an application of a fuzzy WTC for the above problem. Coupled FP results has experienced rapid development in the contemporary time through works of [21][22][23][24][25][26][27]. In particular, the corresponding fuzzy coupled FP and related results are presented in [14,22].
In Hilbert spaces, the concept of weak contraction is an extension of Banach's contraction principle which was initiated by Alber et al. [28]. It can be said that a weak contraction lies between a contraction and a non-expansive contraction. By many works FP results involving weak contractions have been considered, for more details see [22,[29][30][31][32].
In 2011, a tripled FP result has been introduced by Berinde and Borcut [24] as an extension of coupled FP result in partially ordered metric spaces, under this space they presented exciting results about tripled FP consequences. For more illustrations about the contributions of researchers in this line, see [33][34][35][36][37][38][39]. We use a WTC in this manuscript for which two control functions are applied. Also, a fuzzy Q−property has been used in FMSs which is important a fuzzified geometric concept of Hilbert space suitable for FMSs. Now, we give some essential mathematical notions of our discussion.

Main Results
We begin this section with the definition below.

Definition 8.
Assume that (Ξ, Θ) is a pair of non-empty subsets of a FMS (Λ, ∆, ). We say that (ϑ, θ, κ) ∈ Ξ 3 is a tripled best proximity point of the mapping : Ξ 3 → Θ if it verifies the stipulation that for all τ > 0, It is clear that if Ξ = Θ, a tripled best proximity point reduces to a tripled FP.
Proof. Only take Ω(e) = e and Υ(e) = e − αe in Theorem 1 for all e ≥ 0. Now, we introduce a non-trivial example to support the results of Theorem 1.

Conclusions
The aim of this manuscript is to consider the global GOP of obtaining the distance between two subsets of an FMS and solve it by FP techniques through the determination of two different pairs of points each of which determines the fuzzy distance for which we use a tripled mapping from one set to the other.