On pairs of fuzzy dominated mappings and applications

The main purpose of this paper is to present some fixed-point results for a pair of fuzzy dominated mappings which are generalized V-contractions in modular-like metric spaces. Some theorems using a partial order are discussed and also some useful results to graphic contractions for fuzzy-graph dominated mappings are developed. To explain the validity of our results, 2D and 3D graphs have been constructed. Also, applications are provided to show the novelty of our obtained results and their usage in engineering and computer science.


Introduction and preliminaries
The fixed-point theory becomes essential in analysis (see ). In [19], Chistyakov firstly introduced the notion of a modular metric and discussed thoroughly its convergence, convexity, relation with metrics, convex cones, and the structure of semigroups on such spaces. The modular metric spaces generalize classical modulars over linear spaces, like Orlicz, Lebesgue, Musielak-Orlicz, Lorentz, Calderon-Lozanovskii, and Orlicz-Lorentz spaces. The main idea behind this new concept is the physical interpretation of the modular. We look at these spaces as the nonlinear version of the classical modular spaces. Padcharoen [42] initiated the idea of rational type F-contractions in modular metric spaces and proved some important results. Additional results in such spaces proved by different authors can be seen in [18,31,33,37]. Nadler [39] presented a fixedpoint theorem for multivalued mappings and generalized its analogues for single-valued mappings. Fixed-point results of multivalued mappings have several applications in engineering, control theory, differential equations, games and economics; see [11,16]. In this paper, we are using multivalued mappings. Wardowski [66] introduced a new type of contractions, named F-contractions, to obtain a fixed-point result. For more results in this direction, see [2,3,6,8,15,32,33,38,55]. Here, we have used a weak family of mappings instead of the function F introduced by Wardowski. In [9] the authors observed that there are mappings which possess fixed points. Namely, they introduced a condition on closed balls to achieve common fixed points for such mappings. For further results on closed balls, see [50,51,63]. In this paper, we are using a sequence instead of a closed ball. Ran and Reurings [49] and Nieto and Rodríguez-López [41] gave fixed-point theory results in partially ordered sets. For more results in ordered spaces, see [20,21,23]. Asl et al. [10] gave the notion of α * -admissible mappings and α-ω-contractive set-valued mappings (see also [5,26,29,56]) and generalized the restriction of order. Rasham et al. [53] introduced the concept of α * -dominated mappings to establish a new condition of order and obtained some results (see also [52,54,59,62]). They proved that there are mappings which are α * -dominated, but not α * -admissible. The notion of fuzzy sets is introduced by Zadeh [67] and then a lot of researchers did their research work in this field. Weiss [68] and Butnariu [17] firstly discussed the concept of fuzzy mappings and showed many related results. Heilpern [16] discussed a result on fuzzy mappings, which was a further generalization of Nadler's set-valued result [39] using a Hausdorff metric. Due to importance of the Heilpern's results, fixed-point theory for fuzzy contractions using a Hausdorff metric has become more important, see [44-48, 51, 61, 62]. In this article, we prove fixed point results for a pair of fuzzy dominated maps which are generalized V -contractions and provide related graphs for 2D and 3D. An application for the solution of electric circuit equations is also presented. Moreover, a fractional differential equation is solved. Our obtained results generalize those presented in [54,57,59,61,66].
We start with the following statements which are helpful to prove our results.
is called a modular-like metric on A if for all a, b, c ∈ A; l, n > 0, and u l (a, b) = u(l, a, b), the following hold: is called a modular-like metric space. If we replace (ii) by u l (a, b) = 0 if and only if a = b, then (A, u) becomes a modular metric space. If we replace (ii) by u l (a, b) = 0 for some l > 0 then a = b, then (A; u) becomes a regular modular-like metric on A. For e ∈ A and > 0, B u l (e, ) = {p ∈ A : |u l (e, p)u l (e, e)| ≤ )} is the closed ball. We abbreviate by "m.l.m. space" a modular-like metric space. (i) A sequence (a n ) n∈N in A is u-Cauchy for some l > 0, if and only if lim n,m→+∞ u l (a m , a n ) exists and is finite. (ii) A sequence (a n ) n∈N in A u-converges to a ∈ A for some l > 0, if and only if lim n→+∞ u l (a n , a) = u l (a, a).
some a ∈ E, so that for some l > 0, lim n→+∞ u l (a n , a) = u l (a, a) = lim n,m→+∞ u l (a m , a n ). If every e ∈ A has a greatest estimate in E, then E is identified as a proximal set. For example, let A = R + ∪ {0} and u l (e, p) = 1 l (e + p) for all l > 0. Define a set E = [4,6], then for each y ∈ A, u l (y, E) = u l y, [4,6] = inf n∈ [4,6] u l (y, n) = u l (y, 4).
Hence, 4 is the finest estimate in E for very y ∈ A. Also, [4,6] is a proximal set. From now on, denote by P(A) the set of compact proximinal subsets in A.  Then K and L are not γ * -admissible, but they are γ * -dominated.   Now, we select a subset of the family F(G) of all fuzzy sets, which is a subfamily with stronger properties, i.e., the subfamily of the approximate quantities, denoted by W (G). Definition 1.12 ([24]) A fuzzy subset U of G is an approximate quantity iff its β-level set is a compact convex subset of G for each β ∈ [0, 1] and sup e∈G U(e) = 1.
Now, we are ready to prove our main theorems for a pair of fuzzy mappings which are a generalized rational type contraction.

Main results
Let (L, u) be an m.l.m. space, x 0 ∈ L and S, T : L → W (L) be fuzzy mappings on L. Moreover, let γ , β : L → [0; 1] be two real functions. Let . Continuing this process, we construct a sequence {x n } of points in L such that We use {TS(x n )} to denote this sequence. We say that {TS(x n )} is a sequence in L generated by x 0 .
for all elements x, g in {TS(x n )}, with either α(x, g) ≥ 1 or α(g, x) ≥ 1. Then, by Theorem 2.2, {TS(x n )} is a sequence in L and {TS(x n )} → x * ∈ L. Now, x n , x * ∈ L and either x n x * , or x * x n implies that either α(x n , x * ), or α(x * , x n ) ≥ 1. So, all requirements of Theorem 2.3 are satisfied. Hence, x * is the common fuzzy fixed point of both S and T in L and u l (x * , x * ) = 0.   This implies that  That is, Then This implies that If we take multivalued α * -dominated mappings from a ground set L to the proximinal subsets of L instead of fuzzy α * -dominated mappings from L to the approximate quantities W (L) in Theorem 2.3, we obtain the following result.   If we take S = T in Corollary 2.8, we obtain the following result.

Applications on graphic contractions
Jachymski [30] proved a relation between graph and fixed point theory by the orientation of graphic contractions. Let A be a nonempty set. Let V (Y ) and L(Y ) denote the set of vertices and the set of edges containing all loops, respectively, for a graph Y .   Hence, α * (t, [St] γ (t) ) = 1, α * (t, [Ty] β(y) ) = 1, for every t ∈ L. So, the mappings are α *dominated on L. Furthermore, inequality (3.1) can be expressed as

Applications to electric circuit equations
In this section, we discuss the solution of the electric circuit equation (see [7]) which is a second-order differential equation. The electric circuit (as in Fig. 4) contains an electromotive force E, a resistor R, an inductor L, a capacitor C, and a voltage V in series. If the current I is the rate of change of q with respect to time t, we have I = dq dt and

Figure 4 Electric circuit
By Kirchhoff 's law, the sum of these voltage drops is equal to supplied voltage, i.e., The Green function associated to (ECE) is given by where the constant τ > 0 is calculated in terms of R and L. Let L = C[0, 1] be the set of all continuous functions defined on [0, 1]. The modular-like metric u on L is defined as Moreover, we define the graph with the partial order relation: for u, g ∈ C[0, 1],   where t ∈ C[0, 1]. Then b * is the solution of (4.1) and (4.2) if and only if b * is a common fixed point of S and T. From condition (ii), it is very easy to show that for every u, g ∈ L, we have u ≤ Su and g ≤ Tg, i.e., b, S(b) ∈ Y (G) = ∅ and g, T(g) ∈ Y (G) = ∅.
Let b, g ∈ L, then from condition (i), we have This implies that Since 1 -2tr + tτ e -τ te -τ t ≤ 1, we get that So, all the requirements of Theorem 2.2 are satisfied for R(f ) = -1 f , f > 0, and u(b, c) = 1 2 b + c τ . Hence, the mappings S and T have a common fixed point. Consequently, the differential equation arising in the electric circuit (ECE) has a solution.

Conclusion
In this article, we have given some new results for a pair of fuzzy mappings which are Ciric and Wardowski type contractions. Dominated mappings are used to prove such fixed-point results. Further, results in ordered modular-like spaces involving graphic contractions equipped with graph dominated mappings are presented. The results have been demonstrated graphically by 2D and 3D graphs. This provides justification for our obtained results. In the end, we applied our results to solve electric circuit equations and fractional differential equations.