Some fixed point results for two families of fuzzy A-dominated contractive mappings on closed ball

Abstract

The purpose of this paper is to find out common fixed point results for two families of fuzzy mappings fulfilling generalized rational type A-dominated contractive conditions on a closed ball in complete dislocated b-metric space. Example is also given which shows the novelty of our results.

Keywords

Fixed point closed ball two families of fuzzy mapping

1 Introduction and preliminaries

Fixed point theory plays a fundamental role in functional analysis. Nadler [17] started the investigation of fixed point results for the setvalued functions. Due to its significance, a large number of authors have proved many interesting multiplications of these results (see [1 –35]).

Nazir et al. [18] showed common fixed point results for the family of generalized multivalued F-contraction mappings in ordered metric spaces. Recently Shoaib et al. [29] discussed some theorems for a family of setvalued functions. Rasham et al. [22] proved multivalued fixed point theorems for new F-contractive functions on dislocated metric spaces.

The notion about fuzzy sets was first presented by Zadeh [35]. Fuzzy sets have many applications in decision making (see [16 , 40]). The idea about fuzzy mappings was given by Weiss [34] and Butnariu [5] and also obtained some interesting fixed point results. Afterward, Heilpern [8] established the idea of fuzzy contraction mappings and proved a fixed point theorem which was a generalization of Nadler’s [17] fixed point theorem for multivalued mappings.

In this paper, we have obtained common fixed point of two families of fuzzy mappings satisfying conditions only on a sequence. We have used more weaker class of strictly increasing mappings A rather than class of mappings F used by Wardowski [33]. Example is given to demonstrate the variety of our results. Moreover, we investigate our results in a more better framework of dislocated b-metric space. New results in ordered spaces, partial b-metric space, dislocated metric space, partial metric space, b-metric space and metric space can be obtained as corollaries of our results. We give the following concepts which will be helpful to understand the paper.

1.1 Definition[12]

Let M be a nonempty set and let d _b : M × M → [0, ∞) be a function, if for any x, y, z ∈ M, the following conditions hold:

(i) d _b (x, y) ≤ b [d _b (x, z) + d _b (z, y)] ,

(where b≥ 1) ;

(ii) d _b (x, y) =0 implies x = y ;

(iii) d _b (x, y) = d _b (y, x) .

Then d _b is called a dislocated b-metric (or simply d _b-metric) and the pair (M, d _b) is called a dislocated b-metric space. It should be noted that every dislocated metric is a dislocated b-metric with b = 1 .

It is clear that if d _b (x, y) =0, then from (ii), x = y. But if x = y, d _b (x, y) may not be 0 . For x ∈ M and ɛ > 0, $\bar{B (x, ɛ)} = {y \in M : d_{b} (x, y) \leq ɛ}$ is a closed ball in (M, d _b) . We use D . B . M space instead dislocated b-metric space.

1.2 Example[31]

Let M = Q ⁺ ∪ {0} and let d _b : M × M → M be the complete D . B . M space defined by $d_{b} (u, v) = (u + v)^{2} for all u, v \in M .$

1.3 Definition[12]

Let (M, d _b) be a D . B . M space.

(i) A sequence {x _n} in (M, d _b) is called Cauchy sequence if given ɛ > 0, there corresponds n ₀ ∈ N such that for all n, m≥ n ₀ we have d _b (x _m, x _n) <ɛ or $lim_{n, m \to \infty} d_{b} (x_{n}, x_{m}) = 0 .$

(ii) A sequence {x _n} dislocated b-converges (for short d _b -converges) to x if $lim_{n \to \infty} d_{b} (x_{n}, x) = 0 .$ In this case x is called a d _b-limit of {x _n} .

(iii) (M, d _b) is called complete if every Cauchy sequence in M converges to a point x ∈ M such that d _b (x, x) =0.

1.4 Definition[31]

Let K be a nonempty subset of D . B . M space of M and let x ∈ M . An element y ₀ ∈ K is called a best approximation in K if $d_{b} (x, K) = d_{b} (x, y_{0}), where d_{b} (x, K) = inf_{y \in K} d_{b} (x, y) .$ If each x ∈ X has at least one best approximation in K, then K is called a proximinal set. We denote P (M) be the set of all closed proximinal subsets of M .

1.5 Example

Let $X = ℝ^{+} \cup {0}$ and let d _b : X × X → X be the complete dislocated b-metric on X defined by $d_{b} (x, y) = (x + y)^{2}, for all x, y \in X$ with parameter b = 2 . Define a set A = [2, 3] , then for each x ∈ X $d_{b} (x, A) = d_{b} (x, [2, 3]) = inf_{u \in [2, 3]} d_{b} (x, u) = d_{b} (x, 2) .$ Hence 2 is a best approximation in A for each x ∈ X . Also [2, 3] is a proximinal set.

1.6 Definition[30]

The function H _{d
_b} : P (M) × P (M) → R ⁺, defined by $H_{d_{b}} (N, R) = max {sup_{n \in N} d_{b} (n, R), sup_{r \in R} d_{b} (N, r)}$ is called dislocated Hausdorff b-metric on P (M) .

Let $X = ℝ^{+} \cup {0}$ and d _b (x, y) = (x + y) ² be the complete dislocated b-metric on X . If N = [3, 5] , R = [7, 8] , then H _{d
_b} (N, R) =144 .

1.7 Definition[22]

Let (M, d _b) be a D . B . M space. Let S : M → P (M) be multivalued mapping, α : M × M → [0, + ∞) and α _∗ (i, Si) = inf {α (i, l) : l ∈ Si}. Let H ⊆ M, then S is said to be α _∗-dominated on H, whenever α _∗ (i, Si) ≥1 for all i ∈ H . If H = M, then we say that the S is α _∗-dominated. If S : M → M is a self mapping, then S is α-dominated on H, whenever α (i, Si) ≥1 for all i ∈ H .

1.8 Example[22]

Let $Z = ℝ .$ Define the mapping α : Z × Z → [0, ∞) by $α (j, k) = {\begin{matrix} 1 if j > k \\ \frac{1}{2} otherwise \end{matrix}} .$ Define S, T : Z → P (Z) by $Sj = [j - 4, j - 3] and Tk = [k - 2, k - 1] .$ Suppose j = 3 and k = 2.5 . As 3 > 2.5, then α (3, 2.5) ≥1 .Now, α _∗ (S3, T2.5) = inf {α (a, b) : a ∈ S3, b ∈ T2.5} $= \frac{1}{2} ≯ 1,$ this means the pair (S, T) is not α _∗-admissible. Also, α _∗ (S3, S2) ≯1 and α _∗ (T3, T2) ≯1 . This implies S and T are not α _∗-admissible individually. Now, α _∗ (j, Sj) = inf {α (j, b) : b ∈ Sj} ≥1, for all j ∈ Z . Hence S is α _∗-dominated mapping. Similarly α _∗ (k, Tk) = inf {α (k, b) : b ∈ Tk} ≥1 . Hence it is clear that S and T are α _∗-dominated but not α _∗-admissible.

1.9 Definition

A function from a nonempty M to [0, 1] is a fuzzy set. Let F (M) be a set of all fuzzy sets in M. If A is a fuzzy set and y ∈ M, then the function values A (y) is grade of membership of y in A . The γ-level set of fuzzy set A, is denoted by [A] _γ, and defined as: $\begin{array}{l} {[A]}_{γ} = {y : A (y) \geq γ} where γ \in (0, 1], \\ {[A]}_{0} = \bar{{y : A (y) > 0}} . \end{array}$ Let N be a metric space and M be any nonempty set. Let S be a mapping called fuzzy mapping, if S : M → F (M). Then, S is a fuzzy subset on M × N with membership function S (c ₁) (c ₂). The function S (c ₁) (c ₂) is the grade of membership of c ₂ in S (c ₁). For simplicity, we represent the γ-level set of S (c) by [Sc] _γ instead of [S (c)] _γ [31].

1.10 Definition[31]

A point c ∈ M is a fuzzy fixed point of a fuzzy mapping S : M → F (M) if there is γ ∈ (0, 1] such as c ∈ [Sc] _γ.

1.11 Example

Let M = {0, 1, 2}. Define a fuzzy mapping S : M → F (M) by $S (0) (z) = S (1) (z) = S (2) (z) = {\begin{matrix} 2 / 3 if z=2 \\ 0 if z=0, 1 \end{matrix} .$ For $γ \in (0, \frac{2}{3}]$ , we have ${[Sx]}_{γ} = {2} for all x \in M .$ Here, 2 ∈ M is a fuzzy fixed point for the fuzzy mapping S.

1.12 Lemma[22]

Let (M, d _b) be a D . B . M space. Let (P (M) , H _{d
_b}) be a dislocated Hausdorff b-metric space on P (M) . Then, for all G, H ∈ P (M) and for each g ∈ G such that d _b (g, H) = d _b (g, h _g), where h _g ∈ H . Then the following holds: $H_{d_{b}} (G, H) \geq d_{b} (g, h_{g}) .$

2 Main results

Let (M, d _b) be a D . B . M space, c ₀ ∈ M, let {S _σ : σ ∈ Ω} and {T _β : β ∈ Φ} be two families of fuzzy mappings from M to F (M). Let c ₁ ∈ [S _a c ₀] _γ(c
₀) be an element such that d _b (c ₀, [S _a c ₀] _γ(c
₀)) = d _b (c ₀, c ₁) . Let c ₂ ∈ [T _b c ₁] _γ(c
₁) be such that d _b (c ₁, [T _b c ₁] _γ(c
₁)) = d _b (c ₁, c ₂) . Let c ₃ ∈ [S _a c ₂] _γ(c
₂) be such that d _b (c ₂, [S _a c ₂] _γ(c
₂)) = d _b (c ₂, c ₃) .

In this way, we get a sequence {T _β S _σ (c _n)} in M, where

c _2n+1 ∈ [S _i c _2n] _{γ(c
_2n)}, c _2n+2 ∈ [T _j c _2n+1] _{γ(c
_2n+1)}, $n \in ℕ,$ i ∈ Ω and j ∈ Φ . Also

d _b (c _2n, [S _i c _2n] _{γ(c
_2n)}) = d _b (c _2n, c _2n+1) ,

d _b (c _2n+1, [T _j c _2n+1] _{γ(c
_2n+1)}) = d _b (c _2n+1, c _2n+2) . Then, {T _β S _σ (c _n)} is said to be a sequence in M generated by c ₀ . If {S _σ : σ ∈ Ω} = {T _β : β ∈ Φ} , then we say {MS _σ (c _n)} instead of {T _β S _σ (c _n)} .

2.1 Theorem

Let (M, d _b) be a complete D . B . M space with constant b ≥ 1. Let r > 0, $c_{0} \in \bar{B_{d_{b}} (c_{0}, r)} \subseteq M,$ α : M × M → [0, ∞) and {S _σ : σ ∈ Ω}, {T _β : β ∈ Φ} be two families of α _∗-dominated fuzzy mappings from M to F (M) on $\bar{B_{d_{b}} (c_{0}, r)} .$ Suppose that the following satisfy:

i) There exist τ, μ ₁, μ ₂, μ ₃, μ ₄ > 0 satisfying bμ ₁ + bμ ₂ + (1 + b) bμ ₃ + μ ₄ < 1 and a strictly increasing mapping A such that

$τ + A (H_{d_{b}} ([S_{σ} e]_{γ (e)}, [T_{β} y]_{γ (y)})) \leq A (μ_{1} d_{b} (e, y) + μ_{2} d_{b} (e, [S_{σ} e]_{γ (e)}) + μ_{3} d_{b} (e, [T_{β} y]_{γ (y)}) + μ_{4} \frac{d_{b} (e, [S_{σ} e]_{γ (e)}) . d_{b} (y, [T_{β} y]_{γ (y)})}{1 + d_{b} (e, y)})$ (2.1)

whenever $e, y \in \bar{B_{d_{b}} (c_{0}, r)} \cap {T_{β} S_{σ} (c_{n})},$ α (e, y) ≥1, σ ∈ Ω, β ∈ Φ, γ (e) ∈ (0, 1] and

H _{d
_b} ([S _σ e] _γ(e), [T _β y] _γ(y)) >0.

ii) If $η = \frac{μ_{1} + μ_{2} + b μ_{3}}{1 - b μ_{3} - μ_{4}},$ then $d_{b} (c_{0}, [S_{a} c_{0}]_{γ (c_{0})}) \leq η (1 - b η) r .$ (2.2) Then, {T _β S _σ (c _n)} is a sequence in $\bar{B_{d_{b}} (c_{0}, r)}$ , α (c _n, c _n+1) ≥1 for all $n \in ℕ \cup {0}$ and ${T_{β} S_{σ} (c_{n})} \to u \in \bar{B_{d_{b}} (c_{0}, r)} .$ Also, if u satisfies (2.1) and either α (c _n, u) ≥1 or α (u, c _n) ≥1 for all $n \in ℕ \cup {0}$ , then S _σ and T _β have common fuzzy fixed point u in $\bar{B_{d_{b}} (c_{0}, r)}$ for all σ ∈ Ω and β ∈ Φ . Moreover, d _b (u, u) =0 .

Proof. Consider a sequence {T _β S _σ (c _n)} . From (2.2), we get $d_{b} (c_{0}, c_{1}) = d_{b} (c_{0}, [S_{a} c_{0}]_{γ (c_{0})}) \leq η (1 - b η) r < r .$ It follows that, $c_{1} \in \bar{B_{d_{b}} (c_{0}, r)} .$ Let $c_{2}, \dots, c_{j} \in \bar{B_{d_{b}} (c_{0}, r)}$ for some $j \in ℕ$ . If j is odd, then $j = 2 \overset{`}{ı} + 1$ for some $\overset{`}{ı} \in ℕ$ . Since {S _σ : σ ∈ Ω} and {T _β : β ∈ Φ} be two families of α _∗-dominated fuzzy mappings on $\bar{B_{d_{b}} (c_{0}, r)}$ , so $α_{*} (c_{2 \overset{`}{ı}}, [S_{σ} c_{2 \overset{`}{ı}}]_{γ (c_{2 \overset{`}{ı}})}) \geq 1$ and $α_{*} (c_{2 \overset{`}{ı} + 1},$ $[T_{β} c_{2 \overset{`}{ı} + 1}]_{γ (c_{2 \overset{`}{ı} + 1})}) \geq 1$ for all σ ∈ Ω and β ∈ Φ . As $α_{*} (c_{2 \overset{`}{ı}}, [S_{σ} c_{2 \overset{`}{ı}}]_{γ (c_{2 \overset{`}{ı}})}) \geq 1,$ this implies $inf {α (c_{2 \overset{`}{ı}}, b) : b \in [S_{σ} c_{2 \overset{`}{ı}}]_{γ (c_{2 \overset{`}{ı}})}} \geq 1 .$ Also $c_{2 \overset{`}{ı} + 1} \in [S_{f} c_{2 \overset{`}{ı}}]_{γ (c_{2 \overset{`}{ı}})}$ for some f ∈ Ω, so $α (c_{2 \overset{`}{ı}}, c_{2 \overset{`}{ı} + 1}) \geq 1 .$ Also $c_{2 \overset{`}{ı} + 2} \in [T_{g} c_{2 \overset{`}{ı} + 1}]_{γ (c_{2 \overset{`}{ı} + 1})}$ for some g ∈ Φ. Now by using Lemma 1.12, we have $\begin{matrix} τ + A (d_{b} (c_{2 \overset{`}{ı} + 1}, c_{2 \overset{`}{ı} + 2})) \\ \leq τ + A (H_{d_{b}} ([S_{f} c_{2 \overset{`}{ı}}]_{γ (c_{2 \overset{`}{ı}})}, [T_{g} c_{2 \overset{`}{ı} + 1}]_{γ (c_{2 \overset{`}{ı} + 1})})) \\ \leq A (μ_{1} d_{b} (c_{2 \overset{`}{ı}}, c_{2 \overset{`}{ı} + 1}) + μ_{2} d_{b} (c_{2 \overset{`}{ı}}, [S_{f} c_{2 \overset{`}{ı}}]_{γ (c_{2 \overset{`}{ı}})})) \\ (+ μ_{3} d_{b} (c_{2 \overset{`}{ı}}, [T_{g} c_{2 \overset{`}{ı} + 1}]_{γ (c_{2 \overset{`}{ı} + 1})})) \\ (+ μ_{4} \frac{d_{b} (c_{2 \overset{`}{ı}}, [S_{f} c_{2 \overset{`}{ı}}]_{γ (c_{2 \overset{`}{ı}})}) . d_{b} (c_{2 \overset{`}{ı} + 1}, [T_{g} c_{2 \overset{`}{ı} + 1}]_{γ (c_{2 \overset{`}{ı} + 1})})}{1 + d_{b} (c_{2 \overset{`}{ı}}, c_{2 \overset{`}{ı} + 1})}) \\ \leq {Ad}_{b} (μ_{1} + μ_{2} + b μ_{3}) (c_{2 \overset{`}{ı}}, c_{2 \overset{`}{ı} + 1}) \\ (+ b μ_{3} d_{b} (c_{2 \overset{`}{ı} + 1}, c_{2 \overset{`}{ı} + 2}) \\ + μ_{4} \frac{d_{b} (c_{2 \overset{`}{ı}}, c_{2 \overset{`}{ı} + 1}) . d_{b} (c_{2 \overset{`}{ı} + 1}, c_{2 \overset{`}{ı} + 2})}{1 + d_{b} (c_{2 \overset{`}{ı}}, c_{2 \overset{`}{ı} + 1})} \\ \leq A ((μ_{1} + μ_{2} + b μ_{3}) d_{b} (c_{2 \overset{`}{ı}}, c_{2 \overset{`}{ı} + 1}) \\ + (b μ_{3} + μ_{4}) d_{b} (c_{2 \overset{`}{ı} + 1}, c_{2 \overset{`}{ı} + 2})) . \end{matrix}$

This implies $\begin{matrix} A (d_{b} (c_{2 \overset{`}{ı} + 1}, c_{2 \overset{`}{ı} + 2})) < A ((μ_{1} + μ_{2} + b μ_{3}) d_{b} \\ (c_{2 \overset{`}{ı}}, c_{2 \overset{`}{ı} + 1}) + (b μ_{3} + μ_{4}) d_{b} (c_{2 \overset{`}{ı} + 1}, c_{2 \overset{`}{ı} + 2})) . \end{matrix}$

As A is strictly increasing. So, we have $\begin{matrix} d_{b} (c_{2 \overset{`}{ı} + 1}, c_{2 \overset{`}{ı} + 2}) < (μ_{1} + μ_{2} + b μ_{3}) d_{b} (c_{2 \overset{`}{ı}}, c_{2 \overset{`}{ı} + 1}) \\ + (b μ_{3} + μ_{4}) d_{b} (c_{2 \overset{`}{ı} + 1}, c_{2 \overset{`}{ı} + 2}) . \end{matrix}$

Which implies

$\begin{matrix} (1 - b μ_{3} - μ_{4}) d_{b} (c_{2 \overset{`}{ı} + 1}, c_{2 \overset{`}{ı} + 2}) \\ < (μ_{1} + μ_{2} + b μ_{3}) d_{b} (c_{2 \overset{`}{ı}}, c_{2 \overset{`}{ı} + 1}) \\ d_{b} (c_{2 \overset{`}{ı} + 1}, c_{2 \overset{`}{ı} + 2}) \\ < (\frac{μ_{1} + μ_{2} + b μ_{3}}{1 - b μ_{3} - μ_{4}}) d_{b} (c_{2 \overset{`}{ı}}, c_{2 \overset{`}{ı} + 1}) . \end{matrix}$

As $η = \frac{μ_{1} + μ_{2} + b μ_{3}}{1 - b μ_{3} - μ_{4}} < 1 .$

Hence $\begin{matrix} d_{b} (c_{2 \overset{`}{ı} + 1}, c_{2 \overset{`}{ı} + 2}) \\ < η d_{b} (c_{2 \overset{`}{ı}}, c_{2 \overset{`}{ı} + 1}) < η^{2} d_{b} (c_{2 \overset{`}{ı} - 1}, c_{2 \overset{`}{ı}}) \\ < \dots < η^{2 i + 1} d_{b} (c_{0}, c_{1}) . \end{matrix}$ Similarly, if j is even, we have $d_{b} (c_{2 \overset{`}{ı} + 2}, c_{2 \overset{`}{ı} + 3}) < η^{2 i + 2} d_{b} (c_{0}, c_{1}) .$ Now, we have $d_{b} (c_{j}, c_{j + 1}) < η^{j} d_{b} (c_{0}, c_{1}) for some j \in ℕ .$ (2.3) Now, $\begin{matrix} d_{b} (c_{0}, c_{j + 1}) \leq \\ {bd}_{b} (c_{0}, c_{1}) + b^{2} d_{b} (c_{1}, c_{2}) + \dots + b^{j + 1} d_{b} (c_{j}, c_{j + 1}) \\ \leq {bd}_{b} (c_{0}, c_{1}) + b^{2} η (d_{b} (c_{0}, c_{1})) + \dots \\ + b^{j + 1} η^{j + 1} (d_{b} (c_{0}, c_{1})), (by (2.3)) \end{matrix}$

μ ₁, μ ₂, μ ₃, μ ₄ > 0, b ≥ 1 and bμ ₁ + bμ ₂ + (1 + b) bμ ₃ + μ ₄ < 1, so |bη| < 1 . Then, we have

$d_{b} (c_{0}, c_{j + 1}) \leq (\frac{b (1 - (b η)^{j + 1})}{1 - b η}) η (1 - b η) r < r,$ which implies $c_{j + 1} \in \bar{B_{d_{b}} (c_{0}, r)} .$ Hence, by induction $c_{n} \in \bar{B_{d_{b}} (c_{0}, r)}$ for all n ∈ N. Also α (c _n, c _n+1) ≥1 for all $n \in ℕ \cup {0}$ . Now, $d_{b} (c_{n}, c_{n + 1}) < η^{n} d_{b} (c_{0}, c_{1}) for all n \in ℕ .$ (2.4) Now, for any positive integers m, n (n > m), we have $\begin{matrix} d_{b} (c_{m}, c_{n}) \leq b (d_{b} (c_{m}, c_{m + 1})) + b^{2} (d_{b} (c_{m + 1}, c_{m + 2})) \\ + \dots + b^{n - m} (d_{b} (c_{n - 1}, c_{n})), \\ < b η^{m} d_{b} (c_{0}, c_{1}) + b^{2} η^{m + 1} d_{b} (c_{0}, c_{1}) + \dots \\ + b^{n - m} η^{n - 1} d_{b} (c_{0}, c_{1}), (by (2.4)) \\ < b η^{m} (1 + b η + \dots) d_{b} (c_{0}, c_{1}) \\ d_{b} (c_{m}, c_{n}) < (\frac{b η^{m}}{1 - b η}) d_{b} (c_{0}, c_{1}) \to 0 as m \to \infty . \end{matrix}$

Hence {T _β S _σ (c _n)} is a Cauchy sequence in $\bar{B_{d_{b}} (c_{0}, r)}$ . Since $(\bar{B_{d_{b}} (c_{0}, r)}, d_{b})$ is a complete metric space, so there exist $u \in \bar{B_{d_{b}} (c_{0}, r)}$ such that {T _β S _σ (c _n)} → u as n → ∞ , then $lim_{n \to \infty} d_{b} (c_{n}, u) = 0 .$ (2.5) By assumption, α (c _n, u) ≥1. Suppose that

d _b (u, [T _β u] _γ(u)) >0, then there exists a positive integer k such that d _b (c _n, [T _β u] _γ(u)) >0 for all n ≥ k. For n ≥ k, we have $\begin{matrix} d_{b} (u, [T_{β} u]_{γ (u)}) \leq d_{b} (u, c_{2 n + 1}) \\ + d_{b} (c_{2 n + 1}, [T_{β} u]_{γ (u)}) . \end{matrix}$

Now, there exists some e ∈ Ω such that c _2n+1 ∈ [S _e c _2n] _{γ(c
_2n)} and d _b (c _2n, [S _e c _2n] _{γ(c
_2n)}) = d _b (c _2n, c _2n+1). By using Lemma 1.12 and inequality (2.1), we have

$\begin{matrix} d_{b} (u, [T_{β} u]_{γ (u)}) \leq \\ {bd}_{b} (u, c_{2 n + 1}) + {bH}_{d_{b}} ([S_{e} c_{2 n}]_{γ (c_{2 n})}, [T_{β} u]_{γ (u)}), \\ for some β \in Φ \\ < {bd}_{b} (u, c_{2 n + 1}) + b μ_{1} d_{b} (c_{2 n}, u) \\ + b μ_{2} d_{b} (c_{2 n}, [S_{e} c_{2 n}]_{γ (c_{2 n})}) \\ + b μ_{3} d_{b} (c_{2 n}, [T_{β} u]_{γ (u)}) \\ + b μ_{4} \frac{d_{b} (c_{2 n}, [S_{e} c_{2 n}]_{γ (c_{2 n})}) . d_{b} (u, [T_{β} u]_{γ (u)})}{1 + d_{b} (c_{2 n}, u)} . \end{matrix}$ Letting n → ∞ , and by using (2.5) we get $\begin{matrix} \begin{matrix} d_{b} (u, [T_{β} u]_{γ (u)}) & < & b μ_{3} d_{b} (u, [T_{β} u]_{γ (u)}) \\ < & d_{b} (u, [T_{β} u]_{γ (u)}), \end{matrix} \end{matrix}$ which is a contradiction. So our supposition is wrong. Hence d _b (u, [T _β u] _γ(u)) =0 or u ∈ [T _β u] _γ(u) for all β ∈ Φ . Similarly, by using Lemma 1.12 and inequality (2.1), we can show that d _b (u, [S _σ u] _γ(u)) =0 or u ∈ [S _σ u] _γ(u) for all σ ∈ Ω . Hence the S _σ and T _β have a common fuzzy fixed point u in $\bar{B_{d_{b}} (c_{0}, r)}$ for all σ ∈ Ω and β ∈ Φ . Now, $d_{b} (u, u) \leq {bd}_{b} (u, [T_{β} u]_{γ (u)}) + {bd}_{b} ([T_{β} u]_{γ (u)}, u) \leq 0 .$ This implies that d _b (u, u) =0 .

2.2 Example

Let M = Q ⁺ ∪ {0} and let d _b : M × M → M be the complete D . B . M space defined by $d_{b} (u, v) = (u + v)^{2} for all u, v \in M .$ with b = 2 . Define S _m, T _n : M × M → F (M) be two families of fuzzy mappings for S _m (u) , T _n (u) ∈ F (M) and λ, μ ∈ (0, 1] by $S_{m} (u) (t) = {\begin{matrix} λ if 0 \leq t < \frac{u}{4 m} \\ \frac{λ}{2} if \frac{u}{4 m} \leq t \leq \frac{u}{3 m} \\ \frac{λ}{3} if \frac{u}{3 m} < t \leq \frac{u}{2 m} \\ 0 if \frac{u}{2 m} < t \leq 1 \end{matrix} where m = 1, 2, 3, \dots$ and

$T_{n} (u) (t) = {\begin{matrix} μ if 0 \leq t < \frac{u}{3 n} \\ \frac{μ}{4} if \frac{u}{3 n} \leq t \leq \frac{u}{2 n} \\ \frac{μ}{5} if \frac{u}{2 n} < t \leq \frac{u}{n} \\ 0 if \frac{u}{n} < t \leq 1 \end{matrix} where n = 1, 2, 3, \dots$ Consider, u ₀ = 1, r = 16, then $\bar{B_{d_{b}} (u_{0}, r)} = [0, 3] .$ Take ${[S_{m} u]}_{\frac{λ}{2}} = [\frac{u}{4 m}, \frac{u}{3 m}] and {[T_{n} u]}_{\frac{μ}{4}} = [\frac{u}{3 n}, \frac{u}{2 n}] .$ Now, $d_{b} (x_{0}, [S_{1} u_{0}]_{\frac{λ}{2}}) = d_{b} (1, [S_{1} 1]_{\frac{λ}{2}}) = d_{b} (1, \frac{1}{4}) .$ So $u_{1} = \frac{1}{4} .$ Now, $d_{b} (u_{1}, [T_{1} u_{1}]_{\frac{μ}{4}}) = d_{b} (\frac{1}{3}, [T_{1} \frac{1}{4}]_{\frac{μ}{4}}) = d_{b} (\frac{1}{4}, \frac{1}{12}) .$ So $u_{2} = \frac{1}{12} .$ Now, $d_{b} (u_{2}, [S_{2} u_{2}]_{\frac{λ}{2}}) = d_{b} (\frac{1}{12}, [S_{2} \frac{1}{12}]_{\frac{λ}{2}}) = d_{b} (\frac{1}{12}, \frac{1}{96}) .$ So $u_{3} = \frac{1}{96} .$ Continuing in this way, we have ${T_{n} S_{m} (u_{n})} = {1, \frac{1}{4}, \frac{1}{12}, \frac{1}{96} . . . .} .$ Take $μ_{1} = \frac{1}{10},$ $μ_{2} = \frac{1}{20},$ $μ_{3} = \frac{1}{80},$ $μ_{4} = \frac{1}{40},$ then bμ ₁ + bμ ₂ + (1 + b) bμ ₃ + μ ₄ < 1 and $η = \frac{7}{38} .$ Now

$d_{b} (u_{0}, [S_{1} u_{0}]_{\frac{λ}{2}}) = \frac{25}{16} < \frac{7}{38} (1 - \frac{14}{38}) 16 = η (1 - b η) r .$

Consider the mapping α : M × M → [0, ∞) by $α (u, v) = {\begin{matrix} 1 if u > v \\ \frac{1}{4} otherwise \end{matrix}$ Now, if $u, v \in \bar{B_{d_{b}} (u_{0}, r)} \cap {T_{n} S_{m} (x_{n})}$ with α (u, v) ≥1, we have $\begin{matrix} H_{d_{b}} ([S_{m} u]_{\frac{λ}{2}}, [T_{n} v]_{\frac{μ}{4}}) \leq \\ \frac{1}{10} d_{b} (u, v) + \frac{1}{20} d_{b} (u, [S_{m} u]_{\frac{λ}{2}}) + \frac{1}{80} d_{b} (u, [T_{n} v]_{\frac{μ}{4}}) + \frac{1}{40} \frac{d_{b} (u, [S_{m} u]_{\frac{λ}{2}}) . d_{b} (v, [T_{n} v]_{\frac{μ}{4}})}{1 + d_{b} (u, v)} = \frac{1}{10} d_{b} (u, v) + \frac{1}{20} d_{b} (u, [\frac{u}{4 m}, \frac{u}{3 m}]) + \frac{1}{80} d_{b} (u, [\frac{v}{3 n}, \frac{v}{2 n}]) + \frac{1}{40} \frac{d_{b} (u, [\frac{u}{4 m}, \frac{u}{3 m}]) . d_{b} (v, [\frac{v}{3 n}, \frac{v}{2 n}])}{1 + d_{b} (u, v)} . H_{d_{b}} ([S_{m} u]_{\frac{λ}{2}}, [T_{n} v]_{\frac{μ}{4}}) \leq \\ \frac{1}{10} (u + v)^{2} + \frac{1}{20} (u + \frac{u}{4 m})^{2} + \frac{1}{80} (u + \frac{v}{3 n})^{2} + \frac{1}{40} \frac{(u + \frac{u}{4 m})^{2} . (v + \frac{v}{3 n})^{2}}{1 + (u + v)^{2}} . \end{matrix}$ $\begin{matrix} H_{d_{b}} ([S_{σ} e]_{γ (e)}, [T_{β} y]_{γ (y)}) < \\ μ_{1} d_{b} (e, y) + μ_{2} d_{b} (e, [S_{σ} e]_{γ (e)}) \end{matrix}$ Thus, $\begin{matrix} \begin{matrix} + μ_{3} d_{b} (e, [T_{β} y]_{γ (y)}) \\ + μ_{4} \frac{d_{b} (e, [S_{σ} e]_{γ (e)}) . d_{b} (y, [T_{β} y]_{γ (y)})}{1 + d_{b} (e, y)} \end{matrix} \end{matrix}$ which implies that, for any $τ \in (0, \frac{6}{47}]$ and for a strictly increasing mapping A (s) = ln s, we have $\begin{matrix} τ + A (H_{d_{b}} ([S_{σ} e]_{γ (e)}, [T_{β} y]_{γ (y)})) \\ \leq A (μ_{1} d_{b} (e, y) + μ_{2} d_{b} (e, [S_{σ} e]_{γ (e)}) \\ + μ_{3} d_{b} (e, [T_{β} y]_{γ (y)}) \\ + μ_{4} \frac{d_{b} (e, [S_{σ} e]_{γ (e)}) . d_{b} (y, [T_{β} y]_{γ (y)})}{1 + d_{b} (e, y)}) . \end{matrix}$ Thus all the conditions of Theorem 2.1 are satisfied. Hence S _σ and T _β have a common fuzzy fixed point for all σ ∈ Ω and β ∈ Φ.

If, we take {S _σ : σ ∈ Ω} = {T _β : β ∈ Φ} in Theorem 2.1, then we have the following result.

2.3 Corollary

Let (M, d _b) be a complete D . B . M space with constant b ≥ 1. Let r > 0, $c_{0} \in \bar{B_{d_{b}} (c_{0}, r)} \subseteq M,$ α : M × M → [0, ∞) and {S _σ : σ ∈ Ω} be a family of α _∗-dominated fuzzy mappings from M to F (M) on $\bar{B_{d_{b}} (c_{0}, r)} .$ Suppose that the following satisfy:

i) There exist τ, μ ₁, μ ₂, μ ₃, μ ₄ > 0 satisfying bμ ₁ + bμ ₂ + (1 + b) bμ ₃ + μ ₄ < 1 and a strictly increasing mapping A such that $\begin{matrix} τ + A (H_{d_{b}} ([S_{σ} e]_{γ (e)}, [S_{β} y]_{γ (y)})) \\ \leq A (μ_{1} d_{b} (e, y) + μ_{2} d_{b} (e, [S_{σ} e]_{γ (e)}) \\ + μ_{3} d_{b} (e, [S_{β} y]_{γ (y)}) \\ + μ_{4} \frac{d_{b} (e, [S_{σ} e]_{γ (e)}) . d_{b} (y, [S_{β} y]_{γ (y)})}{1 + d_{b} (x, y)}) \end{matrix}$ (2.6) whenever $e, y \in \bar{B_{d_{b}} (c_{0}, r)} \cap {S_{σ} (c_{n})}$ , α (e, y) ≥1, σ, β ∈ Ω, γ (e) ∈ (0, 1] and H _{d
_b} ([S _σ e] _γ(e), [S _β y] _γ(y)) >0 .

ii) If $η = \frac{μ_{1} + μ_{2} + b μ_{3}}{1 - b μ_{3} - μ_{4}},$ then $d_{b} (c_{0}, [S_{σ} c_{0}]_{γ (c_{0})}) \leq η (1 - b η) r .$ Then {S _σ (c _n)} is a sequence in $\bar{B_{d_{b}} (c_{0}, r)}$ , α (c _n, c _n+1) ≥1 for all $n \in ℕ \cup {0}$ and ${S_{σ} (c_{n})} \to u \in \bar{B_{d_{b}} (c_{0}, r)} .$ Also, if u satisfies (2.6) and either α (c _n, u) ≥1 or α (u, c _n) ≥1 for all $n \in ℕ \cup {0}$ , then {S _σ : σ ∈ Ω} have common fuzzy fixed point u in $\bar{B_{d_{b}} (c_{0}, r)}$ . Moreover, d _b (u, u) =0 .

If, we take μ ₂ = 0 in Theorem 2.1, then we have the following result.

2.4 Corollary

i) There exist τ, μ ₁, μ ₃, μ ₄ > 0 satisfying bμ ₁ + (1 + b) bμ ₃ + μ ₄ < 1 and a strictly increasing mapping A such that $\begin{matrix} τ + A (H_{d_{b}} ([S_{σ} e]_{γ (e)}, [T_{β} y]_{γ (y)})) \\ \leq A (μ_{1} d_{b} (e, y) + μ_{3} d_{b} (e, [T_{β} y]_{γ (y)}) \\ + μ_{4} \frac{d_{b} (e, [S_{σ} e]_{γ (e)}) . d_{b} (y, [T_{β} y]_{γ (y)})}{1 + d_{b} (e, y)}), \end{matrix}$ (2.7) whenever $e, y \in \bar{B_{d_{b}} (c_{0}, r)} \cap {T_{β} S_{σ} (c_{n})},$ α (e, y) ≥1, σ ∈ Ω, β ∈ Φ, γ (e) ∈ (0, 1] and

H _{d
_b} ([S _σ e] _γ(e), [T _β y] _γ(y)) >0 .

ii) If $η = \frac{μ_{1} + b μ_{3}}{1 - b μ_{3} - μ_{4}},$ then $d_{b} (c_{0}, [S_{σ} c_{0}]_{γ (c_{0})}) \leq η (1 - b η) r .$ Then {TS _σ (c _n)} is a sequence in $\bar{B_{d_{b}} (c_{0}, r)}$ , α (c _n, c _n+1) ≥1 for all $n \in ℕ \cup {0}$ and ${T_{β} S_{σ} (c_{n})} \to u \in \bar{B_{d_{b}} (c_{0}, r)} .$ Also, if u satisfies (2.7) and either α (c _n, u) ≥1 or α (u, c _n) ≥1 for all $n \in ℕ \cup {0}$ , then S _σ and T _β have common fuzzy fixed point u in $\bar{B_{d_{b}} (c_{0}, r)}$ for all σ ∈ Ω and β ∈ Φ . Moreover, d _b (u, u) =0 .

If, we take μ ₃ = 0 in Theorem 2.1, then we have the following result.

2.5 Corollary

i) There exist τ, μ ₁, μ ₂, μ ₄ > 0 satisfying bμ ₁ + bμ ₂ + μ ₄ < 1 and a strictly increasing mapping A such that $\begin{matrix} τ + A (H_{d_{b}} ([S_{σ} e]_{γ (e)}, [T_{β} y]_{γ (y)})) \\ \leq A (μ_{1} d_{b} (e, y) + μ_{2} d_{b} (e, [S_{σ} e]_{γ (e)}) \\ + μ_{4} \frac{d_{b} (e, [S_{σ} e]_{γ (e)}) . d_{b} (y, [T_{β} y]_{γ (y)})}{1 + d_{b} (e, y)}), \end{matrix}$ (2.8) whenever $e, y \in \bar{B_{d_{b}} (c_{0}, r)} \cap {T_{β} S_{σ} (c_{n})}$ , σ ∈ Ω, β ∈ Φ, α (e, y) ≥1, γ (e) ∈ (0, 1] and H _{d
_b} ([S _σ e] _γ(e), [T _β y] _γ(y)) >0 .

ii) If $η = \frac{μ_{1} + μ_{2}}{1 - μ_{4}},$ then $d_{b} (c_{0}, [S_{σ} c_{0}]_{γ (c_{0})}) \leq η (1 - b η) r .$ Then {T _β S _σ (c _n)} is a sequence in $\bar{B_{d_{b}} (c_{0}, r)}$ , α (c _n, c _n+1) ≥1 for all $n \in ℕ \cup {0}$ and ${T_{β} S_{σ} (c_{n})} \to u \in \bar{B_{d_{b}} (c_{0}, r)} .$ Also, if u satisfies (2.8) and either α (c _n, u) ≥1 or α (u, c _n) ≥1 for all $n \in ℕ \cup {0}$ , then S _σ and T _β have common fuzzy fixed point u in $\bar{B_{d_{b}} (c_{0}, r)}$ for all σ ∈ Ω and β ∈ Φ . Moreover, d _b (u, u) =0 .

If, we take μ ₄ = 0 in Theorem 2.1, then we have the following result.

2.6 Corollary

i) There exist τ, μ ₁, μ ₂, μ ₃ > 0 satisfying bμ ₁ + bμ ₂ + (1 + b) bμ ₃ < 1 and a strictly increasing mapping A such that $\begin{matrix} τ + A (H_{d_{b}} ([S_{σ} e]_{γ (e)}, [T_{β} y]_{γ (y)})) \\ \leq A (μ_{1} d_{b} (e, y) + μ_{2} d_{b} (e, [S_{σ} e]_{γ (e)}) \\ + μ_{3} d_{b} (e, [T_{β} y]_{γ (y)})) \end{matrix}$ (2.9) whenever $e, y \in \bar{B_{d_{b}} (c_{0}, r)} \cap {T_{β} S_{σ} (c_{n})},$ α (e, y) ≥1, σ ∈ Ω, β ∈ Φ, γ (y) ∈ M and H _{d
_b} ([S _σ e] _γ(e), [T _β y] _γ(y)) >0 .

ii) If $η = \frac{μ_{1} + μ_{2} + b μ_{3}}{1 - b μ_{3}},$ then $d_{b} (c_{0}, [S_{σ} c_{0}]_{γ (c_{0})}) \leq η (1 - b η) r .$ Then, {T _β S _σ (c _n)} is a sequence in $\bar{B_{d_{b}} (c_{0}, r)}$ , α (c _n, c _n+1) ≥1 for all $n \in ℕ \cup {0}$ and ${T_{β} S_{σ} (c_{n})} \to u \in \bar{B_{d_{b}} (c_{0}, r)} .$ Also, if u satisfies (2.9) and either α (c _n, u) ≥1 or α (u, c _n) ≥1 for all $n \in ℕ \cup {0}$ , then S _σ and T _β have common fuzzy fixed point u in $\bar{B_{d_{b}} (c_{0}, r)}$ for all σ ∈ Ω and β ∈ Φ . Moreover, d _b (u, u) =0 .

3 Fixed Point Results For Multivalued Mappings

In this part of the paper, we proved that fixed point for multivalued mappings can be derived by utilizing Theorem 2.1 in dislocated b-metric space.

3.1 Theorem

Let (M, d _b) be a complete D . B . M space with constant b ≥ 1. Let r > 0, $c_{0} \in \bar{B_{d_{b}} (c_{0}, r)} \subseteq M,$ α : M × M → [0, ∞) and {Q _σ : σ ∈ Ω}, {R _β : β ∈ Φ} be two families of α _∗-dominated multivalued mappings from M to P (M) on $\bar{B_{d_{b}} (c_{0}, r)} .$ Suppose that the following satisfy:

i) There exist τ, μ ₁, μ ₂, μ ₃, μ ₄ > 0 satisfying bμ ₁ + bμ ₂ + (1 + b) bμ ₃ + μ ₄ < 1 and a strictly increasing mapping A such that

$\begin{matrix} τ + A (H_{d_{b}} (Q_{σ} e, R_{β} y)) \\ \leq A (μ_{1} d_{b} (e, y) + μ_{2} d_{b} (e, Q_{σ} e) + μ_{3} d_{b} (e, R_{β} y) \\ + μ_{4} \frac{d_{b} (e, Q_{σ} e) . d_{b} (y, R_{β} y)}{1 + d_{b} (e, y)}), \end{matrix}$ (3.1) whenever $e, y \in \bar{B_{d_{b}} (c_{0}, r)} \cap {R_{β} Q_{σ} (c_{n})},$ α (e, y) ≥1, σ ∈ Ω, β ∈ Φ and H _{d
_b} (Q _σ e, R _β y) >0.

ii) If $η = \frac{μ_{1} + μ_{2} + b μ_{3}}{1 - b μ_{3} - μ_{4}},$ then $d_{b} (c_{0}, Q_{a} c_{0}) \leq η (1 - b η) r .$ (3.2) Then, {R _β Q _σ (c _n)} is a sequence in $\bar{B_{d_{b}} (c_{0}, r)}$ , α (c _n, c _n+1) ≥1 for all $n \in ℕ \cup {0}$ and ${R_{β} Q_{σ} (c_{n})} \to u \in \bar{B_{d_{b}} (c_{0}, r)} .$ Also, if u satisfies (3.1) and either α (c _n, u) ≥1 or α (u, c _n) ≥1 for all $n \in ℕ \cup {0}$ , then Q _σ and R _β have common fixed point u in $\bar{B_{d_{b}} (c_{0}, r)}$ for all σ ∈ Ω and β ∈ Φ . Moreover, d _b (u, u) =0 .

Proof. Consider an arbitrary mapping θ : M → (0, 1]. Consider {S _σ : σ ∈ Ω} and {T _β : β ∈ Φ} be two family of fuzzy mappings from M to F (M) defined as $(S_{σ} e) (t) = {\begin{matrix} θ (e), if t \in Q_{σ} e \\ 0, if t \notin Q_{σ} e \end{matrix}$ and $(T_{β} e) (t) = {\begin{matrix} θ (e), if t \in R_{β} e \\ 0, if t \notin R_{β} e \end{matrix} .$ We obtain $[S_{σ} e]_{γ (e)} = {t : S_{σ} (e) (t) \geq θ (e)} = Q_{σ} e$ and $[T_{β} e]_{α (e)} = {t : T_{β} (e) (t) \geq θ (e)} = R_{β} e .$ So, the condition (3.1) reduced to (2.1) of Theorem 2.1. So, there is $u \in [S_{σ} u]_{γ (u)} \cap [T_{β} u]_{α (u)} = Q_{σ} u \cap R_{β} u .$

4 Application to the systems of integral equations

4.1 Theorem

Let (X, d _b) be a complete D . B . M space with constant b ≥ 1. Let c ₀ ∈ X and {S _β : β∈ Ω } be a family of mappings from X to X . Assume that, There exist τ, μ ₁, μ ₂, μ ₃, μ ₄ > 0 satisfying bμ ₁ + bμ ₂ + μ ₃ + μ ₄ < 1 and a strictly increasing mapping A such that the following holds:

$\begin{matrix} τ + A (d_{b} (S_{α} x, S_{β} y)) \\ \leq A (\begin{matrix} μ_{1} d_{b} (x, y) + μ_{2} d_{b} (x, S_{α} x) \\ + μ_{3} d_{b} (y, S_{β} y) + μ_{4} \frac{d_{b}^{2} (x, S_{α} x) . d_{b} (y, S_{β} y)}{1 + d_{b}^{2} (x, y)} \end{matrix}), \end{matrix}$

for all x, y ∈ X and d (S _α x, S _β y) >0 where α, β ∈ Ω with α ≠ β . Also if the inequality (4.1) holds for u, then the family {S _β : β∈ Ω } has a unique common fixed point m in X.

Proof: The proof of this Theorem is similar as Theorem 2.1. We have to prove the uniqueness only. Let w be another common fixed point of the family {S _β : β ∈ Ω } . Suppose d _b (S _α m, S _β w) >0. Then, we have

$\begin{matrix} τ + A (d_{b} (S_{α} m, S_{β} w)) \\ \leq A (\begin{matrix} μ_{1} d_{b} (m, w) + μ_{2} d_{b} (m, S_{α} m) \\ + μ_{3} d_{b} (w, S_{β} w) + μ_{4} \frac{d_{b} (m, S_{α} m) . d_{b} (w, S_{β} w)}{1 + d_{b} (m, w)} \end{matrix}) . \end{matrix}$

This implies that $d_{b} (m, w) < μ_{1} d_{b} (m, w) < d_{b} (m, w),$ which is a contradiction. So d _b (S _α m, S _β w) =0 . Hence m = w .

In this section, we discuss the application of Theorem 4.1. Consider a system of Volterra type integral equations. $m (k) = \int_{0}^{q} H_{α} (q, r, m (r)) dr,$ (4.2) for all q, r ∈ [0, 1] and α ∈ Ω . We discuss about the unique common solution of (4.2). Let $X = C ([0, 1], ℝ)$ be the set of all real valued continuous functions on [0, 1], endowed with the complete dislocated b-metric. For $m \in C ([0, 1], ℝ),$ define supremum norm as: $| | m | |_{τ} = sup_{q \in [0, 1]} {| m (q) | e^{- τ q}}$ , where τ > 0 is taken arbitrary. Then define $d_{τ} (m, g) = {[sup_{q \in [0, 1]} {| m (q) + g (q) | e^{- τ q}}]}^{2} = | | m + g | |_{τ}^{2}$ for all $m, g \in C ([0, 1], ℝ),$ with these settings, $(C ([0, 1], ℝ), d_{τ})$ becomes a complete D . B . M . S.

Now we prove the following theorem to ensure the existence of solution of integral equation.

4.2 Theorem

Assume the following conditions are satisfied:

(i) $H_{α} : [0, 1] \times [0, 1] \times ℝ \to ℝ$ ;

(ii) S _α : X → P (X) where α ∈ Ω defined as $S_{α} m (q) = \int_{0}^{q} H_{α} (q, r, m (r)) dr .$ Suppose there exist τ > 0, such that

$\begin{matrix} | H_{α} (q, r, m (r)) + H_{β} (q, r, g (r)) | \\ \leq \frac{τ N_{(α, β)} (m (r), g (r))}{τ | | N_{(α, β)} (m, g) | |_{τ} + 1} \end{matrix}$

for all q, r ∈ [0, 1] and $m, g \in C ([0, 1], ℝ),$ where

$\begin{matrix} N_{(α, β)} (m (r), g (r)) \\ = μ_{1} [| m (r) + g (r) |]^{2} + μ_{2} [| m (r) + S_{α} m (r) |]^{2} \\ + μ_{3} [| g (r) + S_{β} g (r) |]^{2} \\ + μ_{4} \frac{[| m (r) + S_{α} m (r) |]^{2} . [| g (r) + S_{β} g (r) |]^{2}}{1 + [| m (r) + g (r) |]^{2}}, \end{matrix}$

where μ ₁, μ ₂, μ _3, μ ₄ ≥ 0, and bμ ₁ + bμ ₂ + μ ₃ + μ ₄ < 1 . Then integral equations given in (4.2) have a unique common solution.

Proof: By assumption (ii) $\begin{matrix} | S_{α} m (q) + S_{β} g (q) | \\ = \int_{0}^{q} | H_{α} (q, r, m (r) + H_{β} (q, r, g (r))) | dr, \\ \leq \int_{0}^{q} \frac{τ}{τ | | N_{(α, β)} (m, g) | |_{τ} + 1} ([N_{(α, β)} (m (r), g (r))] e^{- τ r}) e^{τ r} dr \\ \leq \int_{0}^{q} \frac{τ}{τ | | N_{(α, β)} (m, g) | |_{τ} + 1} | | N_{(α, β)} (m, g) | |_{τ} e^{τ r} dr \end{matrix}$

$\begin{matrix} \leq \frac{τ | | N_{(α, β)} (m, g) | |_{τ}}{τ} | | \\ N_{(α, β)} (m, g) | |_{τ} + 1 \int_{0}^{q} e^{τ r} dr \\ \leq \frac{| | N_{(α, β)} (m, g) | |_{τ}}{τ | | N_{(α, β)} (m, g) | |_{τ} + 1} e^{τ q} . \end{matrix}$

$τ + \frac{1}{| | N_{(α, β)} (m, g) | |_{τ}} \leq \frac{1}{| | S_{α} m (q) + S_{β} g (q) | |_{τ}} .$ Which further implies $τ - \frac{1}{| | S_{α} m (q) + S_{β} g (q) | |_{τ}} \leq \frac{- 1}{| | N_{(α, β)} (m, g) | |_{τ}} .$ So all the conditions of Theorem 4.1 are satisfied for $F (g) = \frac{- 1}{\sqrt{g}}; g$ >0 and $d_{τ} (m, g) = | | m + g | |_{τ}^{2}$ . Hence integral equations given in (4.2) have a unique common solution.

5 Conclusion and comparison

In the present paper, we have achieved fixed point results for a pair of families of fuzzy generalized A- dominated contractive mappings on an intersection of a closed ball and a sequence, which is a weaker class rather than class of mappings F used by Wardowski [33]. We have used α _∗-dominated mappings which are more general than α _∗-admissible mappings and showed that fixed point exists even if the contractive condition holds on subspaces rather than whole spaces. Moreover, we investigate our results in a more better framework. Examples and application are given to demonstrate the variety of our results. New results in ordered spaces, partial b-metric space, dislocated metric space, partial metric space, b-metric space and metric space can be obtained as corollaries of our results. Many fixed point results for families of multivalued and singlevalued contractive mappings can also be obtained as corollaries of our results. One can further extend our results to L-fuzzy mappings, bipolar fuzzy mappings and fuzzy neutrosophic soft mappings. Moreover applications in decision-making and fuzzy functional inclusions can be investigated (see [14 , 36–40]).

Footnotes

Acknowledgements

The authors would like to thank the Editor, the Associate Editor and the anonymous referees for sparing their valuable time for reviewing this article.

References

Acar

Ö.

, Durmaz

, Minak

, Generalized multivalued F–contractions on complete metric spaces, Bull Iranian Math Society 40 (2014), 1469–1478.

Ameer

, Arshad

, Two new generalization for F-contraction on closed ball and fixed point theorem with application, J Mathematical Exten 11 (2017), 1–24.

Arshad

, Khan

S.U.

, Ahmad

, Fixed point results for F-contractions involving some new rational expressions, JP Journal of Fixed Point Theory and Appl 11 (2016), 79–97.

Boriceanu

, Fixed Point theory for multivalued generalized contraction on a set with two b-metrics, studia Univ Babes, Bolya: Math LIV 3 (2009), 1–14.

Butnariu

, Fixed point for fuzzy mapping, Fuzzy Sets Systems 7 (1982), 191–207.

Chen

, Wen

, Dong

, Gu

, Fixed point theorems for generalized F-contractions in b-metric-like spaces, J Nonlinear Sci Appl 9 (2016), 2161–2174.

Czerwik

, Contraction mappings in b-metric spaces, Acta Math Inform Univ Ostraviensis 1 (1993).

Heilpern

, Fuzzy mappings and fixed point theorem, Journal of Mathematical Analysis and Applications 83 (1981), 566–569.

Hussain

, Ahmad

, Azam

, On Suzuki-Wardowski type fixed point theorems, J Nonlinear Sci Appl 8 (2015), 1095–1111.

10.

Hussain

, Al-Mezel

, Salimi

, Fixed points for ψ-graphic contractions with application to integral equations, Abst Applied Anal 2013.

11.

Hussain

, Salimi

, Suzuki-Wardowski type fixed point theorems for α-GF-contractions, Taiwanese J Math 20 (2014).

12.

Hussain

, Roshan

J.R.

, Paravench

, Abbas

, Common Fixed Point results for weak contractive mappings in ordered dislocated b-metric space with applications, J Ineq Appl (2013), 1–21.

13.

Jachymski

, The contraction principle for mappings on a metric space with a graph, Proc Amer Math Soc 1(136) (2008), 1359–1373.

14.

Jiang

, Zhan

, Chen

, Covering based variable precision (I,T)-fuzzy rough sets with applications to multi-attribute decision-making, IEEE Transactions on Fuzzy Systems. doi: 10.1109/TFUZZ.2018.2883023

15.

Kumamt

, Sukprasert

, Kumam

, Shoaib

, Shahzad

, Mahmood

, Some fuzzy fixed point results for fuzzy mappings in b-metric spaces, Cogent Mathematics & Statistics 5 (2018), 1–12.

16.

, Zhan

, Ali

M.I.

, Mehmood

, A survey of decision making methods based on two classes of hybrid soft set models, Artificial Intelligence Review 49(4) (2018), 511–529.

17.

Nadler

, Multivalued contraction mappings, Pac J Math 30 (1969), 475–488.

18.

Nazir

, Silvestrov

, Common fixed point results for family of generalized multivalued F-contraction mappings in ordered metric spaces, arXiv: 1606.05299vi [math GM].

19.

Piri

, Kumam

, Some fixed point theorems concerning F-contraction in complete metric spaces, Fixed Point Theory Appl (2014), 210.

20.

Piri

, Rahrovi

, Morasi

, Kumam

, Fixed point theorem for F-Khan-contractions on complete metric spaces and application to the integral equations, J Nonlinear Sci Appl 10 (2017), 4564–4573.

21.

Al Rawashdeh

, Mehmood

and Rashid

, Coincidence and common fixed points of integral contractions for L-fuzzy maps with applications in fuzzy functional inclusions, Journal of Intelligent & Fuzzy Systems 35(2) (2018), 2173–2187.

22.

Rasham

, Shoaib

, Alamri

B.S.

, Arshad

, Multivalued fixed point results for new generalized F-Dominted contractive mappings on dislocated metric space with application, Journal of Function Spaces (2018).

23.

Rasham

, Shoaib

, Hussain

, Arshad

, Khan

S.U.

, Common fixed point results for new Ciric-type rational multivalued F–contraction with an application, J Fixed Point Theory Appl 20(1) (2018).

24.

Rasham

, Shoaib

, Park

, Arshad

, Fixed point results for a pair of multi dominated mappings on a smallest subset in k-sequentially dislocated quasi metric space with application, J Comput Analysis and Appl 25(5) (2018), 975–986.

25.

Riaz

, Hashmi

M.R.

, Fixed points of fuzzy neutrosophic soft mapping with decision-making, Fixed point theory and applications 7 (2018), 1–10.

26.

Sgroi

, Vetro

, Multi-valued F–contractions and the solution of certain functional and integral equations, Filomat 27(7) (2013), 1259–1268.

27.

Shoaib

, Fixed point results for α _*-ψmultivalued mappings, Bulletin of Mathematical Analysis and Applications 8(4) (2016), 43–55.

28.

Shoaib

, Azam

, Arshad

, Shahzad

, Fixed point results for the multivalued mapping on closed ball in dislocated fuzzy metric space, J Mathematical Anal 8(2) (2017), 98–106.

29.

Shoaib

, Azam

, Shahzad

, Common fixed point results for the family of multivalued mappings satisfying contraction on a sequence in Hausdorff fuzzy metric space, J Comp Anal Appl 24(4) (2018), 692–699.

30.

Shoaib

, Hussain

, Arshad

, Azam

, Fixed point results for α _*-ψCiric type multivalued mappings on an intersection of a closed ball and a sequence with graph, J Math Anal 7(3) (2016), 41–50.

31.

Shoaib

, Kumam

, Shahzad

, Phiangsungnoen

, Mahmood

, Fixed point results for fuzzy mappings in a b-metric space, Fixed Point Theory and Applications (2018).

32.

Shahzad

, Shoaib

, Mahmood

, Fixed point theorems for fuzzy mappings in b- Metric space, Italian Journal of Pure and Applied Mathematics 38 (2017), 419–427.

33.

Wardowski

, Fixed point theory of a new type of contractive mappings in complete metric spaces, Fixed Point Theory Appl (2012).

34.

Weiss

M.D.

, Fixed points and induced fuzzy topologies for fuzzy sets, J Math Anal Appl 50 (1975), 142–150.

35.

Zadeh

L.A.

, Fuzzy sets, Information and Control 8(3) (1965), 338–353.

36.

Zhan

, Sun

, Alcantud

J.C.R.

, Covering based multigranulation (I,T)-fuzzy rough set models and applications in multi-attribute group decision-making, Information Sciences 476 (2019), 290–318.

37.

Zhang

, Zhan

, Fuzzy soft β-covering based fuzzy rough sets and corresponding decision-making applications, Int J Mach Learn Cybern (2018), https://doi/10.1007/s13042-018-0828-3

38.

Zhang

, Zhan

, Xu

Z.X.

, Covering-based generalized IF rough sets with applications to multi-attribute decision-making, Information Sciences 478 (2019), 275–302.

39.

Zhan

, Xu

, Two types of coverings based multigranulation rough fuzzy sets and applications to decision making, Artificial Intelligence Review (2018). https://doi.org/10.1007/s10462-018-9649-8

40.

Zhan

, Wang

, Certain types of soft coverings based rough sets with applications, Int J Mach Learn Cybern (2018). https://doi/10.1007/s13042-018-0785-x.