New fixed-circle results on fuzzy metric spaces with an application to dynamic market equilibrium

. In this study, the fixed point theory on fuzzy metric spaces has been generalized to the fixed-circle theory by making a geometric interpretation. The necessary conditions to exist the fixed circles of a self-mapping have been investigated and the uniqueness of the circle is examined under suitable conditions. We present some illustrative examples of obtained results and also offer an application to confirm the utility of our established result for finding the unique solution of an integral equation appearing in the dynamic market equilibrium aspects of economics.


Introduction and Preliminaries
The fuzzy set theory is one of the most valuable theories in solving uncertainty-related problems. Zadeh put forward this theory in 1965 [27]. Basic definitions and theorems of general topology have been generalized to fuzzy topological spaces. After that, one of this space's primary problems is obtaining an appropriate notion of fuzzy metric spaces. Many authors studied this problem. Significantly, the authors in [1,3,8,11] have presented the concept of fuzzy metric space in various ways. George and Veeramani [4] have defined and studied a concept of fuzzy metric space with the help of continuous t-norm. The study of the relationship between fuzzy metric spaces and metric spaces constitutes a natural and interesting question in the theory of fuzzy metric spaces. In [4], it is proved that every metric space can induce a standard fuzzy metric spaces.
Tas and Özgür [16] have recently pioneered a new trend by approaching the fixed point theory from a different perspective. Using the notion of a fixed circle, they and many researchers obtained valuable results on metric spaces and some generalized metric spaces [10, 12-15, 17-19, 22-25]. Also, Gopal et. al. [7] introduced the notion of a fixed circle in fuzzy metric spaces.
This paper presents some existence theorems for fixed circles on fuzzy metric spaces with continuous t-norms. Then, since the fixed circles obtained in these theorems do not have to be unique, we give theorems for uniqueness using some contraction conditions. An application to dynamic market equilibrium is presented to validate the main result.
Firstly, we recall some essential background on fuzzy metric spaces.
The next inequalities are satisfied: for each continuous t-norm * .
A circle and a fixed circle are defined on a fuzzy metric space as follows.

Definition 3 ([7]
). Let (χ, M, * ) be a fuzzy metric space. A circle of center η 0 ∈ χ and radius r ∈ (0, 1) is defined as follows This equality can rewrite as follows: Example 6. If we take the function M d defined in Example 2 with the usual metric space and choose η 0 = 4, r = 1 3 and t = 2, then we obtain ). Let C F (η 0 , r, t) be a circle in a fuzzy metric space (χ, M, * ) and τ : χ → χ be a self-mapping. If τ η = η for all η ∈ C F (η 0 , r, t), then the circle C F (η 0 , r, t) is called as the fixed circle of τ.
The purpose of this work is to present novel fixed-circle results on fuzzy metric spaces taking into account the research and literature mentioned above. Because a fuzzy metric space is a generalization of a metric and there are certain examples of fuzzy metric that are not metric, this is significant (see the Example 4).

Main results
2.1. Some existence conditions for fixed-circles on fuzzy metric spaces. In order to obtain following existence theorems for a fixed-circle, we used some contractive conditions. Theorem 1. Let (χ, M, * M ) be a fuzzy metric space where * M is a minimum t-norm and C F (η 0 , r, t) be any circle on χ. Define the mapping for all η ∈ χ. If the self-mapping τ : χ → χ is a function such that for all η ∈ C F (η 0 , r, t), the following inequalities are satisfied: then, the circle C F (η 0 , r, t) is a fixed-circle of τ.
Proof. Let η ∈ C F (η 0 , r, t). Assume that η ̸ = τ η. Then, take into account the inequalities (2) and (3), we obtain that Notice that the fuzzy set is defined on [0, 1]. Hence, it should be M(η, τ η, t) = 1. As a result, we get η = τ η and C F (η 0 , r, t) is a fixed-circle of τ. □ Example 8. Let χ = R endowed with the standard fuzzy metric space with the usual metric space. Think the circle C F (0, 2 5 , 3) and define τ : R → R by 7η + 5 for each η ∈ R. Then, τ verifies the hypotheses of Theorem 1. So, τ fixes the circle C F (0, 2 5 , 3). Theorem 2. Let (χ, M, * L ) be a fuzzy metric space where * L is a Lukasiewicz t-norm and C F (η 0 , r, t) be any circle on χ. Let the mapping φ be defined as Theorem 1. If the self-mapping τ : χ → χ is a function such that for all η ∈ C F (η 0 , r, t) and some h ∈ [0, 1) the following inequalities are fulfilled: then, the circle C F (η 0 , r, t) is a fixed circle of τ.
Proof. By the mapping φ defined in (1), it suffices to show that τ η = η where η ∈ C F (η 0 , r, t) is an arbitrary point. Taking into consideration the inequalities (4), (5) and (6), we find This gives a contradiction with h ∈ [0, 1) and shows that η = τ η. Thus, C F (η 0 , r, t) is a fixed-circle of τ. □ Theorem 3. Let (χ, M, * L ) be a fuzzy metric space where * L is a Lukasiewicz t-norm and C F (η 0 , r, t) be any circle on χ. Let the mapping φ be defined as Theorem 1. If the self-mapping τ : χ → χ is a function such that for all η ∈ C F (η 0 , r, t), the following inequalities are satisfied: then, the circle C F (η 0 , r, t) is a fixed circle of τ.
Proof. Let C F (η 0 , r, t) and C F (η 1 , p, s) be two fixed-circles of τ. Let u ∈ C F (η 0 , r, t) and v ∈ C F (η 1 , p, s) be arbitrary points. It suffices to show that u = v. Using the inequality (9) and Lemma 1, we obtain This gives a contradiction since k ∈ (0, 1). As a result, C F (η 0 , r, t) is the unique fixed-circle of τ. □ Notice that the contractive inequality (9) τ given in the Proof of Proposition 1 is not satisfied.
Gregori and Sapena [6] proposed a contraction in a fuzzy metric space. Using the contractive condition, we obtain a uniqueness theorem as follows.
Proof. The proof is similar to Proof of Theorem 4. □ Remark 2. The choice of used contractive condition in the uniqueness theorem is not unique. Any contractive condition used to derive the fixed point theorem can also be selected.

Application
This section presents an application to dynamic market equilibrium to support our work. In many markets, current prices and price trends affect supply and demand (i.e., whether prices are rising or falling at an increasing or decreasing rate). So, the economist wants to know the current price P (t), the first derivative dP (t) dt and second derivative d 2 (P (t)) dt 2 . Suppose a 1 , a 2 , b 1 , b 2 , x 1 and x 2 are constants. Comment on the dynamic stability of the market if price clears the market at each point in time. In equilibrium, Q s = Q d [2]. For this reason, .
, y = y 1 −y 2 , a = a 1 −a 2 and dividing through by y, P (t) is expressed by the following initial value problem where x 2 y = 4b y and b x = µ is a continuous function. One can easily show that problem (10) is equivalent to the integral equation: where Γ(t, s) is Green's function given by To prove the existence part of a solution integral equation, we use Theorem 4: min{η, ζ} + t max{η, ζ} + t for each t > 0, and η, ζ ∈ χ with continuous t-norm * such that η * P ζ = ηζ.
One can easily verify that (χ, M, * ) is a fuzzy metric space. Consider the mapping τ : χ → χ is defined by τ P (t) = I 0 G(t, s, P (s))ds.

Conclusions
In this study, the geometric properties of the fixed point set are investigated when the fixed point of the mappings on fuzzy metric spaces is more than one. For this, the fixed-circle problem, which is a generalization of the fixed point theory, is used. By this new approach, new geometric generalizations of known fixed point theorems can study on fuzzy metric and generalized metric spaces.