Metric Spaces

Hindawi Journal of Function Spaces Volume 2018, Article ID 5650242, 9 pages https://doi.org/10.1155/2018/5650242

Research Article Fixed Points of L-Fuzzy Mappings in Ordered 𝑏-Metric Spaces Ismat Beg

,1 Mohamed A. Ahmed,2 and Hatem A. Nafadi3

1

Lahore School of Economics, Lahore, Pakistan Assiut University, Assiut, Egypt 3 Port Said University, Port Said, Egypt 2

Correspondence should be addressed to Ismat Beg; [email protected] Received 26 October 2017; Revised 20 December 2017; Accepted 11 January 2018; Published 1 March 2018 Academic Editor: Tomonari Suzuki Copyright © 2018 Ismat Beg et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. We prove common fixed point theorems for weakly commuting and occasionally coincidentally idempotent L-fuzzy mappings in ordered 𝑏-metric spaces. We also obtain common fixed point for pair of mapping satisfying (JCLR) property. An application to integral type and usual contractive condition is given.

1. Introduction In 1967, Goguen [1] introduced the notion of L-fuzzy sets as a generalization of fuzzy sets [2]. Afterward, Heilpern [3] gave the concept of fuzzy mappings and proved fixed point theorems for fuzzy contractive mappings in metric linear spaces as a generalization of Nadler [4] contraction principle. Subsequently, several authors studied the generalizations/extensions and applications of these results; in many papers sufficient conditions for the existence of fixed point for fuzzy and L-fuzzy contractive mappings in metric spaces and 𝑏-metric spaces are obtained (see [5–8]). Rashid et al. [9] proved common L-fuzzy fixed point theorem in complete metric spaces. They extended the results of Heilpern [3] into L-fuzzy mappings in metric spaces. In this paper, we prove some common fixed point theorems for L-fuzzy mappings in ordered 𝑏-metric spaces. An application to integral type and usual contractive condition is also given, we will generalize and extend results [10, 11].

2. Preliminaries Definition 1 (see [12]). Let 𝑋 be a nonempty set. A mapping 𝑑 : 𝑋 × 𝑋 → [0, ∞) is called 𝑏-metric if there exists a real number 𝑏 ≥ 1 such that, for every 𝑥, 𝑦, 𝑧 ∈ 𝑋, we have (𝑑1 ) 𝑑(𝑥, 𝑦) = 0 ⇔ 𝑥 = 𝑦,

(𝑑2 ) 𝑑(𝑥, 𝑦) = 𝑑(𝑦, 𝑥), (𝑑3 ) 𝑑(𝑥, 𝑧) ≤ 𝑏{𝑑(𝑥, 𝑦) + 𝑑(𝑦, 𝑧)}. In this case, the pair (𝑋, 𝑑) is called a 𝑏-metric space. Definition 2 (see [13]). Let (𝑋, 𝑑) be a 𝑏-metric space. A sequence {𝑥𝑛 } in 𝑋 is called (i) convergent if and only if there exists 𝑥 ∈ 𝑋 such that 𝑑(𝑥𝑛 , 𝑥) → 0 as 𝑛 → ∞, (ii) Cauchy if and only if 𝑑(𝑥𝑛 , 𝑥𝑚 ) → 0 as 𝑚, 𝑛 → ∞. A 𝑏-metric space is said to be complete if and only if each Cauchy sequence in this space is convergent. Let (𝑋, 𝑑) be a 𝑏-metric space, let 𝐶𝐿(𝑋) denote the collection of all nonempty closed subsets of 𝑋, let 𝑊(𝑋) be the set of all fuzzy sets of 𝑋, where its 𝛼-level sets are nonempty compact subsets of 𝑋, and let R+ be the set of nonnegative real numbers; define the metrics 𝑑 : 𝑋 × 𝐶𝐿(𝑋) → R+ and 𝐻 : 𝐶𝐿(𝑋) × 𝐶𝐿(𝑋) → R+ as 𝑑(𝑥, 𝐴) = inf 𝑦∈𝐴𝑑(𝑥, 𝑦) and 𝐻(𝐴, 𝐵) = max{sup𝑎∈𝐴 𝑑(𝑎, 𝐵), sup𝑏∈𝐵 𝑑(𝐴, 𝑏)}. Notice that 𝑑(𝑎, 𝐵) ≤ 𝑑(𝑎, 𝑏) and 𝑑(𝑎, 𝐵) ≤ 𝐻(𝐴, 𝐵), for each 𝑎 ∈ 𝐴 and 𝑏 ∈ 𝐵. Let 𝑓 : 𝑋 → 𝑋, 𝐹 : 𝑋 → 𝐶𝐿(𝑋), and 𝐶(𝑓, 𝐹) be the set of the coincidence points of (𝑓, 𝐹). Lemma 3 (see [14]). Let (𝑋, 𝑑) be a 𝑏-metric space, 𝐴, 𝐵 ∈ 𝐶𝐿(𝑋); then 𝑑(𝑎, 𝐵) ≤ 𝐻(𝐴, 𝐵) for all 𝑎 ∈ 𝐴.

2

Journal of Function Spaces

Definition 4 (see [15]). A partially ordered set consists of a set 𝑋 and a binary relation ⪯ on 𝑋 which satisfies the following conditions for all 𝑥, 𝑦, 𝑧 ∈ 𝑋:

{𝑦2𝑛+1 } = {𝑔𝑥2𝑛+1 } ⊂ 𝐹𝑥2𝑛 ,

(1) 𝑥 ⪯ 𝑥 (reflexivity).

{𝑦2𝑛+2 } = {𝑓𝑥2𝑛+2 } ⊂ 𝐺𝑥2𝑛+1 ,

(2) If 𝑥 ⪯ 𝑦 and 𝑦 ⪯ 𝑥, then 𝑥 = 𝑦 (antisymmetry).

A set with a partial order ⪯ is called a partially ordered set. Let (𝑋, ⪯) be a partially ordered set and 𝑥, 𝑦 ∈ 𝑋. Elements 𝑥 and 𝑦 are said to be comparable elements of 𝑋 if either 𝑥 ⪯ 𝑦 or 𝑦 ⪯ 𝑥. Definition 5. Let 𝐴 and 𝐵 be two nonempty subsets of (𝑋, ⪯); the relation ⪯1 between 𝐴 and 𝐵 is defined as 𝐴 ⪯1 𝐵: if for every 𝑎 ∈ 𝐴 there exists 𝑏 ∈ 𝐵 such that 𝑎 ⪯ 𝑏. Definition 6 (see [1]). A function 𝐴 : 𝑋 → 𝐿 is said to be an L -fuzzy set if 𝐴 is a mapping from a nonempty set 𝑋 into a complete distributive lattice 𝐿. Definition 7 (see [9, 10, 16]). Let 𝐴 be an L-fuzzy set in 𝑋. The 𝛼L -level set of 𝐴 (denoted by 𝐴 𝛼L ) is defined as if 𝛼L ∈ 𝐿 \ {0L } ,

𝐴 0L = {𝑥 ∈ 𝑋 : 0L ⪯𝐿 𝐴 (𝑥)},

(2)

then 𝑂(𝐹, 𝐺, 𝑓, 𝑔, 𝑥0 ) is called the orbit for the mappings (𝐹, 𝐺, 𝑓, 𝑔).

(3) If 𝑥 ⪯ 𝑦 and 𝑦 ⪯ 𝑧, then 𝑥 ⪯ 𝑧 (transitivity).

𝐴 𝛼L = {𝑥 ∈ 𝑋 : 𝛼L ⪯𝐿 𝐴 (𝑥)}

Definition 13 (see [19]). Let (𝑋, 𝑑) be a metric space, 𝑓, 𝑔 : 𝑋 → 𝑋, and 𝐹, 𝐺 : 𝑋 → 𝑊(𝑋). Let 𝑥0 ∈ 𝑋; if there exists a sequence {𝑦𝑛 } in 𝑋 such that

(1)

where 𝐵 denotes the closure of the set 𝐵. Definition 8 (see [9]). Let 𝑋 and 𝑌 be two arbitrary nonempty sets and IL (𝑌) denote the collection of all L-fuzzy sets in 𝑌. A mapping 𝐹 is called L-fuzzy mapping if 𝐹 is a mapping from 𝑋 into IL (𝑌). An L-fuzzy mapping 𝐹 is an L-fuzzy subset on 𝑋 × 𝑌 with membership function 𝐹(𝑥)(𝑦). The function 𝐹(𝑥)(𝑦) is the grade of membership of 𝑦 in 𝐹(𝑥). Definition 9 (see [9]). Let 𝐹, 𝐺 be L -fuzzy mappings from an arbitrary nonempty set 𝑋 into IL (𝑋). A point 𝑧 ∈ 𝑋 is called an L-fuzzy fixed point of 𝐹 if 𝑧 ∈ {𝐹𝑧}𝛼L , where 𝛼L ∈ 𝐿 \ {0L }. The point 𝑧 ∈ 𝑋 is called a common L-fuzzy fixed point of 𝐹 and 𝐺 if 𝑧 ∈ {𝐹𝑧}𝛼L ∩ {𝐺𝑧}𝛼L . Remark 10 (see [10, 16]). If 𝛼L = 1L then the L-fuzzy fixed point is just the fixed point for the 𝐿-fuzzy mapping. Definition 11 (see [17]). Let (𝑋, 𝑑) be a metric space, 𝑓 : 𝑋 → 𝑋, and 𝐹 : 𝑋 → 𝐶𝐿(𝑋). Then the pair (𝑓, 𝐹) is said to have (CLR) property if there exists a sequence {𝑥𝑛 } in 𝑋 and 𝐴 ∈ 𝐶𝐿(𝑋) such that lim𝑛→∞ 𝑓𝑥𝑛 = 𝑢 ∈ 𝐴 = lim𝑛→∞ 𝐹𝑥𝑛 , with 𝑢 = 𝑓V, for some 𝑢, V ∈ 𝑋. Definition 12 (see [18]). Let (𝑋, 𝑑) be a metric space, 𝑌 ⊆ 𝑋, 𝑓 : 𝑋 → 𝑋, and 𝐹 : 𝑋 → 𝐶𝐿(𝑋). Two mappings (𝑓, 𝐹) are said to be occasionally coincidentally idempotent if 𝑓2 𝑥 = 𝑓𝑥 for some 𝑥 ∈ 𝐶(𝑓, 𝐹).

Definition 14 (see [19]). Metric space 𝑋 is called 𝑥0 joint orbitally complete, if every Cauchy sequence of each orbit at 𝑥0 is convergent in 𝑋. Definition 15 (see [20]). Let 𝑓 be a mapping from a subset 𝑌 of a metric space (𝑋, 𝑑) into 𝑋 and 𝐹 : 𝑌 → 𝐶𝐿(𝑋); then 𝑓 is said to be 𝐹-weakly commuting at 𝑥 ∈ 𝑌 if 𝑓2 𝑥 ∈ 𝐹𝑓𝑥.

3. Definitions in 𝑏-Metric Spaces Let (𝑋, 𝑑) be a 𝑏-metric space, 𝑌 ⊆ 𝑋, 𝑓, 𝑔 : 𝑌 → 𝑋 and 𝐹, 𝐺 are two L-fuzzy mappings from 𝑌 into IL (𝑋) such that, for each 𝑥 ∈ 𝑌, 𝛼L ∈ 𝐿 \ {0L }, {𝐹𝑥}𝛼L and {𝐺𝑥}𝛼L are nonempty closed subsets of 𝑋. Similar to [11], first we rewrite some notions in 𝑏-metric spaces as follows. Definition 16. The pairs (𝑓, 𝐹) and (𝑔, 𝐺) are said to have (JCLR) property if there exist two sequences {𝑥𝑛 } and {𝑦𝑛 } in 𝑌 and 𝐴, 𝐵 ∈ 𝐶𝐿(𝑋) such that lim 𝑓𝑥𝑛 = lim 𝑔𝑦𝑛 = 𝑢,

𝑛→∞

𝑛→∞

lim {𝐹𝑥𝑛 }𝛼L = 𝐴,

𝑛→∞

(3)

lim {𝐺𝑦𝑛 }𝛼L = 𝐵,

𝑛→∞

with 𝑢 = 𝑓V = 𝑔𝑤, 𝑢 ∈ 𝐴 ∩ 𝐵, for some 𝑢, V, 𝑤 ∈ 𝑌. Definition 17. Two mappings 𝑓 and 𝐹 are said to be occasionally coincidentally idempotent if 𝑓2 𝑥 = 𝑓𝑥 for some 𝑥 ∈ 𝐶(𝑓, 𝐹). Definition 18. Two mappings 𝑓 and 𝐹 are said to be 𝐹-weakly commuting at 𝑥 ∈ 𝑌 if 𝑓𝑓𝑥 ∈ {𝐹𝑓𝑥}𝛼L . Let Φ be the family of all continuous mappings 𝜙 : [0, ∞)6 → [0, ∞) satisfying the following properties: (𝜙1 ) 𝐹 is nondecreasing in the 1st variable and nonincreasing in the 3rd, 4th, 5th, and 6th coordinate variables. (𝜙2 ) There exists ℎ ∈ (0, 1) such that for every 𝑢, V ≥ 0, 𝑏 ≥ 1 with 𝜙(𝑢, V, V, 𝑢, 𝑏(𝑢+V), 0) ≤ 0 or 𝜙(𝑢, V, 𝑢, V, 0, 𝑏(𝑢+ V)) ≤ 0 implies 𝑢 ≤ ℎV. Example 19. (i) 𝜙(𝑡1 , 𝑡2 , 𝑡3 , 𝑡4 , 𝑡5 , 𝑡6 ) = 𝑡1 − ℎ max{𝑡2 , (𝑡3 + 𝑡4 )𝑡5 𝑡6 }. (ii) 𝜙(𝑡1 , 𝑡2 , 𝑡3 , 𝑡4 , 𝑡5 , 𝑡6 ) = 𝑡1 − ℎ min{𝑡2 , (𝑡3 + 𝑡4 )/𝑏, (𝑡5 + 𝑡6 )/𝑏2 }, 𝑏 ≥ 2. (iii) 𝜙(𝑡1 , 𝑡2 , 𝑡3 , 𝑡4 , 𝑡5 , 𝑡6 ) = 𝑡1 − ℎ𝑡2 .

Journal of Function Spaces

3

Let Ψ be the family of continuous mappings 𝜓 : [0, ∞)3 → R, which is nondecreasing in the first coordinate and satisfying the following condition for each 𝑢, V ≥ 0: if 𝜓(𝑢, V, V) ≤ 0 or 𝜓(𝑢, V, 𝑢) ≤ 0, then 𝑢 ≤ V.

but 𝜙 (𝑑 (𝑦1 , 𝑦2 ) , 𝑑 (𝑦0 , 𝑦1 ) , 𝑑 (𝑦0 , 𝑦1 ) , 𝑑 (𝑦1 , 𝑦2 ) , 𝑏 (𝑑 (𝑦0 , 𝑦1 ) + 𝑑 (𝑦1 , 𝑦2 )) , 0)

Example 20. (i) 𝜓(𝑡1 , 𝑡2 , 𝑡3 ) = 𝑡1 − min{𝑡2 , (𝑡2 + 𝑡3 )/𝑛}, 𝑛 ≥ 2. (ii) 𝜓(𝑡1 , 𝑡2 , 𝑡3 ) = 𝑡1 − 𝑡2 . (iii) 𝜓(𝑡1 , 𝑡2 , 𝑡3 ) = 𝑡1 − (𝑡2 + 𝑡3 )/𝑛, 𝑛 ≥ 2.

≤ 𝜙 (𝐻 ({𝐹𝑥0 }𝛼L , {𝐺𝑥1 }𝛼L ) , 𝑑 (𝑓𝑥0 , 𝑔𝑥1 ) , 𝑑 (𝑓𝑥0 , {𝐹𝑥0 }𝛼L ) , 𝑑 (𝑔𝑥1 , {𝐺𝑥1 }𝛼L ) , 𝑑 (𝑓𝑥0 , {𝐺𝑥1 }𝛼L ) , 0) ≤ 0;

4. Common Fixed Points in Ordered 𝑏-Metric Spaces Theorem 21. Let (𝑋, 𝑑) be a 𝑏-metric space, 𝑌 ⊆ 𝑋, and ⪯ be a partial order defined on 𝑌, 𝑓, 𝑔 : 𝑌 → 𝑋 such that 𝑓(𝑌) and 𝑔(𝑌) are closed. Suppose that 𝐹, 𝐺 are two L-fuzzy mappings from 𝑌 into IL (𝑋) such that for each 𝑥 ∈ 𝑌 and 𝛼L ∈ 𝐿\{0L }, {𝐹𝑥}𝛼L and {𝐺𝑥}𝛼L are nonempty closed subsets of 𝑋 which satisfy the following conditions:

then from the property 𝜙(𝑢, V, V, 𝑢, 𝑏(𝑢 + V), 0) ≤ 0 which implies 𝑢 ≤ ℎV, there exists ℎ ∈ (0, 1) such that 𝑑(𝑦1 , 𝑦2 ) ≤ ℎ𝑑(𝑦0 , 𝑦1 ). Again, 𝑦2 = 𝑓𝑥2 ∈ {𝐺𝑥1 }𝛼L , 𝑦3 = 𝑔𝑥3 ∈ {𝐹𝑥2 }𝛼L , 𝑥2 ⪯ 𝑥3 , and 𝑑(𝑦3 , 𝑦2 ) = 𝑑(𝑔𝑥3 , 𝑓𝑥2 ) ≤ 𝐻({𝐹𝑥2 }𝛼L , {𝐺𝑥1 }𝛼L ); then 𝜙 (𝐻 ({𝐹𝑥2 }𝛼L , {𝐺𝑥1 }𝛼L ) , 𝑑 (𝑓𝑥2 , 𝑔𝑥1 ) ,

(1) {𝐹𝑥}𝛼L ⪯1 𝑔(𝑌) and {𝐺𝑥}𝛼L ⪯1 𝑓(𝑌).

𝑑 (𝑓𝑥2 , {𝐹𝑥2 }𝛼L ) , 𝑑 (𝑔𝑥1 , {𝐺𝑥1 }𝛼L ) , 0,

(2) 𝑔𝑦 ∈ {𝐹𝑥}𝛼L or 𝑓𝑦 ∈ {𝐺𝑥}𝛼L implies 𝑥 ⪯ 𝑦.

𝑑 (𝑔𝑥1 , {𝐹𝑥2 }𝛼L )) ≤ 0.

(3) If 𝑦𝑛 → 𝑦, then 𝑦𝑛 ⪯ 𝑦 for all 𝑛 ∈ N.

(8)

Also

(4) (𝑓, 𝐹) and (𝑔, 𝐺) are weakly commuting and occasionally coincidentally idempotent. (5) For all comparable elements 𝑥, 𝑦 ∈ 𝑌 and 𝐸 > 0, there exists 𝜙 ∈ Φ such that 𝜙 (𝑄) + 𝐸 min (𝑄) ≤ 0,

(7)

(4)

𝜙 (𝑑 (𝑦2 , 𝑦3 ) , 𝑑 (𝑦1 , 𝑦2 ) , 𝑑 (𝑦2 , 𝑦3 ) , 𝑑 (𝑦1 , 𝑦2 ) , 0, 𝑏 (𝑑 (𝑦1 , 𝑦2 ) + 𝑑 (𝑦2 , 𝑦3 ))) ≤ 𝜙 (𝐻 ({𝐹𝑥2 }𝛼L , {𝐺𝑥1 }𝛼L ) , 𝑑 (𝑓𝑥2 , 𝑔𝑥1 ) ,

(9)

𝑑 (𝑓𝑥2 , {𝐹𝑥2 }𝛼L ) , 𝑑 (𝑔𝑥1 , {𝐺𝑥1 }𝛼L ) , 0, 𝑑 (𝑔𝑥1 , {𝐹𝑥2 }𝛼L )) ≤ 0;

where 𝑄 = (𝐻 ({𝐹𝑥}𝛼L , {𝐺𝑦}𝛼L ) , 𝑑 (𝑓𝑥, 𝑔𝑦) , 𝑑 (𝑓𝑥, {𝐹𝑥}𝛼L ) , 𝑑 (𝑔𝑦, {𝐺𝑦}𝛼L ) , 𝑑 (𝑓𝑥, {𝐺𝑦}𝛼L ) , 𝑑 (𝑔𝑦, {𝐹𝑥}𝛼L )) ,

(5)

then from the property 𝜙(𝑢, V, 𝑢, V, 0, 𝑏(𝑢 + V)) ≤ 0 which implies 𝑢 ≤ ℎV, there exists ℎ ∈ (0, 1) such that 𝑑(𝑦2 , 𝑦3 ) ≤ ℎ𝑑(𝑦1 , 𝑦2 ) ≤ ℎ2 𝑑(𝑦0 , 𝑦1 ). By induction we obtain 𝑑(𝑦𝑛 , 𝑦𝑛+1 ) ≤ ℎ𝑛 𝑑(𝑦0 , 𝑦1 ). Continuing in this way, we obtain an orbit 𝑂(𝐹, 𝐺, 𝑓, 𝑔, 𝑥0 ) such that 𝑦2𝑛+1 = 𝑔𝑥2𝑛+1 ∈ {𝐹𝑥2𝑛 }𝛼L and 𝑦2𝑛+2 = 𝑓𝑥2𝑛+2 ∈ {𝐺𝑥2𝑛+1 }𝛼L . Further,

(6) One of 𝑓(𝑌) or 𝑔(𝑌) is 𝑥0 joint orbitally complete for some 𝑥0 ∈ 𝑌;

𝑑 (𝑦𝑛 , 𝑦𝑚 ) ≤ 𝑏𝑑 (𝑦𝑛 , 𝑦𝑛+1 ) + 𝑏2 𝑑 (𝑦𝑛+1 , 𝑦𝑛+2 ) + ⋅ ⋅ ⋅

then 𝑓 and 𝑔 have a common fixed point, while 𝐹 and 𝐺 have a common 𝐿-fuzzy fixed point

≤ 𝑏𝑑 (𝑦𝑛 , 𝑦𝑛+1 ) + 𝑏2 𝑑 (𝑦𝑛+1 , 𝑦𝑛+2 ) + ⋅ ⋅ ⋅

Proof. Let 𝑥0 ∈ 𝑋 and 𝑦0 = 𝑓𝑥0 ; then by (1), there exist 𝑥1 , 𝑥2 ∈ 𝑌 such that 𝑦1 = 𝑔𝑥1 ∈ {𝐹𝑥0 }𝛼L and 𝑦2 = 𝑓𝑥2 ∈ {𝐺𝑥1 }𝛼L , from (2), 𝑥0 ⪯ 𝑥1 ⪯ 𝑥2 . Now, 𝑑(𝑦1 , 𝑦2 ) = 𝑑(𝑔𝑥1 , 𝑓𝑥2 ) ≤ 𝐻({𝐹𝑥0 }𝛼L , {𝐺𝑥1 }𝛼L ); then 𝜙 (𝐻 ({𝐹𝑥0 }𝛼L , {𝐺𝑥1 }𝛼L ) , 𝑑 (𝑓𝑥0 , 𝑔𝑥1 ) , 𝑑 (𝑓𝑥0 , {𝐹𝑥0 }𝛼L ) , 𝑑 (𝑔𝑥1 , {𝐺𝑥1 }𝛼L ) , 𝑑 (𝑓𝑥0 , {𝐺𝑥1 }𝛼L ) , 0) ≤ 0,

+ 𝑏𝑚−𝑛−1 𝑑 (𝑦𝑚−1 , 𝑦𝑚 )

+ 𝑏𝑚−𝑛 𝑑 (𝑦𝑚−1 , 𝑦𝑚 ) 𝑛

≤ 𝑏ℎ 𝑑 (𝑦0 , 𝑦1 ) 𝑡 + 𝑏 ℎ

(10) 𝑑 (𝑦0 , 𝑦1 ) + ⋅ ⋅ ⋅

+ 𝑏𝑚−𝑛 ℎ𝑚−1 𝑑 (𝑦0 , 𝑦1 ) =

(6)

2 𝑛+1

𝑏ℎ𝑛 𝑑 (𝑦0 , 𝑦1 ) . 1 − 𝑏ℎ

Therefore, lim𝑛,𝑚→∞ 𝑑(𝑦𝑛 , 𝑦𝑚 ) = 0. Hence, {𝑦𝑛 } is a Cauchy sequence. As {𝑦2𝑛+2 } is a Cauchy sequence in 𝑓(𝑌), and 𝑓(𝑌) is joint orbitally complete, therefore, there exists 𝑧 ∈ 𝑋 such

4

Journal of Function Spaces

that 𝑦2𝑛+2 → 𝑧 = 𝑓𝑢, for 𝑢 ∈ 𝑌. Next, we show that 𝑧 ∈ {𝐹𝑢}𝛼L . Since 𝜙 (𝑄) + 𝐸 min (𝑄) ≤ 0,

each 𝑥 ∈ 𝑋, there exists 𝑦 ∈ 𝑌 such that 𝑥 ≤ 𝑦. Let 𝐿 = [0, 1] and define the maps 𝑓, 𝑔, 𝐹, 𝐺 on 𝑋 as 𝑔𝑥 = 𝑥/2, 𝑓𝑥 = 𝑥/3, 0, { { { { { {1 (𝐹𝑥) (𝑦) = { , {3 { { { {2 , {3

(11)

where 𝑄 = (𝐻 ({𝐹𝑢}𝛼L , {𝐺𝑥2𝑛+1 }𝛼L ) , 𝑑 (𝑓𝑢, 𝑔𝑥2𝑛+1 ) , 𝑑 (𝑓𝑢, {𝐹𝑢}𝛼L ) , 𝑑 (𝑔𝑥2𝑛+1 , {𝐺𝑥2𝑛+1 }𝛼L ) ,

𝑥 , 5 𝑥 𝑥 if < 𝑦 < , 5 4 𝑥 if ≤ 𝑦 ≤ 1, 4 if 0 ≤ 𝑦 ≤

𝑥 0, if 0 ≤ 𝑦 ≤ , { { 7 { { { {1 𝑥 𝑥 (𝐺𝑥) (𝑦) = { , if < 𝑦 < , { 6 7 6 { { { 𝑥 {1 , if ≤ 𝑦 ≤ 1. {4 6

(12)

𝑑 (𝑓𝑢, {𝐺𝑥2𝑛+1 }𝛼L ) , 𝑑 (𝑔𝑥2𝑛+1 , {𝐹𝑢}𝛼L )) ,

(15)

then Define the sequences 𝑥2𝑛 , 𝑥2𝑛+1 , and 𝑥2𝑛+2 in 𝑌 as 𝑥2𝑛 = {1/(2𝑛 + 1)}, 𝑥2𝑛+1 = {1/(2𝑛 + 2)}, and 𝑥2𝑛+2 = {1/(2𝑛 + 3)}, 𝑛 ∈ N; then 𝑦2𝑛+1 = 𝑔𝑥2𝑛+1 = 1/2(2𝑛 + 2) and 𝑦2𝑛+2 = 𝑓𝑥2𝑛+2 = 1/3(2𝑛 + 3). Now, {𝐹𝑥2𝑛 }2/3 = [1/4(2𝑛 + 1), 1] and {𝐺𝑥2𝑛+1 }1/4 = [1/6(2𝑛 + 2), 1]; then 𝑔𝑥2𝑛+1 ∈ {𝐹𝑥2𝑛 }2/3 and 𝑓𝑥2𝑛+2 ∈ {𝐺𝑥2𝑛+1 }1/4 . Further, lim𝑛→∞ 𝑓𝑥2𝑛+2 = lim𝑛→∞ 𝑔𝑥2𝑛+1 = 0 ∈ [0, 1] = lim𝑛→∞ {𝐹𝑥2𝑛 }2/3 = lim𝑛→∞ {𝐺𝑥2𝑛+1 }1/4 . Let 𝜙(𝑡1 , . . . , 𝑡6 ) = 𝑡1 − ℎ𝑡2 ; then we have

𝜙 (𝐻 ({𝐹𝑢}𝛼L , {𝐺𝑥2𝑛+1 }𝛼L ) , 𝑑 (𝑓𝑢, 𝑔𝑥2𝑛+1 ) , 𝑑 (𝑓𝑢, {𝐹𝑢}𝛼L ) , 𝑑 (𝑔𝑥2𝑛+1 , {𝐺𝑥2𝑛+1 }𝛼L ) , 𝑑 (𝑓𝑢, {𝐺𝑥2𝑛+1 }𝛼L ) , 𝑑 (𝑔𝑥2𝑛+1 , {𝐹𝑢}𝛼L )) ≤ 0, 𝜙 (𝑑 (𝑦2𝑛+2 , {𝐹𝑢}𝛼L ) , 𝑑 (𝑧, 𝑦2𝑛+1 ) , 𝑑 (𝑧, {𝐹𝑢}𝛼L ) ,

𝜙 (𝑄) + 𝐸 min (𝑄) = 0,

(13)

𝑑 (𝑦2𝑛+1 , 𝑦2𝑛+2 ) , 𝑑 (𝑧, 𝑦2𝑛+2 ) , 𝑏𝑑 (𝑦2𝑛+1 , {𝐹𝑢}𝛼L )) where

≤ 𝜙 (𝐻 ({𝐹𝑢}𝛼L , {𝐺𝑥2𝑛+1 }𝛼L ) , 𝑑 (𝑓𝑢, 𝑔𝑥2𝑛+1 ) ,

𝑄 = (𝐻 ({𝐹𝑥2𝑛 }𝛼𝐿 , {𝐺𝑥2𝑛+1 }𝛼𝐿 ) , 𝑑 (𝑓𝑥2𝑛+2 , 𝑔𝑥2𝑛+1 ) ,

𝑑 (𝑓𝑢, {𝐹𝑢}𝛼L ) , 𝑑 (𝑔𝑥2𝑛+1 , {𝐺𝑥2𝑛+1 }𝛼L ) ,

𝑑 (𝑓𝑥2𝑛+2 , {𝐹𝑥2𝑛 }𝛼𝐿 ) , 𝑑 (𝑔𝑥2𝑛+1 , {𝐺𝑥2𝑛+1 }𝛼𝐿 ) ,

𝑑 (𝑓𝑢, {𝐺𝑥2𝑛+1 }𝛼L ) , 𝑑 (𝑔𝑥2𝑛+1 , {𝐹𝑢}𝛼L )) ≤ 0.

(17)

𝑑 (𝑓𝑥2𝑛+2 , {𝐺𝑥2𝑛+1 }𝛼𝐿 ) , 𝑑 (𝑔𝑥2𝑛+1 , {𝐹𝑥2𝑛 }𝛼𝐿 )) .

As 𝑛 → ∞ 𝜙 (𝑑 (𝑧, {𝐹𝑢}𝛼L ) , 0, 𝑑 (𝑧, {𝐹𝑢}𝛼L ) , 0, 0, 𝑏𝑑 (𝑧, {𝐹𝑢}𝛼L ))

(16)

(14)

≤ 0.

By 𝜙(𝑢, V, 𝑢, V, 0, 𝑏(𝑢 + V)) ≤ 0 which implies 𝑢 ≤ ℎV, we have 𝑑(𝑧, {𝐹𝑢}𝛼L ) ≤ ℎ.0 = 0. Thus, 𝑧 ∈ {𝐹𝑢}𝛼L . As 𝑧 = 𝑓𝑢 ∈ {𝐹𝑢}𝛼L ∈ 𝑔(𝑌), therefore, there exists 𝜐 ∈ 𝑌 such that 𝑧 = 𝑔𝜐. Similarly, 𝑧 = 𝑔𝜐 ∈ {𝐺𝜐}𝛼L . Since the pair (𝐹, 𝑓) are weakly commuting and occasionally coincidentally idempotent and 𝑧 = 𝑓𝑢 ∈ {𝐹𝑢}𝛼L , then 𝑧 = 𝑓𝑢 = 𝑓𝑓𝑢 = 𝑓𝑧; therefore, 𝑧 = 𝑓𝑧 = 𝑓𝑓𝑢 ∈ {𝑓𝐹𝑢}𝛼L ⊆ {𝐹𝑓𝑢}𝛼L = {𝐹𝑧}𝛼L . Also 𝑧 = 𝑔𝜐 ∈ {𝐺𝜐}𝛼L ; then 𝑧 = 𝑔𝜐 = 𝑔𝑔𝜐 = 𝑔𝑧; therefore, 𝑔𝑧 = 𝑔𝑔𝜐 ∈ {𝑔𝐺𝜐}𝛼L ⊆ {𝐺𝑔𝜐}𝛼L = {𝐺𝑧}𝛼L . Example 22. Let 𝑌 ⊆ 𝑋 = [0, 1] and 𝑑(𝑥, 𝑦) = |𝑥 − 𝑦|2 ; then (𝑋, 𝑑) is a 𝑏-metric space with 𝑏 = 2. Define the partial order 𝑥 ⪯ 𝑦 as 𝑥 ≤ 𝑦 for each 𝑥, 𝑦 ∈ 𝑋 and 𝑋 ⪯1 𝑌 as follows: for

Finally, 𝑓0 = 𝑓𝑓0 ∈ [0, 1] = 𝐹𝑓0 and 𝑔𝑔0 = 𝑔0 ∈ [0, 1] = 𝐺𝑔0; that is, (𝑓, 𝐹) and (𝑔, 𝐺) are weakly commuting and occasionally coincidentally idempotent (𝑓, 𝐹) and (𝑔, 𝐺) are weakly commuting and occasionally coincidentally idempotent. Now, 𝑓, 𝑔, 𝐹, 𝐺 satisfy all conditions of Theorem 21 and 𝑓0 = 𝑔0 = 0 ∈ [0, 1] = {𝐹0}2/3 = {𝐺0}1/4 is a common fixed point. Corollary 23. Let (𝑋, 𝑑) be a complete 𝑏-metric space, 𝑌 ⊆ 𝑋, and ⪯ be a partial order defined on 𝑌, 𝑓, 𝑔 : 𝑌 → 𝑋 such that 𝑓(𝑌) and 𝑔(𝑌) are closed. Suppose that 𝐹, 𝐺 are two Lfuzzy mappings from 𝑌 into IL (𝑋) such that, for each 𝑥 ∈ 𝑌 and 𝛼L ∈ 𝐿 \ {0L }, {𝐹𝑥}𝛼L and {𝐺𝑥}𝛼L are nonempty closed subsets of 𝑋 which satisfy conditions ((1)–(5)); then 𝑓 and 𝑔 have a common fixed point, while 𝐹 and 𝐺 have a common 𝐿fuzzy fixed point. Remark 24. (1) Theorem 21 is a generalization of Theorem 2.2 [6], Theorem 14 [9], and Theorem 3.1 [8].

Journal of Function Spaces

5

(2) In Theorem 21, one may put one of 𝑓 = 𝑔 or 𝐹 = 𝐺 or both; in this case, condition (4) has the following versions:

𝜙 (𝐻 ({𝐹𝑥}𝛼L , {𝐺𝑦}𝛼L ) , 𝑑 (𝑓𝑥, 𝑓𝑦) , 𝑑 (𝑓𝑥, {𝐹𝑥}𝛼L ) , 𝑑 (𝑓𝑦, {𝐺𝑦}𝛼L ) , 𝑑 (𝑓𝑥, {𝐺𝑦}𝛼L ) , 𝑑 (𝑓𝑦, {𝐹𝑥}𝛼L )) + 𝐸 min (𝐻 ({𝐹𝑥}𝛼L , {𝐺𝑦}𝛼L ) , 𝑑 (𝑓𝑥, 𝑓𝑦) , 𝑑 (𝑓𝑥, {𝐹𝑥}𝛼L ) , 𝑑 (𝑓𝑦, {𝐺𝑦}𝛼L ) , 𝑑 (𝑓𝑥, {𝐺𝑦}𝛼L ) , 𝑑 (𝑓𝑦, {𝐹𝑥}𝛼L )) ≤ 0, 𝜙 (𝐻 ({𝐹𝑥}𝛼L , {𝐹𝑦}𝛼L ) , 𝑑 (𝑓𝑥, 𝑔𝑦) , 𝑑 (𝑓𝑥, {𝐹𝑥}𝛼L ) , 𝑑 (𝑔𝑦, {𝐹𝑦}𝛼L ) , 𝑑 (𝑓𝑥, {𝐹𝑦}𝛼L ) , 𝑑 (𝑔𝑦, {𝐹𝑥}𝛼L ))

(18)

+ 𝐸 min (𝐻 ({𝐹𝑥}𝛼L , {𝐹𝑦}𝛼L ) , 𝑑 (𝑓𝑥, 𝑔𝑦) , 𝑑 (𝑓𝑥, {𝐹𝑥}𝛼L ) , 𝑑 (𝑔𝑦, {𝐹𝑦}𝛼L ) , 𝑑 (𝑓𝑥, {𝐹𝑦}𝛼L ) , 𝑑 (𝑔𝑦, {𝐹𝑥}𝛼L )) ≤ 0, 𝜙 (𝐻 ({𝐹𝑥}𝛼L , {𝐹𝑦}𝛼L ) , 𝑑 (𝑓𝑥, 𝑓𝑦) , 𝑑 (𝑓𝑥, {𝐹𝑥}𝛼L ) , 𝑑 (𝑓𝑦, {𝐹𝑦}𝛼L ) , 𝑑 (𝑓𝑥, {𝐹𝑦}𝛼L ) , 𝑑 (𝑔𝑦, {𝐹𝑥}𝛼L )) + 𝐸 min (𝐻 ({𝐹𝑥}𝛼L , {𝐹𝑦}𝛼L ) , 𝑑 (𝑓𝑥, 𝑔𝑦) , 𝑑 (𝑓𝑥, {𝐹𝑥}𝛼L ) , 𝑑 (𝑓𝑦, {𝐹𝑦}𝛼L ) , 𝑑 (𝑓𝑥, {𝐹𝑦}𝛼L ) , 𝑑 (𝑓𝑦, {𝐹𝑥}𝛼L )) ≤ 0.

Theorem 25. Let (𝑋, 𝑑) be a 𝑏-metric space, 𝑌 ⊆ 𝑋, and ⪯ be a partial order defined on 𝑌, 𝑓, 𝑔 : 𝑌 → 𝑋 such that 𝑓(𝑌) and 𝑔(𝑌) are closed. Suppose that 𝐹𝑛 is a sequence of L-fuzzy mappings from 𝑌 into IL (𝑋) such that, for each 𝑥 ∈ 𝑌 and 𝛼L ∈ 𝐿\{0L }, {𝐹𝑛 𝑥}𝛼L are nonempty closed subsets of 𝑋 which satisfy the following conditions:

(1) (𝑓, 𝐹) and (𝑔, 𝐺) satisfy (JCLR) property. (2) (𝑓, 𝐹) and (𝑔, 𝐺) are weakly commuting and occasionally coincidentally idempotent.

(1) {𝐹𝑘 𝑥}𝛼L ⪯1 𝑔(𝑌) and {𝐹𝑙 𝑥}𝛼L ⪯1 𝑓(𝑌). (2) 𝑔𝑦 ∈ {𝐹𝑘 𝑥}𝛼L or 𝑓𝑦 ∈ {𝐹𝑙 𝑥}𝛼L implies 𝑥 ⪯ 𝑦. (3) If 𝑦𝑛 → 𝑦, then 𝑦𝑛 ⪯ 𝑦 for all 𝑛 ∈ 𝑁. (4) (𝑓, 𝐹𝑘 ) and (𝑔, 𝐹𝑙 ) are weakly commuting and occasionally coincidentally idempotent. (5) For all comparable elements 𝑥, 𝑦 ∈ 𝑌, 𝑘 = 2𝑛 + 1 and 𝑙 = 2𝑛 + 2, 𝑛 ∈ N and 𝐸 > 0, there exists 𝜙 ∈ Φ such that 𝜙 (𝑄) + 𝐸 min (𝑄) ≤ 0,

𝑓(𝑌) and 𝑔(𝑌) are closed. Suppose that 𝐹, 𝐺 are two L-fuzzy mappings from 𝑌 into IL (𝑋) such that, for each 𝑥 ∈ 𝑌 and 𝛼L ∈ 𝐿\{0L }, {𝐹𝑥}𝛼L and {𝐺𝑥}𝛼L are nonempty closed subsets of 𝑋 which satisfy the following conditions:

(19)

where 𝑄 = (𝐻 ({𝐹𝑘 𝑥}𝛼L , {𝐹𝑙 𝑦}𝛼L ) , 𝑑 (𝑓𝑥, 𝑔𝑦) , 𝑑 (𝑓𝑥, {𝐹𝑘 𝑥}𝛼L ) , 𝑑 (𝑔𝑦, {𝐹𝑙 𝑦}𝛼L ) , 𝑑 (𝑓𝑥, {𝐹𝑙 𝑦}𝛼L ) , (20) 𝑑 (𝑔𝑦, {𝐹𝑘 𝑥}𝛼L )) .

(6) One of 𝑓(𝑌) or 𝑔(𝑌) is 𝑥0 joint orbitally complete for some 𝑥0 ∈ 𝑌, then 𝑓 and 𝑔 have a common fixed point while 𝐹𝑘 and 𝐹𝑙 have a common 𝐿-fuzzy fixed point. Proof. Put 𝐹 = 𝐹𝑘 and 𝐺 = 𝐹𝑙 in Theorem 21. This concludes the proof.

(3) For all comparable elements 𝑥, 𝑦 ∈ 𝑌 and 𝐸 > 0 there exists 𝜙 ∈ Φ such that 𝜙 (𝑄) + 𝐸 min (𝑄) ≤ 0,

(21)

where 𝑄 = (𝐻 ({𝐹𝑥}𝛼L , {𝐺𝑦}𝛼L ) , 𝑑 (𝑓𝑥, 𝑔𝑦) , 𝑑 (𝑓𝑥, {𝐹𝑥}𝛼L ) , 𝑑 (𝑔𝑦, {𝐺𝑦}𝛼L ) , 𝑑 (𝑓𝑥, {𝐺𝑦}𝛼L ) ,

(22)

𝑑 (𝑔𝑦, {𝐹𝑥}𝛼L )) ; then 𝑓 and 𝑔 have a common fixed point, while 𝐹 and 𝐺 have a common 𝐿-fuzzy fixed point. Proof. Since (𝑓, 𝐹) and (𝑔, 𝐺) satisfy (JCLR) property, there exist two sequences {𝑥𝑛 }, {𝑦𝑛 } in 𝑌 and 𝐴, 𝐵 ∈ 𝐶𝐿(𝑋) such that lim 𝑓𝑥𝑛 = lim 𝑔𝑦𝑛 = 𝑢,

𝑛→∞

𝑛→∞

lim {𝐹𝑥𝑛 }𝛼L = 𝐴,

𝑛→∞

(23)

lim {𝐺𝑦𝑛 }𝛼L = 𝐵.

𝑛→∞

5. Common Fixed Points with (JCLR) Property

With 𝑢 = 𝑓V = 𝑔𝑤 and 𝑢 ∈ 𝐴 ∩ 𝐵, for some 𝑢, V, 𝑤 ∈ 𝑌, we show that 𝐴 = 𝐵, since

Theorem 26. Let (𝑋, 𝑑) be a 𝑏-metric space, 𝑌 ⊆ 𝑋, and ⪯ be a partial order defined on 𝑌, 𝑓, 𝑔 : 𝑌 → 𝑋 such that

𝜙 (𝑄) + 𝐸 (𝑄) ≤ 0,

(24)

6

Journal of Function Spaces Theorem 27. In Theorem 26, we may replace condition (3) by another: for all 𝑥, 𝑦 ∈ 𝑌 and 𝐸 > 0 there exist 𝜓 ∈ Ψ such that

where 𝑄 = (𝐻 ({𝐹𝑥𝑛 }𝛼L , {𝐺𝑦𝑛 }𝛼L ) , 𝑑 (𝑓𝑥𝑛 , 𝑔𝑦𝑛 ) , 𝑑 (𝑓𝑥𝑛 , {𝐹𝑥𝑛 }𝛼L ) , 𝑑 (𝑔𝑦𝑛 , {𝐺𝑦𝑛 }𝛼L ) ,

(25)

+ 𝐸 min (𝐻 ({𝐹𝑥}𝛼L , {𝐺𝑦}𝛼L ) , 𝑑 (𝑓𝑥, 𝑔𝑦) ,

𝑑 (𝑓𝑥𝑛 , {𝐺𝑦𝑛 }𝛼L ) , 𝑑 (𝑔𝑦𝑛 , {𝐹𝑥𝑛 }𝛼L )) .

𝑑 (𝑓V, 𝐴)) ≤ 0,

(26)

which gives 𝜙 (𝐻 (𝐴, 𝐵) , 0, 0, 𝐻 (𝐴, 𝐵) , 𝑏 (𝐻 (𝐴, 𝐵) + 0) , 0) ≤ 𝜙 (𝐻 (𝐴, 𝐵) , 0, 𝑑 (𝑓V, 𝐴) , 𝑑 (𝑓V, 𝐵) , 𝑑 (𝑓V, 𝐵) ,

(32)

𝑑 (𝑓𝑥, {𝐹𝑥}𝛼L )) ≤ 0;

As 𝑛 → ∞, 𝜙 (𝐻 (𝐴, 𝐵) , 0, 𝑑 (𝑓V, 𝐴) , 𝑑 (𝑓V, 𝐵) , 𝑑 (𝑓V, 𝐵) ,

𝜓 (𝐻 ({𝐹𝑥}𝛼L , {𝐺𝑦}𝛼L ) , 𝑑 (𝑓𝑥, 𝑔𝑦) , 𝑑 (𝑓𝑥, {𝐹𝑥}𝛼L ))

(27)

then 𝑓 and 𝑔 have a common fixed point, while 𝐹 and 𝐺 have a common 𝐿-fuzzy fixed point. Proof. Since (𝑓, 𝐹) and (𝑔, 𝐺) satisfy (JCLR) property, there exist two sequences {𝑥𝑛 }, {𝑦𝑛 } in 𝑌 and 𝐴, 𝐵 ∈ 𝐶𝐿(𝑋) such that lim𝑛→∞ 𝑓𝑥𝑛 = lim𝑛→∞ 𝑔𝑦𝑛 = 𝑢, lim𝑛→∞ {𝐹𝑥𝑛 }𝛼L = 𝐴, lim𝑛→∞ {𝐺𝑦𝑛 }𝛼L = 𝐵, with 𝑢 = 𝑓V = 𝑔𝑤 and 𝑢 ∈ 𝐴 ∩ 𝐵, for some 𝑢, V, 𝑤 ∈ 𝑌. Now, we show that 𝑓V ∈ {𝐹V}𝛼L ; otherwise, since 𝜓 (𝐻 ({𝐹V}𝛼L , {𝐺𝑦𝑛 }𝛼L ) , 𝑑 (𝑓V, 𝑔𝑦𝑛 ) , 𝑑 (𝑓V, {𝐹V}𝛼L ))

𝑑 (𝑓V, 𝐴)) ≤ 0. From the property 𝜙(𝑢, V, 𝑢, V, 𝑏(𝑢 + V), 0) ≤ 0 which implies 𝑢 ≤ ℎV, there exists ℎ ∈ (0, 1) such that 𝐻(𝐴, 𝐵) ≤ ℎ.0 = 0, so that 𝐴 = 𝐵. Now, 𝑓V ∈ {𝐹V}𝛼L ; to prove this, since 𝑓V ∈ 𝐴, we show that 𝐴 = {𝐹V}𝛼L . As 𝜙 (𝑄) + 𝐸 (𝑄) ≤ 0,

(28)

where

𝑑 (𝑓V, {𝐹V}𝛼L ) , 𝑑 (𝑔𝑦𝑛 , {𝐺𝑦𝑛 }𝛼L ) , 𝑑 (𝑓V, {𝐺𝑦𝑛 }𝛼L ) , (29) 𝑑 (𝑔𝑦𝑛 , {𝐹V}𝛼L )) ,

when 𝑛 → ∞, we have that

𝑑 (𝑓V, 𝐴) , 𝑑 (𝑓V, {𝐹V}𝛼L )) ≤ 0.

(30)

≤ 𝜓 (𝐻 ({𝐹V}𝛼L , 𝐵) , 0, 𝑑 (𝑓V, {𝐹V}𝛼L )) ≤ 0.

(34)

By 𝜓(𝑢, V, 𝑢) ≤ 0 ⇒ 𝑢 ≤ V, this gives 𝑑({𝐹V}𝛼L , 𝑓V) ≤ 0 which is a contradiction of 𝑑(𝑥, 𝑦) ≥ 0, ∀𝑥, 𝑦 ∈ 𝑌; then 𝑑(𝑓V, {𝐹V}𝛼L ) = 0 and 𝑓V ∈ {𝐹V}𝛼L . Also, 𝑔𝑤 ∈ {𝐺𝑤}𝛼L for 𝑤 ∈ 𝑌. We prove this by contradiction; otherwise, since 𝜓 (𝐻 ({𝐹𝑥𝑛 }𝛼L , {𝐺𝑤}𝛼L ) , 𝑑 (𝑓𝑥𝑛 , 𝑔𝑤) , 𝑑 (𝑓𝑥𝑛 , (35)

𝑑 (𝑓𝑥𝑛 , 𝑔𝑤) , 𝑑 (𝑓𝑥𝑛 , {𝐹𝑥𝑛 }𝛼L )) ≤ 0.

𝜙 (𝐻 ({𝐹V}𝛼L , 𝐴) , 0, 𝐻 (𝐴, {𝐹V}𝛼L ) , 0, 0, 𝑏 (0 + 𝐻 (𝐴, {𝐹V}𝛼L ))) ≤ 𝜙 (𝐻 ({𝐹V}𝛼L , 𝐴) , 0,

When 𝑛 → ∞, we have 𝜓(𝐻({𝐹V}𝛼L , 𝐵), 0, 𝑑(𝑓V, {𝐹V}𝛼L )) ≤ 0. But 𝑓V ∈ 𝐵 implies 𝑑(𝑓V, {𝐹V}𝛼L ) = 𝑑({𝐹V}𝛼L , 𝑓V) ≤ 𝐻({𝐹V}𝛼L , 𝐵); then

{𝐹𝑥𝑛 }𝛼L )) + 𝐸 min (𝐻 ({𝐹𝑥𝑛 }𝛼L , {𝐺𝑤}𝛼L ) ,

So that

(33)

𝑑 (𝑓V, {𝐹V}𝛼L )) ≤ 0.

𝜓 (𝑑 ({𝐹V}𝛼L , 𝑓V) , 0, 𝑑 (𝑓V, {𝐹V}𝛼L ))

𝑄 = (𝐻 ({𝐹V}𝛼L , {𝐺𝑦𝑛 }𝛼L ) , 𝑑 (𝑓V, 𝑔𝑦𝑛 ) ,

𝜙 (𝐻 ({𝐹V}𝛼L , 𝐴) , 0, 𝑑 (𝑓V, {𝐹V}𝛼L ) , 𝑑 (𝑓V, 𝐴) ,

+ 𝐸 min (𝐻 ({𝐹V}𝛼L , {𝐺𝑦𝑛 }𝛼L ) , 𝑑 (𝑓V, 𝑔𝑦𝑛 ) ,

Letting 𝑛 → ∞, we have 𝜓(𝐻(𝐴, {𝐺𝑤}𝛼L ), 0, 𝑑(𝑓V, 𝐴)) ≤ 0, but 𝑔𝑤 ∈ 𝐴 implies 𝑑(𝑔𝑤, {𝐺𝑤}𝛼L ) ≤ 𝐻(𝐴, {𝐺𝑤}𝛼L ); then (31)

𝑑 (𝑓V, {𝐹V}𝛼L ) , 𝑑 (𝑓V, 𝐴) , 𝑑 (𝑓V, 𝐴) , 𝑑 (𝑓V, {𝐹V}𝛼L )) ≤ 0.

From the property 𝜙(𝑢, V, 𝑢, V, 0, 𝑏(𝑢 + V)) ≤ 0 which implies 𝑢 ≤ ℎV, there exists ℎ ∈ (0, 1) such that 𝐻({𝐹V}𝛼L , 𝐴) ≤ ℎ.0 = 0; this gives 𝐻(𝐴, {𝐹V}𝛼L ) = 0; then 𝐴 = {𝐹V}𝛼L , and 𝑓V ∈ {𝐹V}𝛼L . By a similar way, one can find 𝑔𝑤 ∈ {𝐺𝑤}𝛼L for 𝑤 ∈ 𝑌. Further, 𝑓𝑓V = 𝑓V and 𝑓𝑓V ∈ {𝐹𝑓V}𝛼L so that 𝑢 = 𝑓𝑢 ∈ {𝐹𝑢}𝛼L . Also, 𝑔𝑔𝑤 = 𝑔𝑤 and 𝑔𝑔𝑤 ∈ {𝐺𝑔𝑤}𝛼L imply 𝑢 = 𝑔𝑢 ∈ {𝐺𝑢}𝛼L . Then 𝑓, 𝑔, 𝐹, 𝐺 have a common fixed point.

𝜓 (𝑑 (𝑔𝑤, {𝐺𝑤}𝛼L ) , 0, 0) ≤ 𝜓 (𝐻 (𝐴, {𝐺𝑤}𝛼L ) , 0, 0) ≤ 0.

(36)

From 𝜓(𝑢, V, V) ≤ 0 ⇒ 𝑢 ≤ V, this gives 𝑑(𝑔𝑤, {𝐺𝑤}𝛼L ) ≤ 0 which is a contradiction; then 𝑔𝑤 ∈ {𝐺𝑤}𝛼L . Further, being weakly commuting and occasionally coincidentally idempotent imply that, respectively, 𝑢 = 𝑓V = 𝑓𝑓V = 𝑓𝑢 and 𝑓𝑓V ∈ {𝐹𝑓V}𝛼L , so that 𝑢 = 𝑓𝑢 ∈ {𝐹𝑢}𝛼L . Also, 𝑔𝑔𝑤 = 𝑔𝑤 and 𝑔𝑔𝑤 ∈ {𝐺𝑔𝑤}𝛼L imply 𝑢 = 𝑔𝑢 ∈ {𝐺𝑢}𝛼L . Then 𝑓, 𝑔, 𝐹, 𝐺 have a common fixed point.

Journal of Function Spaces

7

6. An Application to Integral Contractive Condition

where 𝑄 = (𝐻 ({𝐹𝑘 𝑥𝑘𝑛 }𝛼L , {𝐹𝑙 𝑦𝑘𝑛 }𝛼L ) , 𝑑 (𝑓𝑥𝑘𝑛 , 𝑔𝑦𝑘𝑛 ) ,

Suppose that 𝛼, 𝛽 : [0, ∞) → [0, ∞) are summable nonnegative Lebesgue integrable functions such that for each 𝜖 > 0, 𝜖 𝜖 ∫0 𝛼(𝑠)𝑑𝑠 ≥ 0 and ∫0 𝛽(𝑒)𝑑𝑒 ≥ 0. As a simple application of Theorems 21, 25, and 26, we give the following result. Theorem 28. Let (𝑋, 𝑑) be a 𝑏-metric space, 𝑌 ⊆ 𝑋, and let ⪯ be a partial order defined on 𝑌, 𝑓, 𝑔 : 𝑌 → 𝑋 such that 𝑓(𝑌) and 𝑔(𝑌) are closed. Suppose that 𝐹, 𝐺 are two L-fuzzy mappings from 𝑌 into IL (𝑋) such that for each 𝑥 ∈ 𝑌 and 𝛼L ∈ 𝐿\{0L }, {𝐹𝑥}𝛼L and {𝐺𝑥}𝛼L are nonempty closed subsets of 𝑋 and satisfying the following conditions

(2) (𝑓, 𝐹𝑘 ) and (𝑔, 𝐹𝑙 ) are weakly commuting and occasionally coincidentally idempotent. (3) For all comparable elements 𝑥, 𝑦 ∈ 𝑌 and 𝐸 > 0, there exist 𝜙 ∈ Φ such that ∫

0

min(𝑄)

𝛼 (𝑠) 𝑑𝑠 + 𝐸 ∫

0

𝑑 (𝑓𝑥𝑘𝑛 , {𝐹𝑙 𝑦𝑘𝑛 }𝛼L ) , 𝑑 (𝑔𝑦𝑘𝑛 , {𝐹𝑘 𝑥𝑘𝑛 }𝛼L )) . As 𝑛 → ∞, we find that 𝜙(𝐻(𝐴 𝑘 ,𝐵𝑘 ),0,0,𝑑(𝑓V𝑘 ,𝐵𝑘 ),𝑑(𝑓V𝑘 ,𝐵𝑘 ),0)

∫

0

𝛼 (𝑠) 𝑑𝑠 ≤ 0.

(42)

𝜙 (𝐻 (𝐴 𝑘 , 𝐵𝑘 ) , 0, 0, 𝑑 (𝑓V𝑘 , 𝐵𝑘 ) , 𝑑 (𝑓V𝑘 , 𝐵𝑘 ) , 0) ≤ 0, (43) which give 𝜙(𝐻(𝐴 𝑘 ,𝐵𝑘 ),0,0,𝐻(𝐴 𝑘 ,𝐵𝑘 ),𝑏(𝐻(𝐴 𝑘 ,𝐵𝑘 )+0),0)

∫

𝛽 (𝑒) 𝑑𝑒 ≤ 0,

(41)

So that

(1) (𝑓, 𝐹𝑘 ) and (𝑔, 𝐹𝑙 ) satisfy (JCLR) property.

𝜙(𝑄)

𝑑 (𝑓𝑥𝑘𝑛 , {𝐹𝑘 𝑥𝑘𝑛 }𝛼L ) , 𝑑 (𝑔𝑦𝑘𝑛 , {𝐹𝑙 𝑦𝑘𝑛 }𝛼L ) ,

(37)

0

≤∫

𝛼 (𝑠) 𝑑𝑠

𝜙(𝐻(𝐴 𝑘 ,𝐵𝑘 ),0,𝑑(𝑓V𝑘 ,𝐴 𝑘 ),𝑑(𝑓V𝑘 ,𝐵𝑘 ),𝑑(𝑓V𝑘 ,𝐵𝑘 ),𝑑(𝑓V𝑘 ,𝐴 𝑘 ))

0

where

𝛼 (𝑠) 𝑑𝑠

(44)

≤ 0.

𝑄 = (𝐻 ({𝐹𝑘 𝑥}𝛼L , {𝐹𝑙 𝑦}𝛼L ) , 𝑑 (𝑓𝑥, 𝑔𝑦) , 𝑑 (𝑓𝑥, {𝐹𝑘 𝑥}𝛼L ) , 𝑑 (𝑔𝑦, {𝐹𝑙 𝑦}𝛼L ) , 𝑑 (𝑓𝑥, {𝐹𝑙 𝑦}𝛼L ) , (38)

By (𝜙), we have 𝐻(𝐴 𝑘 , 𝐵𝑘 ) = 0; that is, 𝐴 𝑘 = 𝐵𝑘 . As 𝑢𝑘 ∈ 𝐴 𝑘 , we show that {𝐹𝑘 V𝑘 }𝛼L = 𝐴 𝑘 . As ∫

𝑑 (𝑔𝑦, {𝐹𝑘 𝑥}𝛼L )) ,

𝜙(𝑄)

0

𝛼 (𝑠) 𝑑𝑠 + 𝐸 ∫

min(𝑄)

0

𝛽 (𝑒) 𝑑𝑒 ≤ 0,

(45)

where where 𝑘 = 2𝑛 + 1, 𝑙 = 2𝑛 + 2, 𝑛 ∈ N; then 𝑓 and 𝑔 have a common fixed point, while 𝐹𝑘 and 𝐹𝑙 have a common 𝐿-fuzzy fixed point.

𝑄 = (𝐻 ({𝐹𝑘 V𝑘 }𝛼L , {𝐹𝑙 𝑦𝑘𝑛 }𝛼L ) , 𝑑 (𝑓V𝑘 , 𝑔𝑦𝑘𝑛 ) , 𝑑 (𝑓V𝑘 , {𝐹𝑘 V𝑘 }𝛼L ) , 𝑑 (𝑔𝑦𝑘𝑛 , {𝐹𝑙 𝑦𝑘𝑛 }𝛼L ) ,

Proof. Since (𝑓, 𝐹𝑘 ) and (𝑔, 𝐹𝑙 ) satisfy the property (JCLR), there exist two sequences {𝑥𝑘𝑛 }, {𝑦𝑘𝑛 } in 𝑌 and 𝐴 𝑘 , 𝐵𝑘 ∈ 𝐶𝐿(𝑋) such that lim 𝑓𝑥𝑘𝑛 = lim 𝑔𝑦𝑘𝑛 = 𝑢𝑘 ,

𝑛→∞

lim {𝐹𝑘 𝑥𝑘𝑛 }𝛼L = 𝐴 𝑘 ,

(39)

lim {𝐹𝑙 𝑦𝑘𝑛 }𝛼L = 𝐵𝑘 ,

0

𝛼 (𝑠) 𝑑𝑠 + 𝐸 ∫

min(𝑄)

0

𝜙(𝐻({𝐹𝑘 V𝑘 }𝛼L ,𝐴 𝑘 ),0,0,𝑑({𝐹𝑘 V𝑘 }𝛼L ,𝐴 𝑘 ),𝑑({𝐹𝑘 V𝑘 }𝛼L ,𝐴 𝑘 ),0)

𝛼 (𝑠) 𝑑𝑠 (47)

≤ 0.

with 𝑢𝑘 = 𝑓V𝑘 = 𝑔𝑤𝑙 , 𝑢𝑘 ∈ 𝐴 𝑘 ∩ 𝐵𝑘 , for some 𝑢𝑘 , V𝑘 , 𝑤𝑘 ∈ 𝑌. Now, we show that 𝐴 𝑘 = 𝐵𝑘 . As 𝜙(𝑄)

∫

0

𝑛→∞

∫

𝑑 (𝑓V𝑘 , {𝐹𝑙 𝑦𝑘𝑛 }𝛼L ) , 𝑑 (𝑔𝑦𝑘𝑛 , {𝐹𝑘 V𝑘 }𝛼L )) , which on making 𝑛 → ∞ reduces to

𝑛→∞

𝑛→∞

(46)

𝛽 (𝑒) 𝑑𝑒 ≤ 0,

(40)

By a similar way, we have {𝐹𝑘 V𝑘 }𝛼L = 𝐴 𝑘 . The remaining parts are easy to prove. This concludes the proof. Now, we may apply Theorem 28 with the following example.

8

Journal of Function Spaces

Example 29. Let 𝑋 = [0, 1], 𝑌 ⊆ 𝑋, and 𝑑(𝑥, 𝑦) = |𝑥 − 𝑦|2 ; then (𝑋, 𝑑) is a 𝑏-metric space with 𝑏 = 2. Define the partial order 𝑥 ⪯ 𝑦 as 𝑥 ≤ 𝑦 for each 𝑥, 𝑦 ∈ 𝑋 and 𝑋 ⪯1 𝑌 as follows: for each 𝑥 ∈ 𝑋, there exist 𝑦 ∈ 𝑌 such that 𝑥 ≤ 𝑦. Let 𝐿 = [0, 1] and define the maps 𝑓, 𝑔, 𝐹, 𝐺 on 𝑋 as

3 , { { { 4 { { {1 (𝐺𝑥) (𝑦) = { , { 2 { { { {1 {4, ∫

1 , 10 1 𝑥+1 if 0, there exists 𝜙 ∈ Φ such that

𝑑 (𝑓𝑥𝑛 , {𝐺𝑦𝑛 }𝛼L ) , 𝑑 (𝑔𝑦𝑛 , {𝐹𝑥𝑛 }𝛼L )) .

min(𝑄)

(51)

𝑑 (𝑓𝑥𝑛 , {𝐺𝑦𝑛 }𝛼L ) , 𝑑 (𝑔𝑦𝑛 , {𝐹𝑥𝑛 }𝛼L )) = 0,

𝑄 = (𝐻 ({𝐹𝑥𝑛 }𝛼L , {𝐺𝑦𝑛 }𝛼L ) , 𝑑 (𝑓𝑥𝑛 , 𝑔𝑦𝑛 ) ,

𝜙 (𝑄) + 𝐸 ∫

(50)

𝜙 (𝐻 ({𝐹𝑥𝑛 }𝛼L , {𝐺𝑦𝑛 }𝛼L ) , 𝑑 (𝑓𝑥𝑛 , 𝑔𝑦𝑛 ) ,

where

𝑑 (𝑓𝑥𝑛 , {𝐹𝑥𝑛 }𝛼L ) , 𝑑 (𝑔𝑦𝑛 , {𝐺𝑦𝑛 }𝛼L ) ,

(49)

Since lim𝑛→∞ 𝑓𝑥𝑛 = lim𝑛→∞ 𝑔𝑦𝑛 = 1/6 ∈ [1/6, 1] = lim𝑛→∞ 𝐹𝑥𝑛 = lim𝑛→∞ 𝐺𝑦𝑛 , the hybrid pairs (𝑓, 𝐹) and (𝑔, 𝐺) satisfy the property (JCLR). Also 𝑓(1/6) = 𝑓𝑓(1/6) = 1/6 and 𝑔(1/6) = 𝑔𝑔(1/6) = 1/6 (occasionally coincidentally idempotent). Finally, 𝑓𝑓(1/6) = 1/6 ∈ [7/54, 1] = {𝐹𝑓(1/6)}2/3 and 𝑔𝑔(1/6) = 1/6 ∈ [7/48, 1] = {𝐺𝑔(1/6)}2/3 (weakly commuting). Let 𝜙(𝑡1 , . . . , 𝑡6 ) = 𝑡1 − ℎ max{𝑡2 , (𝑡3 + 𝑡4 )𝑡5 𝑡6 }. Further,

𝜙(𝐻({𝐹𝑥𝑛 }𝛼L ,{𝐺𝑦𝑛 }𝛼L ),𝑑(𝑓𝑥𝑛 ,𝑔𝑦𝑛 ),𝑑(𝑓𝑥𝑛 ,{𝐹𝑥𝑛 }𝛼L ),𝑑(𝑔𝑦𝑛 ,{𝐺𝑦𝑛 }𝛼L ),𝑑(𝑓𝑥𝑛 ,{𝐺𝑦𝑛 }𝛼L ),𝑑(𝑔𝑦𝑛 ,{𝐹𝑥𝑛 }𝛼L ))

𝛼 (𝑠) 𝑑𝑠 + 𝐸 ∫

𝛽 (𝑒) 𝑑𝑒 ≤ 0,

𝑑 (𝑔𝑦, {𝐹𝑥}𝛼L )) .

Then ∫

0

𝑑 (𝑓𝑥, {𝐹𝑥}𝛼L ) , 𝑑 (𝑔𝑦, {𝐺𝑦}𝛼L ) , 𝑑 (𝑓𝑥, {𝐺𝑦}𝛼L ) ,

1 if 0 ≤ 𝑦 < , 9 1 𝑥+1 if ≤ 𝑦 < , 9 8 𝑥+1 if ≤ 𝑦 ≤ 1, 8

min(𝑄)

min(𝑄)

𝑄 = (𝐻 ({𝐹𝑥}𝛼L , {𝐺𝑦}𝛼L ) , 𝑑 (𝑓𝑥, 𝑔𝑦) ,

0

𝜙(𝑄)

𝛼 (𝑠) 𝑑𝑠 + 𝐸 ∫

where

1 1 { { { 6 , if 0 ≤ 𝑥 < 3 , 𝑔𝑥 = { { { 𝑥 , if 1 ≤ 𝑥 ≤ 1, 3 {2 if 0 ≤ 𝑦 ≤

∫

0

1 1 { { { 6 , if 0 ≤ 𝑥 < 2 , 𝑓𝑥 = { { { 𝑥 , if 1 ≤ 𝑥 ≤ 1, 2 {3

5 { , { { 6 { { { {4 (𝐹𝑥) (𝑦) = { , { 5 { { { { {2, {3

for all 𝑥, 𝑦 ∈ 𝑋. Define two sequences {𝑥𝑛 } and {𝑦𝑛 } in 𝑌 such that {𝑥𝑛 } = {1/2 + 1/4𝑛}, {𝑦𝑛 } = {1/3 + 1/6𝑛}, 𝑛 ∈ N. Now, we show that for 𝑥, 𝑦 ∈ 𝑌 there exist 𝜙 ∈ Φ such that

min(𝑄)

𝛼 (𝑠) 𝑑𝑠 + 𝐸 ∫

0

𝛽 (𝑒) 𝑑𝑒 ≤ 0,

(56)

where 𝑄 = (𝐻 ({𝐹𝑥}𝛼L , {𝐺𝑦}𝛼L ) , 𝑑 (𝑓𝑥, 𝑔𝑦) , 𝑑 (𝑓𝑥, {𝐹𝑥}𝛼L )) .

(57)

If (𝑓, 𝐹) and (𝑔, 𝐺) satisfy (JCLR) property, weakly commuting and occasionally coincidentally idempotent, then 𝑓 and 𝑔 have a common fixed point, while 𝐹 and 𝐺 have a common 𝐿-fuzzy fixed point. Proof. It is similar to Theorem 27.

Journal of Function Spaces

Conflicts of Interest The authors declare that there are no conflicts of interest regarding the publication of this paper.

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