Fixed Point Theory and Applications
Volume 2010 (2010), Article ID 125082, 13 pages
doi:10.1155/2010/125082
Research Article

Common Fixed Point Theorem in Partially Ordered ℒ -Fuzzy Metric Spaces

S. Shakeri,1 L. J. B. Ćirić,2 and R. Saadati3

1Young Research Club, Islamic Azad University-Ayatollah Amoli Branch, P.O. Box 678, Amol, Iran
2Faculty of Mechanical Engineering, Kraljice Marije 16, 11 000 Belgrade, Serbia
3Faculty of Sciences, Islamic Azad University-Ayatollah Amoli Branch, P.O. Box 678, Amol, Iran

Received 29 October 2009; Accepted 27 January 2010

Academic Editor: Juan Jose Nieto

Copyright © 2010 S. Shakeri 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.

Abstract

We introduce partially ordered ℒ -fuzzy metric spaces and prove a common fixed point theorem in these spaces.

1. Introduction

The Banach fixed point theorem for contraction mappings has been generalized and extended in many directions [1–43]. Recently Nieto and Rodríguez-López [27–29] and Ran and Reurings [33] presented some new results for contractions in partially ordered metric spaces. The main idea in [27–33] involves combining the ideas of iterative technique in the contraction mapping principle with those in the monotone technique.

Recall that if ( 𝑋 , ≤ ) is a partially ordered set and 𝐹 ∶ 𝑋 → 𝑋 is such that for 𝑥 , 𝑦 ∈ 𝑋 , 𝑥 ≤ 𝑦 implies 𝐹 ( 𝑥 ) ≤ 𝐹 ( 𝑦 ) , then a mapping 𝐹 is said to be nondecreasing. The main result of Nieto and Rodríguez-López [27–33] and Ran and Reurings [33] is the following fixed point theorem.

Theorem 1.1. Let ( 𝑋 , ≤ ) be a partially ordered set and suppose that there is a metric 𝑑 on 𝑋 such that ( 𝑋 , 𝑑 ) is a complete metric space. Suppose that 𝐹 is a nondecreasing mapping with 𝑑 ( 𝐹 ( 𝑥 ) , 𝐹 ( 𝑦 ) ) ≤ 𝑘 𝑑 ( 𝑥 , 𝑦 ) ( 1 . 1 ) for all 𝑥 , 𝑦 ∈ 𝑋 , 𝑥 ≤ 𝑦 , where 0 < 𝑘 < 1 . Also suppose the following. (a) 𝐹 is continuous.(b)If { 𝑥 𝑛 } ⊂ 𝑋 is a nondecreasing sequence with 𝑥 𝑛 → 𝑥 in 𝑋 , then 𝑥 𝑛 ≤ 𝑥 for all 𝑛 hold.
If there exists an 𝑥 0 ∈ 𝑋 with 𝑥 0 ≤ 𝐹 ( 𝑥 0 ) , then 𝐹 has a fixed point.

The works of Nieto and Rodríguez-López [27, 28] and Ran and Reurings [33] have motivated Agarwal et al. [1], Bhaskar and Lakshmikantham [3], and Lakshmikantham and Ćirić [23] to undertake further investigation of fixed points in the area of ordered metric spaces. We prove the existence and approximation results for a wide class of contractive mappings in intuitionistic metric space. Our results are an extension and improvement of the results of Nieto and Rodríguez-López [27, 28] and Ran and Reurings [33] to more general class of contractive type mappings and include several recent developments.

2. Preliminaries

The notion of fuzzy sets was introduced by Zadeh [44]. Various concepts of fuzzy metric spaces were considered in [15, 16, 22, 45]. Many authors have studied fixed point theory in fuzzy metric spaces; see, for example, [7, 8, 25, 26, 39, 46–48]. In the sequel, we will adopt the usual terminology, notation, and conventions of ℒ -fuzzy metric spaces introduced by Saadati et al. [36] which are a generalization of fuzzy metric sapces [49] and intuitionistic fuzzy metric spaces [32, 37].

Definition (see [46]). Let ℒ = ( 𝐿 , ≤ 𝐿 ) be a complete lattice, and 𝑈 a nonempty set called a universe. An ℒ -fuzzy set 𝒜 on 𝑈 is defined as a mapping 𝒜 ∶ 𝑈 → 𝐿 . For each 𝑢 in 𝑈 , 𝒜 ( 𝑢 ) represents the degree (in 𝐿 ) to which 𝑢 satisfies 𝒜 .

Lemma (see [13, 14]). Consider the set 𝐿 ∗ and the operation ≤ 𝐿 ∗ defined by 𝐿 ∗ = 𝑥 1 , 𝑥 2 ∶ 𝑥 1 , 𝑥 2 ∈ [ ] 0 , 1 2 , 𝑥 1 + 𝑥 2 , ≤ 1 ( 2 . 1 ) ( 𝑥 1 , 𝑥 2 ) ≤ 𝐿 ∗ ( 𝑦 1 , 𝑦 2 ) ⇔ 𝑥 1 ≤ 𝑦 1 , and 𝑥 2 ≥ 𝑦 2 , for every ( 𝑥 1 , 𝑥 2 ) , ( 𝑦 1 , 𝑦 2 ) ∈ 𝐿 ∗ . Then ( 𝐿 ∗ , ≤ 𝐿 ∗ ) is a complete lattice.

Classically, a triangular norm 𝑇 on ( [ 0 , 1 ] , ≤ ) is defined as an increasing, commutative, associative mapping 𝑇 ∶ [ 0 , 1 ] 2 → [ 0 , 1 ] satisfying 𝑇 ( 1 , 𝑥 ) = 𝑥 , for all 𝑥 ∈ [ 0 , 1 ] . These definitions can be straightforwardly extended to any lattice ℒ = ( 𝐿 , ≤ 𝐿 ) . Define first 0 ℒ = i n f 𝐿 and 1 ℒ = s u p 𝐿 .

Definition. A negation on ℒ is any strictly decreasing mapping 𝒩 ∶ 𝐿 → 𝐿 satisfying 𝒩 ( 0 ℒ ) = 1 ℒ and 𝒩 ( 1 ℒ ) = 0 ℒ . If 𝒩 ( 𝒩 ( 𝑥 ) ) = 𝑥 , for all 𝑥 ∈ 𝐿 , then 𝒩 is called an involutive negation.

In this paper the negation 𝒩 ∶ 𝐿 → 𝐿 is fixed.

Definition. A triangular norm ( 𝑡 -norm) on ℒ is a mapping 𝒯 ∶ 𝐿 2 → 𝐿 satisfying the following conditions: (i) ( for all 𝑥 ∈ 𝐿 ) ( 𝒯 ( 𝑥 , 1 ℒ ) = 𝑥 ) (boundary condition); (ii) ( for all ( 𝑥 , 𝑦 ) ∈ 𝐿 2 ) ( 𝒯 ( 𝑥 , 𝑦 ) = 𝒯 ( 𝑦 , 𝑥 ) ) (commutativity); (iii) ( for all ( 𝑥 , 𝑦 , 𝑧 ) ∈ 𝐿 3 ) ( 𝒯 ( 𝑥 , 𝒯 ( 𝑦 , 𝑧 ) ) = 𝒯 ( 𝒯 ( 𝑥 , 𝑦 ) , 𝑧 ) ) (associativity); (iv) ( for all ( 𝑥 , 𝑥 ′ , 𝑦 , 𝑦 ′ ) ∈ 𝐿 4 ) ( 𝑥 ≤ 𝐿 𝑥 ′ and 𝑦 ≤ 𝐿 𝑦 ′ ⇒ 𝒯 ( 𝑥 , 𝑦 ) ≤ 𝐿 𝒯 ( 𝑥 ′ , 𝑦 ′ ) ) (monotonicity).

A 𝑡 -norm 𝒯 on ℒ is said to be continuous if for any 𝑥 , 𝑦 ∈ ℒ and any sequences { 𝑥 𝑛 } and { 𝑦 𝑛 } which converge to 𝑥 and 𝑦 we have

l i m 𝑛 𝒯 𝑥 𝑛 , 𝑦 𝑛 . = 𝒯 ( 𝑥 , 𝑦 ) ( 2 . 2 ) For example, 𝒯 ( 𝑥 , 𝑦 ) = m i n ( 𝑥 , 𝑦 ) and 𝒯 ( 𝑥 , 𝑦 ) = 𝑥 𝑦 are two continuous 𝑡 -norms on [ 0 , 1 ] . A 𝑡 -norm can also be defined recursively as an ( 𝑛 + 1 ) -ary operation ( 𝑛 ∈ ℕ ) by 𝒯 1 = 𝒯 and

𝒯 𝑛 𝑥 1 , … , 𝑥 𝑛 + 1 𝒯 = 𝒯 𝑛 − 1 𝑥 1 , … , 𝑥 𝑛 , 𝑥 𝑛 + 1 ( 2 . 3 ) for 𝑛 ≥ 2 and 𝑥 𝑖 ∈ 𝐿 .

A 𝑡 -norm 𝒯 is said to be of Hadžić type if the family { 𝒯 𝑛 } 𝑛 ∈ ℕ is equicontinuous at 𝑥 = 1 ℒ , that is,

0 ∀ 𝜀 ∈ 𝐿 ⧵ ℒ , 1 ℒ 0 ∃ 𝛿 ∈ 𝐿 ⧵ ℒ , 1 ℒ ∶ 𝑎 > 𝐿 𝒩 ( 𝛿 ) ⟹ 𝒯 𝑛 ( 𝑎 ) > 𝐿 𝒩 ( 𝜀 ) ( 𝑛 ≥ 1 ) . ( 2 . 4 )

𝒯 𝑀 is a trivial example of a 𝑡 -norm of Hadžić type, but there exist 𝑡 -norms of Hadžić type weaker than 𝒯 𝑀 [50] where

𝒯 𝑀 ( 𝑥 , 𝑦 ) = 𝑥 , i f 𝑥 ≤ 𝐿 𝑦 , 𝑦 , i f 𝑦 ≤ 𝐿 𝑥 . ( 2 . 5 )

Definition. The 3-tuple ( 𝑋 , ℳ , 𝒯 ) is said to be an ℒ -fuzzy metric space if 𝑋 is an arbitrary (nonempty) set, 𝒯 is a continuous 𝑡 -norm on ℒ and ℳ is an ℒ -fuzzy set on 𝑋 2 × ] 0 , + ∞ [ satisfying the following conditions for every 𝑥 , 𝑦 , 𝑧 in 𝑋 and 𝑡 , 𝑠 in ] 0 , + ∞ [ : (a) ℳ ( 𝑥 , 𝑦 , 𝑡 ) > 𝐿 0 ℒ ;(b) ℳ ( 𝑥 , 𝑦 , 𝑡 ) = 1 ℒ for all 𝑡 > 0 if and only if 𝑥 = 𝑦 ; (c) ℳ ( 𝑥 , 𝑦 , 𝑡 ) = ℳ ( 𝑦 , 𝑥 , 𝑡 ) ; (d) 𝒯 ( ℳ ( 𝑥 , 𝑦 , 𝑡 ) , ℳ ( 𝑦 , 𝑧 , 𝑠 ) ) ≤ 𝐿 ℳ ( 𝑥 , 𝑧 , 𝑡 + 𝑠 ) ; (e) ℳ ( 𝑥 , 𝑦 , ⋅ ) ∶ ] 0 , ∞ [ → 𝐿 is continuous. If the ℒ -fuzzy metric space ( 𝑋 , ℳ , 𝒯 ) satisfies the condition: ( 𝑓 ) l i m 𝑡 → ∞ ℳ ( 𝑥 , 𝑦 , 𝑡 ) = 1 ℒ , ( 2 . 6 ) then ( 𝑋 , ℳ , 𝒯 ) is said to be Menger ℒ -fuzzy metric space or for short a 𝐌 ℒ -fuzzy metric space.

Let ( 𝑋 , ℳ , 𝒯 ) be an ℒ -fuzzy metric space. For 𝑡 ∈ ] 0 , + ∞ [ , we define the open ball 𝐵 ( 𝑥 , 𝑟 , 𝑡 ) with center 𝑥 ∈ 𝑋 and radius 𝑟 ∈ 𝐿 ⧵ { 0 ℒ , 1 ℒ } , as

𝐵 ( 𝑥 , 𝑟 , 𝑡 ) = 𝑦 ∈ 𝑋 ∶ ℳ ( 𝑥 , 𝑦 , 𝑡 ) > 𝐿 𝒩 . ( 𝑟 ) ( 2 . 7 ) A subset 𝐴 ⊆ 𝑋 is called open if for each 𝑥 ∈ 𝐴 , there exist 𝑡 > 0 and 𝑟 ∈ 𝐿 ⧵ { 0 ℒ , 1 ℒ } such that 𝐵 ( 𝑥 , 𝑟 , 𝑡 ) ⊆ 𝐴 . Let 𝜏 ℳ denote the family of all open subsets of 𝑋 . Then 𝜏 ℳ is called the topology induced by the ℒ -fuzzy metric ℳ .

Example (see [38]). Let ( 𝑋 , 𝑑 ) be a metric space. Denote 𝒯 ( 𝑎 , 𝑏 ) = ( 𝑎 1 𝑏 1 , m i n ( 𝑎 2 + 𝑏 2 , 1 ) ) for all 𝑎 = ( 𝑎 1 , 𝑎 2 ) and 𝑏 = ( 𝑏 1 , 𝑏 2 ) in 𝐿 ∗ and let 𝑀 and 𝑁 be fuzzy sets on 𝑋 2 × ( 0 , ∞ ) defined as follows: ℳ 𝑀 , 𝑁 𝑡 ( 𝑥 , 𝑦 , 𝑡 ) = ( 𝑀 ( 𝑥 , 𝑦 , 𝑡 ) , 𝑁 ( 𝑥 , 𝑦 , 𝑡 ) ) = , 𝑡 + 𝑑 ( 𝑥 , 𝑦 ) 𝑑 ( 𝑥 , 𝑦 ) . 𝑡 + 𝑑 ( 𝑥 , 𝑦 ) ( 2 . 8 ) Then ( 𝑋 , ℳ 𝑀 , 𝑁 , 𝒯 ) is an intuitionistic fuzzy metric space.

Example. Let 𝑋 = ℕ . Define 𝒯 ( 𝑎 , 𝑏 ) = ( m a x ( 0 , 𝑎 1 + 𝑏 1 − 1 ) , 𝑎 2 + 𝑏 2 − 𝑎 2 𝑏 2 ) for all 𝑎 = ( 𝑎 1 , 𝑎 2 ) and 𝑏 = ( 𝑏 1 , 𝑏 2 ) in 𝐿 ∗ , and let ℳ ( 𝑥 , 𝑦 , 𝑡 ) on 𝑋 2 × ( 0 , ∞ ) be defined as follows: ⎧ ⎪ ⎨ ⎪ ⎩ 𝑥 ℳ ( 𝑥 , 𝑦 , 𝑡 ) = 𝑦 , 𝑦 − 𝑥 𝑦 i f 𝑦 𝑥 ≤ 𝑦 , 𝑥 , 𝑥 − 𝑦 𝑥 i f 𝑦 ≤ 𝑥 ( 2 . 9 ) for all 𝑥 , 𝑦 ∈ 𝑋 and 𝑡 > 0 . Then ( 𝑋 , ℳ , 𝒯 ) is an ℒ -fuzzy metric space.

Lemma (see [49]). Let ( 𝑋 , ℳ , 𝒯 ) be an ℒ -fuzzy metric space. Then, ℳ ( 𝑥 , 𝑦 , 𝑡 ) is nondecreasing with respect to 𝑡 , for all 𝑥 , 𝑦 in 𝑋 .

Definition. A sequence { 𝑥 𝑛 } 𝑛 ∈ ℕ in an ℒ -fuzzy metric space ( 𝑋 , ℳ , 𝒯 ) is called a Cauchy sequence, if for each 𝜀 ∈ 𝐿 ⧵ { 0 ℒ } and 𝑡 > 0 , there exists 𝑛 0 ∈ ℕ such that for all 𝑚 ≥ 𝑛 ≥ 𝑛 0 ( 𝑛 ≥ 𝑚 ≥ 𝑛 0 ) , ℳ 𝑥 𝑚 , 𝑥 𝑛 > , 𝑡 𝐿 𝒩 ( 𝜀 ) . ( 2 . 1 0 ) The sequence { 𝑥 𝑛 } 𝑛 ∈ ℕ is said to be convergent to 𝑥 ∈ 𝑋 in the ℒ -fuzzy metric space ( 𝑋 , ℳ , 𝒯 ) (denoted by 𝑥 𝑛 ℳ → 𝑥 ) if ℳ ( 𝑥 𝑛 , 𝑥 , 𝑡 ) = ℳ ( 𝑥 , 𝑥 𝑛 , 𝑡 ) → 1 ℒ whenever 𝑛 → + ∞ for every 𝑡 > 0 . A ℒ -fuzzy metric space is said to be complete if and only if every Cauchy sequence is convergent.

Definition. Let ( 𝑋 , ℳ , 𝒯 ) be an ℒ -fuzzy metric space. ℳ is said to be continuous on 𝑋 × 𝑋 × ] 0 , ∞ [ if l i m 𝑛 → ∞ ℳ 𝑥 𝑛 , 𝑦 𝑛 , 𝑡 𝑛 = ℳ ( 𝑥 , 𝑦 , 𝑡 ) ( 2 . 1 1 ) whenever a sequence { ( 𝑥 𝑛 , 𝑦 𝑛 , 𝑡 𝑛 ) } in 𝑋 × 𝑋 × ] 0 , ∞ [ converges to a point ( 𝑥 , 𝑦 , 𝑡 ) ∈ 𝑋 × 𝑋 × ] 0 , ∞ [ , that is, l i m 𝑛 ℳ ( 𝑥 𝑛 , 𝑥 , 𝑡 ) = l i m 𝑛 ℳ ( 𝑦 𝑛 , 𝑦 , 𝑡 ) = 1 ℒ and l i m 𝑛 ℳ ( 𝑥 , 𝑦 , 𝑡 𝑛 ) = ℳ ( 𝑥 , 𝑦 , 𝑡 ) .

Lemma. Let ( 𝑋 , ℳ , 𝒯 ) be an ℒ -fuzzy metric space. Then ℳ is continuous function on 𝑋 × 𝑋 × ] 0 , ∞ [ .

Proof. The proof is the same as that for fuzzy spaces (see [35, Proposition 1 ]).

Lemma. If an 𝐌 ℒ -fuzzy metric space ( 𝑋 , ℳ , 𝒯 ) satisfies the following condition: ℳ ( 𝑥 , 𝑦 , 𝑡 ) = 𝐶 , ∀ 𝑡 > 0 , ( 2 . 1 2 ) then one has 𝐶 = 1 ℒ and 𝑥 = 𝑦 .

Proof. Let ℳ ( 𝑥 , 𝑦 , 𝑡 ) = 𝐶 for all 𝑡 > 0 . Then by ( 𝑓 ) of Definition 2.5, we have 𝐶 = 1 ℒ and by ( 𝑏 ) of Definition 2.5, we conclude that 𝑥 = 𝑦 .

Lemma (see [50]). Let ( 𝑋 , ℳ , 𝒯 ) be an 𝐌 ℒ -fuzzy metric space in which 𝒯 is Hadži c ' type. Suppose ℳ 𝑥 𝑛 , 𝑥 𝑛 + 1 ≥ , 𝑡 𝐿 ℳ 𝑥 0 , 𝑥 1 , 𝑡 𝑘 𝑛 ( 2 . 1 3 ) for some 0 < 𝑘 < 1 and 𝑛 ∈ ℕ . Then { 𝑥 𝑛 } is a Cauchy sequence.

3. Main Results

Definition. Suppose that ( 𝑋 , ≤ ) is a partially ordered set and 𝐹 , ℎ ∶ 𝑋 → 𝑋 are mappings of 𝑋 into itself. We say that 𝐹 is ℎ -nondecreasing if for 𝑥 , 𝑦 ∈ 𝑋 , ℎ ( 𝑥 ) ≤ ℎ ( 𝑦 ) i m p l i e s 𝐹 ( 𝑥 ) ≤ 𝐹 ( 𝑦 ) . ( 3 . 1 )

Now we present the main result in this paper.

Theorem. Let ( 𝑋 , ≤ ) be a partially ordered set and suppose that there is an ℒ -fuzzy metric ℳ on 𝑋 such that ( 𝑋 , ℳ , 𝒯 ) is a complete 𝐌 ℒ -fuzzy metric space in which 𝒯 is Hadži c ' type. Let 𝐹 , ℎ ∶ 𝑋 → 𝑋 be two self-mappings of 𝑋 such that there exist 𝑘 ∈ ( 0 , 1 ) and 𝑞 ∈ ( 0 , 1 ) such that 𝐹 ( 𝑋 ) ⊆ ℎ ( 𝑋 ) , 𝐹 is a ℎ -nondecreasing mapping and ℳ ( 𝐹 ( 𝑥 ) , 𝐹 ( 𝑦 ) , 𝑘 𝑡 ) ≥ 𝐿 𝒯 𝑀 ℳ { ℳ ( ℎ ( 𝑥 ) , ℎ ( 𝑦 ) , 𝑡 ) , ℳ ( ℎ ( 𝑥 ) , 𝐹 ( 𝑥 ) , 𝑡 ) , ℳ ( ℎ ( 𝑦 ) , 𝐹 ( 𝑦 ) , 𝑡 ) , ( ℎ ( 𝑥 ) , 𝐹 ( 𝑦 ) , ( 1 + 𝑞 ) 𝑡 ) , ℳ ( ℎ ( 𝑦 ) , 𝐹 ( 𝑥 ) , ( 1 − 𝑞 ) 𝑡 ) } ( 3 . 2 ) for all 𝑥 , 𝑦 ∈ 𝑋 for which ℎ ( 𝑥 ) ≤ ℎ ( 𝑦 ) and all 𝑡 > 0 .
Also suppose that
i f ℎ 𝑥 𝑛 ⊂ 𝑋 i s a n o n d e c r e a s i n g s e q u e n c e w i t h ℎ 𝑥 𝑛 ⟶ ℎ ( 𝑧 ) i n ℎ ( 𝑋 ) , t h e n ℎ ( 𝑧 ) ≤ ℎ ( ℎ ( 𝑧 ) ) a n d ℎ 𝑥 𝑛 ≤ ℎ ( 𝑧 ) ∀ 𝑛 h o l d . ( 3 . 3 ) Also suppose that ℎ ( 𝑋 ) is closed. If there exists an 𝑥 0 ∈ 𝑋 with ℎ ( 𝑥 0 ) ≤ 𝐹 ( 𝑥 0 ) , then 𝐹 and ℎ have a coincidence. Further, if 𝐹 and ℎ commute at their coincidence points, then 𝐹 and ℎ have a common fixed point.

Proof. Let 𝑥 0 ∈ 𝑋 be such that ℎ ( 𝑥 0 ) ≤ 𝐹 ( 𝑥 0 ) . Since 𝐹 ( 𝑋 ) ⊆ ℎ ( 𝑋 ) , we can choose 𝑥 1 ∈ 𝑋 such that ℎ ( 𝑥 1 ) = 𝐹 ( 𝑥 0 ) . Again from 𝐹 ( 𝑋 ) ⊆ ℎ ( 𝑋 ) we can choose 𝑥 2 ∈ 𝑋 such that ℎ ( 𝑥 2 ) = 𝐹 ( 𝑥 1 ) . Continuing this process we can choose a sequence { 𝑥 𝑛 } in 𝑋 such that ℎ 𝑥 𝑛 + 1 𝑥 = 𝐹 𝑛 ∀ 𝑛 ≥ 0 . ( 3 . 4 ) Since ℎ ( 𝑥 0 ) ≤ 𝐹 ( 𝑥 0 ) and ℎ ( 𝑥 1 ) = 𝐹 ( 𝑥 0 ) , we have ℎ ( 𝑥 0 ) ≤ ℎ ( 𝑥 1 ) . Then from (3.1), 𝐹 𝑥 0 𝑥 ≤ 𝐹 1 , ( 3 . 5 ) that is, by (3.4), ℎ ( 𝑥 1 ) ≤ ℎ ( 𝑥 2 ) . Again from (3.1), 𝐹 𝑥 1 𝑥 ≤ 𝐹 2 , ( 3 . 6 ) that is, ℎ ( 𝑥 2 ) ≤ ℎ ( 𝑥 3 ) . Continuing we obtain 𝐹 𝑥 0 𝑥 ≤ 𝐹 1 𝑥 ≤ 𝐹 2 𝑥 ≤ 𝐹 3 𝑥 ≤ ⋯ ≤ 𝐹 𝑛 𝑥 ≤ 𝐹 𝑛 + 1 ≤ ⋯ . ( 3 . 7 )
Now we will show that a sequence { ℳ ( 𝐹 ( 𝑥 𝑛 ) , 𝐹 ( 𝑥 𝑛 + 1 ) , 𝑡 ) } converges to 1 ℒ for each 𝑡 > 0 . If ℳ ( 𝐹 ( 𝑥 𝑛 ) , 𝐹 ( 𝑥 𝑛 + 1 ) , 𝑡 ) = 1 ℒ for some 𝑛 and for each 𝑡 > 0 , then it is easily to show that ℳ ( 𝐹 ( 𝑥 𝑛 + 𝑘 ) , 𝐹 ( 𝑥 𝑛 + 𝑘 + 1 ) , 𝑡 ) = 1 ℒ for all 𝑘 ≥ 0 . So we suppose that ℳ ( 𝐹 ( 𝑥 𝑛 ) , 𝐹 ( 𝑥 𝑛 + 1 ) , 𝑡 ) < 𝐿 1 ℒ for all 𝑛 . We show that for each 𝑡 > 0 ,
ℳ 𝐹 𝑥 𝑛 𝑥 , 𝐹 𝑛 + 1 ≥ , 𝑘 𝑡 𝐿 ℳ 𝐹 𝑥 𝑛 − 1 𝑥 , 𝐹 𝑛 , 𝑡 ∀ 𝑛 ≥ 1 . ( 3 . 8 ) Since from (3.4) and (3.7) we have ℎ ( 𝑥 𝑛 − 1 ) ≤ ℎ ( 𝑥 𝑛 ) , from (3.1) with 𝑥 = 𝑥 𝑛 and 𝑦 = 𝑥 𝑛 + 1 , ℳ 𝐹 𝑥 𝑛 𝑥 , 𝐹 𝑛 + 1 ≥ , 𝑘 𝑡 𝐿 𝒯 𝑀 ℳ ℎ 𝑥 𝑛 𝑥 , ℎ 𝑛 + 1 ℎ 𝑥 , 𝑡 , ℳ 𝑛 𝑥 , 𝐹 𝑛 ℎ 𝑥 , 𝑡 , ℳ 𝑛 + 1 𝑥 , 𝐹 𝑛 + 1 , ℳ ℎ 𝑥 , 𝑡 𝑛 𝑥 , 𝐹 𝑛 + 1 ℎ 𝑥 , ( 1 + 𝑞 ) 𝑡 , ℳ 𝑛 + 1 𝑥 , 𝐹 𝑛 . , ( 1 − 𝑞 ) 𝑡 ( 3 . 9 ) So by (3.4), ℳ 𝐹 𝑥 𝑛 𝑥 , 𝐹 𝑛 + 1 ≥ , 𝑘 𝑡 𝐿 𝒯 𝑀 ℳ 𝐹 𝑥 𝑛 − 1 𝑥 , 𝐹 𝑛 𝐹 𝑥 , 𝑡 , ℳ 𝑛 − 1 𝑥 , 𝐹 𝑛 𝐹 𝑥 , 𝑡 , ℳ 𝑛 𝑥 , 𝐹 𝑛 + 1 , ℳ 𝐹 𝑥 , 𝑡 𝑛 − 1 𝑥 , 𝐹 𝑛 + 1 , ( 1 + 𝑞 ) 𝑡 , 1 ℒ . ( 3 . 1 0 ) Since by (d) of Definition 2.5 ℳ 𝐹 𝑥 𝑛 − 1 𝑥 , 𝐹 𝑛 + 1 , ≥ ( 1 + 𝑞 ) 𝑡 𝐿 𝒯 𝑀 ℳ 𝐹 𝑥 𝑛 − 1 𝑥 , 𝐹 𝑛 𝐹 𝑥 , 𝑡 , ℳ 𝑛 𝑥 , 𝐹 𝑛 + 1 , , 𝑞 𝑡 ( 3 . 1 1 ) we have ℳ 𝐹 𝑥 𝑛 𝑥 , 𝐹 𝑛 + 1 ≥ , 𝑘 𝑡 𝐿 𝒯 𝑀 ℳ 𝐹 𝑥 𝑛 − 1 𝑥 , 𝐹 𝑛 𝐹 𝑥 , 𝑡 , ℳ 𝑛 𝑥 , 𝐹 𝑛 + 1 , ℳ 𝐹 𝑥 , 𝑡 𝑛 𝑥 , 𝐹 𝑛 + 1 . , 𝑞 𝑡 ( 3 . 1 2 ) As 𝑡 -norm is continuous, letting 𝑞 → 1 ℒ we get ℳ 𝐹 𝑥 𝑛 𝑥 , 𝐹 𝑛 + 1 ≥ , 𝑘 𝑡 𝐿 𝒯 𝑀 ℳ 𝐹 𝑥 𝑛 − 1 𝑥 , 𝐹 𝑛 𝐹 𝑥 , 𝑡 , ℳ 𝑛 𝑥 , 𝐹 𝑛 + 1 . , 𝑡 ( 3 . 1 3 ) Consequently, ℳ 𝐹 𝑥 𝑛 𝑥 , 𝐹 𝑛 + 1 ≥ , 𝑡 𝐿 𝒯 𝑀 ℳ 𝐹 𝑥 𝑛 − 1 𝑥 , 𝐹 𝑛 , 1 𝑘 𝑡 𝐹 𝑥 , ℳ 𝑛 𝑥 , 𝐹 𝑛 + 1 , 1 𝑘 𝑡 . ( 3 . 1 4 ) By repeating the above inequality, we obtain ℳ 𝐹 𝑥 𝑛 𝑥 , 𝐹 𝑛 + 1 ≥ , 𝑡 𝐿 𝒯 𝑀 ℳ 𝐹 𝑥 𝑛 − 1 𝑥 , 𝐹 𝑛 , 1 𝑘 𝑡 𝐹 𝑥 , ℳ 𝑛 𝑥 , 𝐹 𝑛 + 1 , 1 𝑘 𝑝 𝑡 . ( 3 . 1 5 ) Since ℳ ( 𝐹 ( 𝑥 𝑛 ) , 𝐹 ( 𝑥 𝑛 + 1 ) , ( 1 / 𝑘 𝑝 ) 𝑡 ) → 1 ℒ as 𝑝 → ∞ , it follows that ℳ 𝐹 𝑥 𝑛 𝑥 , 𝐹 𝑛 + 1 ≥ , 𝑡 𝐿 ℳ 𝐹 𝑥 𝑛 − 1 𝑥 , 𝐹 𝑛 , 1 𝑘 𝑡 . ( 3 . 1 6 ) Thus we proved (3.7). By repeating the above inequality (3.7), we get ℳ 𝐹 𝑥 𝑛 𝑥 , 𝐹 𝑛 + 1 ≥ , 𝑡 𝐿 ℳ 𝐹 𝑥 0 𝑥 , 𝐹 1 , 1 𝑘 𝑛 𝑡 . ( 3 . 1 7 ) Since ℳ ( 𝑥 , 𝑦 , 𝑡 ) → 1 ℒ as 𝑡 → + ∞ and 𝑘 < 1 , letting 𝑛 → ∞ in (3.17) we get l i m 𝑛 → ∞ ℳ 𝐹 𝑥 𝑛 𝑥 , 𝐹 𝑛 + 1 , 𝑡 = 1 ℒ f o r e a c h 𝑡 > 0 . ( 3 . 1 8 )
Now we will prove that { 𝐹 ( 𝑥 𝑛 ) } is a Cauchy sequence which means that for every 𝛿 > 0 and 𝜖 ∈ 𝐿 ⧵ { 0 ℒ , 1 ℒ } there exists 𝑛 ( 𝛿 , 𝜖 ) ∈ ℕ such that
𝑀 𝐹 𝑥 𝑛 𝑥 , 𝐹 𝑛 + 𝑝 > , 𝛿 𝐿 𝒩 ( 𝜖 ) f o r e v e r y 𝑛 ≥ 𝑛 ( 𝛿 , 𝜖 ) a n d e v e r y 𝑝 ∈ ℕ . ( 3 . 1 9 ) Let 𝜖 ∈ 𝐿 ⧵ { 0 ℒ , 1 ℒ } and 𝛿 > 0 be arbitrary. For any 𝑝 ≥ 1 we have 𝛿 = 𝛿 ( 1 − 𝑘 ) ( 1 + 𝑘 + ⋯ + 𝑘 𝑝 + ⋯ ) > 𝛿 ( 1 − 𝑘 ) 1 + 𝑘 + ⋯ + 𝑘 𝑝 − 1 . ( 3 . 2 0 ) Since 𝑀 ( 𝑥 , 𝑦 , 𝑡 ) is nondecreasing with respect to 𝑡 , for all 𝑥 , 𝑦 in 𝑋 , ℳ 𝐹 𝑥 𝑛 𝑥 , 𝐹 𝑛 + 𝑝 ≥ , 𝛿 𝐿 ℳ 𝐹 𝑥 𝑛 𝑥 , 𝐹 𝑛 + 𝑝 , 𝛿 ( 1 − 𝑘 ) 1 + 𝑘 𝑛 + ⋯ + 𝑘 𝑝 − 1 ( 3 . 2 1 ) and hence, by (d) of Definition 2.5, ℳ 𝐹 𝑥 𝑛 𝑥 , 𝐹 𝑛 + 𝑝 ≥ , 𝛿 𝐿 𝒯 𝑀 𝑝 − 2 ℳ 𝐹 𝑥 𝑛 𝑥 , 𝐹 𝑛 + 1 𝐹 𝑥 , ( 1 − 𝑘 ) 𝛿 , ℳ 𝑛 + 1 𝑥 , 𝐹 𝑛 + 2 𝐹 𝑥 , ( 1 − 𝑘 ) 𝛿 𝑘 , … , ℳ 𝑛 + 𝑝 − 1 𝑥 , 𝐹 𝑛 + 𝑝 , ( 1 − 𝑘 ) 𝛿 𝑘 𝑝 − 1 . ( 3 . 2 2 ) From (3.17) it follows that ℳ 𝐹 𝑥 𝑛 + 𝑖 𝑥 , 𝐹 𝑛 + 𝑖 + 1 ≥ , 𝑡 𝐿 ℳ 𝐹 𝑥 𝑛 𝑥 , 𝐹 𝑛 + 1 , 𝑡 𝑘 𝑖 f o r e a c h 𝑖 ≥ 𝐿 1 ℒ . ( 3 . 2 3 ) From (3.23) with 𝑡 = ( 1 − 𝑘 ) 𝛿 𝑘 𝑖 we get ℳ 𝐹 𝑥 𝑛 + 𝑖 𝑥 , 𝐹 𝑛 + 𝑖 + 1 , ( 1 − 𝑘 ) 𝛿 𝑘 𝑖 ≥ 𝐿 ℳ 𝐹 𝑥 𝑛 𝑥 , 𝐹 𝑛 + 1 , . ( 1 − 𝑘 ) 𝛿 ( 3 . 2 4 ) Thus by (3.22), ℳ 𝐹 𝑥 𝑛 𝑥 , 𝐹 𝑛 + 𝑝 ≥ , 𝛿 𝐿 𝒯 𝑛 𝑀 ℳ 𝐹 𝑥 𝑛 𝑥 , 𝐹 𝑛 + 1 , 𝐹 𝑥 ( 1 − 𝑘 ) 𝛿 , ℳ 𝑛 𝑥 , 𝐹 𝑛 + 1 , 𝐹 𝑥 ( 1 − 𝑘 ) 𝛿 , … , ℳ 𝑛 𝑥 , 𝐹 𝑛 + 1 . , ( 1 − 𝑘 ) 𝛿 ( 3 . 2 5 ) Hence we get ℳ 𝐹 𝑥 𝑛 𝑥 , 𝐹 𝑛 + 𝑝 ≥ , 𝛿 𝐿 ℳ 𝐹 𝑥 𝑛 𝑥 , 𝐹 𝑛 + 1 , . ( 1 − 𝑘 ) 𝛿 ( 3 . 2 6 ) From (3.26) and (3.17), ℳ 𝐹 𝑥 𝑛 𝑥 , 𝐹 𝑛 + 𝑝 ≥ , 𝛿 𝐿 ℳ 𝐹 𝑥 0 𝑥 , 𝐹 1 , ( 1 − 𝑘 ) 𝛿 𝑘 𝑛 . ( 3 . 2 7 ) Hence we conclude, as ℳ ( 𝑥 , 𝑦 , 𝑡 ) → 1 ℒ as 𝑡 → + ∞ and 𝑘 < 1 , that there exists 𝑛 ( 𝛿 , 𝜖 ) ∈ ℕ such that ℳ 𝐹 𝑥 𝑛 𝑥 , 𝐹 𝑛 + 𝑝 > , 𝛿 𝐿 𝒩 ( 𝜖 ) f o r e v e r y 𝑛 ≥ 𝑛 ( 𝛿 , 𝜖 ) a n d e v e r y 𝑝 ∈ ℕ . ( 3 . 2 8 ) Thus we proved that { 𝐹 ( 𝑥 𝑛 ) } is a Cauchy sequence.
Since ℎ ( 𝑋 ) is closed and as 𝐹 ( 𝑥 𝑛 ) = ℎ ( 𝑥 𝑛 + 1 ) , there is some 𝑧 ∈ 𝑋 such that
l i m 𝑛 → ∞ ℎ 𝑥 𝑛 = ℎ ( 𝑧 ) . ( 3 . 2 9 )
Now we show that 𝑧 is a coincidence of 𝐹 and ℎ . Since from (3.3) and (3.29) we have ℎ ( 𝑥 𝑛 ) ≤ ℎ ( 𝑧 ) for all 𝑛 , then from (3.2) and by (d) of Definition 2.5 we have
ℳ 𝐹 𝑥 𝑛 ≥ , 𝐹 ( 𝑧 ) , 𝑘 𝑡 𝐿 𝒯 𝑀 ℳ ℎ 𝑥 𝑛 ℎ 𝑥 , ℎ ( 𝑧 ) , 𝑡 , ℳ 𝑛 𝑥 , 𝐹 𝑛 ℳ ℎ 𝑥 , 𝑡 , ℳ ( ℎ ( 𝑧