q-ROF mappings and Suzuki type common ﬁxed point results in b -metric spaces with application

In the present paper the concepts of q-rung orthopair fuzzy mappings (q-ROF mapping) and q-rung ( α , β )-cuts are introduced. Some common ﬁxed point results for q-ROF mappings are presented in b-metric spaces using Suzuki-type contractive conditions. Examples in support of obtained results are also presented. We have also presented an application of our result for the existence of solution of nonlinear fractional integral inclusion. The results are of their own kind in the literature of q-ROF sets and will pave the way for further research in the area


Introduction
The concept of fuzzy sets was introduced by Zadeh [55] to pave a path for the better interpretation of data in real life problems.The key concept given by him was to award a membership grade from [0, 1] to the specific attribute.Since then various ideas and applications of fuzzy sets towards decision making, game theory, control systems, engineering, robotics, image processing, optimization theory, etc. have been initiated.There are situations where just membership grade is not enough to deal with, and on this account a grade against the membership of an attribute to a specific trait was introduced.Such sets are defined as orthopair fuzzy sets represented by μ A , ν A , where μ A stands for grade of membership while ν A for nonmembership.Generalizations of orthopair fuzzy sets have been introduced as intuitionistic fuzzy sets (IFS) and Pythagorean fuzzy sets [11,12].The difference between these two types is that for the first case, the sum of membership and nonmembership grades is bounded by 1, while for the second case the sum of squares of membership and nonmembership grades is bounded by 1.
Yager then, in 2017, gave a further generalization of orthopair fuzzy sets known as qrung orthopair fuzzy sets (q-ROF sets) where (μ A ) q + (ν A ) q ≤ 1 [54].The main advantage of a q-ROF set is that it increases the bounding space of selection of belongingness and non-belongingness grade of a trait for a given set.Several mathematicians have further studied q-ROF sets and have applied the concept in decision making problems and artificial intelligence, especially in the filed of medicine and agriculture.Multi-attribute decision making is an important aspect of decision sciences.It is a process that can give the ranking results for the finite alternatives according to the attribute values of different alternatives.The concept of q-ROF sets is combined with many existing aggregation operators for the improved management of evaluating information and decision making [40,45,53,56].
Ever since, in the history of fixed point theory, mathematicians have introduced several contractive conditions and mappings for more improved fixed point results.In this regard several contractions have been developed like Banach contraction, Chatterjea contraction [26], Kannan contraction [36], α-ψ type contractions [51], (θ , L)-weak contraction [22], etc. Suzuki [52] in 2008 introduced Suzuki-type contractive condition, which generalizes Banach contraction and characterizes the metric completeness of the underlying space.Since then the concept has been extended in various directions, and fixed point, common fixed point results along with applications have been presented, for example, [1,6,7,21,23,34,38,39,41,42,50].In 2015 Saleem et al. [49] presented fixed point results for Suzukitype contractive conditions utilizing multivalued mappings in fuzzy metric spaces with applications.Recently Gopal and Moreno [31] presented the concept of Suzuki-type fuzzy Z-contractive mappings, which is a generalization of Fuzzy Z-contractive mappings, and obtained fixed point results.
In the following article we introduce the notion of q-rung orthopair fuzzy mapping as a generalization of fuzzy mapping and q-rung (α, β)-level sets and hence prove some common fixed point results for a pair of q-rung orthopair fuzzy mapping in b-metric space.

Preliminaries
Consider (X, d) to be a metric space and CB(X) denotes the family of all closed and bounded subsets of X.Consider that H denotes the Hausdorff metric induced by d defined as where A, B ∈ CB(X) and d(x, B) = inf y∈B d(x, y).

Lemma 1 ([43]) If A, B ∈ CB(X) and x ∈ A, then for any real number l ≥ 1 there exists y ∈ B such that d(x, y) ≤ l.H(A, B). Also d(x, B) ≤ H(A, B).
Czerik introduced the generalized notion of b-metric space by changing the triangular inequality in a metric space.Definition 1 ([28]) Consider X = ∅ and s ≥ 1.A function d : X × X → [0, ∞) will be bmetric on X if for all x, y, z ∈ X the following hold: (i) d(x, y) = 0 if and only if x = y (indistancy); Definition 2 ([35]) Consider {x n } to be a sequence in a b-metric space (X, d).Then, (a) Then (X, d) is a b-metric space.If τ > 2, the ordinary triangular inequality does not hold and (X, d) is not a metric space.
Zadeh [55] introduced fuzzy sets by defining the membership grade of an instinct to a trait which in real life does not have precisely defined criteria of membership for that particular trait.

Definition 3
Let X be a nonempty set.Then μ : X → [0, 1] is a fuzzy set defining the grades of membership of elements of X.
Definition 4 An α-level set of a fuzzy set μ is defined as A fuzzy set μ is convex if and only if the sets μ α are convex.
Atanassov [11] presented intuitionistic fuzzy sets as a generalized notion of fuzzy sets.Intuitionistic fuzzy set depicts the grade of membership of an element for a set and its grade of nonmembership.Atanassov [12] then in 1993 introduced another type of orthopair fuzzy sets known as Pythagorean fuzzy sets in which the sum of squares of grades of membership and nonmembership of element is bounded by 1. Yager [54] in 2017 generalized the class of orthopair fuzzy sets called q-rung orthopair fuzzy sets or q-ROF sets.
Following concepts are defined by Yager in [54].
Definition 5 A q-rung orthopair fuzzy subset A of X, denoted as a q-ROF set, is an orthopair.
Definition 6 A fuzzy subset A of X is called an approximate quantity if and only if its αlevel set is a compact convex subset of X for each α ∈ [0, 1] and sup A(x) = 1 for all x ∈ X. W (X) denotes the collection of approximate quantities of X.When A ∈ W (X) and A(x 0 ) = 1 for some x 0 ∈ X, then A is identified as an approximation of x 0 .
Let A, B ∈ W (X).An approximate quantity A is more accurate than B, denoted by A ⊂ B, if and only if A(x) ≤ B(x) for all x ∈ X.
Let A, B ∈ W (X), α ∈ [0, 1], then the distance between A and B is defined as follows: Let X be a set and Y be a metric linear space.F is called fuzzy mapping if F is a mapping from the set X into W (Y ), i.e., F(x) is an approximate quantity.
Lemma 2 Let x ∈ X, A ∈ W (X) and {x} be a fuzzy set with membership function equal to the characteristic function of set {x}.If {x} ⊂ A, then p α (x, A) = 0.
Lemma 3 For any x, y ∈ X, In 2008, Suzuki [52] presented a fixed point theorem generalizing the Banach contraction theorem and characterizing the metric completeness.
Theorem 1 Consider (X, d) to be a complete metric space and T : X → X.A nonincreasing function θ : [0, 1) → ( 1 2 , 1] is given by Suppose that there is r ∈ [0, 1) so that θ (r)d(x, Tx) ≤ d(x, y) implies d(Tx, Ty) ≤ rd(x, y) for all x, y ∈ X.Then there exists a unique fixed point of T.Moreover, lim n T n x = z for all x ∈ X.
Theorem 2 Consider a nonincreasing function defined as ϕ : [0, 1) → (0, 1]: Consider (X, d) to be a complete metric space and T : X → CB(X).Suppose that there is r ∈ [0, 1) so that ϕ(r for all x, y ∈ X.Then z ∈ X so that z ∈ Tz.

q-ROF mappings and level sets
On the basis of well-known fuzzy notions existing in the literature, we have dedicated the following section to some new concepts defined for q-ROF sets such as q-rung α-level sets, q-rung (α, β)-level sets, and q-rung orthopair fuzzy mappings.A common fixed point result for a pair of q-rung orthopair fuzzy mappings in the settings of b-metric space is also presented using Suzuki-type contractive condition.An example in the support of our main result is also given.Throughout this article the class of all q-ROF subsets of X will be denoted by F q (X) and ϕ : [0, 1) → (0, 1] is a nonincreasing function defined as Definition 9 Consider X to be an arbitrary set, Y a metric space.A mapping T : X → F q (Y ) is called q-rung orthopair fuzzy mapping.
Since we claim that q-rung orthopair fuzzy mapping is a generalization of intuitionistic fuzzy mapping, so below is an example in support of the claim.
Example 2 Consider X = [0, 1] and let T : X → F q (X) be defined as ) ) ) ) Clearly T is a q-rung orthopair fuzzy mapping for q = 6 and is not an intuitionistic fuzzy mapping.
Definition 10 Let X be a metric space.A point x * ∈ X is called fixed point of a q-rung orthopair fuzzy mapping T : for some x * ∈ X.
Definition 12 Consider (X, d) to be a b-metric space with a constant s ≥ 1.For A, B ∈ K , (X) and α, β ∈ [0, 1], define The following results are the generalizations of the results defined in [33] which will be helpful in proving fixed point theorems for q-ROF mappings in b-metric spaces.Lemma 4 Let x ∈ X, A ∈ K , (X), and {x} be a q-ROF set as its membership function is equal to χ {x} (the characteristic function of {x}) and nonmembership function is equal to 1χ {x} defined as for some e ∈ {x}.Clearly, (μ(x) Lemma 7 Consider (X, d) to be a complete b-metric linear space, s ≥ 1, and let T : X → K , (X) be a q-rung orthopair fuzzy mapping.Consider that for each x ∈ X and each pair Theorem 3 Consider (X, d) to a complete b-metric linear space, s ≥ 1, and T : X → K , (X) be a q-rung orthopair fuzzy mapping.For each element x ∈ X and each pair (α, for all x, y ∈ X.Then there exists z ∈ X such that z Then from (3.2) we have This implies {x n } is a Cauchy sequence.As X is complete, z ∈ X so that lim n→∞ x n = z.
Next it is proved that Then from (3.4) we get 2) we have Therefore, by Lemma 7, we obtain A contradiction, hence z ∈ [Tz] q (α,β) Tz .Now, for the case 1  2 ≤ r < 1, we will first prove If x = z, then (3.6) holds.Let x = z, then for every n belonging to natural numbers, there is a sequence Using (3.2) we get (3.6).
and T : X → F q (X) is a q-ROF mapping defined as follows: If x = 0, then we have ) ) ) 1 q t > 1 60 .
Then, for q = 5, r = 0.999, s = 4 all the conditions of Theorem 3 are satisfied.
Corollary 1 Consider (X, d) to be a complete b-metric linear space, s ≥ 1, and let T : X → K , (X) be an intuitionistic fuzzy mapping.Consider T to satisfy the same contractive conditions as in Theorem 3, then T has a fixed point.
Corollary 2 Consider (X, d) to be a complete b-metric linear space, s ≥ 1, and let T : X → K , (X) be a fuzzy mapping.Consider T to satisfy the same contractive conditions as in Theorem 3, then T has a fixed point.
Corollary 3 Consider (X, d) to be a complete b-metric linear space, s ≥ 1, S, T : X → K , (X) be intuitionistic fuzzy mappings.Then S and T have a common fixed point under the contractive conditions as in Theorem 4.
Corollary 4 Consider (X, d) to be a complete b-metric linear space, s ≥ 1, S, T : X → K , (X) be fuzzy mappings.Then S and T have a common fixed point under the contractive conditions as in Theorem 4.
Next it will be proved that for all y ∈ X -{ω}. Since

Application
There are known applications of fuzzy sets for the solution of integral equations (for example, see [18,19]).In the present section, with the help of completeness property of function space C[a, b] and by applying Theorem 6, we have presented an existence theorem for the solution of the class of nonlinear integral equations.
We will use Theorem 6 to show the existence of common solutions of two nonlinear integral inclusions defined as where . By a common solution of system 1, we mean a continuous function x such that where Theorem 6 Consider the system of nonlinear integral inclusions in (2.1).Assume that the following conditions hold: (i) The operators F 1 (σ , τ , x(τ )), x(σ )z(σ ) p .
Proof Let X = C[a, b] and define d : X × X → R by d(x, y) = |x(σ )y(σ )| p for all x, y ∈ X.Then (X, d) is a complete b-metric space with s = 2 p-1 where p > 1. Assume that U, V , E, Z : X → (0, 1] are four arbitrary mappings.Now, define a pair of qth rung fuzzy mappings A, ξ : X → F q (X) as follows: A : X → F q (X) : Hence, by Theorem 6, there exists a common fixed point of mappings A and B.

Conclusion
The concept of q-ROF mapping, as a generalization of fuzzy mappings, is introduced.Also the concept of q-rung (α, β)-level sets is presented and some common fixed point results utilizing this concept for q-ROF mappings are obtained in b-metric space via Suzuki-type contractive conditions.We have also presented examples in support of our results.An application of obtained results for the existence of solution of nonlinear fractional integral inclusion is also presented.