Fixed Point Theory and Applications
Volume 2009 (2009), Article ID 546273, 11 pages
doi:10.1155/2009/546273
Research Article

Common Fixed Point Theorems in Menger Probabilistic Quasimetric Spaces

Shaban Sedghi,1 Tatjana Žikić-Došenović,2 and Nabi Shobe3

1Department of Mathematics, Islamic Azad University-Babol Branch, P.O. Box 163, Ghaemshahr, Iran
2Faculty of Technology, University of Novi Sad, Bulevar Cara Lazara 1, 21000 Novi Sad, Serbia
3Department of Mathematics, Islamic Azad University-Babol Branch, Babol, Iran

Received 21 November 2008; Accepted 19 April 2009

Academic Editor: Massimo Furi

Copyright © 2009 Shaban Sedghi 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 consider complete Menger probabilistic quasimetric space and prove common fixed point theorems for weakly compatible maps in this space.

1. Introduction and Preliminaries

K. Menger introduced the notion of a probabilistic metric space in 1942 and since then the theory of probabilistic metric spaces has developed in many directions [1]. The idea of K . Menger was to use distribution functions instead of nonnegative real numbers as values of the metric. The notion of a probabilistic metric space corresponds to the situations when we do not know exactly the distance between two points, we know only probabilities of possible values of this distance. Such a probabilistic generalization of metric spaces appears to be well adapted for the investigation of physiological thresholds and physical quantities particularly in connections with both string and 𝐸 - i n fi n i t y theory; see [2–5]. It is also of fundamental importance in probabilistic functional analysis, nonlinear analysis and applications [6–10].

In the sequel, we will adopt usual terminology, notation, and conventions of the theory of Menger probabilistic metric spaces, as in [7, 8, 10]. Throughout this paper, the space of all probability distribution functions (in short, dfs) is denoted by Δ + = { 𝐹 ∶ ℝ ∪ { − ∞ , + ∞ } → [ 0 , 1 ] ∶ 𝐹 is left-continuous and nondecreasing on ℝ , 𝐹 ( 0 ) = 0 and 𝐹 ( + ∞ ) = 1 } , and the subset 𝐷 + ⊆ Δ + is the set 𝐷 + = { 𝐹 ∈ Δ + ∶ 𝑙 − 𝐹 ( + ∞ ) = 1 } . Here 𝑙 − 𝑓 ( 𝑥 ) denotes the left limit of the function 𝑓 at the point 𝑥 , 𝑙 − 𝑓 ( 𝑥 ) = l i m 𝑡 → 𝑥 − 𝑓 ( 𝑡 ) . The space Δ + is partially ordered by the usual pointwise ordering of functions, that is, 𝐹 ≤ 𝐺 if and only if 𝐹 ( 𝑡 ) ≤ 𝐺 ( 𝑡 ) for all 𝑡 in ℝ . The maximal element for Δ + in this order is the df given by 𝜀 0 ( 𝑡 ) = 0 , i f 𝑡 ≤ 0 , 1 , i f 𝑡 > 0 . ( 1 . 1 )

Definition 1.1 (see [1]). A mapping 𝑇 ∶ [ 0 , 1 ] × [ 0 , 1 ] → [ 0 , 1 ] is 𝑡 - n o r m if 𝑇 is satisfying the following conditions:(a) 𝑇 is commutative and associative;(b) 𝑇 ( 𝑎 , 1 ) = 𝑎 for all 𝑎 ∈ [ 0 , 1 ] ;(d) 𝑇 ( 𝑎 , 𝑏 ) ≤ 𝑇 ( 𝑐 , 𝑑 ) , whenever 𝑎 ≤ 𝑐 and 𝑏 ≤ 𝑑 , and 𝑎 , 𝑏 , 𝑐 , 𝑑 ∈ [ 0 , 1 ] .
The following are the four basic 𝑡 - n o r m s :
𝑇 𝑀 𝑇 ( 𝑥 , 𝑦 ) = m i n ( 𝑥 , 𝑦 ) , 𝑃 𝑇 ( 𝑥 , 𝑦 ) = 𝑥 ⋅ 𝑦 , 𝐿 𝑇 ( 𝑥 , 𝑦 ) = m a x ( 𝑥 + 𝑦 − 1 , 0 ) , 𝐷 ( 𝑥 , 𝑦 ) = m i n ( 𝑥 , 𝑦 ) , i f m a x ( 𝑥 , 𝑦 ) = 1 , 0 , o t h e r w i s e . ( 1 . 2 )

Each 𝑡 - n o r m    𝑇 can be extended [11] (by associativity) in a unique way to an 𝑛 -ary operation taking for ( 𝑥 1 , … , 𝑥 𝑛 ) ∈ [ 0 , 1 ] 𝑛 the values 𝑇 1 ( 𝑥 1 , 𝑥 2 ) = 𝑇 ( 𝑥 1 , 𝑥 2 ) and 𝑇 𝑛 𝑥 1 , … , 𝑥 𝑛 + 1 𝑇 = 𝑇 𝑛 − 1 𝑥 1 , … , 𝑥 𝑛 , 𝑥 𝑛 + 1 ( 1 . 3 ) for 𝑛 ≥ 2 and 𝑥 𝑖 ∈ [ 0 , 1 ] , for all 𝑖 ∈ { 1 , 2 , … , 𝑛 + 1 } .

We also mention the following families of 𝑡 - n o r m s .

Definition 1.2. It is said that the 𝑡 -norm 𝑇 is of Hadžić-type ( 𝐻 - t y p e for short) and 𝑇 ∈ ℋ if the family { 𝑇 𝑛 } 𝑛 ∈ ℕ of its iterates defined, for each 𝑥 in [ 0 , 1 ] , by 𝑇 0 ( 𝑥 ) = 1 , 𝑇 𝑛 + 1 ( 𝑥 ) = 𝑇 ( 𝑇 𝑛 ( 𝑥 ) , 𝑥 ) , ∀ 𝑛 ≥ 0 , ( 1 . 4 ) is equicontinuous at 𝑥 = 1 , that is, ∀ 𝜖 ∈ ( 0 , 1 ) ∃ 𝛿 ∈ ( 0 , 1 ) s u c h t h a t 𝑥 > 1 − 𝛿 ⟹ 𝑇 𝑛 ( 𝑥 ) > 1 − 𝜖 , ∀ 𝑛 ≥ 1 . ( 1 . 5 )
There is a nice characterization of continuous 𝑡 - n o r m    𝑇 of the class ℋ [12].
(i)If there exists a strictly increasing sequence ( 𝑏 𝑛 ) 𝑛 ∈ 𝐍 in [ 0 , 1 ] such that l i m 𝑛 → ∞ 𝑏 𝑛 = 1 and 𝑇 ( 𝑏 𝑛 , 𝑏 𝑛 ) = 𝑏 𝑛 ∀ 𝑛 ∈ ℕ , then 𝑇 is of Hadžić-type.(ii)If 𝑇 is continuous and 𝑇 ∈ ℋ , then there exists a sequence ( 𝑏 𝑛 ) 𝑛 ∈ ℕ as in (i).The 𝑡 - n o r m    𝑇 𝑀 is an trivial example of a 𝑡 - n o r m of 𝐻 - t y p e , but there are 𝑡 - n o r m s    𝑇 of Hadžić-type with 𝑇 ≠ 𝑇 𝑀 (see, e.g., [13]).

Definition 1.3 (see [13]). If 𝑇 is a 𝑡 - n o r m and ( 𝑥 1 , 𝑥 2 , … , 𝑥 𝑛 ) ∈ [ 0 , 1 ] 𝑛 ( 𝑛 ∈ ℕ ) , then 𝑇 𝑛 𝑖 = 1 𝑥 𝑖 is defined recurrently by 1, if 𝑛 = 0 and 𝑇 𝑛 𝑖 = 1 𝑥 𝑖 = 𝑇 ( 𝑇 𝑛 − 1 𝑖 = 1 𝑥 𝑖 , 𝑥 𝑛 ) for all 𝑛 ≥ 1 . If ( 𝑥 𝑖 ) 𝑖 ∈ ℕ is a sequence of numbers from [ 0 , 1 ] , then 𝑇 ∞ 𝑖 = 1 𝑥 𝑖 is defined as l i m 𝑛 → ∞ 𝑇 𝑛 𝑖 = 1 𝑥 𝑖 (this limit always exists) and 𝑇 ∞ 𝑖 = 𝑛 𝑥 𝑖 as 𝑇 ∞ 𝑖 = 1 𝑥 𝑛 + 𝑖 . In fixed point theory in probablistic metric spaces there are of particular interest the 𝑡 -norms 𝑇 and sequences ( 𝑥 𝑛 ) ⊂ [ 0 , 1 ] such that l i m 𝑛 → ∞ 𝑥 𝑛 = 1 and l i m 𝑛 → ∞ 𝑇 ∞ 𝑖 = 1 𝑥 𝑛 + 𝑖 = 1 . Some examples of 𝑡 - n o r m s with the above property are given in the following proposition.

Proposition 1.4 (see [13]). (i) For 𝑇 ≥ 𝑇 𝐿 the following implication holds:
l i m 𝑛 → ∞ 𝑇 ∞ 𝑖 = 1 𝑥 𝑛 + 𝑖 = 1 ⟺ ∞ 𝑛 = 1 0 𝑥 0 2 0 0 𝑑 1 − 𝑥 𝑛 < ∞ . ( 1 . 6 )
(ii) If 𝑇 ∈ ℋ , then for every sequence ( 𝑥 𝑛 ) 𝑛 ∈ ℕ in I such that l i m 𝑛 → ∞ 𝑥 𝑛 = 1 , one has l i m 𝑛 → ∞ 𝑇 ∞ 𝑖 = 1 𝑥 𝑛 + 𝑖 = 1 .

Note [14, Remark 13] that if 𝑇 is a 𝑡 - n o r m for which there exists ( 𝑥 𝑛 ) ⊂ [ 0 , 1 ] such that l i m 𝑛 → ∞ 𝑥 𝑛 = 1 and l i m 𝑛 → ∞ 𝑇 ∞ 𝑖 = 1 𝑥 𝑛 + 𝑖 = 1 , then s u p 𝑡 < 1 𝑇 ( 𝑡 , 𝑡 ) = 1 . Important class of 𝑡 - n o r m s is given in the following example.

Example 1.5. (i) The Dombi family of 𝑡 - n o r m s   ( 𝑇 𝐷 𝜆 ) 𝜆 ∈ [ 0 , ∞ ] is defined by
𝑇 𝐷 𝜆 ⎧ ⎪ ⎪ ⎨ ⎪ ⎪ ⎩ 𝑇 ( 𝑥 , 𝑦 ) = 𝐷 𝑇 ( 𝑥 , 𝑦 ) , 𝜆 = 0 , 𝑀 ( 1 𝑥 , 𝑦 ) , 𝜆 = ∞ , ( 1 + ( 1 − 𝑥 ) / 𝑥 ) 𝜆 + ( ( 1 − 𝑦 ) / 𝑦 ) 𝜆 1 / 𝜆 , 𝜆 ∈ ( 0 , ∞ ) . ( 1 . 7 )
( i i ) The Aczél-Alsina family of 𝑡 - n o r m s    ( 𝑇 𝜆 𝐴 𝐴 ) 𝜆 ∈ [ 0 , ∞ ] is defined by
𝑇 𝜆 𝐴 𝐴 ( ⎧ ⎪ ⎪ ⎨ ⎪ ⎪ ⎩ 𝑇 𝑥 , 𝑦 ) = 𝐷 𝑇 ( 𝑥 , 𝑦 ) , 𝜆 = 0 , 𝑀 ( 𝑒 𝑥 , 𝑦 ) , 𝜆 = ∞ , − ( − l o g 𝑥 ) 𝜆 + ( − l o g 𝑦 ) 𝜆 1 / 𝜆 , 𝜆 ∈ ( 0 , ∞ ) . ( 1 . 8 )
( i i i ) Sugeno-Weber family of 𝑡 - n o r m s    ( 𝑇 𝜆 𝑆 𝑊 ) 𝜆 ∈ [ − 1 , ∞ ] is defined by
𝑇 𝜆 𝑆 𝑊 ⎧ ⎪ ⎪ ⎨ ⎪ ⎪ ⎩ 𝑇 ( 𝑥 , 𝑦 ) = 𝐷 𝑇 ( 𝑥 , 𝑦 ) , 𝜆 = − 1 , 𝑃 ( 𝑥 , 𝑦 ) , 𝜆 = ∞ , m a x 0 , 𝑥 + 𝑦 − 1 + 𝜆 𝑥 𝑦 1 + 𝜆 , 𝜆 ∈ ( − 1 , ∞ ) . ( 1 . 9 )

In [13] the following results are obtained.

(a)If ( 𝑇 𝐷 𝜆 ) 𝜆 ∈ ( 0 , ∞ ) is the Dombi family of 𝑡 - n o r m s and ( 𝑥 𝑛 ) 𝑛 ∈ ℕ is a sequence of elements from ( 0 , 1 ] such that l i m 𝑛 → ∞ 𝑥 𝑛 = 1 then we have the following equivalence: ∞ 𝑖 = 1 0 𝑥 0 2 0 0 𝑑 ( 1 − 𝑥 𝑖 ) 𝜆 < ∞ ⟺ l i m 𝑛 → ∞ ( 𝑇 𝐷 𝜆 ) ∞ 𝑖 = 𝑛 𝑥 𝑖 = 1 . ( 1 . 1 0 ) (b)Equivalence (1.10) holds also for the family ( 𝑇 𝜆 𝐴 𝐴 ) 𝜆 ∈ ( 0 , ∞ ) , that is, ∞ 𝑖 = 1 0 𝑥 0 2 0 0 𝑑 ( 1 − 𝑥 𝑖 ) 𝜆 < ∞ ⟺ l i m 𝑛 → ∞ ( 𝑇 𝜆 𝐴 𝐴 ) ∞ 𝑖 = 𝑛 𝑥 𝑖 = 1 . ( 1 . 1 1 ) (c)If ( 𝑇 𝜆 𝑆 𝑊 ) 𝜆 ∈ ( − 1 , ∞ ] is the Sugeno-Weber family of 𝑡 - n o r m s and ( 𝑥 𝑛 ) 𝑛 ∈ ℕ is a sequence of elements from ( 0 , 1 ] such that l i m 𝑛 → ∞ 𝑥 𝑛 = 1 then we have the following equivalence: ∞ 𝑖 = 1 0 𝑥 0 2 0 0 𝑑 1 − 𝑥 𝑖 < ∞ ⟺ l i m 𝑛 → ∞ ( 𝑇 𝜆 𝑆 𝑊 ) ∞ 𝑖 = 𝑛 𝑥 𝑖 = 1 . ( 1 . 1 2 )

Proposition 1.6. Let ( 𝑥 𝑛 ) 𝑛 ∈ ℕ be a sequence of numbers from [ 0 , 1 ] such that l i m 𝑛 → ∞ 𝑥 𝑛 = 1 and 𝑡 -norm 𝑇 is of 𝐻 - t y p e . Then l i m 𝑛 → ∞ 𝐓 ∞ 𝑖 = 𝑛 𝑥 𝑖 = l i m 𝑛 → ∞ 𝐓 ∞ 𝑖 = 𝑛 𝑥 𝑛 + 𝑖 = 1 . ( 1 . 1 3 )

Definition 1.7. A Menger Probabilistic Quasimetric space (briefly, Menger PQM space) is a triple ( 𝑋 , ℱ , 𝑇 ) , where 𝑋 is a nonempty set, 𝑇 is a continuous 𝑡 - n o r m , and ℱ is a mapping from 𝑋 × 𝑋 into 𝐷 + , such that, if 𝐹 𝑝 , 𝑞 denotes the value of ℱ at the pair ( 𝑝 , 𝑞 ) , then the following conditions hold, for all 𝑝 , 𝑞 , 𝑟 in 𝑋 ,(PQM1) 𝐹 𝑝 , 𝑞 ( 𝑡 ) = 𝐹 𝑞 , 𝑝 ( 𝑡 ) = 𝜀 0 ( 𝑡 ) for all 𝑡 > 0 if and only if 𝑝 = 𝑞 ;(PQM2) 𝐹 𝑝 , 𝑞 ( 𝑡 + 𝑠 ) ≥ 𝑇 ( 𝐹 𝑝 , 𝑟 ( 𝑡 ) , 𝐹 𝑟 , 𝑞 ( 𝑠 ) ) for all 𝑝 , 𝑞 , 𝑟 ∈ 𝑋 and 𝑡 , 𝑠 ≥ 0 .

Definition 1.8. Let ( 𝑋 , ℱ , 𝑇 ) be a Menger PQM space.(1)A sequence { 𝑥 𝑛 } 𝑛 in 𝑋 is said to be convergent to 𝑥 in 𝑋 if, for every 𝜖 > 0 and 𝜆 > 0 , there exists positive integer 𝑁 such that 𝐹 𝑥 𝑛 , 𝑥 ( 𝜖 ) > 1 − 𝜆 whenever 𝑛 ≥ 𝑁 .(2)A sequence { 𝑥 𝑛 } 𝑛 in 𝑋 is called Cauchy sequence [15] if, for every 𝜖 > 0 and 𝜆 > 0 , there exists positive integer 𝑁 such that 𝐹 𝑥 𝑛 , 𝑥 𝑚 ( 𝜖 ) > 1 − 𝜆 whenever 𝑛 ≥ 𝑚 ≥ 𝑁 ( 𝑚 ≥ 𝑛 ≥ 𝑁 ).(3)A Menger PQM space ( 𝑋 , ℱ , 𝑇 ) is said to be complete if and only if every Cauchy sequence in 𝑋 is convergent to a point in 𝑋 .

In 1998, Jungck and Rhoades [16] introduced the following concept of weak compatibility.

Definition 1.9. Let 𝐴 and 𝑆 be mappings from a Menger PQM space ( 𝑋 , ℱ , 𝑇 ) into itself. Then the mappings are said to be weak compatible if they commute at their coincidence point, that is, 𝐴 𝑥 = 𝑆 𝑥 implies that 𝐴 𝑆 𝑥 = 𝑆 𝐴 𝑥 .

2. The Main Result

Throughout this section, a binary operation 𝑇 ∶ [ 0 , 1 ] × [ 0 , 1 ] → [ 0 , 1 ] is a continuous 𝑡 -norm and satisfies the condition l i m 𝑛 → ∞ 𝑇 ∞ 𝑖 = 𝑛 1 − 𝑎 𝑖 ( 𝑡 ) = 1 , ( 2 . 1 ) where 𝑎 ∶ ℝ + → ( 0 , 1 ) . It is easy to see that this condition implies l i m 𝑛 → ∞ 𝑎 𝑛 ( 𝑡 ) = 0 .

Lemma 2.1. Let ( 𝑋 , ℱ , 𝑇 ) be a Menger PQM space. If the sequence { 𝑥 𝑛 } in X is such that for every 𝑛 ∈ ℕ , 𝐹 𝑥 𝑛 , 𝑥 𝑛 + 1 ( 𝑡 ) ≥ 1 − 𝑎 𝑛 ( 𝑡 ) 1 − 𝐹 𝑥 0 , 𝑥 1 ( 𝑡 ) ( 2 . 2 ) for very 𝑡 > 0 , where 𝑎 ∶ ℝ + → ( 0 , 1 ) is a monotone increasing functions.Then the sequence { 𝑥 𝑛 } is a Cauchy sequence.

Proof. For every 𝑚 > 𝑛 and 𝑥 𝑛 , 𝑥 𝑚 ∈ 𝑋 , we have 𝐹 𝑥 𝑛 , 𝑥 𝑚 𝑇 ( 𝑡 ) ≥ 𝑇 𝑚 − 2 𝐹 𝑥 𝑛 , 𝑥 𝑛 + 1 𝑡 𝑚 − 𝑛 , … , 𝐹 𝑥 𝑚 − 2 , 𝑥 𝑚 − 1 𝑡 𝑚 − 𝑛 , 𝐹 𝑥 𝑚 − 1 , 𝑥 𝑚 𝑡 𝑚 − 𝑛 ≥ 𝑇 𝑚 − 1 1 − 𝑎 𝑛 𝑡 𝑚 − 𝑛 1 − 𝐹 𝑥 0 , 𝑥 1 𝑡 𝑚 − 𝑛 , 1 − 𝑎 𝑛 + 1 𝑡 × 𝑚 − 𝑛 1 − 𝐹 𝑥 0 , 𝑥 1 𝑡 𝑚 − 𝑛 , … , 1 − 𝑎 𝑚 − 1 𝑡 𝑚 − 𝑛 1 − 𝐹 𝑥 0 , 𝑥 1 𝑡 𝑚 − 𝑛 ≥ 𝑇 𝑚 − 1 1 − 𝑎 𝑛 𝑡 𝑚 − 𝑛 , 1 − 𝑎 𝑛 + 1 𝑡 𝑚 − 𝑛 , … , 1 − 𝑎 𝑚 − 1 𝑡 𝑚 − 𝑛 ≥ 𝑇 𝑚 − 1 1 − 𝑎 𝑛 ( 𝑡 ) , 1 − 𝑎 𝑛 + 1 ( 𝑡 ) , … , 1 − 𝑎 𝑚 − 1 ( 𝑡 ) = 𝑇 𝑚 − 1 𝑖 = 𝑛 1 − 𝑎 𝑖 ( 𝑡 ) ≥ 𝑇 ∞ 𝑖 = 𝑛 1 − 𝑎 𝑖 ( 𝑡 ) > 1 − 𝜆 ( 2 . 3 ) for each 0 < 𝜆 < 1 and 𝑡 > 0 . Hence sequence { 𝑥 𝑛 } is Cauchy sequence.

Theorem 2.2. Let ( 𝑋 , ℱ , 𝑇 ) be a complete Menger PQM space and let 𝑓 , 𝑔 , ℎ ∶ 𝑋 → 𝑋 be maps that satisfy the following conditions: (a) 𝑔 ( 𝑋 ) ∪ ℎ ( 𝑋 ) ⊆ 𝑓 ( 𝑋 ) ; (b)the pairs ( 𝑓 , 𝑔 ) and ( 𝑓 , ℎ ) are weak compatible, 𝑓 ( 𝑋 ) is closed subset of 𝑋 ; (c) m i n { 𝐹 𝑔 ( 𝑥 ) , ℎ ( 𝑦 ) ( 𝑡 ) , 𝐹 ℎ ( 𝑥 ) , 𝑔 ( 𝑦 ) ( 𝑡 ) } ≥ 1 − 𝑎 ( 𝑡 ) ( 1 − 𝐹 𝑓 ( 𝑥 ) , 𝑓 ( 𝑦 ) ( 𝑡 ) ) for all 𝑥 , 𝑦 ∈ 𝑋 and every 𝑡 > 0 , where 𝑎 ∶ ℝ + → ( 0 , 1 ) is a monotone increasing function. If l i m 𝑛 → ∞ 𝑇 ∞ 𝑖 = 𝑛 1 − 𝑎 𝑖 ( 𝑡 ) = 1 , ( 2 . 4 ) then 𝑓 , 𝑔 , and ℎ have a unique common fixed point.

Proof. Let 𝑥 0 ∈ 𝑋 . By (a), we can find 𝑥 1 such that 𝑓 ( 𝑥 1 ) = 𝑔 ( 𝑥 0 ) and ℎ ( 𝑥 1 ) = 𝑓 ( 𝑥 2 ) . By induction, we can define a sequence { 𝑥 𝑛 } such that 𝑓 ( 𝑥 2 𝑛 + 1 ) = 𝑔 ( 𝑥 2 𝑛 ) and ℎ ( 𝑥 2 𝑛 + 1 ) = 𝑓 ( 𝑥 2 𝑛 + 2 ) . By induction again, 𝐹 𝑓 ( 𝑥 2 𝑛 ) , 𝑓 ( 𝑥 2 𝑛 + 1 ) ( 𝑡 ) = 𝐹 ℎ ( 𝑥 2 𝑛 − 1 ) , 𝑔 ( 𝑥 2 𝑛 ) 𝐹 ( 𝑡 ) ≥ m i n ℎ ( 𝑥 2 𝑛 − 1 ) , 𝑔 ( 𝑥 2 𝑛 ) ( 𝑡 ) , 𝐹 𝑔 ( 𝑥 2 𝑛 − 1 ) , ℎ ( 𝑥 2 𝑛 ) ( 𝑡 ) ≥ 1 − 𝑎 ( 𝑡 ) 1 − 𝐹 𝑓 ( 𝑥 2 𝑛 − 1 ) , 𝑓 ( 𝑥 2 𝑛 ) . ( 𝑡 ) ( 2 . 5 )
Similarly, we have
𝐹 𝑓 ( 𝑥 2 𝑛 − 1 ) , 𝑓 ( 𝑥 2 𝑛 ) ( 𝑡 ) = 𝐹 𝑔 ( 𝑥 2 𝑛 − 2 ) , ℎ ( 𝑥 2 𝑛 − 1 ) 𝐹 ( 𝑡 ) ≥ m i n ℎ ( 𝑥 2 𝑛 − 2 ) , 𝑔 ( 𝑥 2 𝑛 − 1 ) ( 𝑡 ) , 𝐹 𝑔 ( 𝑥 2 𝑛 − 2 ) , ℎ ( 𝑥 2 𝑛 − 1 ) ( 𝑡 ) ≥ 1 − 𝑎 ( 𝑡 ) 1 − 𝐹 𝑓 ( 𝑥 2 𝑛 − 2 ) , 𝑓 ( 𝑥 2 𝑛 − 1 ) . ( 𝑡 ) ( 2 . 6 )
Hence, it follows that
𝐹 𝑓 ( 𝑥 𝑛 ) , 𝑓 ( 𝑥 𝑛 + 1 ) ( 𝑡 ) ≥ 1 − 𝑎 ( 𝑡 ) 1 − 𝐹 𝑓 ( 𝑥 𝑛 − 1 ) , 𝑓 ( 𝑥 𝑛 ) ( 𝑡 ) ≥ 1 − 𝑎 ( 𝑡 ) 1 − 1 − 𝑎 ( 𝑡 ) 1 − 𝐹 𝑓 ( 𝑥 𝑛 − 2 ) , 𝑓 ( 𝑥 𝑛 − 1 ) ( 𝑡 ) = 1 − 𝑎 2 ( 𝑡 ) 1 − 𝐹 𝑓 ( 𝑥 𝑛 − 2 ) , 𝑓 ( 𝑥 𝑛 − 1 ) ⋮ ( 𝑡 ) ≥ 1 − 𝑎 𝑛 ( 𝑡 ) 1 − 𝐹 𝑓 ( 𝑥 0 ) , 𝑓 ( 𝑥 1 ) . ( 𝑡 ) ( 2 . 7 ) for 𝑛 = 1 , 2 , … .
Now by Lemma 2.1, { 𝑓 ( 𝑥 𝑛 ) } is a Cauchy sequence. Since the space 𝑓 ( 𝑋 ) is complete, there exists a point 𝑦 ∈ 𝑋 such that
l i m 𝑛 → ∞ 𝑓 𝑥 𝑛 = l i m 𝑛 → ∞ 𝑔 𝑥 2 𝑛 = l i m 𝑛 → ∞ ℎ 𝑥 2 𝑛 + 1 = 𝑦 ∈ 𝑓 ( 𝑋 ) . ( 2 . 8 ) It follows that, there exists 𝑣 ∈ 𝑋 such that 𝑓 ( 𝑣 ) = 𝑦 . We prove that 𝑔 ( 𝑣 ) = ℎ ( 𝑣 ) = 𝑦 . From (c), we get 𝐹 𝑔 ( 𝑥 2 𝑛 ) , ℎ ( 𝑣 ) 𝐹 ( 𝑡 ) ≥ m i n 𝑔 ( 𝑥 2 𝑛 ) , ℎ ( 𝑣 ) ( 𝑡 ) , 𝐹 ℎ ( 𝑥 2 𝑛 ) , 𝑔 ( 𝑣 ) ( 𝑡 ) ≥ 1 − 𝑎 ( 𝑡 ) 1 − 𝐹 𝑓 ( 𝑥 2 𝑛 ) , 𝑓 ( 𝑣 ) ( 𝑡 ) ( 2 . 9 ) as 𝑛 → ∞ , we have 𝐹 𝑦 , ℎ ( 𝑣 ) ( 𝑡 ) ≥ 1 − 𝑎 ( 𝑡 ) 1 − 𝐹 𝑦 , 𝑦 ( 𝑡 ) = 1 ( 2 . 1 0 ) which implies that, ℎ ( 𝑣 ) = 𝑦 . Moreover, 𝐹 𝑔 ( 𝑣 ) , ℎ ( 𝑥 2 𝑛 + 1 ) 𝐹 ( 𝑡 ) ≥ m i n 𝑔 ( 𝑣 ) , ℎ ( 𝑥 2 𝑛 + 1 ) ( 𝑡 ) , 𝐹 ℎ ( 𝑣 ) , 𝑔 ( 𝑥 2 𝑛 + 1 ) ( 𝑡 ) ≥ 1 − 𝑎 ( 𝑡 ) 1 − 𝐹 𝑓 ( 𝑣 ) , 𝑓 ( 𝑥 2 𝑛 + 1 ) ( 𝑡 ) ( 2 . 1 1 ) as 𝑛 → ∞ , we have 𝐹 𝑔 ( 𝑣 ) , 𝑦 ( 𝑡 ) ≥ 1 − 𝑎 ( 𝑡 ) 1 − 𝐹 𝑦 , 𝑦 ( 𝑡 ) = 1 ( 2 . 1 2 ) which implies that 𝑔 ( 𝑣 ) = 𝑦 . Since, the pairs ( 𝑓 , 𝑔 ) and ( 𝑓 , ℎ ) are weak compatible, we have 𝑓 ( 𝑔 ( 𝑣 ) ) = 𝑔 ( 𝑓 ( 𝑣 ) ) , hence it follows that 𝑓 ( 𝑦 ) = 𝑔 ( 𝑦 ) . Similarly, we get 𝑓 ( 𝑦 ) = ℎ ( 𝑦 ) . Now, we prove that 𝑔 ( 𝑦 ) = 𝑦 . Since, from (c) we have 𝐹 𝑔 ( 𝑦 ) , ℎ ( 𝑥 2 𝑛 + 1 ) 𝐹 ( 𝑡 ) ≥ m i n 𝑔 ( 𝑦 ) , ℎ ( 𝑥 2 𝑛 + 1 ) ( 𝑡 ) , 𝐹 ℎ ( 𝑦 ) , 𝑔 ( 𝑥 2 𝑛 + 1 ) ( 𝑡 ) ≥ 1 − 𝑎 ( 𝑡 ) 1 − 𝐹 𝑓 ( 𝑦 ) , 𝑓 ( 𝑥 2 𝑛 + 1 ) ( 𝑡 ) ( 2 . 1 3 ) as 𝑛 → ∞ , we have 𝐹 𝑔 ( 𝑦 ) , 𝑦 ( 𝑡 ) ≥ 1 − 𝑎 ( 𝑡 ) 1 − 𝐹 𝑓 ( 𝑦 ) , 𝑦 ( 𝑡 ) = 1 − 𝑎 ( 𝑡 ) 1 − 𝐹 𝑔 ( 𝑦 ) , 𝑦 ( 𝑡 ) ≥ 1 − 𝑎 ( 𝑡 ) 1 − 1 − 𝑎 ( 𝑡 ) 1 − 𝐹 𝑔 ( 𝑦 ) , 𝑦 ( 𝑡 ) = 1 − 𝑎 2 ( 𝑡 ) 1 − 𝐹 𝑔 ( 𝑦 ) , 𝑦 ⋮ ( 𝑡 ) ≥ 1 − 𝑎 𝑛 ( 𝑡 ) 1 − 𝐹 𝑔 ( 𝑦 ) , 𝑦 ( 𝑡 ) ⟶ 1 . ( 2 . 1 4 ) It follows that 𝑔 ( 𝑦 ) = 𝑦 . Therefore, ℎ ( 𝑦 ) = 𝑓 ( 𝑦 ) = 𝑔 ( 𝑦 ) = 𝑦 . That is 𝑦 is a common fixed point of 𝑓 , 𝑔 , and ℎ .
If 𝑦 and 𝑧 are two fixed points common to 𝑓 , 𝑔 , and ℎ , then
𝐹 𝑦 , 𝑧 ( 𝑡 ) = 𝐹 𝑔 ( 𝑦 ) , ℎ ( 𝑧 ) 𝐹 ( 𝑡 ) ≥ m i n 𝑔 ( 𝑦 ) , ℎ ( 𝑧 ) ( 𝑡 ) , 𝐹 ℎ ( 𝑦 ) , 𝑔 ( 𝑧 ) ( 𝑡 ) ≥ 1 − 𝑎 ( 𝑡 ) 1 − 𝐹 𝑓 ( 𝑦 ) , 𝑓 ( 𝑧 ) ( 𝑡 ) = 1 − 𝑎 ( 𝑡 ) 1 − 𝐹 𝑦 , 𝑧 ( 𝑡 ) ≥ 1 − 𝑎 ( 𝑡 ) 1 − 1 − 𝑎 ( 𝑡 ) 1 − 𝐹 𝑦 , 𝑧 ⋮ ( 𝑡 ) ≥ 1 − 𝑎 𝑛 ( 𝑡 ) 1 − 𝐹 𝑦 , 𝑧 ( 𝑡 ) ⟶ 1 ( 2 . 1 5 ) as 𝑛 → ∞ , which implies that 𝑦 = 𝑧 and so the uniqueness of the common fixed point.

Corollary 2.3. Let ( 𝑋 , ℱ , 𝑇 ) be a complete Menger PQM space and let 𝑓 , 𝑔 ∶ 𝑋 → 𝑋 be maps that satisfy the following conditions: (a) 𝑔 ( 𝑋 ) ⊆ 𝑓 ( 𝑋 ) ; (b)the pair ( 𝑓 , 𝑔 ) is weak compatible, 𝑓 ( 𝑋 ) is closed subset of 𝑋 ; (c) 𝐹 𝑔 ( 𝑥 ) , 𝑔 ( 𝑦 ) ( 𝑡 ) ≥ 1 − 𝑎 ( 𝑡 ) ( 1 − 𝐹 𝑓 ( 𝑥 ) , 𝑓 ( 𝑦 ) ( 𝑡 ) ) for all 𝑥 , 𝑦 ∈ 𝑋 and 𝑡 > 0 , where 𝑎 ∶ ℝ + → ( 0 , 1 ) is monotone increasing function. If l i m 𝑛 → ∞ 𝑇 ∞ 𝑖 = 𝑛 1 − 𝑎 𝑖 ( 𝑡 ) = 1 , ( 2 . 1 6 ) then 𝑓 and 𝑔 have a unique common fixed point.

Proof. It is enough, set ℎ = 𝑔 in Theorem 2.2.

Corollary 2.4. Let ( 𝑋 , ℱ , 𝑇 ) be a complete Menger PQM space and let 𝑓 1 , 𝑓 2 , … , 𝑓 𝑛 , 𝑔 ∶ 𝑋 → 𝑋 be maps that satisfy the following conditions: (a) 𝑔 ( 𝑋 ) ⊆ 𝑓 1 𝑓 2 ⋯ 𝑓 𝑛 ( 𝑋 ) ; (b)the pair ( 𝑓 1 𝑓 2 ⋯ 𝑓 𝑛 , 𝑔 ) is weak compatible, 𝑓 1 𝑓 2 ⋯ 𝑓 𝑛 ( 𝑋 ) is closed subset of 𝑋 ; (c) 𝐹 𝑔 ( 𝑥 ) , 𝑔 ( 𝑦 ) ( 𝑡 ) ≥ 1 − 𝑎 ( 𝑡 ) ( 1 − 𝐹 𝑓 1 𝑓 2 ⋯ 𝑓 𝑛 ( 𝑥 ) , 𝑓 1 𝑓 2 ⋯ 𝑓 𝑛 ( 𝑦 ) ( 𝑡 ) ) for all 𝑥 , 𝑦 ∈ 𝑋 and 𝑡 > 0 , where 𝑎 ∶ ℝ + → ( 0 , 1 ) is monotone increasing function; (d) 𝑔 𝑓 2 ⋯ 𝑓 𝑛 = 𝑓 2 ⋯ 𝑓 𝑛 𝑔 𝑓 𝑔 , 3 ⋯ 𝑓 𝑛 = 𝑓 3 ⋯ 𝑓 𝑛 ⋮ 𝑔 , 𝑔 𝑓 𝑛 = 𝑓 𝑛 𝑓 𝑔 , 1 𝑓 2 ⋯ 𝑓 𝑛 = 𝑓 2 ⋯ 𝑓 𝑛 𝑓 1 , 𝑓 1 𝑓 2 𝑓 3 ⋯ 𝑓 𝑛 = 𝑓 3 ⋯ 𝑓 𝑛 𝑓 1 𝑓 2 , ⋮ 𝑓 1 ⋯ 𝑓 𝑛 − 1 𝑓 𝑛 = 𝑓 𝑛 𝑓 1 ⋯ 𝑓 𝑛 − 1 . ( 2 . 1 7 ) If l i m 𝑛 → ∞ 𝑇 ∞ 𝑖 = 𝑛 1 − 𝑎 𝑖 ( 𝑡 ) = 1 , ( 2 . 1 8 ) then 𝑓 1 , 𝑓 2 , … , 𝑓 𝑛 , 𝑔 have a unique common fixed point.

Proof. By Corollary 2.3, if set 𝑓 1 𝑓 2 ⋯ 𝑓 𝑛 = 𝑓 then 𝑓 , 𝑔 have a unique common fixed point in 𝑋 . That is, there exists 𝑥 ∈ 𝑋 , such that 𝑓 1 𝑓 2 ⋯ 𝑓 𝑛 ( 𝑥 ) = 𝑔 ( 𝑥 ) = 𝑥 . We prove that 𝑓 𝑖 ( 𝑥 ) = 𝑥 , for 𝑖 = 1 , 2 , … . From (c), we have 𝐹 𝑔 ( 𝑓 2 ⋯ 𝑓 𝑛 𝑥 ) , 𝑔 ( 𝑥 ) ( 𝑡 ) ≥ 1 − 𝑎 ( 𝑡 ) 1 − 𝐹 𝑓 1 𝑓 2 ⋯ 𝑓 𝑛 ( 𝑓 2 ⋯ 𝑓 𝑛 𝑥 ) , 𝑓 1 𝑓 2 ⋯ 𝑓 𝑛 ( 𝑥 ) ( 𝑡 ) . ( 2 . 1 9 ) By (d), we get 𝐹 𝑓 2 ⋯ 𝑓 𝑛 ( 𝑥 ) , 𝑥 ( 𝑡 ) ≥ 1 − 𝑎 ( 𝑡 ) 1 − 𝐹 𝑓 2 ⋯ 𝑓 𝑛 ( 𝑥 ) , 𝑥 ( 𝑡 ) ( 2 . 2 0 ) Hence, 𝑓 2 ⋯ 𝑓 𝑛 ( 𝑥 ) = 𝑥 . Thus , 𝑓 1 ( 𝑥 ) = 𝑓 1 𝑓 2 ⋯ 𝑓 𝑛 ( 𝑥 ) = 𝑥 .
Similarly, we have 𝑓 2 ( 𝑥 ) = ⋯ 𝑓 𝑛 ( 𝑥 ) = 𝑥 .

Corollary 2.5. Let ( 𝑋 , ℱ , 𝑇 ) be a complete PQM space and let 𝑓 , 𝑔 , ℎ ∶ 𝑋 → 𝑋 satisfy conditions (a), (b), and (c) of Theorem 2.2. If 𝑇 is a 𝑡 - n o r m of 𝐻 - t y p e then there exists a unique common fixed point for the mapping 𝑓 , 𝑔 , and ℎ .

Proof. By Proposition 1.6 all the conditions of the Theorem 2.2 are satisfied.

Corollary 2.6. Let ( 𝑋 , ℱ , 𝑇 𝐷 𝜆 ) for some 𝜆 > 0 be a complete PQM space and let 𝑓 , 𝑔 , ℎ ∶ 𝑋 → 𝑋 satisfy conditions (a), (b), and (c) of Theorem 2.2. If ∑ ∞ 𝑖 = 1 0 𝑥 0 2 0 0 𝑑 ( 𝑎 𝑖 ( 𝑡 ) ) 𝜆 < ∞ then there exists a unique common fixed point for the mapping 𝑓 , 𝑔 , and ℎ .

Proof. From equivalence (1.10) we have ∞ 𝑖 = 1 𝑎 0 𝑥 0 2 0 0 𝑑 𝑖 ( 𝑡 ) 𝜆 < ∞ ⟺ l i m 𝑛 → ∞ ( 𝑇 𝐷 𝜆 ) ∞ 𝑖 = 𝑛 1 − 𝑎 𝑖 ( 𝑡 ) = 1 . ( 2 . 2 1 )

Corollary 2.7. Let ( 𝑋 , ℱ , 𝑇 𝜆 𝐴 𝐴 ) for some 𝜆 > 0 be a complete PQM space and let 𝑓 , 𝑔 , ℎ ∶ 𝑋 → 𝑋 satisfy conditions (a), (b), and (c) of Theorem 2.2. If ∑ ∞ 𝑖 = 1 0 𝑥 0 2 0 0 𝑑 ( 𝑎 𝑖 ( 𝑡 ) ) 𝜆 < ∞ then there exists a unique common fixed point for the mapping 𝑓 , 𝑔 , and ℎ .