Fixed point results for fuzzy mappings on an intersection of an open ball and a sequence

Abstract

In this paper, we have achieved fixed point results for pair of fuzzy mappings satisfying Ciric type contraction on a sequence contained in an open ball in ordered left (right) K-sequentially complete dislocated quasi metric space. An example and an application are presented to demonstrate the novelty of the results. Our results generalize and extend some recent results in literature.

Keywords

Fixed point open ball fuzzy mapping generalized contraction 46S40 47H10 54H25

1 Introduction and preliminaries

Fixed point theory has a wide range of applications in various fields of analysis. The most important tool in fixed point theory is the Banach Contraction Principle. Many authors obtained different fixed point results in various metric spaces under certain contractive conditions (see [1 –28]). A set-valued mapping R from a nonempty set S to the subsets of S has a fixed point s ∈ S, if s ∈ Rs. If we take singleton subsets of S instead of subsets of S, then R represents a self-valued mapping from S to S. A self-valued mapping R : S → S has a fixed point s ∈ S, if s = Rs. The results of set-valued mappings generalize the results of self-valued mappings.

Zadeh [29] introduced a notion of fuzzy sets. Fuzzy sets have many applications in decision making (see [12 , 31]). Later on Weiss [27] and Butnariu [9] initially studied the fixed points of fuzzy mappings. Heilpern [10] presented the idea of fuzzy contraction mappings and proved a fixed point result which was the fuzzy analogue of Nadler’s fixed point result for set-valued mappings [13]. Afterward a lot of interesting results were proved in different metric spaces in the framework of fuzzy mappings, many of them useful, as can be seen in [11 , 25]. Recently very interesting results for family of fuzzy mappings are proved by Rasham et al. [14] in dislocated b-metric spaces. In this paper, they obtained common fixed point of two families of fuzzy mappings satisfying conditions only on a sequence. They also used more weaker class of strictly increasing mappings A rather than class of mappings F used by Wardowski [26].

In literature, there are many results about fixed point of certain mappings that are contractive over an entire space. Arshad et al. [4] observed that sometimes it happened that if R : S → S is not a contraction however R : S → U is a contraction, where U is a closed subset in S. They introduced a necessary and sufficient condition on closed ball to achieve common fixed point for such mappings. For more results on closed ball can be seen in (see [21 –24]).

In this paper, we extend the result of Altun et al. [2] in four different ways by using

fuzzy mappings instead of single-valued mappings;

open ball instead of whole space;

Ciric type contraction instead of Banach type contraction;

left (right) K-sequentially complete dislocated quasi metric space instead of complete metric space.

This paper extends and generalizes several comparable results in the existing literature. We give the following definitions and results which will be needed in this sequel.

Definition 1.1: [2] Let Ψ denotes the set of functions ψ : [0, ∞) → [0, ∞) satisfying the conditions:

₁) ψ is non-decreasing.

₂) For all t > 0, we have

μ_{0} (t) = \sum_{k = 0}^{\infty} ψ^{k} (t) < \infty,

where, ψ^k is the k^th iterate of ψ. Any function ψ ∈ Ψ is called a (c)-comparison function.

Lemma 1.2: [2] Let ψ ∈ Ψ . Then

i) ψ (t) < t, for all t > 0,

ii) ψ (0) = 0,

iii) ψ is continuous at t = 0,

iv) μ₀ is nondecreasing and continuous at 0.

Definition 1.3. [20] Suppose W is a nonempty set and d_lq : W × W → [0, ∞) is called a dislocated quasi metric (or simply d_lq-metric) if the following conditions hold for all c, y, z ∈ W

If d_lq (c, y) = d_lq (y, c) =0, then c = y ;

d_lq (c, y) ≤ d_lq (c, z) + d_lq (z, y).

The pair (W, d_lq) is called a dislocated quasi metric space.

It is observed that if d_lq (c, y) = d_lq (y, c) for all c, y ∈ W, then (W, d_lq) becomes a dislocated metric space (metric-like space) (W, d_l). For any c ∈ W and ɛ > 0, B_{d
_lq} (c, ɛ) = {y ∈ W : d_lq (c, y) < ɛ and d_lq (y, c) < ɛ} and $\bar{B_{d_{lq}} (c, ɛ)} = {y \in W : d_{lq} (c, y) \leq ɛ$ and d_lq (y, c) ≤ ɛ} are open and closed ball in (W, d_lq) respectively. Also B_{d
_l} (c, ɛ) = {y ∈ W : d_l (c, y) ≤ ɛ} is a closed ball in (W, d_l).

Example 1.4. [20] Let W = [0, ∞) and d_lq (c, y) = c + max {c, y} for any c, y ∈ W, then (W, d_lq) is called dislocated quasi metric space.

Definition 1.5. [20] Let (W, d_lq) be a dislocated quasi metric space. Then

(i) A sequence {c_n} in (W, d_lq) is called left (right) K-Cauchy if for all ɛ > 0, there exists n₀ ∈N such that for all n > m ≥ n₀ (respectively for all m > n ≥ n₀) , d_lq (c_m, c_n) < ɛ .

(ii) A sequence {c_n} dislocated quasi-convergent (for short d_lq -convergent) to c if $lim_{n \to \infty} d_{lq} (c_{n}, c) = lim_{n \to \infty} d_{lq} (c, c_{n}) = 0$ or for any ɛ > 0, there exists n₀ ∈N, such that for all n > n₀, d_lq (c, c_n) < ɛ and d_lq (c_n, c) < ɛ. In this case c is called a d_lq-limit of {c_n} .

(iii) A dislocated quasi metric space (W, d_lq) is called left (right) K-sequentially complete if every left (right) K-Cauchy sequence in W converges to a point c ∈ W such that d_lq (c, c) =0.

Definition 1.6. [4] Let (W, ≤ , d_q) is called an ordered dislocated quasi metric space, if

(W, d_lq) is dislocated quasi metric space

≤ is a partial order on W.

Definition 1.7: [20] Let K be a nonempty subset of a dislocated quasi metric space (W, d_lq) and let c ∈ W. An element y₀ ∈ K is called a best approximation in K if $\begin{matrix} d_{lq} (c, K) & = & d_{lq} (c, y_{0}), where d_{lq} (c, K) \\ = & inf_{y \in K} d_{lq} (c, y) \\ and d_{lq} (K, c) & = & d_{lq} (y_{0}, c), where d_{lq} (K, c) \\ = & inf_{y \in K} d_{q} (y, c) . \end{matrix}$ If each c ∈ W has at least one best approximation in K, then K is called a proximinal set. We denote P (W) is a collection of all proximinal subsets in W .

Definition 1.8: [20] The function H_{d
_lq} : P (W) × P (W) → W, defined by $H_{d_{lq}} (A, B) = max {sup_{a \in A} d_{lq} (a, B), sup_{b \in B} d_{lq} (A, b)},$ is called dislocated quasi Hausdorff metric on P (W) . Also (P (W) , H_{d
_lq}) is known as dislocated quasi Hausdorff metric space.

Definition 1.9: [15] A fuzzy set S is a function from W to [0, 1], M_q (W) is a collection of all fuzzy sets in W. If S is a fuzzy set and c ∈ W, then the function values S (c) is called the grade of membership of c in S . The γ-level set of fuzzy set S, is denoted by [S] _γ, and defined as: $\begin{array}{l} {[S]}_{γ} = {c : S (c) \geq γ} where γ \in (0, 1), \\ {[S]}_{0} = \bar{{c : S (c) > 0}} . \end{array}$ Let W be any nonempty set and Z be a metric space, then T : W → M_q (Z) is called a fuzzy mapping. A fuzzy mapping T is a fuzzy subset on W × Z with membership function T (c) (z). The function T (c) (z) is the grade of membership of z in T (c). For convenience, we denote the γ-level set of T (c) by [Tc] _γ instead of [T (c)] _γ.

Definition 1.10: [15] A point w ∈ W is called a fuzzy fixed point of a fuzzy mapping T : W → M_q (W) if there exists γ ∈ (0, 1] such that w ∈ [Tw] _γ.

Lemma 1.11: [20] Suppose (W, d_lq) is a dislocated quasi metric space and (P (W) , H_{d
_lq}) is a dislocated quasi Hausdorff metric space on P (W) . Then for all A, B ∈ P (W) and for each a ∈ A, there exists b_a ∈ B, such that H_{d
_lq} (A, B) ≥ d_lq (a, b_a) and H_{d
_lq} (B, A) ≥ d_lq (b_a, a).

Lemma 1.12: [20] Every closed ball Y in a left (right) K-sequentially complete dislocated quasi metric space W is left (right) K-sequentially complete.

2 Main result

Let (W, d_lq) be a dislocated quasi metric space, c₀ ∈ W and T : W → M_q (W) be a fuzzy mapping on W. Moreover, let α : W → [0, 1] be a real function. As [Tc₀] _α(c₀) is a proximinalset, then there exists c₁ ∈ [Tc₀] _α(c₀) such that d_lq (c₀, [Tc₀] _α(c₀)) = d_lq (c₀, c₁) and d_lq ([Tc₀] _α(c₀),c₀) = d_lq (c₁, c₀) . Now, for c₁ ∈ W, there exist c₂ ∈ [Tc₁] _α(c₁) be such that d_lq (c₁, [Tc₁] _α(c₁)) = d_lq (c₁, c₂) and d_lq ([Tc₁] _α(c₁), c₁) = d_lq (c₂, c₁) . Continuing this process, we construct a sequencec_n of points in W such that c_n+1 ∈ [Tc_n] _{α(c_n)}, d_lq (c_n, [Tc_n] _{α(c_n)}) = d_lq (c_n, c_n+1) and d_lq ([Tc_n] _{α(c_n)},c_n) = d_lq (c_n+1, c_n) . We denote this iterative sequence {WT (c_n)} and say that {WT (c_n)} is a sequence in W generated by c₀.

Theorem 2.1. Let (W, ⪯ , d_lq) be an ordered left (right) K-sequentially complete dislocated quasi metric space and S, T : W → M_q (W) be two fuzzy mappings. Suppose the following assertions hold:

(i) There exists a function μ ∈ Ψ, c₀ ∈ W, α (c) , α (y) ∈ (0, 1] and r > 0, we have $\begin{matrix} max {H_{d_{lq}} ([Tc]_{α (c)}, [Ty]_{α (y)}), H_{d_{lq}} ([Ty]_{α (y)}, \\ [Tc]_{α (c)})} \leq μ (D_{lq} (c, y)), \end{matrix}$ for all c, y ∈ B_{d
_lq} (c₀, r) ∩ {WT (c_n)} with c ≽ [Sc] _α(c), y ⪯ [Sy] _α(y), where $\begin{matrix} D_{lq} (c, y) = max {d_{lq} (c, y), d_{lq} (c, [Tc]_{α (c)}), \\ d_{lq} (y, [Ty]_{α (y)})} . \end{matrix}$

(ii) If c ∈ B_{d
_lq} (c₀, r), d_lq (c, [Tc] _α(c)) = d_lq (c, y) and d_lq ([Tc] _α(c), c) = d_lq (y, c) , then $\begin{matrix} (a) If c ⪯ [Sc]_{α (c)}, implies y ≽ [Sy]_{α (y)} \\ (b) If c ≽ [Sc]_{α (c)}, implies y ⪯ [Sy]_{α (y)} \end{matrix}$

(iii) The set G (S) = {c : c ⪯ [Sc] _α(c) and c ∈ B_{d
_lq} (c₀, r)} is closed and contains c₀.

(iv) $\begin{matrix} \sum_{i = 0}^{n} max {μ^{i} (d_{lq} (c_{1}, c_{0}), μ^{i} (d_{lq} (c_{0}, c_{1}))} \\ < r for all n \in ℕ . \end{matrix}$ Then a subsequence {c_2n} of {WT (c_n)} is a sequence in G (S) and {c_2n} → c^∗ ∈ G (S) and d_lq (c^∗, c^∗) =0 . Also, if the inequality (i) holds for c^∗. Then S and T have a common fuzzy fixed point c^∗ in B_{d
_lq} (c₀, r).

Proof: As c₀ is an element of G (S) , from condition (iii) c₀ ⪯ [Sc₀] _α(c₀) . Consider the sequence {WT (c_n)}, then there exists c₁ ∈ [Tc₀] _α(c₀) such that $\begin{matrix} d_{lq} (c_{0}, [{Tc}_{0}]_{α (c_{0})}) = d_{lq} (c_{0}, c_{1}) and \\ d_{lq} ([{Tc}_{0}]_{α (c_{0})}, c_{0}) = d_{lq} (c_{1}, c_{0}) . \end{matrix}$ From condition (ii) c₁ ≽ [Sc₁] _α(c₁) . From condition (iv), we have $\begin{matrix} max {d_{lq} (c_{1}, c_{0}), d_{lq} (c_{0}, c_{1})} \leq \sum_{i = 0}^{j} \\ max {μ^{i} (d_{lq} (c_{1}, c_{0}), μ^{i} (d_{lq} (c_{0}, c_{1}))} < r . \end{matrix}$ It follows that, d_lq (c₁, c₀) < r and d_lq (c₀, c₁) < r. So, we have c₁ ∈ B_{d
_lq} (c₀, r). Also, $\begin{matrix} d_{lq} (c_{1}, [{Tc}_{1}]_{α (c_{1})}) = d_{lq} (c_{1}, c_{2}) and \\ d_{lq} ([{Tc}_{1}]_{α (c_{1})}, c_{1}) = d_{lq} (c_{2}, c_{1}) . \end{matrix}$ As c₁ ≽ [Sc₁] _α(c₁), so from condition (ii), we have c₂ ⪯ [Sc₂] _α(c₂). By triangular inequality, we have $d_{lq} (c_{0}, c_{2}) \leq d_{lq} (c_{0}, c_{1}) + d_{lq} (c_{1}, c_{2}) .$ (2.1) Now, by Lemma 1.11, we have $\begin{matrix} d_{lq} (c_{1}, c_{2}) & \leq & H_{d_{lq}} ([{Tc}_{0}]_{α (c_{0})}, [{Tc}_{1}]_{α (c_{1})}) \\ \leq & max {H_{dq} ([{Tc}_{0}]_{α (c_{0})}, [{Tc}_{1}]_{α (c_{1})}), \\ H_{dq} ([{Tc}_{1}]_{α (c_{1})}, [{Tc}_{0}]_{α (c_{0})})} . \end{matrix}$ As c₀, c₁ ∈ B_{d
_lq} (c₀, r) ∩ {WT (c_n)} , c₁ ≽ [Sc₁] _α(c₁) and c₀ ⪯ [Sc₀] _α(c₀), then by (i), we have $\begin{matrix} d_{lq} (c_{1}, c_{2}) & \leq & μ (D_{lq} (c_{1}, c_{0})) (by (i)) \\ d_{lq} (c_{1}, c_{2}) & \leq & μ (max {d_{lq} (c_{1}, c_{0}), d_{lq} (c_{1}, c_{2}), \\ d_{lq} (c_{0}, c_{1})}) . \end{matrix}$ If max { d_lq (c₁, c₀) , d_lq (c₁, c₂) , d_lq (c₀, c₁) } = d_lq (c₁, c₂) , then a contradiction arise by the fact μ (t) < t, so we have $d_{lq} (c_{1}, c_{2}) \leq μ (max {d_{lq} (c_{1}, c_{0}), d_{lq} (c_{0}, c_{1})}) .$ (2.2) Now, inequality (2.1) implies $\begin{matrix} d_{lq} (c_{0}, c_{2}) \leq d_{lq} (c_{0}, c_{1}) + μ (max {d_{lq} (c_{1}, c_{0}), \\ d_{lq} (c_{0}, c_{1})}) \\ \leq max {d_{lq} (c_{1}, c_{0}), d_{lq} (c_{0}, c_{1})} \\ + μ (max {d_{lq} (c_{1}, c_{0}), d_{lq} (c_{0}, c_{1})}) \\ = \sum_{i = 0}^{1} max {μ^{i} (d_{lq} (c_{1}, c_{0}), μ^{i} (d_{lq} (c_{0}, c_{1}))} . \end{matrix}$ By using (iv), we have $d_{lq} (c_{0}, c_{2}) < r .$ (2.3) Now, by triangular inequality, we have $d_{lq} (c_{2}, c_{0}) \leq d_{lq} (c_{2}, c_{1}) + d_{lq} (c_{1}, c_{0}) .$ (2.4) Now, by Lemma 1.11, we have $\begin{matrix} d_{lq} (c_{2}, c_{1}) & \leq & H_{d_{lq}} ([{Tc}_{1}]_{α (c_{1})}, [{Tc}_{0}]_{α (c_{0})}) \\ \leq & max {H_{dq} ([{Tc}_{1}]_{α (c_{1})}, [{Tc}_{0}]_{α (c_{0})}), \\ H_{dq} ([{Tc}_{0}]_{α (c_{0})}, [{Tc}_{1}]_{α (c_{1})})} . \end{matrix}$ As c₁, c₀ ∈ B_{d
_lq} (c₀, r) ∩ {WT (c_n)} , c₁ ≽ [Sc₁] _α(c₁) and c₀ ⪯ [Sc₀] _α(c₀), then by (i), we have $\begin{matrix} d_{lq} (c_{2}, c_{1}) \leq μ (max {d_{lq} (c_{1}, c_{0}), \\ d_{lq} (c_{1}, c_{2}), d_{lq} (c_{0}, c_{1})}) . \end{matrix}$ If max { d_lq (c₁, c₀) , d_lq (c₁, c₂) , d_lq (c₀, c₁) } = d_lq (c₁, c₂) , then by (2.2), we have $\begin{matrix} d_{lq} (c_{2}, c_{1}) \leq μ (max {d_{lq} (c_{1}, c_{0}), \\ μ (max {d_{lq} (c_{1}, c_{0}), d_{lq} (c_{0}, c_{1})}), d_{lq} (c_{0}, c_{1})}) . \end{matrix}$ If max { d_lq (c₁, c₀) , d_lq (c₀, c₁) } = d_lq (c₀, c₁) , then, we have $\begin{matrix} d_{lq} (c_{2}, c_{1}) & \leq & μ (max {d_{lq} (c_{1}, c_{0}), μ (d_{lq} (c_{0}, c_{1})), \\ d_{lq} (c_{0}, c_{1})}) \\ \leq & μ (max {d_{lq} (c_{1}, c_{0}), d_{lq} (c_{0}, c_{1})}) . \end{matrix}$ Similarly, if max { d_lq (c₁, c₀) , d_lq (c₀, c₁) } = d_lq (c₁, c₀) , then, we have $\begin{matrix} d_{lq} (c_{2}, c_{1}) \leq μ (max {d_{lq} (c_{1}, c_{0}), d_{lq} (c_{0}, c_{1})}) . \end{matrix}$ Now, by (2.4) $\begin{matrix} d_{lq} (c_{2}, c_{0}) & \leq & d_{lq} (c_{1}, c_{0}) + μ (max {d_{lq} (c_{1}, c_{0}), \\ d_{lq} (c_{0}, c_{1})}) \\ \leq & \sum_{i = 0}^{1} max {μ^{i} (d_{lq} (c_{1}, c_{0}), \\ μ^{i} (d_{lq} (c_{0}, c_{1}))} < r . \end{matrix}$ It follows that, d_lq (c₂, c₀) < r . By (2.3), d_lq (c₀, c₂) < r . So, c₂ ∈ B_{d
_lq} (c₀, r) . Also, $\begin{matrix} d_{lq} (c_{2}, [{Tc}_{2}]_{α (c_{2})}) = d_{lq} (c_{2}, c_{3}) and \\ d_{lq} ([{Tc}_{2}]_{α (c_{2})}, c_{2}) = d_{lq} (c_{3}, c_{2}) . \end{matrix}$ As c₂ ⪯ [Sc₂] _α(c₂), so from condition (ii), we have c₃ ≽ [Sc₃] _α(c₃) . Let c₃, …, c_j ∈ B_{d
_lq} (c₀, r) ∩ {WT (c_n)}, c_j ⪯ [Sc_j] _{α(c_j)} and c_j+1 ≽ [Sc_j+1] _{α(c_j+1)} for some $j \in ℕ,$ where j = 2i, $i = 2, 3, . . ., \frac{j}{2} .$ Now, by Lemma 1.11, we obtain $\begin{matrix} d_{lq} (c_{2 i}, c_{2 i + 1}) & \leq & H_{d_{lq}} ([{Tc}_{2 i - 1}]_{α (c_{2 i - 1})}, [{Tc}_{2 i}]_{α (c_{2 i})}) \\ \leq & max {\begin{matrix} H_{d_{lq}} ([{Tc}_{2 i - 1}]_{α (c_{2 i - 1})}, [{Tc}_{2 i}]_{α (c_{2 i})}), \\ H_{d_{lq}} ([{Tc}_{2 i}]_{α (c_{2 i})}, [{Tc}_{2 i - 1}]_{α (c_{2 i - 1})}) \end{matrix}} . \end{matrix}$ As c_2i-1, c_2i ∈ B_{d
_lq} (c₀, r) ∩ {WT (c_n)} , c_2i-1 ≽ [Sc_2i-1] _{α(c_2i-1)}, c_2i ⪯ [Sc_2i] _{α(c_2i)}, then, we have $\begin{matrix} d_{lq} (c_{2 i}, c_{2 i + 1}) & \leq & μ (max {d_{lq} (c_{2 i - 1}, c_{2 i}), \\ d_{lq} (c_{2 i - 1}, c_{2 i}), d_{lq} (c_{2 i}, c_{2 i + 1}}) \\ d_{lq} (c_{2 i}, c_{2 i + 1}) & \leq & μ (max {d_{lq} (c_{2 i - 1}, c_{2 i}), \\ d_{lq} (c_{2 i}, c_{2 i + 1})}) . \end{matrix}$ If μ (max {d_lq (c_2i-1, c_2i) , d_lq (c_2i, c_2i+1)}) = d_lq (c_2i,c_2i+1) , then d_lq (c_2i, c_2i+1) ≤ μ (d_lq (c_2i, c_2i+1)) , which is contradiction to the fact μ (t) < t . Therefore, $\begin{matrix} max {d_{lq} (c_{2 i - 1}, c_{2 i}), d_{lq} (c_{2 i}, c_{2 i + 1})} = d_{lq} (c_{2 i - 1}, c_{2 i}) . \end{matrix}$ Then, we have $d_{lq} (c_{2 i}, c_{2 i + 1}) \leq μ (d_{lq} (c_{2 i - 1}, c_{2 i})),$ (2.5) which implies that $\begin{matrix} d_{lq} (c_{2 i}, c_{2 i + 1}) \leq max {μ (d_{lq} (c_{2 i - 1}, c_{2 i}), \end{matrix}$

$μ (d_{lq} (c_{2 i}, c_{2 i - 1})} .$ (2.6) Now, by Lemma 1.11 $\begin{matrix} d_{lq} (c_{2 i - 1}, c_{2 i}) & \leq & H_{d_{lq}} ([{Tc}_{2 i - 2}]_{α (c_{2 i - 2})}, [{Tc}_{2 i - 1}]_{α (c_{2 i - 1})}) \\ \leq & max {\begin{matrix} H_{d_{lq}} ([{Tc}_{2 i - 2}]_{α (c_{2 i - 2})}, [{Tc}_{2 i - 1}]_{α (c_{2 i - 1})}), \\ H_{d_{lq}} ([{Tc}_{2 i - 1}]_{α (c_{2 i - 1})}, [{Tc}_{2 i - 2}]_{α (c_{2 i - 2})}) \end{matrix}} . \end{matrix}$ As c_2i-1, c_2i-2 ∈ B_{d
_lq} (c₀, r) ∩ {WT (c_n)} , c_2i-1 ≽ [Sc_2i-1] _{α(c_2i-1)} and c_2i-2 ⪯ [Sc_2i-2] _{α(c_2i-2)}, then by (i), we have $\begin{matrix} d_{lq} (c_{2 i - 1}, c_{2 i}) \leq μ (max d_{lq} (c_{2 i - 1}, c_{2 i - 2}), \\ (d_{lq} (c_{2 i - 1}, c_{2 i}), d_{lq} (c_{2 i - 2}, c_{2 i - 1})} . \end{matrix}$ If max {d_lq (c_2i-1, c_2i-2) , (d_lq (c_2i-1, c_2i) , d_lq (c_2i-2,c_2i-1)} = d_lq (c_2i-1, c_2i) , then contradiction arise to the fact μ (t) < t . Now, $\begin{matrix} d_{lq} (c_{2 i - 1}, c_{2 i}) \leq μ (max {d_{lq} (c_{2 i - 1}, c_{2 i - 2}), \\ d_{lq} (c_{2 i - 2}, c_{2 i - 1})}) . \end{matrix}$ As μ is non decreasing function. So, $\begin{matrix} μ (d_{lq} (c_{2 i - 1}, c_{2 i})) \leq μ^{2} (max {d_{lq} (c_{2 i - 1}, c_{2 i - 2}), \\ d_{lq} (c_{2 i - 2}, c_{2 i - 1})}) \\ = max {μ^{2} (d_{lq} (c_{2 i - 1}, c_{2 i - 2})), \end{matrix}$

$μ^{2} (d_{lq} (c_{2 i - 2}, c_{2 i - 1}))}$ (2.7) Using (2.7) in (2.5), we have $\begin{matrix} d_{lq} (c_{2 i}, c_{2 i + 1}) \leq max {μ^{2} (d_{lq} (c_{2 i - 1}, c_{2 i - 2})), \end{matrix}$

$μ^{2} (d_{lq} (c_{2 i - 2}, c_{2 i - 1}))},$ (2.8) Now by Lemma 1.11 $\begin{matrix} d_{lq} (c_{2 i - 2}, c_{2 i - 1}) \leq H_{d_{lq}} ([{Tc}_{2 i - 3}]_{α (c_{2 i - 3})},) \\ [{Tc}_{2 i - 2}]_{α (c_{2 i - 2})}) \\ \leq max {\begin{matrix} H_{d_{lq}} ([{Tc}_{2 i - 3}]_{α (c_{2 i - 3})}, [{Tc}_{2 i - 2}]_{α (c_{2 i - 2})}), \\ H_{d_{lq}} ([{Tc}_{2 i - 2}]_{α (c_{2 i - 2})}, [{Tc}_{2 i - 3}]_{α (c_{2 i - 3})}) \end{matrix}} . \end{matrix}$ As c_2i-3, c_2i-2 ∈ B_{d
_lq} (c₀, r) ∩ {WT (c_n)} , c_2i-3 ≽ [Sc_2i-3] _{α(c_2i-3)} and c_2i-2 ⪯ [Sc_2i-2] _{α(c_2i-2)}, then by (i), we have $\begin{matrix} d_{lq} (c_{2 i - 2}, c_{2 i - 1}) \leq μ (max {d_{lq} (c_{2 i - 3}, c_{2 i - 2}), \\ d_{lq} (c_{2 i - 2}, c_{2 i - 1})} . \end{matrix}$ If max {d_lq (c_2i-3, c_2i-2) , d_lq (c_2i-2, c_2i-1)} = d_lq(c_2i-2, c_2i-1) , then contradiction arise to the fact μ (t) < t . Therefore $d_{lq} (c_{2 i - 2}, c_{2 i - 1}) \leq μ (d_{lq} (c_{2 i - 3}, c_{2 i - 2}))$ (2.9)

$\begin{matrix} \leq μ (max {d_{lq} (c_{2 i - 3}, c_{2 i - 2}), d_{lq} (c_{2 i - 2}, c_{2 i - 3})}) \end{matrix}$

$\begin{matrix} μ^{2} d_{lq} (c_{2 i - 2}, c_{2 i - 1}) \leq μ^{3} (max {d_{lq} (c_{2 i - 3}, c_{2 i - 2}), \end{matrix}$

$d_{lq} (c_{2 i - 2}, c_{2 i - 3})}) .$ (2.10) Now by Lemma 1.11 $\begin{matrix} d_{lq} (c_{2 i - 1}, c_{2 i - 2}) \leq H_{d_{lq}} ([{Tc}_{2 i - 2}]_{α (c_{2 i - 2})}, \\ [{Tc}_{2 i - 3}]_{α (c_{2 i - 3})}) \\ \leq max {\begin{matrix} H_{d_{lq}} ([{Tc}_{2 i - 3}]_{α (c_{2 i - 3})}, [{Tc}_{2 i - 2}]_{α (c_{2 i - 2})}), \\ H_{d_{lq}} ([{Tc}_{2 i - 2}]_{α (c_{2 i - 2})}, [{Tc}_{2 i - 3}]_{α (c_{2 i - 3})}) \end{matrix}} . \end{matrix}$ As c_2i-3, c_2i-2 ∈ B_{d
_lq} (c₀, r) ∩ {WT (c_n)} , c_2i-3 ≽ [Sc_2i-3] _{α(c_2i-3)} and c_2i-2 ⪯ [Sc_2i-2] _{α(c_2i-2)}, then by (i), we have $\begin{matrix} d_{lq} (c_{2 i - 1}, c_{2 i - 2}) \leq μ (max {d_{lq} (c_{2 i - 3}, c_{2 i - 2}), \\ d_{lq} (c_{2 i - 2}, c_{2 i - 1})}) . \end{matrix}$ By using Inequality (2.9), we have $\begin{matrix} d_{lq} (c_{2 i - 1}, c_{2 i - 2}) & \leq & μ (max {d_{lq} (c_{2 i - 3}, c_{2 i - 2}), \\ μ (d_{lq} (c_{2 i - 3}, c_{2 i - 2}))} . \\ d_{lq} (c_{2 i - 1}, c_{2 i - 2}) & \leq & μ (d_{lq} (c_{2 i - 3}, c_{2 i - 2})) . \end{matrix}$ Which implies that $\begin{matrix} μ^{2} d_{lq} (c_{2 i - 1}, c_{2 i - 2}) \leq μ^{2} (μ (max {d_{lq} (c_{2 i - 3}, c_{2 i - 2}), \end{matrix}$

$d_{lq} (c_{2 i - 2}, c_{2 i - 3})}) .$ (2.11) Combining (2.8), (2.10) and (2.7), we have $\begin{matrix} d_{lq} (c_{2 i}, c_{2 i + 1}) \leq max {μ^{3} (d_{lq} (c_{2 i - 3}, c_{2 i - 2})), \end{matrix} μ^{3} (d_{lq} (c_{2 i - 2}, c_{2 i - 3}))} .$ (2.12) Following (2.6), (2.8) and (2.12), we have $\begin{matrix} d_{lq} (c_{2 i}, c_{2 i + 1}) \leq max {μ^{2 i} (d_{lq} (c_{0}, c_{1})), \end{matrix}$

$μ^{2 i} (d_{lq} (c_{1}, c_{0}))} .$ (2.13) Now by Lemma 1.11 $\begin{matrix} d_{lq} (c_{2 i + 1}, c_{2 i}) \leq H_{d_{lq}} ([{Tc}_{2 i}]_{α (c_{2 i})}, [{Tc}_{2 i - 1}]_{α (c_{2 i - 1})}) \\ \leq max {\begin{matrix} H_{d_{lq}} ([{Tc}_{2 i - 1}]_{α (c_{2 i - 1})}, [{Tc}_{2 i}]_{α (c_{2 i})}), \\ H_{d_{lq}} ([{Tc}_{2 i}]_{α (c_{2 i})}, [{Tc}_{2 i - 1}]_{α (c_{2 i - 1})}) \end{matrix}} . \end{matrix}$ As c_2i-1, c_2i ∈ B_{d
_lq} (c₀, r) ∩ {WT (c_n)} , c_2i-1≽ [Sc_2i-1] _{α(c_2i-1)} and c_2i ⪯ [Sc_2i] _{α(c_2i)} then by (i), we have $d_{lq} (c_{2 i + 1}, c_{2 i}) \leq μ (max {d_{lq} (c_{2 i - 1}, c_{2 i}), d_{lq} (c_{2 i}, c_{2 i + 1})}) .$ (2.14) By Inequality (2.5) we have $\begin{matrix} d_{lq} (c_{2 i + 1}, c_{2 i}) \leq μ (max {d_{lq} (c_{2 i - 1}, c_{2 i}), \\ μ (d_{lq} (c_{2 i - 1}, c_{2 i})}) . \end{matrix}$ As μ (t) < t, we have $d_{lq} (c_{2 i + 1}, c_{2 i}) \leq μ (d_{lq} (c_{2 i - 1}, c_{2 i})) .$ (2.15) Now, $\begin{matrix} d_{lq} (c_{2 i + 1}, c_{2 i}) \leq max {μ (d_{lq} (c_{2 i - 1}, c_{2 i})), \end{matrix}$

$μ (d_{lq} (c_{2 i}, c_{2 i - 1})} .$ (2.16) Now, using (2.7) in (2.15), we have $\begin{matrix} d_{lq} (c_{2 i + 1}, c_{2 i}) \leq max {μ^{2} d_{lq} (c_{2 i - 1}, c_{2 i - 2}), \end{matrix}$

$μ^{2} d_{lq} (c_{2 i - 2}, c_{2 i - 1})} .$ (2.17) Combining (2.10), (2.11) and (2.17), we have $\begin{matrix} d_{lq} (c_{2 i + 1}, c_{2 i}) \leq max {μ^{3} (d_{lq} (c_{2 i - 3}, c_{2 i - 2})), \end{matrix} μ^{3} (d_{lq} (c_{2 i - 2}, c_{2 i - 3}))} .$ (2.18) Following (2.16), (2.17) and (2.18), we have $\begin{matrix} d_{lq} (c_{2 i + 1}, c_{2 i}) \leq max {μ^{2 i} (d_{lq} (c_{1}, c_{0})), \\ μ^{2 i} (d_{lq} (c_{0}, c_{1}))} . \end{matrix}$ Similarly, we have $\begin{matrix} d_{lq} (c_{2 i}, c_{2 i - 1}) \leq max {μ^{2 i - 1} (d_{lq} (c_{0}, c_{1})), \\ μ^{2 i - 1} (d_{lq} (c_{1}, c_{0}))} . \end{matrix}$ Combining the above two inequality $\begin{matrix} d_{lq} (c_{j + 1}, c_{j}) \leq max {μ^{j} (d_{lq} (c_{0}, c_{1})), \end{matrix}$

$μ^{j} (d_{lq} (c_{1}, c_{0}))} .$ (2.19) Now, by (2.13), (iv) and triangular inequality, we have $\begin{matrix} d_{lq} (c_{0}, c_{j + 1}) \leq d_{lq} (c_{0}, c_{1}) + d_{lq} (c_{1}, c_{2}) + \dots + \\ d_{lq} (c_{j}, c_{j + 1}) \leq d_{lq} (c_{0}, c_{1}) + \dots + \\ max {μ^{j} (d_{lq} (c_{1}, c_{0})), μ^{j} (d_{lq} (c_{0}, c_{1}))} \end{matrix}$

$\leq \sum_{i = 0}^{j} max {μ^{i} (d_{lq} (c_{1}, c_{0}), μ^{i} (d_{lq} (c_{0}, c_{1}))} < r .$ (2.20) Similarly, by (2.19), (iv) and triangular inequality, we have $\begin{matrix} d_{lq} (c_{j + 1}, c_{0}) \leq \sum_{i = 0}^{j} max {μ^{i} (d_{lq} (c_{1}, c_{0}), \end{matrix}$

$μ^{i} (d_{lq} (c_{0}, c_{1}))} < r .$ (2.21) By Inequality (2.20) and (2.21), we have c_j+1 ∈ B_{d
_lq} (c₀, r) . Also, $\begin{matrix} d_{lq} (c_{ji + 1}, [{Tc}_{j + 1}]_{α (c_{j + 1})}) = d_{lq} (c_{j + 1}, c_{j + 2}) \end{matrix}$ and $\begin{matrix} d_{lq} ([{Tc}_{j + 1}]_{α (c_{j + 1})}, c_{j + 1}) = d_{lq} (c_{j + 2}, c_{j + 1}) . \end{matrix}$ As c_j+1 ≽ [Sc_j+1] _{α(c_j+1)}, so from condition (ii), we have c_j+2 ⪯ [Sc_j+2] _{α(c_j+2)}. Similarly,we get $\begin{matrix} d_{lq} (c_{j + 1}, c_{j + 2}) \leq max {μ^{j + 1} (d_{lq} (c_{1}, c_{0}))), \end{matrix}$

$μ^{j + 1} (d_{lq} (c_{0}, c_{1}))}$ (2.22) and $\begin{matrix} d_{lq} (c_{j + 2}, c_{j + 1}) \leq max {μ^{j + 1} (d_{lq} (c_{1}, c_{0}))), \end{matrix}$

$μ^{j + 1} (d_{lq} (c_{0}, c_{1}))} .$ (2.23) Also, $\begin{matrix} d_{lq} (c_{0}, c_{j + 2}) < r and d_{lq} (c_{j + 2}, c_{0}) < r . \end{matrix}$ It follows that c_j+2 ∈ B_{d
_lq} (c₀, r). Also, $\begin{matrix} d_{lq} (c_{j + 2}, [{Tc}_{j + 2}]_{α (c_{2 i + 2})}) = d_{lq} (c_{j + 2}, c_{j + 3}) \end{matrix}$ and $\begin{matrix} d_{lq} ([{Tc}_{j + 2}]_{α (c_{j + 2})}, c_{j + 2}) = d_{lq} (c_{j + 3}, c_{j + 2}) . \end{matrix}$ As c_j+2 ⪯ [Sc_j+2] _{α(c_j+2)}, so from condition (ii), we have c_j+3 ≽ [Sc_j+3] _{α(c_j+3)}. Hence by mathematical induction c_n ∈ B_{d
_lq} (c₀, r) , c_2n ⪯ [Sc_2n] _{α(c_2n)} and c_2n+1 ≽ [Sc_2n+1] _{α(c_2n+1)} for all $n \in ℕ$ . Also c_2n ∈ G (S) . Now inequalities (2.13), (2.19), (2.22) and (2.23) can be merged as, $d_{lq} (c_{n}, c_{n + 1}) \leq max {μ^{n} (d_{lq} (c_{1}, c_{0})), μ^{n} (d_{lq} (c_{0}, c_{1}))}$ (2.24) $d_{lq} (c_{n + 1}, c_{n}) \leq max {μ^{n} (d_{lq} (c_{1}, c_{0})), μ^{n} (d_{lq} (c_{0}, c_{1}))},$ (2.25) for all n ∈ N . Fix ɛ > 0 and let $k_{1} (ɛ) \in ℕ$ such that $\begin{matrix} \sum_{k \geq k_{1} (ɛ)} max {μ^{k} (d_{lq} (c_{1}, c_{0})), μ^{k} (d_{lq} (c_{0}, c_{1}))} < ɛ . \end{matrix}$ Let $n, m \in ℕ$ with m > n > k₁ (ɛ) , then $\begin{matrix} d_{lq} (c_{n}, c_{m}) & \leq & \sum_{k = n}^{m - 1} d_{lq} (c_{k}, c_{k + 1}) \\ \leq & \sum_{k = n}^{m - 1} max {μ^{k} (d_{lq} (c_{1}, c_{0})), \\ μ^{k} (d_{lq} (c_{0}, c_{1}))}, (by (2.24)) \\ d_{lq} (c_{n}, c_{m}) & \leq & \sum_{k \geq k_{1} (ɛ)} max {μ^{n} (d_{lq} (c_{1}, c_{0})), \\ μ^{n} (d_{lq} (c_{0}, c_{1}))} < ɛ . \end{matrix}$ Thus we proved that {WT (c_n)} is a left K-Cauchy sequence in (B_{d
_lq} (c₀, r) , d_lq). Similarly, by using (2.25), we have $\begin{matrix} d_{lq} (c_{m}, c_{n}) \leq \sum_{k = n}^{m - 1} d_{lq} (c_{k + 1}, c_{k}) < ɛ \end{matrix}$ Hence, {WT (x_n)} is a right K-Cauchy sequence in (B_{d
_lq} (c₀, r) , d_lq) . As every closed set in left(right) K-sequentially complete dislocated quasi metric space is left(right) K-sequentially complete and G (S) is closed set, so G (S) is left(right) K-sequentially complete. As {c_2n} is a left(right) K-Cauchy sequence in G (S) , so there exists c^∗ ∈ G (S) such that {c_2n} → c^∗, that is $lim_{n \to \infty} d_{lq} (c_{2 n}, c^{*}) = lim_{n \to \infty} d_{lq} (c^{*}, c_{2 n}) = 0$ (2.26) Also, $c^{*} ⪯ [{Sc}^{*}]_{α (c^{*})} .$ (2.27) Now, $\begin{matrix} d_{lq} (c^{*}, c^{*}) \leq d_{lq} (c^{*}, c_{2 n}) + d_{lq} (c_{2 n}, c^{*}) . \end{matrix}$ This implies d_lq (c^∗, c^∗) =0 as n → ∞ . Now by Lemma 1.9 $\begin{matrix} d_{lq} (c^{*}, [{Tc}^{*}]_{α (c^{*})}) \leq d_{lq} (c^{*}, c_{2 n + 2}) + d_{lq} (c_{2 n + 2}, [{Tc}^{*}]_{α (c^{*})}) \\ \leq d_{lq} (c^{*}, c_{2 n + 2}) + H_{d_{lq}} ([{Tc}_{2 n + 1}]_{α (c_{2 n + 1})}, [{Tc}^{*}]_{α (c^{*})}), \\ \leq d_{lq} (c^{*}, c_{2 n + 2}) + max {\begin{matrix} H_{d_{lq}} ([{Tc}_{2 n + 1}]_{α (c_{2 n + 1})}, [{Tc}^{*}]_{α (c^{*})}), \\ H_{d_{lq}} ([{Tc}^{*}]_{α (c^{*})}, [{Tc}_{2 n + 1}]_{α (c_{2 n + 1})}) \end{matrix}} . \end{matrix}$ By assumption, inequality (i) holds for c^∗ . Also c_2n+1 ≽ [Sc_2n+1] _{α(c_2n+1)} and c^∗ ⪯ [Sc^∗] _α(c^∗), so $\begin{matrix} d_{lq} (c^{*}, [{Tc}^{*}]_{α (c^{*})}) \leq d_{lq} (c^{*}, c_{2 n + 2}) \\ + μ (max {\begin{matrix} d_{lq} (c_{2 n + 1}, c^{*}), d_{lq} (c_{2 n + 1}, c_{2 n + 2}), \\ d_{lq} (c^{*}, [{Tc}^{*}]_{α (c^{*})}) \end{matrix}}) \\ \leq d_{lq} (c^{*}, c_{2 n + 2}) + μ (max {d_{lq} (c_{2 n + 1}, c_{2 n + 2}) \\ + d_{lq} (c_{2 n + 2}, c^{*}), d_{lq} (c_{2 n + 1}, c_{2 n + 2}), d_{lq} (c^{*}, [{Tc}^{*}]_{α (c^{*})})}) . \end{matrix}$ Letting n⟶ ∞ and by using inequalities (2.24) and (2.26), we obtain $\begin{matrix} d_{lq} (c^{*}, [{Tc}^{*}]_{α (c^{*})}) \leq μ (d_{lq} (c^{*}, [{Tc}^{*}]_{α (c^{*})})) . \end{matrix}$ This implies that $d_{lq} (c^{*}, [{Tc}^{*}]_{α (c^{*})}) = 0 .$ (2.28) Now, $\begin{matrix} d_{lq} ([{Tc}^{*}]_{α (c^{*})}, c^{*}) \leq d_{lq} ([{Tc}^{*}]_{α (c^{*})}, c_{2 n + 2}) \\ + d_{lq} (c_{2 n + 2}, c^{*}) \leq H_{d_{lq}} ([{Tc}^{*}]_{α (c^{*})}, \\ [{Tc}_{2 n + 1}]_{α (c_{2 n + 1})}) + d_{lq} (c_{2 n + 2}, c^{*}) \\ \leq max {\begin{matrix} H_{d_{lq}} ([{Tc}_{2 n + 1}]_{α (c_{2 n + 1})}, [{Tc}^{*}]_{α (c^{*})}), \\ H_{d_{lq}} ([{Tc}^{*}]_{α (c^{*})}, [{Tc}_{2 n + 1}]_{α (c_{2 n + 1})}) \end{matrix}} \\ + d_{lq} (c_{2 n + 2}, c^{*}) . \end{matrix}$ As inequality (i) holds for c^∗, c^∗ ⪯ [Sc^∗] _α(c^∗) and c_2n+1 ≽ [Sc_2n+1] _{α(c_2n+1)}, then, we obtain $\begin{matrix} d_{lq} ([{Tc}^{*}]_{α (c^{*})}, c^{*}) \leq μ (max {d_{lq} (c_{2 n + 1}, c^{*}), \\ d_{lq} (c_{2 n + 1}, c_{2 n + 2}), d_{lq} (c^{*}, [{Tc}^{*}]_{α (c^{*})})}) \\ + d_{lq} (c_{2 n + 2}, c^{*}) . \end{matrix}$ Taking n⟶ ∞ and by using inequalities (2.24), (2.26) and (2.28), we have $d_{lq} ([{Tc}^{*}]_{α (c^{*})}, c^{*}) = 0 .$ (2.29) From inequalities (2.28) and (2.29), we have c^∗ ∈ [Tc^∗] _α(c^∗) . As c^∗ ⪯ [Sc^∗] _α(c^∗) and d_lq (c^∗, [Tc^∗] _α(c^∗)) = d_lq ([Tc^∗] _α(c^∗), c^∗) =0 = d_lq (c^∗, c^∗) , then from (ii) $c^{*} ≽ [{Sc}^{*}]_{α (c^{*})} .$ (2.30) From (2.27) and (2.30), we have c^∗ ⪯ [Sc^∗] _α(c^∗) ⪯ c^∗ . This implies c^∗ ⪯ y ⪯ c^∗, for all y ∈ [Sc^∗] _α(c^∗) . Therefore c^∗ = y, for all y ∈ [Sc^∗] _α(c^∗) or [Sc^∗] _α(c^∗) = { c^∗ } . Hence, c^∗ is a common fuzzy fixed point for S and T .

Example 2.2: Let W = [0, ∞) and $\begin{matrix} d_{lq} (c, y) = c + 2 y, (c, y) \in W \times W . \end{matrix}$ Then (W, ⪯ , d_lq) be an ordered left(right) K sequentially complete dislocated quasi metric space. Let $μ (t) = \frac{99}{100} t$ and $R$ be a binary relation on W defined by $\begin{matrix} R = {(c, c) : c \in W} \\ \cup {(c, \frac{c}{2}) : c \in {1, \frac{1}{4}, \frac{1}{16}, \dots}} \\ \cup {(\frac{c}{2}, c) : c \in {\frac{1}{2}, \frac{1}{8}, \frac{1}{32}, \dots}} . \end{matrix}$ Consider the partial order on W defined by $\begin{matrix} (c, y) \in W \times W, c ⪯ y if and only (c, y) \in R . \end{matrix}$ Define a pair of fuzzy mappings S, T : W → M_q (W) by $(T c) (s) = {\begin{matrix} \frac{α}{4} & if & \frac{c}{4} & \leq & s & \leq \frac{c}{2} \\ \frac{α}{2} & if & \frac{c}{2} & < & s & \leq c \end{matrix}$ and $(S c) (t) = {\begin{matrix} \frac{β}{4} & if t \in {\frac{c}{2}} for c \in [0, 1] \\ β & t \in {\frac{c + 2}{2}} f o r c \notin [0, 1] \end{matrix} .$ Now, for α, β ∈ (0, 1] we define $\begin{matrix} {[Tc]}_{α / 4} = [\frac{c}{4}, \frac{c}{2}], and \\ {[Sc]}_{β / 4} = {\frac{c}{2}} for c \in [0, 1] . \end{matrix}$ Let $\begin{matrix} A & = & {c : c ⪯ {[Sc]}_{β / 4}} = {0, 1, \frac{1}{4}, \frac{1}{16}, \dots}, \\ B & = & {y : y ≽ {[Sy]}_{β / 4}} = {0, \frac{1}{2}, \frac{1}{8}, \frac{1}{32}, \dots} . \end{matrix}$ Let c₀ = 1 and r = 275, then $\begin{matrix} B_{d_{lq}} (c_{0}, r) & = {y : d_{lq} (1, y) < 275 \land d_{lq} (y, 1) < 275} \\ = [0, 137) . \end{matrix}$ Then $\begin{matrix} G (S) & = & {c : c ⪯ {[Sc]}_{β / 4} and c \in B_{d_{lq}} (c_{0}, r)} \\ = & {0, 1, \frac{1}{4}, \frac{1}{16}, \frac{1}{64}, \dots} \cap [0, 137) \\ = & {0, 1, \frac{1}{4}, \frac{1}{16}, \frac{1}{64}, \dots} . \end{matrix}$ Now, as $\frac{1}{4^{n - 1}} \in B_{d_{lq}} (c_{0}, r)$ $d_{l q} (\frac{1}{4^{n - 1}}, {[T \frac{1}{4^{n - 1}}]}_{α / 4}) = d_{l q} (\frac{1}{{4.4}^{n - 1}}, \frac{1}{4^{n - 1}}) .$ and $d_{l q} ({[T \frac{1}{4^{n - 1}}]}_{α / 4}, \frac{1}{4^{n - 1}}) = d_{l q} (\frac{1}{{4.4}^{n - 1}}, \frac{1}{4^{n - 1}}) .$ Also, $(\frac{1}{4^{n - 1}}, \frac{1}{4 . 4^{n - 1}}) \in R$ , so $\frac{1}{4^{n - 1}} ⪯ [S \frac{1}{4^{n - 1}}]_{β / 4}$ . As $(\frac{1}{8 . 4^{n - 1}}, \frac{1}{2 . 4^{n - 1}}) \in R,$ so $\frac{1}{2 . 4^{n - 1}} ≽ [S \frac{1}{{2.4}^{n - 1}}]_{β / 4} .$ Hence, condition (ii)(a) is satisfied. Now, as $\frac{1}{2 . 4^{n - 1}} \in B_{d_{lq}} (c_{0}, r)$ , $\begin{matrix} d_{l q} (\frac{1}{{2.4}^{n - 1}}, {[T \frac{1}{{2.4}^{n - 1}}]}_{α / 4}) = \\ d_{lq} (\frac{1}{2 . 4^{n - 1}}, \frac{1}{8 . 4^{n - 1}}) \end{matrix}$ and $\begin{matrix} d_{l q} ({[T \frac{1}{{2.4}^{n - 1}}]}_{α / 4}, \frac{1}{{2.4}^{n - 1}}) = \\ d_{lq} (\frac{1}{8 . 4^{n - 1}}, \frac{1}{2 . 4^{n - 1}}) . \end{matrix}$ Also, $\frac{1}{2 . 4^{n - 1}} ≽ {[S \frac{1}{{2.4}^{n - 1}}]}_{β / 4},$ implies $\frac{1}{8 . 4^{n - 1}} ⪯ {[S \frac{1}{{8.4}^{n - 1}}]}_{β / 4} .$ Hence, condition (ii)(b) is satisfied. Now $\begin{matrix} B_{d_{lq}} (c_{0}, r) \cap {WT (c_{n})} \\ = {1, \frac{1}{2}, \frac{1}{4}, \frac{1}{8}, \frac{1}{16}, \frac{1}{32}, \frac{1}{64}, \frac{1}{128}, \dots} . \end{matrix}$ Now for c, y ∈ B_{d
_q} (c₀, r) ∩ {WT (c_n)} with c ≽ [Sc] _α(c) and y ⪯ [Sy] _α(y), then c ∈ B and y ∈ A. In general for some $n, m \in ℕ$ $c = \frac{1}{{2.4}^{m - 1}}, and y = \frac{1}{4^{n - 1}} .$ Case i. Let n < m, we have $H_{d_{lq}} ([Ty]_{α / 4}, [Tc]_{α / 4})$ $\begin{matrix} = & H_{d_{lq}} ([\frac{1}{4 . 4^{n - 1}}, \frac{1}{2 . 4^{n - 1}}], \\ [\frac{1}{8 . 4^{m - 1}}, \frac{1}{4 . 4^{m - 1}}]) \end{matrix}$ $= max {\frac{1}{2 . 4^{n - 1}} + \frac{2}{8 . 4^{m - 1}},$ (2.31) $\begin{matrix} \frac{1}{4 . 4^{n - 1}} + \frac{2}{4 . 4^{m - 1}}} \\ = & max {\frac{4 . 4^{m - n} + 2}{8 . 4^{m - 1}}, \frac{4^{m - n} + 2}{4 . 4^{m - 1}}} \\ = & \frac{4 . 4^{m - n} + 2}{8 . 4^{m - 1}} . \end{matrix}$ Now $\begin{matrix} H_{d_{lq}} ([Tc]_{α / 4}, [Ty]_{α / 4}) \\ = H_{d_{lq}} ([\frac{1}{8 . 4^{m - 1}}, \frac{1}{4 . 4^{m - 1}}], [\frac{1}{4 . 4^{n - 1}}, \frac{1}{2 . 4^{n - 1}}]) \end{matrix}$ $= max {\frac{1}{4 . 4^{m - 1}} + \frac{2}{4 . 4^{n - 1}}, \frac{1}{8 . 4^{m - 1}} + \frac{2}{2 . 4^{n - 1}}}$ (2.32) $\begin{matrix} = max {\frac{1 + 2 . 4^{m - n}}{4 . 4^{m - 1}}, \frac{1 + 8 . 4^{m - n}}{8 . 4^{m - 1}}} = \frac{1 + 8 . 4^{m - n}}{8 . 4^{m - 1}} . \end{matrix}$ Also $\begin{matrix} max {H_{d_{lq}} ([Ty]_{α / 4}, [Tc]_{α / 4}), H_{d_{lq}} ([Tc]_{α / 4}, [Ty]_{α / 4})} \\ = max {\frac{4 . 4^{m - n} + 2}{8 . 4^{m - 1}}, \frac{1 + 8 . 4^{m - n}}{8 . 4^{m - 1}}} \\ = \frac{1 + 8 . 4^{m - n}}{8 . 4^{m - 1}} . \end{matrix}$ Now, we have

$\begin{matrix} D_{lq} (c, y) & = & max {\begin{matrix} d_{lq} (\frac{1}{2 . 4^{m - 1}}, \frac{1}{4^{n - 1}}), d_{lq} (\frac{1}{2 . 4^{m - 1}}, {[T \frac{1}{2 . 4^{m - 1}}]}_{α / 4}), \\ d_{lq} (\frac{1}{4^{n - 1}}, {[T \frac{1}{4^{n - 1}}]}_{α / 4}) \end{matrix}} \\ = & max {\frac{1 + 2 . 4^{m - n}}{2 . 4^{m - 1}}, \frac{1}{2 . 4^{m - 1}} + \frac{2}{8 . 4^{m - 1}}, \frac{1}{4^{n - 1}} + \frac{2}{4 . 4^{n - 1}}} (2.33) \\ = & max {\frac{1 + 2 . 4^{m - n}}{2 . 4^{m - 1}}, \frac{6}{8 . 4^{m - 1}}, \frac{6}{4 . 4^{n - 1}}} = \frac{1 + 2 . 4^{m - n}}{2 . 4^{m - 1}} . \end{matrix}$ (2.33) As $\begin{matrix} \frac{1 + 8 . 4^{m - n}}{8 . 4^{m - 1}} < \frac{99}{100} (\frac{1 + 2 . 4^{m - n}}{2 . 4^{m - 1}}) \end{matrix}$ As $μ (t) = \frac{99}{100} t,$ so $\begin{matrix} max {H_{d_{lq}} ([Tc]_{α / 4}, [Ty]_{α / 4}), H_{d_{lq}} ([Ty]_{α / 4}, \\ [Tc]_{α / 4})} < μ (D_{lq} (c, y)) \end{matrix}$ Case ii: Let n > m, then by using (2.31) , we have $\begin{matrix} H_{d_{lq}} ([Ty]_{α / 4}, [Tc]_{α / 4})) \\ = max {\frac{4 + 2 . 4^{n - m}}{8 . 4^{n - 1}}, \frac{1 + 2 . 4^{n - m}}{4 . 4^{n - 1}}} \\ = \frac{1 + 2 . 4^{n - m}}{4 . 4^{n - 1}} \end{matrix}$ Similarly, by using (2.32) , we have $\begin{matrix} H_{d_{lq}} ([Tc]_{α / 4}, [Ty]_{α / 4}) \\ = max {\frac{4^{n - m} + 2}{4 . 4^{n - 1}}, \frac{4^{n - m} + 8}{8 . 4^{n - 1}}} \\ = \frac{4^{n - m} + 2}{4 . 4^{n - 1}} \end{matrix}$ Now, $\begin{matrix} max {H_{d_{lq}} ([Ty]_{α / 4}, [Tc]_{α / 4})), \\ = H_{d_{lq}} ([Tc]_{α / 4}, [Ty]_{α / 4})} \\ = & max {\frac{1 + 2 . 4^{n - m}}{4 . 4^{n - 1}}, \frac{4^{n - m} + 2}{4 . 4^{n - 1}}} \\ = & \frac{1 + 2 . 4^{n - m}}{4 . 4^{n - 1}} \end{matrix}$ Now, by (2.33) , we have $\begin{matrix} D_{lq} (c, y) & = & max {\frac{4^{n - m} + 4}{2 . 4^{n - 1}}, \frac{6}{8 . 4^{m - 1}}, \frac{6}{4 . 4^{n - 1}}} \\ = & \frac{4^{n - m} + 4}{2 . 4^{n - 1}} . \end{matrix}$ As $\begin{matrix} \frac{1 + 2 . 4^{n - m}}{4 . 4^{n - 1}} \leq \frac{99}{100} (\frac{4^{n - m} + 4}{2 . 4^{n - 1}} .) \\ max {H_{d_{lq}} ([Ty]_{α / 4}, [Tc]_{α / 4})), \\ H_{d_{lq}} ([Tc]_{α / 4}, [Ty]_{α / 4})} \leq μ (D_{lq} (c, y)) \end{matrix}$ Case iii: For $\begin{matrix} c = 0, and y = \frac{1}{4^{n - 1}} \end{matrix}$ We have $\begin{matrix} H_{d_{lq}} ([Tc]_{α / 4}, [Ty]_{α / 4}) \\ = max {[0, 0], [\frac{1}{4 . 4^{n - 1}}, \frac{1}{2 . 4^{n - 1}}]} \\ = max {\frac{2}{4 . 4^{n - 1}}, \frac{1}{4^{n - 1}}} = \frac{1}{4^{n - 1}} . \end{matrix}$ Also $\begin{matrix} H_{d_{lq}} ([Ty]_{α / 4}, [Tc]_{α / 4})) \\ = max {[\frac{1}{4 . 4^{n - 1}}, \frac{1}{2 . 4^{n - 1}}], [0, 0]} = \frac{1}{2 . 4^{n - 1}} . \end{matrix}$ Now, $\begin{matrix} max {H_{d_{lq}} ([Tc]_{α / 4}, [Ty]_{α / 4}), \\ H_{d_{lq}} ([Ty]_{α / 4}, [Tc]_{α / 4}))} = \frac{1}{4^{n - 1}} . \end{matrix}$ Also $\begin{matrix} D_{lq} (c, y) = max {\frac{2}{4^{n - 1}}, \frac{4 + 2}{4 . 4^{n - 1}}} = \frac{2}{4^{n - 1}} . \end{matrix}$ So, $\begin{matrix} max {H_{d_{lq}} ([Tc]_{α / 4}, [Ty]_{α / 4}), \\ H_{d_{lq}} ([Ty]_{α / 4}, [Tc]_{α / 4}))} < μ (D_{lq} (c, y)) . \end{matrix}$ Case iv: For $\begin{matrix} c = \frac{1}{2 . 4^{n - 1}} and y = 0 \end{matrix}$ We have $\begin{matrix} H_{d_{lq}} ([Tc]_{α / 4}, [Ty]_{α / 4}) = max {\frac{1}{4 . 4^{n - 1}}, \frac{1}{8 . 4^{n - 1}}} \\ = \frac{1}{4 . 4^{n - 1}} . \end{matrix}$ Also $\begin{matrix} H_{d_{lq}} ([Ty]_{α / 4}, [Tc]_{α / 4})) = max {\frac{2}{8 . 4^{n - 1}}, \frac{2}{4 . 4^{n - 1}}} \\ = \frac{2}{4 . 4^{n - 1}} . \end{matrix}$ Now, $\begin{matrix} max {H_{d_{lq}} ([Tc]_{α / 4}, [Ty]_{α / 4}), \\ H_{d_{lq}} ([Ty]_{α / 4}, [Tc]_{α / 4}))} = \frac{2}{4 . 4^{n - 1}} . \end{matrix}$ Also $\begin{matrix} D_{lq} (c, y) = \frac{6}{8 . 4^{n - 1}} . \end{matrix}$ Clearly $\begin{matrix} max {H_{d_{lq}} ([Tc]_{α / 4}, [Ty]_{α / 4}), \\ H_{d_{lq}} ([Ty]_{α / 4}, [Tc]_{α / 4}))} \leq μ (D_{lq} (c, y)) . \end{matrix}$ Case v: The contraction trivially holds for x = 0 and y = 0. Also $\begin{matrix} \sum_{i = 0}^{n} max {μ^{i} (d_{q} (c_{1}, c_{0}), μ^{i} (d_{q} (c_{0}, c_{1}))} \\ = \sum_{i = 0}^{n} max {μ^{i} (\frac{5}{2}), μ^{i} (2)} \\ = \frac{5}{2} + \frac{99}{40} + \frac{{(99)}^{2}}{40 \times 100} + \frac{{(99)}^{3}}{40 \times {(100)}^{2}} + . . . . . . . . \\ = 250 < 275 = r . \end{matrix}$ Thus all the conditions of Theorem 2.1 are satisfied. Hence S and T have a common fixed point 0 in B_{d
_q} (x₀, r).

By taking D_lq (c, y) = d_lq (c, y) , we obtain the following result.

Corollary 2.3: Let (W, ⪯ , d_lq) be an ordered left (right) K-sequentially complete dislocated quasi metric space, S, T : W → M_q (W) be the two fuzzy mappings. Suppose that the following assertions hold:

(i) There exists a function μ ∈ Ψ, c₀ ∈ W, α (c) , α (y) ∈ (0, 1] and r > 0, we have $\begin{matrix} max {H_{d_{lq}} ([Tc]_{α (c)}, [Ty]_{α (y)}), H_{d_{lq}} ([Ty]_{α (y)}, \\ [Tc]_{α (c)})} \leq μ (d_{lq} (c, y)), \end{matrix}$ for all c, y ∈ B_{d
_lq} (c₀, r) ∩ {WT (c_n)} with c ≽ [Sc] _α(c), y ⪯ [Sy] _α(y) .

By taking complete metric space instead of left (right) K-sequentially complete dislocated quasi metric space, we obtain the following result.

Corollary 2.4: Let (W, ⪯ , d_lq) be an ordered complete dislocated quasi metric space, S, T : W → M_q (W) be the two fuzzy mappings. Suppose that the following assertions hold:

(i) There exists a function μ ∈ Ψ, c₀ ∈ W, α (c) , α (y) ∈ (0, 1] and r > 0, we have $\begin{matrix} H_{d_{lq}} ([Tc]_{α (c)}, [Ty]_{α (y)} \leq μ (max {d_{lq} (c, y), \\ d_{lq} (c, [Tc]_{α (c)}), d_{lq} (y, [Ty]_{α (y)})}), \end{matrix}$

for all c, y ∈ B_{d
_lq} (c₀, r) ∩ {WT (c_n)} with c ≽ [Sc] _α(c), y ⪯ [Sy] _α(y) .

(iii) The set G (S) = {c : c ⪯ [Sc] _α(c) and c ∈ B_{d
_lq} (c₀, r)} is closed and contains c₀.

(iv) $\begin{matrix} \sum_{i = 0}^{n} μ^{i} (d_{lq} (c_{0}, c_{1})) < r for all n \in ℕ . \end{matrix}$ Then a subsequence {c_2n} of {WT (c_n)} is a sequence in G (S) and {c_2n} → c^∗ ∈ G (S) and d_lq (c^∗, c^∗) =0 . Also, if the inequality (i) holds for c^∗. Then S and T have a common fuzzy fixed point c^∗ in B_{d
_lq} (c₀, r).

By excluding open ball, we obtain the following result.

Corollary 2.5: Let (W, ⪯ , d_lq) be an ordered left (right) K-sequentially complete dislocated quasi metric space, S, T : W → M_q (W) be the two fuzzy mappings. Suppose that the following assertions hold:

(i) There exists a function μ ∈ Ψ, c₀ ∈ W, α (c) , α (y) ∈ (0, 1] and r > 0 such that for every (c, y) ∈ W × W, we have $\begin{matrix} max {H_{d_{lq}} ([Tc]_{α (c)}, [Ty]_{α (y)}), H_{d_{lq}} ([Ty]_{α (y)}, \\ [Tc]_{α (c)})} \leq μ (D_{lq} (c, y)), \end{matrix}$

with c ≽ [Sc] _α(c), y ⪯ [Sy] _α(y), where $\begin{matrix} D_{lq} (c, y) = max {d_{lq} (c, y), d_{lq} (c, [Tc]_{α (c)}), \\ d_{lq} (y, [Ty]_{α (y)})} . \end{matrix}$

(ii) If d_lq (c, [Tc] _α(c)) = d_lq (c, y) and d_lq ([Tc] _α(c),c) = d_lq (y, c) , then $\begin{matrix} (a) If c ⪯ [Sc]_{α (c)}, implies y ≽ [Sy]_{α (y)} \\ (b) If c ≽ [Sc]_{α (c)}, implies y ⪯ [Sy]_{α (y)} \end{matrix}$

(iii) The set G (S) ={ c : c ⪯ [Sc] _α(c) } is closed and contains c₀.

Then a subsequence {c_2n} of {WT (c_n)} is a sequence in G (S) and {c_2n} → c^∗ ∈ G (S) and d_lq (c^∗, c^∗) =0 . Also, if the inequality (i) holds for c^∗. Then S and T have a common fuzzy fixed point c^∗ in W.

3 Application

In this section, we show that Theorem 2.1 can be utilized to derive a common fixed point for a set-valued mappings in a dislocated quasi metric space.

Theorem 3.1. Let (W, ⪯ , d_q) be an ordered left (right) K-sequentially complete dislocated quasi metric space, F, R : X → P (W) be the multivalued mappings. Suppose that the following assertions hold:

(i) There exists a function μ ∈ Ψ, c₀ ∈ W and r > 0, we have $\begin{matrix} max {H_{q} (Rc, Ry), H_{q} (Ry, Rc)} \leq μ (D_{q} (c, y)), \end{matrix}$ for all c, y ∈ B_{d
_q} (c₀, r) ∩ {WR (c_n)} with c ≽ Fc, y ⪯ Fy, where $\begin{matrix} D_{q} (c, y) = max {d_{q} (c, y), d_{q} (c, Rc), d_{q} (y, Ry)} . \end{matrix}$

(ii) If c ∈B_{d
_q} (c₀, r), d_q (c, Rc) = d_q (c, y) and d_q (Rc, c) = d_q (y, c) , then $\begin{matrix} (a) c ≼ F_{c}, implies y ≽ F_{y} \\ (b) c ≽ F_{c}, implies y ≼ F_{y} \end{matrix}$

(iii) The set G (F) = {c : c ⪯ Fc and c ∈ B_{d
_q}(c₀, r)} is closed and contains c₀.

(iv) $\begin{matrix} \sum_{i = 0}^{n} max {μ^{i} (d_{q} (c_{1}, c_{0}), μ^{i} (d_{q} (c_{0}, c_{1}))} \\ < r for all n \in ℕ . \end{matrix}$ Then a subsequence {c_2n} of {WR (c_n)} is a sequence in G (F) and {c_2n} → c^∗ ∈ G (F) and d_q (c^∗, c^∗) =0 . Also, if the inequality (i) holds for c^∗, then F and R have a common fixed point c^∗ in B_{d
_q} (c₀, r).

Proof. Let α : W → (0, 1] be any arbitrary mapping. Consider two fuzzy mappings S, T : W → M_q (W) defined by $(S c) (t) = {\begin{matrix} α (c), & t \in F c \\ 0, & t \notin F c \end{matrix}$ and $(T c) (t) = {\begin{matrix} α (c), & t \in R c \\ 0, & t \notin R c . \end{matrix}$ We get that $\begin{matrix} {[Sc]}_{α (c)} = {t : Sc (t) \geq α (c)} = Fc \end{matrix}$ and $\begin{matrix} {[Tc]}_{α (c)} = {t : Tc (t) \geq α (c)} = Rc . \end{matrix}$ Hence, the inequality (i) of Theorem 3.1 becomes the inequality (i) of Theorem 2.1. This implies that there exists c^∗ ∈ [Sc] _α(c) ∩ [Tc] _α(c) = Fc ∩ Rc.

4 Conclusion

In the present paper, we have achieved fixed point results for a pair of fuzzy mappings on an intersection of an open ball and a sequence in left (right) K-sequentially complete dislocated quasi metric spaces. We generalized the result of Altun et al. [2] in different ways. We also used an open ball instead of a whole space and showed that fixed point exists even if the contractive condition holds on the subspaces instead of the whole spaces. Moreover, we investigate our results in a better framework of dislocated quasi metric space and also show that fuzzy fixed point results can be obtained from the fixed point theorems of set-valued mappings. New results in partial metric space, partial quasi metric space, dislocated metric space, quasi metric space and metric space can be obtained as corollaries of our results. Many fixed point results of multivalued and single-valued contractive mappings can also be obtained as corollaries of our results. One can further extend our results to L-fuzzy mappings, intuitionistic fuzzy mappings and bipolar fuzzy mappings.

Conflict of interests

The authors declare that they have no competing interests.

Authors contributions

Each author equally contributed to this paper, read and approved the final manuscript.

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