Common fixed point theorem for a sequence of Mappings in G-metric spaces

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[Volume 2 issue 5 May 2014]

Page No.403-407 ISSN :2320-7167 INTERNATIONAL JOURNAL OF MATHEMATICS AND COMPUTER RESEARCH

Common fixed point theorem for a sequence of Mappings in G- metric spaces

Gopal Meena¹ , Dharmendra Nema²

Assistant Professor¹ , Department of Applied Mathematics Jabalpur Engg. College Jabalpur (M.P.), India

[email protected] PG Student² ,Department of Applied Mathematics

Jabalpur Engg. College Jabalpur(M.P.), India

Abstract:

In the present paper we have established a Common fixed point theorem for a sequence of mappings on a closed subset of Complete 𝐺𝐺-metric spaces. Also we introduce a example to support the usability of our result.

Kew words : Common fixed point , 𝐺𝐺-metric space 1. Introduction .

Throughout this paper , unless otherwise stated 𝑆𝑆 is a closed subset of Complete 𝐺𝐺 -metric space (𝑋𝑋, 𝐺𝐺). In 1991 Koparde and Waghmode[3] have proved common fixed point theorem for the sequence {𝑇𝑇_𝑛𝑛} of mappings . After that Pendhare and Waghmode[6], Veerapandi and kumar[9], Badshah and Meena[1] , proved many results for sequence of mappings in Hilbert spaces.

In 2005, Mustafa and Sims introduced a new class of generalized metric spaces(see[4,5]),which are called 𝐺𝐺 -metric space, as generalization of a metric space (𝑋𝑋, 𝑑𝑑). Subsequently, many fixed point results on such spaces appeared (see, for example[7,8]).

Here we present the necessary definitions and results in 𝐺𝐺-metric spaces , which will be useful for the rest of the paper.

Definition1.1. Let 𝑋𝑋 be a nonempty set. Suppose that 𝐺𝐺 ∶ 𝑋𝑋 × 𝑋𝑋 × 𝑋𝑋 → 𝑅𝑅+ is a function satisfying the following conditions :

(1) 𝐺𝐺(𝑥𝑥, 𝑦𝑦, 𝑧𝑧) = 0 if and only if 𝑥𝑥 = 𝑦𝑦 = 𝑧𝑧 ; (2) 0 < 𝐺𝐺(𝑥𝑥, 𝑥𝑥, 𝑦𝑦) for all 𝑥𝑥, 𝑦𝑦 𝜖𝜖 𝑋𝑋 with 𝑥𝑥 ≠ 𝑦𝑦 ;

(3) 𝐺𝐺(𝑥𝑥, 𝑥𝑥, 𝑦𝑦) ≤ 𝐺𝐺(𝑥𝑥, 𝑦𝑦, 𝑧𝑧) for all 𝑥𝑥, 𝑦𝑦 , 𝑧𝑧 𝜖𝜖 𝑋𝑋 with 𝑦𝑦 ≠ 𝑧𝑧 ;

(4) 𝐺𝐺(𝑥𝑥, 𝑦𝑦, 𝑧𝑧) = 𝐺𝐺(𝑥𝑥, 𝑧𝑧, 𝑦𝑦) = 𝐺𝐺(𝑦𝑦, 𝑧𝑧, 𝑥𝑥) = ... (symmetry in all three variables) ;

IJMCR www.ijmcr.in| 2:5 |May|2014|403-407 | 403

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(5) 𝐺𝐺(𝑥𝑥, 𝑦𝑦, 𝑧𝑧) ≤ 𝐺𝐺(𝑥𝑥, 𝑎𝑎, 𝑎𝑎) + 𝐺𝐺(𝑎𝑎, 𝑦𝑦, 𝑧𝑧) for all 𝑥𝑥, 𝑦𝑦 , 𝑧𝑧, 𝑎𝑎 𝜖𝜖 𝑋𝑋 .

Then 𝐺𝐺 is called a 𝐺𝐺-metric on 𝑋𝑋 and (𝑋𝑋, 𝐺𝐺) is called a 𝐺𝐺-metric space.

Definition1.2.A 𝐺𝐺-metric space (𝑋𝑋, 𝐺𝐺) is said to be symmetric if 𝐺𝐺(𝑥𝑥, 𝑦𝑦, 𝑦𝑦) = 𝐺𝐺(𝑦𝑦, 𝑥𝑥, 𝑥𝑥) for all 𝑥𝑥, 𝑦𝑦 𝜖𝜖 𝑋𝑋.

Definition 1.3 Let (𝑋𝑋, 𝐺𝐺) be a 𝐺𝐺-metric space. We say that {𝑥𝑥𝑛𝑛} is

(1) a 𝐺𝐺-cauchy sequence if, for any 𝜀𝜀 > 0, there is 𝑀𝑀𝜖𝜖 𝑁𝑁 (the set of all positive integers) such that for all 𝑛𝑛, 𝑚𝑚, 𝑙𝑙 ≥ 𝑀𝑀 , 𝐺𝐺(𝑥𝑥_𝑛𝑛, 𝑥𝑥_𝑚𝑚 , 𝑥𝑥_𝑙𝑙) < 𝜀𝜀 .

(2) a 𝐺𝐺-convergent sequence to 𝑥𝑥𝜖𝜖 𝑋𝑋 if, for any 𝜀𝜀 > 0 ,there is 𝑀𝑀𝜖𝜖 𝑁𝑁 (the set of all positive integers) such that for all 𝑛𝑛, 𝑚𝑚 ≥ 𝑀𝑀 , 𝐺𝐺(𝑥𝑥, 𝑥𝑥𝑛𝑛 , 𝑥𝑥𝑚𝑚) < 𝜀𝜀 .

A 𝐺𝐺-metric space (𝑋𝑋, 𝐺𝐺) is said to be complete if every 𝐺𝐺-Cauchy sequence in 𝑋𝑋 is 𝐺𝐺-convergent in 𝑋𝑋.

Proposition 1.1 Let (𝑋𝑋, 𝐺𝐺) be a 𝐺𝐺-metric space . The following are equivalent : (1) {𝑥𝑥𝑛𝑛} is 𝐺𝐺-convergent to 𝑥𝑥;

(2) 𝐺𝐺(𝑥𝑥_𝑛𝑛, 𝑥𝑥_𝑛𝑛 , 𝑥𝑥) → 0 as 𝑛𝑛 → +∞ ; (3) 𝐺𝐺(𝑥𝑥_𝑛𝑛, 𝑥𝑥 , 𝑥𝑥) → 0 as 𝑛𝑛 → +∞ ; (4) 𝐺𝐺(𝑥𝑥_𝑛𝑛, 𝑥𝑥_𝑚𝑚 , 𝑥𝑥) → 0 as 𝑛𝑛 , 𝑚𝑚 → +∞ .

Proposition 1.2 Let (𝑋𝑋, 𝐺𝐺) be a 𝐺𝐺-metric space . The following are equivalent : (1) the sequence {𝑥𝑥_𝑛𝑛} is 𝐺𝐺-Cauchy ;

(2) 𝐺𝐺(𝑥𝑥_𝑛𝑛, 𝑥𝑥_𝑚𝑚 , 𝑥𝑥_𝑚𝑚) → 0 as 𝑛𝑛, 𝑚𝑚 → +∞ .

In this paper we illustrate a common fixed point theorem for sequence of mappings in complete 𝐺𝐺-metric spaces also give an example to support the usability of our result. Our result is a generalization of the results in Hilbert spaces to 𝐺𝐺-metric spaces . To present this work we also see Jaleli and Samet[2].

2.Main Result.

Theorem 2.1. Let 𝑆𝑆 be a closed subset of Complete 𝐺𝐺-metric space (𝑋𝑋, 𝐺𝐺) and {𝑇𝑇𝑛𝑛}: S → S be a sequence of mappings which satisfies the condition :

for all 𝑥𝑥, 𝑦𝑦, 𝑧𝑧 𝜖𝜖 𝑆𝑆

𝐺𝐺�𝑇𝑇𝑖𝑖𝑥𝑥 , 𝑇𝑇𝑗𝑗𝑦𝑦 , 𝑇𝑇𝑘𝑘𝑧𝑧� ≤ 𝑎𝑎𝐺𝐺(𝑥𝑥 , 𝑇𝑇𝑖𝑖𝑥𝑥 , 𝑇𝑇𝑖𝑖𝑥𝑥) + 𝑏𝑏𝐺𝐺�𝑦𝑦 , 𝑇𝑇𝑗𝑗𝑦𝑦 , 𝑇𝑇𝑗𝑗𝑦𝑦� + 𝑐𝑐𝐺𝐺(𝑧𝑧 , 𝑇𝑇𝑘𝑘𝑧𝑧 , 𝑇𝑇𝑘𝑘𝑧𝑧) + 𝑑𝑑𝐺𝐺(𝑥𝑥, 𝑦𝑦, 𝑧𝑧)

where 𝑎𝑎, 𝑏𝑏, 𝑐𝑐, 𝑑𝑑 are positive Constants such that 𝑎𝑎 + 𝑏𝑏 + 𝑐𝑐 + 𝑑𝑑 < 1. Then {𝑇𝑇_𝑛𝑛} has a unique Common fixed point.

Proof. Let 𝑥𝑥₀ 𝜖𝜖 𝑆𝑆 be any arbitrary point . Defined a Sequence {𝑥𝑥_𝑛𝑛} in 𝑆𝑆 as 𝑥𝑥_𝑛𝑛+1 = 𝑇𝑇_𝑛𝑛+1𝑥𝑥_𝑛𝑛 for 𝑛𝑛 = 0,1,2,3 …

Now we shall prove that {𝑥𝑥𝑛𝑛} is a Cauchy sequence , so consider

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𝐺𝐺(𝑥𝑥_𝑛𝑛, 𝑥𝑥_𝑛𝑛+1 , 𝑥𝑥_𝑛𝑛+2) = 𝐺𝐺(𝑇𝑇_𝑛𝑛𝑥𝑥_𝑛𝑛−1, 𝑇𝑇_𝑛𝑛+1𝑥𝑥_𝑛𝑛 , 𝑇𝑇_𝑛𝑛+2𝑥𝑥_𝑛𝑛+1)

≤ 𝑎𝑎𝐺𝐺(𝑥𝑥_𝑛𝑛−1, 𝑇𝑇_𝑛𝑛𝑥𝑥_𝑛𝑛−1 , 𝑇𝑇_𝑛𝑛𝑥𝑥_𝑛𝑛−1) + 𝑏𝑏𝐺𝐺(𝑥𝑥𝑛𝑛 , 𝑇𝑇_𝑛𝑛+1𝑥𝑥_𝑛𝑛 , 𝑇𝑇_𝑛𝑛+1𝑥𝑥_𝑛𝑛) + 𝑐𝑐𝐺𝐺(𝑥𝑥𝑛𝑛+1 , 𝑇𝑇_𝑛𝑛+2𝑥𝑥_𝑛𝑛+1 , 𝑇𝑇_𝑛𝑛+2𝑥𝑥_𝑛𝑛+1) + 𝑑𝑑𝐺𝐺(𝑥𝑥𝑛𝑛−1 , 𝑥𝑥𝑛𝑛 , 𝑥𝑥𝑛𝑛+1)

≤ 𝑎𝑎𝐺𝐺(𝑥𝑥𝑛𝑛−1 , 𝑥𝑥𝑛𝑛 , 𝑥𝑥𝑛𝑛) + 𝑏𝑏𝐺𝐺(𝑥𝑥𝑛𝑛 , 𝑥𝑥𝑛𝑛+1 , 𝑥𝑥𝑛𝑛+1) +𝑐𝑐𝐺𝐺(𝑥𝑥_𝑛𝑛+1, 𝑥𝑥_𝑛𝑛+2 , 𝑥𝑥_𝑛𝑛+2) + 𝑑𝑑𝐺𝐺(𝑥𝑥_𝑛𝑛−1, 𝑥𝑥_𝑛𝑛 , 𝑥𝑥_𝑛𝑛+1) ≤ 𝑎𝑎𝐺𝐺(𝑥𝑥_𝑛𝑛−1, 𝑥𝑥_𝑛𝑛 , 𝑥𝑥_𝑛𝑛+1) + 𝑏𝑏𝐺𝐺(𝑥𝑥𝑛𝑛 , 𝑥𝑥_𝑛𝑛+1 , 𝑥𝑥_𝑛𝑛+2) +𝑐𝑐𝐺𝐺(𝑥𝑥𝑛𝑛 , 𝑥𝑥𝑛𝑛+1 , 𝑥𝑥𝑛𝑛+2) + 𝑑𝑑𝐺𝐺(𝑥𝑥𝑛𝑛−1 , 𝑥𝑥𝑛𝑛 , 𝑥𝑥𝑛𝑛+1)

≤ (𝑎𝑎 + 𝑑𝑑)𝐺𝐺(𝑥𝑥𝑛𝑛−1 , 𝑥𝑥𝑛𝑛 , 𝑥𝑥𝑛𝑛+1) + (𝑏𝑏 + 𝑐𝑐)𝐺𝐺(𝑥𝑥𝑛𝑛 , 𝑥𝑥𝑛𝑛+1 , 𝑥𝑥𝑛𝑛+2) 𝐺𝐺(𝑥𝑥𝑛𝑛 , 𝑥𝑥𝑛𝑛+1 , 𝑥𝑥𝑛𝑛+2) ≤_{1−(𝑏𝑏+𝑐𝑐)}^{(𝑎𝑎+𝑑𝑑)} 𝐺𝐺(𝑥𝑥𝑛𝑛−1 , 𝑥𝑥𝑛𝑛 , 𝑥𝑥𝑛𝑛+1)

i.e 𝐺𝐺(𝑥𝑥_𝑛𝑛, 𝑥𝑥_𝑛𝑛+1 , 𝑥𝑥_𝑛𝑛+2) ≤ 𝑘𝑘𝐺𝐺(𝑥𝑥𝑛𝑛−1 , 𝑥𝑥_𝑛𝑛 , 𝑥𝑥_𝑛𝑛+1) where 𝑘𝑘 =_{1−(𝑏𝑏+𝑐𝑐)}^{(𝑎𝑎+𝑑𝑑)} < 1 .

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𝐺𝐺(𝑥𝑥_𝑛𝑛, 𝑥𝑥_𝑛𝑛+1 , 𝑥𝑥_𝑛𝑛+2) ≤ 𝑘𝑘^𝑛𝑛𝐺𝐺(𝑥𝑥₀, 𝑥𝑥₁ , 𝑥𝑥₂) for all n Now for any positive integers 𝑙𝑙 ≥ 𝑚𝑚 ≥ 𝑛𝑛 ≥ 1. We Consider

𝐺𝐺(𝑥𝑥_𝑛𝑛, 𝑥𝑥_𝑚𝑚 , 𝑥𝑥_𝑙𝑙) ≤ 𝐺𝐺(𝑥𝑥𝑛𝑛 , 𝑥𝑥_𝑛𝑛+1 , 𝑥𝑥_𝑛𝑛+2) + 𝐺𝐺(𝑥𝑥𝑛𝑛+1 , 𝑥𝑥_𝑛𝑛+2 , 𝑥𝑥_𝑛𝑛+3) + ⋯ + 𝐺𝐺(𝑥𝑥𝑚𝑚−1 , 𝑥𝑥_𝑚𝑚 , 𝑥𝑥_𝑚𝑚+1) + +𝐺𝐺(𝑥𝑥𝑚𝑚, 𝑥𝑥𝑚𝑚+1 , 𝑥𝑥𝑚𝑚+2) + 𝐺𝐺(𝑥𝑥𝑚𝑚+1 , 𝑥𝑥𝑚𝑚+2 , 𝑥𝑥𝑚𝑚+3) + ⋯ + 𝐺𝐺(𝑥𝑥𝑙𝑙−2 , 𝑥𝑥𝑙𝑙−1 , 𝑥𝑥𝑙𝑙) ≤ 𝑘𝑘^𝑛𝑛𝐺𝐺(𝑥𝑥₀, 𝑥𝑥₁ , 𝑥𝑥₂) + 𝑘𝑘^𝑛𝑛+1𝐺𝐺(𝑥𝑥₀, 𝑥𝑥₁ , 𝑥𝑥₂) + … + 𝑘𝑘^𝑚𝑚−1𝐺𝐺(𝑥𝑥₀, 𝑥𝑥₁ , 𝑥𝑥₂) +

+ 𝑘𝑘^𝑚𝑚𝐺𝐺(𝑥𝑥₀, 𝑥𝑥₁ , 𝑥𝑥₂) + 𝑘𝑘^𝑚𝑚+1𝐺𝐺(𝑥𝑥₀, 𝑥𝑥₁ , 𝑥𝑥₂) + … + 𝑘𝑘^𝑙𝑙−2𝐺𝐺(𝑥𝑥₀, 𝑥𝑥₁ , 𝑥𝑥₂) 𝐺𝐺(𝑥𝑥_𝑛𝑛, 𝑥𝑥_𝑚𝑚 , 𝑥𝑥_𝑙𝑙) ≤_1−𝑘𝑘^𝑘𝑘^𝑛𝑛 𝐺𝐺(𝑥𝑥₀, 𝑥𝑥₁ , 𝑥𝑥₂) → 0 𝑎𝑎𝑎𝑎 𝑛𝑛 → ∞.

i.e. {𝑥𝑥𝑛𝑛} is a 𝐺𝐺-Cauchy sequence . Since 𝑆𝑆 is a closed subset of Complete 𝐺𝐺-metric space X , so {𝑥𝑥𝑛𝑛} converges to a point 𝑢𝑢 in 𝑆𝑆.

Now we shall prove that 𝑢𝑢 is a common fixed point of the sequence {𝑇𝑇_𝑛𝑛} of mappings from 𝑆𝑆 into itself. Let 𝑇𝑇𝑛𝑛𝑢𝑢 ≠ 𝑢𝑢 for all n , and Consider,

𝐺𝐺(𝑢𝑢, 𝑇𝑇_𝑛𝑛𝑢𝑢 , 𝑇𝑇_𝑚𝑚𝑢𝑢) ≤ 𝐺𝐺(𝑢𝑢, 𝑥𝑥_𝑛𝑛−1 , 𝑥𝑥_𝑛𝑛−1) + 𝐺𝐺(𝑥𝑥_𝑛𝑛−1, 𝑇𝑇_𝑛𝑛𝑢𝑢 , 𝑇𝑇_𝑚𝑚𝑢𝑢) ≤ 𝐺𝐺(𝑢𝑢, 𝑥𝑥_𝑛𝑛−1 , 𝑥𝑥_𝑛𝑛−1) + 𝐺𝐺(𝑇𝑇_𝑛𝑛−1𝑥𝑥_𝑛𝑛−2, 𝑇𝑇_𝑛𝑛𝑢𝑢 , 𝑇𝑇_𝑚𝑚𝑢𝑢)

≤ 𝐺𝐺(𝑢𝑢, 𝑥𝑥𝑛𝑛−1 , 𝑥𝑥𝑛𝑛−1) + 𝑎𝑎𝐺𝐺(𝑥𝑥𝑛𝑛−2 , 𝑇𝑇𝑛𝑛−1𝑥𝑥𝑛𝑛−2, 𝑇𝑇𝑛𝑛−1𝑥𝑥𝑛𝑛−2) + 𝑏𝑏𝐺𝐺(𝑢𝑢, 𝑇𝑇𝑛𝑛𝑢𝑢 , 𝑇𝑇𝑛𝑛𝑢𝑢) +𝑐𝑐𝐺𝐺(𝑢𝑢, 𝑇𝑇𝑚𝑚𝑢𝑢 , 𝑇𝑇𝑚𝑚𝑢𝑢) + 𝑑𝑑𝐺𝐺(𝑥𝑥𝑛𝑛−2, 𝑢𝑢 , 𝑢𝑢)

≤ 𝐺𝐺(𝑢𝑢, 𝑥𝑥_𝑛𝑛−1 , 𝑥𝑥_𝑛𝑛−1) + 𝑎𝑎𝐺𝐺(𝑥𝑥_𝑛𝑛−2, 𝑥𝑥_𝑛𝑛−1, 𝑥𝑥_𝑛𝑛−1) + 𝑏𝑏𝐺𝐺(𝑢𝑢, 𝑇𝑇_𝑚𝑚𝑢𝑢 , 𝑇𝑇_𝑛𝑛𝑢𝑢) +𝑐𝑐𝐺𝐺(𝑢𝑢, 𝑇𝑇_𝑛𝑛𝑢𝑢 , 𝑇𝑇_𝑚𝑚𝑢𝑢) + 𝑑𝑑𝐺𝐺(𝑥𝑥_𝑛𝑛−2, 𝑢𝑢 , 𝑢𝑢)

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𝐺𝐺(𝑢𝑢, 𝑇𝑇_𝑛𝑛𝑢𝑢 , 𝑇𝑇_𝑚𝑚𝑢𝑢) ≤ (𝑏𝑏 + 𝑐𝑐)𝐺𝐺(𝑢𝑢, 𝑇𝑇_𝑛𝑛𝑢𝑢 , 𝑇𝑇_𝑚𝑚𝑢𝑢)

i.e. 𝐺𝐺(𝑢𝑢, 𝑇𝑇_𝑛𝑛𝑢𝑢 , 𝑇𝑇_𝑚𝑚𝑢𝑢) < 𝐺𝐺(𝑢𝑢, 𝑇𝑇_𝑛𝑛𝑢𝑢 , 𝑇𝑇_𝑚𝑚𝑢𝑢) , which is a contradiction . Thus 𝑇𝑇_𝑛𝑛𝑢𝑢 = 𝑇𝑇_𝑚𝑚𝑢𝑢 = 𝑢𝑢 , for all 𝑚𝑚, 𝑛𝑛 .

Hence 𝑢𝑢 is a Common fixed point of the sequence {𝑇𝑇𝑛𝑛} of mappings.

Uniqueness:

Suppose 𝑢𝑢 ≠ 𝑣𝑣 such that 𝑇𝑇𝑛𝑛𝑣𝑣 = 𝑣𝑣 for all n . Consider,

𝐺𝐺(𝑢𝑢, 𝑢𝑢 , 𝑣𝑣) = 𝐺𝐺�𝑇𝑇_𝑛𝑛𝑢𝑢 , 𝑇𝑇_𝑚𝑚𝑢𝑢 , 𝑇𝑇_𝑝𝑝𝑣𝑣�

≤ 𝑎𝑎𝐺𝐺(𝑢𝑢 , 𝑇𝑇_𝑛𝑛𝑢𝑢 , 𝑇𝑇_𝑛𝑛𝑢𝑢 ) + 𝑏𝑏𝐺𝐺(𝑢𝑢 , 𝑇𝑇_𝑚𝑚𝑢𝑢 , 𝑇𝑇_𝑚𝑚𝑢𝑢 ) + 𝑐𝑐𝐺𝐺�𝑣𝑣 , 𝑇𝑇_𝑝𝑝𝑣𝑣 , 𝑇𝑇_𝑝𝑝𝑣𝑣 � + 𝑑𝑑𝐺𝐺(𝑢𝑢 , 𝑢𝑢 , 𝑣𝑣 ) i.e. 𝐺𝐺(𝑢𝑢, 𝑢𝑢 , 𝑣𝑣) < 𝐺𝐺(𝑢𝑢, 𝑢𝑢 , 𝑣𝑣) , which is again a contradiction .

Thus 𝑢𝑢 = 𝑣𝑣 . Hence 𝑢𝑢 is a unique common fixed point of sequence {𝑇𝑇_𝑛𝑛} of mappings.

Example.2.1. Let 𝑋𝑋 = [0,1] , and 𝐺𝐺 = 𝑋𝑋 × 𝑋𝑋 × 𝑋𝑋 → 𝑅𝑅₊ be defined by :

𝐺𝐺(𝑥𝑥, 𝑦𝑦, 𝑧𝑧) ={

Then (𝑋𝑋, 𝐺𝐺) is a Complete 𝐺𝐺-metric space . Let {𝑇𝑇_𝑛𝑛}: 𝑋𝑋 → 𝑋𝑋 be a sequence of mappings defined by 𝑇𝑇_𝑛𝑛𝑥𝑥 =₄^𝑥𝑥_𝑛𝑛 for all 𝑥𝑥 𝜖𝜖 𝑋𝑋 , then {𝑇𝑇_𝑛𝑛} satisfies the inequality of theorem 2.1. Hence the sequence {𝑇𝑇_𝑛𝑛}of mappings has a unique common fixed point in 𝑋𝑋.

3. References.

[1] Badshah V.H. and Meena G., Common Fixed Point theorem of an infinite sequence of mappings, Chh. J. Sci. Tech. Vol. 2(2005), 87- 90.

[2] Jleli Mohamed and Samet Bessem.,Remarks on G-metric spaces and fixed point theorems, Fixed Point Theory and Application..2012 doi: 10.1186/1687-1812-2012-210.

[3] Koparde P.V. and Waghmode B.B., On sequence of mappings in Hilbert space, The Mathematics Education , XXV (1991), 197.

[4] Mustafa Z., A new structure for generalized metric spaces-with applications to fixed point theory.

PhD thesis, the University of Newcastle, Australia (2005).

[5] Mustafa Z. Sims B., A new approach to generalized metric spaces, J. Nonlinear Convex Anal. 7(2), (2006), 289-297.

[6] Pendhare D.M. and Waghmode B.B., On sequence of mappings in Hilbert space, The Mathematics Education , XXXII (1998), 61.

[7] Shatanawi W., Fixed point theory for contractive mappings satiafying Ф-maps in G-metric spaces.

Fixed Point Theory Appl. 2010, Article ID 181650 (2010).

[8] Shatanawi W., Some fixed point theorems in ordered G-metric spaces and applications. Abstr. Appl.

Anal. 2011, Article ID 126205 (2011).

[9] Veerapandi T. and Kumar Anil S., Common Fixed Point theorems of a sequence of mappings in Max{ 𝑥𝑥, 𝑦𝑦, 𝑧𝑧} ,otherwise

0 , if 𝑥𝑥 = 𝑦𝑦 = 𝑧𝑧

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Hilbert space, Bull. Cal. Math. Soc. 91(4),(1999), 299-308.