Communications on Applied Nonlinear Analysis 

ISSN: 1074-133X 

Vol 32 No. 9s (2025) 

 

77 
https://internationalpubls.com 

Results on Fixed Points in Vector Valued 𝑮-Metric Spaces 

Pooja Yadav1, Mamta Kamra2 

[1, 2] "Department of Mathematics, Indira Gandhi University 

Meerpur(Rewari), Haryana-122502, India." 

Article History: 

Received: 12-11-2024 

Revised:24-12-2024 

Accepted:09-01-2025 

Abstract:  

Introduction: Fixed point theory furnishes a robust framework in mathematics 

for examining the function and system behavior by centering on fixed points. 

Kirk et al. brought forth the concept of cyclic transformation in 2003. Cyclic 

contraction broadens the standard contraction transformations in metric space. 

By relaxing the requirement of the triangle inequality, 𝐺-metric space extends 

the scope of metric spaces. Vector 𝐺-metric space is GMS in which the metric is 

lattice valued. 

Objectives: We give the concept of a vector 𝐺 metric space and define some 

related concepts. We establish some fixed point results on vector 𝐺-metric space 

by using cyclic contraction. To underscore the practical implications and 

importance of our findings, we offer a range of examples and corollaries.  

Methods: This research paper outlines fixed point theorems for self-

transformations in vector 𝐺-metric space, with the help of cyclic contraction. 

Conclusions: Within this manuscript, we illustrate fixed point theorems for self-

transformations in V-complete vector 𝐺-metric space by utilizing cyclic 

contraction. These outcomes are expected to inspire researchers to explore 

problem-solving opportunities in diverse areas such as differential equations and 

functional analysis. 

      Keywords: Cyclic contraction, Vector 𝐺-metric space, Vector lattice. 

     Subject Classification:(2010) 47H10, 47H07  

 

1. Introduction.  

Fixed point theory furnishes a robust framework in mathematics for examining the 

function and system behavior by centering on fixed points. Kirk et al. 

Error! Reference source not found. brought forth the concept of cyclic transformation in 

2003. Cyclic contraction(CC) broadens the standard contraction transformations in metric 

space. It is utilized for exploring transformations and fixed points.The exploration of fixed 

point theorems(FPT) for cyclic transformations have been widely studied (defined by 

Error! Reference source not found., Error! Reference source not found., 

Error! Reference source not found., Error! Reference source not found.). By relaxing the 

requirement of the triangle inequality, 𝐺-metric space(GMS) extends the scope of metric 

 
1 poojayadav.math.rs@igu.ac.in 

1 mkhaneja15@gmail.com 

 



Communications on Applied Nonlinear Analysis 

ISSN: 1074-133X 

Vol 32 No. 9s (2025) 

 

78 
https://internationalpubls.com 

spaces. Vector 𝐺-metric space(VGMS) is GMS in which the metric is lattice valued. This 

research paper outlines FPT for self-transformations in VGMS, with the help of CC. 

We lay out a series of definitions and examples that will be applicable in the forthcoming 

portion. For an additional comprehensive analysis of the results regarding the vector lattice 

and GMS, we may refer to (Error! Reference source not found., 

Error! Reference source not found., Error! Reference source not found., 

Error! Reference source not found., Error! Reference source not found., 

Error! Reference source not found., Error! Reference source not found., 

Error! Reference source not found., Error! Reference source not found., 

Error! Reference source not found., Error! Reference source not found., 

Error! Reference source not found., Error! Reference source not found.). 

Definition 1.1[5] Let ℜ be a non-null set and consider {𝜒ℏ}𝜇=1
𝑚  as a collection of non-void 

subsets of ℜ with 𝜇 =∪ℏ=1
𝑚 𝜒𝑗. A transformation ∁: 𝜇 → 𝜇 is termed a cyclic transformation if  

 ∁(𝜒ℏ) ⊆ 𝜒ℏ+1    ℏ = 1,2, … , 𝑚,    𝑤ℎ𝑒𝑟𝑒     𝜒𝑚+1 = 𝜒1. 

 

Definition1.2Error! Reference source not found. let ℜ be a non-null set and 𝑉 be a vector 

lattice, consider a function 𝐺: ℜ × ℜ × ℜ → 𝑉 that obeys the properties mentioned below:   

(i) 𝐺(♭1, ♭2, ♭3) = 0 if ♭1 = ♭2 = ♭3,  

(ii) 0 < 𝐺(♭1, ♭1, ♭2)       ∀   ♭1, ♭2 ∈ ℜ with ♭1 ≠ ♭2,  

(iii) 𝐺(♭1, ♭1, ♭2) ⪯ 𝐺(♭1, ♭2, ♭3)       ∀   ♭1, ♭2, ♭3 ∈ ℜ with ♭3 ≠ ♭2,  

(iv) 𝐺(♭1, ♭2, ♭3) = 𝐺(♭1, ♭3, ♭2) = 𝐺(♭2, ♭3, ♭1) = ⋯      (tri-variable symmetry) 

 (𝑣)𝐺(♭1, ♭2, ♭3) ⪯ 𝐺(♭1, 𝜎, 𝜎) + 𝐺(𝜎, ♭2, ♭3)          ∀   ♭1, ♭2, ♭3, 𝜎 ∈ ℜ.  

The triplet (ℜ, 𝐺, 𝑉) is defined as vector 𝐺-metric space(VGMS). 

Definition 1.3Error! Reference source not found. A VGMS (ℜ, 𝐺, 𝑉) is called symmetric 

VGMS if 𝐺(𝜁, ♭, ♭) = 𝐺(♭, 𝜁, 𝜁)    ∀𝜁, ♭ ∈ ℜ. 

Example 1.4 Let ℜ be a non-null set and take 𝑉 as a vector lattice. Then the function 

𝐺: ℜ × ℜ × ℜ → 𝑉 is determined by  

 𝐺(𝜇, ♭, 𝜁) =
1

2𝑛 (|𝜇 − ♭|, |♭ − 𝜁|, |𝜁 − 𝜇|) 

∀𝜇, ♭, 𝜁 ∈ ℜ and ∀𝑛. Then (ℜ, 𝐺, 𝑉) is a VGMS on ℜ.  

 

Example 1.5 Let ℜ = [1, ∞), 𝑉 be a vector lattice, 𝐺1 and 𝐺2 be two vector 𝐺-metrics on ℜ. 

The transformation 𝐺: ℜ × ℜ × ℜ → 𝑉 is determined by  

 𝐺(𝜇, ♭, 𝜎) = (𝐺1(𝜇, ♭, 𝜎) + 𝐺2(𝜇, ♭, 𝜎))
1

2 

∀    𝜇, ♭, 𝜎 ∈ ℜ. Then (ℜ, 𝐺, 𝑉) is a VGMS on ℜ.  



Communications on Applied Nonlinear Analysis 

ISSN: 1074-133X 

Vol 32 No. 9s (2025) 

 

79 
https://internationalpubls.com 

 

Definition 1.6Error! Reference source not found. In a VGMS (ℜ, 𝐺, 𝑉), 〈𝜍𝑛〉 ∈ ℜ is called 𝑉 

convergent to some 𝜍 ∈ 𝑉 if ∃ 〈𝜇𝑛〉 ∈ 𝑉 satisfying 𝜇𝑛 ↓ 0 and 𝐺(𝜍, 𝜍𝑛, 𝜍𝑚) ≤ 𝜇𝑛 ∀𝑛, 𝑚 ∈ ℤ+. 

 

Definition 1.7Error! Reference source not found. In a VGMS (ℜ, 𝐺, 𝑉), 〈𝜍𝑛〉 ∈ ℜ is called 𝑉-

Cauchy sequence(VCS) if ∃ 〈𝜇𝑛〉 ∈ 𝑉 satisfying 𝜇𝑛 ↓ 0 and 𝐺(𝜍𝑛, 𝜍𝑚, 𝜍𝜇) ≤ 𝜇𝑛 holds for all 

𝑛, 𝑚, 𝜇 ∈ ℤ+ . 

 

Definition 1.8Error! Reference source not found. In a VGMS (ℜ, 𝐺, 𝑉), the term 𝑉-complete 

implies that every VCS in ℜ is 𝑉-convergence to a limit in ℜ. 

 

Definition 1.9[5] A cyclic transformation ∁ on 𝜇 is said to be cyclic contraction if ∃   0 ≤

𝜌 < 1 such that  

 𝐺(∁(𝜇), ∁(♭), ∁(♭)) ⪯ 𝜌𝐺(𝜇, ♭, ♭) 

∀𝜇 ∈ 𝜒𝑞 and ♭ ∈ 𝜒𝑞+1, 𝑞 = 1,2, … 𝑚. 

2  Main Results 

 

In this section, we establish some FPT and corollaries for self-transformations in VGMS 

(ℜ, 𝐺, 𝑉) by employing the methodology of CC. 

Theorem 2.1 Let (ℜ, 𝐺, 𝑉) be symmetric and 𝐺- complete VGMS and {𝜒𝑞}𝑞=1
𝑚  as a 

collection of non-empty subset of ℜ with 𝜇 =∪𝑞=1
𝑚 𝜒𝑞.Let ∁: 𝜇 → 𝜇 be cyclic transformation 

satisfying  

 ∁(𝜒𝑞) ⊆ 𝜒𝑞+1    𝑞 = 1,2, … , 𝑚,    𝑤ℎ𝑒𝑟𝑒     𝜒𝑚+1 = 𝜒1. 

Suppose that ∃ constant 𝜕1, 𝜕2, 𝜕3, 𝜕4, 𝜕5, 𝜕6 and 𝜕7 with 0 ≤ 𝜕1 + 𝜕2 + 𝜕3 + 𝜕4 + 𝜕5 + 𝜕6 +

𝜕7 < 1 such that the transformation ∁ satisfies  

𝐺(∁(𝜇), ∁(♭), ∁(𝑟)) ⪯ 𝜕1𝐺(𝜇, ♭, 𝑟) + 𝜕2𝐺(𝜇, ∁(𝜇), ∁(𝜇)) + 𝜕3𝐺(♭, ∁(♭), ∁(♭)) 

                         +𝜕4𝐺(𝑟, ∁(𝑟), ∁(𝑟)) + 𝜕5𝐺(𝜇, ∁(♭), ∁(♭)) + 𝜕6𝐺(♭, ∁(𝑟), ∁(𝑟)) 

                         +𝜕7𝐺(𝑟, ∁(𝜇), ∁(𝜇)) (1) 

 ∀𝜇 ∈ 𝜒𝑞 and ♭, 𝑟 ∈ 𝜒𝑞+1, 𝑞 = 1,2, … 𝑚. Then 𝜒 has FP in 𝜇 =∩𝑞=1
𝑚 𝜒𝑞 which is unique. 

Proof: We can write eq (1) as  

𝐺(∁(𝜇), ∁(♭), ∁(♭)) ⪯ 𝜕1𝐺(𝜇, ♭, ♭) + 𝜕2𝐺(𝜇, ∁(𝜇), ∁(𝜇)) + 𝜕3𝐺(♭, ∁(♭), ∁(♭)) 



Communications on Applied Nonlinear Analysis 

ISSN: 1074-133X 

Vol 32 No. 9s (2025) 

 

80 
https://internationalpubls.com 

                                 +𝜕4𝐺(♭, ∁(♭), ∁(♭)) + 𝜕5𝐺(𝜇, ∁(♭), ∁(♭)) +

                                                                 𝜕6𝐺(♭, ∁(♭), ∁(♭)) + 𝜕7𝐺(♭, ∁(𝜇), ∁(𝜇)) 

𝐺(∁(𝜇), ∁(♭), ∁(♭)) ⪯ 𝜕1𝐺(𝜇, ♭, ♭) + 𝜕2𝐺(𝜇, ∁(𝜇), ∁(𝜇)) + (𝜕3 + 𝜕4 + 𝜕6)𝐺(♭, ∁(♭), ∁(♭)) 

                                  +𝜕5𝐺(𝜇, ∁(♭), ∁(♭)) + 𝜕7𝐺(♭, ∁(𝜇), ∁(𝜇)) (2) 

 Interchanging the role of 𝜇 and ♭, we get  

𝐺(∁(♭), ∁(𝜇), ∁(𝜇)) ⪯ 𝜕1𝐺(♭, 𝜇, 𝜇) + 𝜕2𝐺(♭, ∁(♭), ∁(♭)) + (𝜕3 + 𝜕4 + 𝜕6)𝐺(𝜇, ∁(𝜇), ∁(𝜇)) 

                    +𝜕5𝐺(♭, ∁(𝜇), ∁(𝜇)) + 𝜕7𝐺(𝜇, ∁(♭), ∁(♭)) 

𝐺(∁(𝜇), ∁(♭), ∁(♭)) ⪯ 𝜕1𝐺(𝜇, ♭, ♭) + 𝜕2𝐺(♭, ∁(♭), ∁(♭)) + (𝜕3 + 𝜕4 + 𝜕6)𝐺(𝜇, ∁(𝜇), ∁(𝜇)) 

                +𝜕5𝐺(♭, ∁(𝜇), ∁(𝜇)) + 𝜕7𝐺(𝜇, ∁(♭), ∁(♭)) (3) 

 Adding eq(2) and eq (3), we get  

2𝐺(∁(𝜇), ∁(♭), ∁(♭)) ⪯ 2𝜕1𝐺(𝜇, ♭, ♭) + (𝜕2 + 𝜕3 + 𝜕4 + 𝜕6)𝐺(𝜇, ∁(𝜇), ∁(𝜇)) + 

                                  (𝜕2 + 𝜕3 + 𝜕4 + 𝜕6)𝐺(♭, ∁(♭), ∁(♭)) + (𝜕5 +

                                                               𝜕7)𝐺(𝜇, ∁(♭), ∁(♭)) + (𝜕5 + 𝜕7)𝐺(♭, ∁(𝜇), ∁(𝜇)) 

 

𝐺(∁(𝜇), ∁(♭), ∁(♭)) ⪯ 𝜕1𝐺(𝜇, ♭, ♭) +
𝜕2 + 𝜕3 + 𝜕4 + 𝜕6

2
[𝐺(𝜇, ∁(𝜇), ∁(𝜇)) + 

                                      𝐺(♭, ∁(♭), ∁(♭))] +
𝜕5+𝜕7

2
[𝐺(𝜇, ∁(♭), ∁(♭)) + 

                                  𝐺(♭, ∁(𝜇), ∁(𝜇))] (4) 

 Putting ♭ = ∁(𝜇) in eq (4)  

𝐺(∁(𝜇), ∁2(𝜇), ∁2(𝜇)) ⪯ 𝜕1𝐺(𝜇, ∁(𝜇), ∁(𝜇)) +
𝜕2 + 𝜕3 + 𝜕4 + 𝜕6

2
[𝐺(𝜇, ∁(𝜇), ∁(𝜇)) + 

                                 𝐺(∁(𝜇), ∁2(𝜇), ∁2(𝜇))] +
𝜕5+𝜕7

2
[𝐺(𝜇, ∁2(𝜇), ∁2(𝜇)) + 

                                       𝐺(∁(𝜇), ∁(𝜇), ∁(𝜇))] (5) 

 By using definition of VGMS, we get  

 𝐺(𝜇, ∁2(𝜇), ∁2(𝜇)) ⪯ 𝐺(𝜇, ∁(𝜇), ∁(𝜇)) + 𝐺(∁(𝜇), ∁2(𝜇), ∁2(𝜇)) (6) 

 Then eq(5) becomes  

                                      𝐺(∁(𝜇), ∁2(𝜇), ∁2(𝜇)) ⪯ 𝜕1𝐺(𝜇, ∁(𝜇), ∁(𝜇)) +
𝜕2 + 𝜕3 + 𝜕4 + 𝜕6

2
 

                                                              [𝐺(𝜇, ∁(𝜇), ∁(𝜇)) +

𝐺(∁(𝜇), ∁2(𝜇), ∁2(𝜇))] + 

                                                              
𝜕5+𝜕7

2
[𝐺(𝜇, ∁(𝜇), ∁(𝜇)) + 



Communications on Applied Nonlinear Analysis 

ISSN: 1074-133X 

Vol 32 No. 9s (2025) 

 

81 
https://internationalpubls.com 

                                                     𝐺(∁(𝜇), ∁2(𝜇), ∁2(𝜇))] 

(1 −
∑7

𝑖=2 𝜕𝑖

2
) 𝐺(∁(𝜇), ∁2(𝜇), ∁2(𝜇)) ⪯ (𝜕1 +

∑7
𝑖=2 𝜕𝑖

2
) 𝐺(𝜇, ∁(𝜇), ∁(𝜇)) 

                                       𝐺(∁(𝜇), ∁2(𝜇), ∁2(𝜇)) ⪯
  𝜕1+

∑7
𝑖=2 𝜕𝑖

2

1−
∑7

𝑖=2 𝜕𝑖
2

𝐺(𝜇, ∁(𝜇), ∁(𝜇)) (7) 

 Putting 𝑞 =
𝜕1+

∑7
𝑖=2 𝜕𝑖

2

1−
∑7

𝑖=2 𝜕𝑖
2

. Since 0 ≤ ∑7
𝑖=1 𝜕𝑖 < 1, so 0 ≤ 𝑞 < 1. Then eq(7) becomes  

 𝐺(∁(𝜇), ∁2(𝜇), ∁2(𝜇)) ⪯ 𝑞𝐺(𝜇, ∁𝜇, ∁𝜇) (8) 

 ∀𝜇 ∈ 𝜒𝑞. Further,  

 

 𝐺(∁𝜌(𝜇), ∁𝜌+1(𝜇), ∁𝜌+1(𝜇)) ⪯ 𝑞𝜌𝐺(𝜇, ∁𝜇, ∁𝜇)    ∀𝜌 ∈ ℕ    𝜇 ∈ 𝜒𝑞 . 

For 𝜌, 𝑗 ∈ ℕ and 𝜇 ∈ 𝜒𝑞  

𝐺(∁𝜌(𝜇), ∁𝜌+𝑗(𝜇), ∁𝜌+𝑗(𝜇)) ⪯ 𝐺(∁𝜌(𝜇), ∁𝜌+1(𝜇), ∁𝜌+1(𝜇)) + 

                                                                 𝐺(∁𝜌+1(𝜇), ∁𝜌+𝑗(𝜇), ∁𝜌+𝑗(𝜇)) 

                                                            ⪯ 𝐺(∁𝜌(𝜇), ∁𝜌+1(𝜇), ∁𝜌+1(𝜇)) + 

                                                        𝐺(∁𝜌+1(𝜇), ∁𝜌+2(𝜇), ∁𝜌+2(𝜇)) 

                                                       +𝐺(∁𝜌+2(𝜇), ∁𝜌+𝑗(𝜇), ∁𝜌+𝑗(𝜇) 

                                                                              ⋮ 

  

                                                                     ⪯ ∑𝑗
𝑖=1 𝐺(∁𝜌+𝑖−1(𝜇), ∁𝜌+𝑖(𝜇), ∁𝜌+𝑖(𝜇)) 

                                                            ⪯ ∑𝑗
𝑖=1 𝑞𝜌+𝑖−1𝐺(𝜇, ∁𝜇, ∁𝜇) 

       𝐺(∁𝜌(𝜇), ∁𝜌+𝑗(𝜇), ∁𝜌+𝑗(𝜇)) ⪯
𝑞𝜌

1−𝑞
𝐺(𝜇, ∁𝜇, ∁𝜇) (9) 

This implies 〈∁𝜌𝜇〉 is a VCS in G-complete VGMS and it converges to 𝜇1 ∈ ℜ. So, ∃ a 

sequence 𝜇𝜌 ∈ 𝑉 such that 𝜇𝜌 ↓ 0 and 𝐺(𝜇1, ∁𝜌(𝜇), ∁𝜌(𝜇)) ⪯ 𝜇𝜌. 

Now  

𝐺(𝜇1, ∁((𝜇1)), ∁(𝜇1)) ⪯ 𝐺(𝜇1, ∁𝜌+1(𝜇), ∁𝜌+1(𝜇)) + 𝐺(∁𝜌+1(𝜇), ∁(𝜇1), ∁(𝜇1)) 

                                   ⪯ 𝐺(𝜇1, ∁𝜌+1(𝜇), ∁𝜌+1(𝜇)) + 𝐺(∁(𝜇1), ∁𝜌+1(𝜇), ∁𝜌+1(𝜇)) 

 By using (4), we get  

𝐺(𝜇1, ∁(𝜇1), ∁(𝜇1)) ⪯ 𝐺(𝜇1, ∁𝜌+1(𝜇), ∁𝜌+1(𝜇)) + 𝜕1𝐺(𝜇1, ∁𝜌(𝜇), ∁𝜌(𝜇)) + 



Communications on Applied Nonlinear Analysis 

ISSN: 1074-133X 

Vol 32 No. 9s (2025) 

 

82 
https://internationalpubls.com 

                                      
𝜕2+𝜕3+𝜕4+𝜕6

2
[𝐺(𝜇1, ∁𝜇1, ∁𝜇1) +

𝐺(∁𝜌(𝜇), ∁𝜌+1(𝜇), ∁𝜌+1(𝜇))] 

                                 +
𝜕5+𝜕7

2
[𝐺(𝜇1, ∁𝜌+1(𝜇), ∁𝜌+1(𝜇)) + 𝐺(∁𝜌(𝜇), ∁𝜇1, ∁𝜇1)] 

                              ⪯ 𝐺(𝜇1, ∁𝜌+1(𝜇), ∁𝜌+1(𝜇)) + 𝜕1𝐺(𝜇1, ∁𝜌(𝜇), ∁𝜌(𝜇)) + 

                                   
𝜕2+𝜕3+𝜕4+𝜕6

2
[𝐺(𝜇1, ∁𝜇1, ∁𝜇1) + 𝐺(∁𝜌(𝜇), 𝜇1, 𝜇1) + 

                                 𝐺(𝜇1, ∁𝜌+1(𝜇), ∁𝜌+1(𝜇))] +
𝜕5+𝜕7

2
[𝐺(𝜇1, ∁𝜌+1(𝜇), ∁𝜌+1(𝜇)) + 

                                     𝐺(∁𝜌(𝜇), 𝜇1, 𝜇1) + 𝐺(𝜇1, ∁𝜇1, ∁𝜇1)] 

  

 𝐺(𝜇1, ∁(𝜇1), ∁(𝜇1)) ⪯ 𝐺(𝜇1, ∁𝜌+1(𝜇), ∁𝜌+1(𝜇)) + 𝜕1𝐺(𝜇1, ∁𝜌(𝜇), ∁𝜌(𝜇)) + 

                                             
𝜕2+𝜕3+𝜕4+𝜕6

2
[𝐺(𝜇1, ∁𝜇1, ∁𝜇1) +

𝐺(𝜇1, ∁𝜌(𝜇), ∁𝜌(𝜇)) + 

                                           𝐺(𝜇1, ∁𝜌+1(𝜇), ∁𝜌+1(𝜇))] +
𝜕5+𝜕7

2
[𝐺(𝜇1, ∁𝜌+1(𝜇), ∁𝜌+1(𝜇)) + 

                                      𝐺(𝜇1, ∁𝜌(𝜇), ∁𝜌(𝜇)) + 𝐺(𝜇1, ∁𝜇1, ∁𝜇1)] 

                                 ⪯ 𝜇𝜌+1 + 𝜕1𝜇𝜌 +
𝜕2+𝜕3+𝜕4+𝜕6

2
[𝐺(𝜇1, ∁𝜇1, ∁𝜇1) + 𝜇𝜌 +

𝜇𝜌+1] 

                                    +
𝜕5+𝜕7

2
[𝜇𝜌+1 + 𝜇𝜌 + 𝐺(𝜇1, ∁𝜇1, ∁𝜇1)] 

                                 ⪯  𝜇𝜌 + 𝜕1𝜇𝜌 +
𝜕2+𝜕3+𝜕4+𝜕6

2
[𝐺(𝜇1, ∁𝜇1, ∁𝜇1) + 𝜇𝜌 + 𝜇𝜌] 

                                            +
𝜕5+𝜕7

2
[𝜇𝜌 + 𝜇𝜌 + 𝐺(𝜇1, ∁𝜇1, ∁𝜇1)]      (10) 

  

 (1 −
∑7

𝑖=2 𝜕𝑖

2
)𝐺(𝜇1, ∁(𝜇1), ∁(𝜇1)) ⪯

2𝜕1+∑7
𝑖=2 𝜕𝑖

2
𝜇𝜌 

                     𝐺(𝜇1, ∁(𝜇1), ∁(𝜇1)) ⪯
2𝜕1+∑7

𝑖=2 𝜕𝑖

2−∑7
𝑖=2 𝜕𝑖

𝜇𝜌 ↓ 0 

It follows that 𝜇1 = ∁𝜇1. 

Now, we assert that 𝜇1 is unique FP. If 𝜕1 is another FP of ∁, then ∀𝜇1 ∈ 𝜒𝑞 and 𝜕1 ∈ 𝜒𝑞+1, 

𝑗 = 1,2, … 𝑚, we have  

    𝐺(𝜇1, 𝜕1, 𝜕1) = 𝐺(∁𝜇1, ∁𝜕1, ∁𝜕1) 

                         ⪯ 𝜕1𝐺(𝜇1, 𝜕1, 𝜕1) + 𝜕2𝐺(𝜇1, ∁𝜇1, ∁𝜇1) + 𝜕3𝐺(𝜕1, ∁𝜕1, ∁𝜕1) 



Communications on Applied Nonlinear Analysis 

ISSN: 1074-133X 

Vol 32 No. 9s (2025) 

 

83 
https://internationalpubls.com 

                             +𝜕4𝐺(𝜕1, ∁𝜕1, ∁𝜕1) + 𝜕5𝐺(𝜇1, ∁𝜕1, ∁𝜕1) + 𝜕6𝐺(𝜕1, ∁𝜕1, ∁𝜕1) 

                              +𝜕7𝐺(𝜕1, ∁𝜇1, ∁𝜇1) 

                         ⪯ 𝜕1𝐺(𝜇1, 𝜕1, 𝜕1) + 𝜕5𝐺(𝜇1, 𝜕1, 𝜕1) + 𝜕7𝐺(𝜕1, 𝜇1, 𝜇1) 

                         ⪯ 𝜕1𝐺(𝜇1, 𝜕1, 𝜕1) + 𝜕5𝐺(𝜇1, 𝜕1, 𝜕1) + 𝜕7𝐺(𝜇1, 𝜕1, 𝜕1) 

                            ⪯ (𝜕1 + 𝜕5 + 𝜕7)𝐺(𝜇1, 𝜕1, 𝜕1). 

Since 0 ≤ 𝜕1 + 𝜕5 + 𝜕7 < ∑7
𝑖=1 𝜕𝑖 < 1. So 𝐺(𝜇1, 𝜕1, 𝜕1) = 0 this implies 𝜇1 = 𝜕1. Thus ∁ 

has FP in 𝜇 which is unique. 

 

Setting 𝜕2 = 𝜕, 𝜕3 = 𝜌 and 𝜕𝑖 = 0 for 𝑖 = 1,3, … ,7 in the previous theorem, we yield the 

subsequent corollary. 

Corollary 2.2: Let (ℜ, 𝐺, 𝑉) be 𝐺 − complete and symmetric VGMS and {𝜒𝑞}𝑞=1
𝑚  be family 

of the non-empty subset of 𝜇 with 𝜇 =∪𝑞=1
𝑚 𝜒𝑞. Let ∁: 𝜇 → 𝜇 be a transformation satisfying  

 ∁(𝜒𝑞) ⊆ 𝜒𝑞+1    𝑞 = 1,2, … , 𝑚,    𝑤ℎ𝑒𝑟𝑒     𝜒𝑚+1 = 𝜒1. 

Suppose that ∃ constant 𝜕, ℜ with 0 < 𝜕 + 𝜌 < 1 such that the transformation ∁ satisfies  

 𝐺(∁𝜇, ∁♭, ∁♭) ⪯ 𝜕𝐺(𝜇, ∁𝜇, ∁𝜇) + 𝜌𝐺(♭, ∁♭, ∁♭) 

∀𝜇 ∈ 𝜒𝑞 and ♭, 𝑟 ∈ 𝜒𝑞+1, 𝑞 = 1,2, … 𝑚. Then 𝜒 has FP in 𝜇 =∩𝑞=1
𝑚 𝜒𝑞 which is unique. 

Example 2.3: The transformation ∁: [0,1] → [0,1] defined by  

 ∁𝜇 = {

𝜇

4
    0 ⪯ 𝜇 <

1

2
𝜇

5
    

1

2
⪯ 𝜇 < 1

 

is a discontinuous transformation with 0 as unique FP. 

Setting 𝜕1 = 𝜕 and 𝜕𝑖 = 0 for 𝑖 = 2, … ,7 in the theorem 2.1, we yield the subsequent 

corollary. 

Corollary 2.4:(Banach Contraction Principle) Let (ℜ, 𝐺, 𝑉) be 𝐺 − complete and symmetric 

VGMS and {𝐴𝑗}𝑗=1
𝑚  be family of non-empty subset of 𝜇 with 𝜇 =∪𝑞=1

𝑚 𝜒𝑞. Let ∁: 𝜇 → 𝜇 be a 

transformation satisfying  

 ∁(𝜒𝑞) ⊆ 𝜒𝑞+1    𝑞 = 1,2, … , 𝑚,    𝑤ℎ𝑒𝑟𝑒     𝜒𝑚+1 = 𝜒1. 

Suppose that ∃ constant 𝜕 with 0 < 𝜕 < 1 such that the transformation ∁ satisfies  

 𝐺(∁𝜇, ∁♭, ∁𝑟) ⪯ 𝜕𝐺(𝜇, ♭, 𝑟) 

∀𝜇 ∈ 𝜒𝑞 and ♭, 𝑟 ∈ 𝜒𝑞+1, 𝑞 = 1,2, … 𝑚. Then 𝜒 has FP in 𝜇 =∩𝑞=1
𝑚 𝜒𝑞 which is unique. 

Theorem 2.5 Let (ℜ, 𝐺, 𝑉) be 𝐺 − complete and symmetric VGMS and {𝜒𝑞}𝑞=1
𝑚  be family of 

non-empty subset of 𝜇 with 𝜇 =∪𝑞=1
𝑚 𝜒𝑞. Let ∁: 𝜇 → 𝜇 be a transformation satisfying  



Communications on Applied Nonlinear Analysis 

ISSN: 1074-133X 

Vol 32 No. 9s (2025) 

 

84 
https://internationalpubls.com 

 ∁(𝜒𝑞) ⊆ 𝜒𝑞+1    𝑞 = 1,2, … , 𝑚,    𝑤ℎ𝑒𝑟𝑒     𝜒𝑚+1 = 𝜒1. 

Suppose that there exist constant 𝜕1, 𝜕2, 𝜕3, 𝜕4 with 0 < 𝜕1 + 𝜕2 + 𝜕3 + 𝜕4 <
1

2
 such that the 

transformation ∁ satisfies  

 𝐺(∁𝜇, ∁♭, ∁♭) ⪯ {𝜕1[𝐺(𝜇, ∁𝜇, ∁𝜇) + 𝐺(♭, ∁♭, ∁♭)], 𝜕2[𝐺(𝜇, ∁♭, ∁♭) + 

                                 𝐺(♭, ∁𝜇, ∁𝜇)], 𝜕3[𝐺(𝜇, ∁𝜇, ∁𝜇) + 𝐺(𝜇, ∁♭, ∁♭)], 

                                 𝜕4[𝐺(♭, ∁𝜇, ∁𝜇) + 𝐺(♭, ∁♭, ∁♭)]} (11) 

 ∀𝜇 ∈ 𝜒𝑞 and ♭, 𝑟 ∈ 𝜒𝑞+1, 𝑞 = 1,2, … 𝑚. Then 𝜒 has FP in 𝜇 =∩𝑞=1
𝑚 𝜒𝑞 which is unique. 

Proof: Interchanging the role of 𝜇 and ♭ in (11), we get  

𝐺(∁♭, ∁𝜇, ∁𝜇) ⪯ {𝜕1[𝐺(♭, ∁♭, ∁♭) + 𝐺(𝜇, ∁𝜇, ∁𝜇)], 𝜕2[𝐺(♭, ∁𝜇, ∁𝜇) + 

                             𝐺(𝜇, ∁♭, ∁♭)], 𝜕3[𝐺(♭, ∁♭, ∁♭) + 𝐺(♭, ∁𝜇, ∁𝜇)], 

                                    𝜕4[𝐺((𝜇, ∁♭, ∁♭) + 𝐺(𝜇, ∁𝜇, ∁𝜇)]} (12) 

 Adding eq(11) and eq(12), we get  

2𝐺(∁𝜇, ∁♭, ∁♭) ⪯ {2𝜕1[𝐺(𝜇, ∁𝜇, ∁𝜇) + 𝐺(♭, ∁♭, ∁♭)],2𝜕2[𝐺(𝜇, ∁♭, ∁♭) + 

                              𝐺(♭, ∁𝜇, ∁𝜇)], (𝜕3 + 𝜕4)[𝐺(𝜇, ∁𝜇, ∁𝜇) + 𝐺(𝜇, ∁♭, ∁♭)], 

                          𝜕3 + 𝜕4)([𝐺(♭, ∁𝜇, ∁𝜇) + 𝐺(♭, ∁♭, ∁♭)]} 

𝐺(∁𝜇, ∁♭, ∁♭) ⪯ {𝜕1[𝐺(𝜇, ∁𝜇, ∁𝜇) + 𝐺(♭, ∁♭, ∁♭)], 𝜕2[𝐺(𝜇, ∁♭, ∁♭) + 

                            𝐺(♭, ∁𝜇, ∁𝜇)],
𝜕3+𝜕4

2
[𝐺(𝜇, ∁𝜇, ∁𝜇) + 𝐺(𝜇, ∁♭, ∁♭)], 

                            
𝜕3+𝜕4

2
[𝐺(♭, ∁𝜇, ∁𝜇) + 𝐺(♭, ∁♭, ∁♭)]} 

                       ⪯ {
2𝜕1+𝜕3+𝜕4

2
𝐺(𝜇, ∁𝜇, ∁𝜇) +

2𝜕1+𝜕3+𝜕4

2
𝐺(♭, ∁♭, ∁♭) 

                         +
2𝜕2+𝜕3+𝜕4

2
𝐺(𝜇, ∁♭, ∁♭) +

2𝜕2+𝜕3+𝜕4

2
𝐺(♭, ∁𝜇, ∁𝜇)} 

                                    ⪯ {
2𝜕1+𝜕3+𝜕4

2
[𝐺(𝜇, ∁𝜇, ∁𝜇) + 𝐺(♭, ∁♭, ∁♭)] + 

                            
2𝜕2+𝜕3+𝜕4

2
[𝐺(𝜇, ∁♭, ∁♭) + 𝐺(♭, ∁𝜇, ∁𝜇)]} (13) 

 Putting ♭ = ∁𝜇 in eq(13), we get  

𝐺(∁𝜇, ∁2𝜇, ∁2𝜇) ⪯ {
2𝜕1 + 𝜕3 + 𝜕4

2
[𝐺(𝜇, ∁𝜇, ∁𝜇) + 𝐺(∁𝜇, ∁2𝜇, ∁2𝜇)] + 

                                 
2𝜕2 + 𝜕3 + 𝜕4

2
[𝐺(𝜇, ∁2𝜇, ∁2𝜇) + 𝐺(∁𝜇, ∁𝜇, ∁𝜇)]} 

                           ⪯ {
2𝜕1 + 𝜕3 + 𝜕4

2
[𝐺(𝜇, ∁𝜇, ∁𝜇) + 𝐺(∁𝜇, ∁2𝜇, ∁2𝜇)] + 



Communications on Applied Nonlinear Analysis 

ISSN: 1074-133X 

Vol 32 No. 9s (2025) 

 

85 
https://internationalpubls.com 

                                                                
2𝜕2+𝜕3+𝜕4

2
𝐺(𝜇, ∁2𝜇, ∁2𝜇)} (14) 

 By using (VGM5), we get  

 𝐺(𝜇, ∁2𝜇, ∁2𝜇) ⪯ 𝐺(𝜇, ∁𝜇, ∁𝜇) + 𝐺(∁𝜇, ∁2𝜇, ∁2𝜇) (15) 

 Then eq(14) becomes  

                   𝐺(∁𝜇, ∁2𝜇, ∁2𝜇) ⪯ {
2𝜕1 + 𝜕3 + 𝜕4

2
[𝐺(𝜇, ∁𝜇, ∁𝜇) + 𝐺(∁𝜇, ∁2𝜇, ∁2𝜇)] + 

                                                   
2𝜕2 + 𝜕3 + 𝜕4

2
𝐺(𝜇, ∁𝜇, ∁𝜇) + 𝐺(∁𝜇, ∁2𝜇, ∁2𝜇)} 

⪯ {∑

4

𝑖=1

𝜕𝑖𝐺(𝜇, ∁𝜇, ∁𝜇) + 

          ∑

4

 𝑖=1

𝜕𝑖𝐺(∁𝜇, ∁2𝜇, ∁2𝜇)} 

                (1 − ∑

4

𝑖=1

𝜕𝑖)𝐺(∁𝜇, ∁2𝜇, ∁2𝜇) ⪯ ∑

4

𝑖=1

𝜕𝑖(𝐺(𝜇, ∁𝜇, ∁𝜇)) 

                            𝐺(∁𝜇, ∁2𝜇, ∁2𝜇) ⪯
∑4

𝑖=1 𝜕𝑖

1−∑4
𝑖=1 𝜕𝑖

𝐺(𝜇, ∁𝜇, ∁𝜇) (16) 

Putting 𝑞 =
∑4

𝑖=1 𝜕𝑖

1−∑4
𝑖=1 𝜕𝑖

. Since 0 ≤ ∑4
𝑖=1 𝜕𝑖 <

1

2
, so 0 ≤ 𝑞 < 1. Then eq(16) becomes  

                                         𝐺(∁𝜇, ∁2𝜇, ∁2𝜇) ⪯ 𝑞𝐺(𝜇, ∁𝜇, ∁𝜇) (17) 

 ∀𝜇 ∈ 𝜒𝑞. Further  

 

 𝐺(∁𝜌𝜇, ∁𝜌+1𝜇, ∁𝜌+1𝜇) ⪯ 𝑞𝜌𝐺(𝜇, ∁𝜇, ∁𝜇)    ∀𝜌 ∈ ℕ    𝜇 ∈ 𝜒𝑞 . 

For 𝜌, 𝑗 ∈ ℕ and 𝜇 ∈ 𝜒𝑞  

                        𝐺(∁𝜌(𝜇), ∁𝜌+𝑗(𝜇), ∁𝜌+𝑗(𝜇)) ⪯ 𝐺(∁𝜌(𝜇), ∁𝜌+1(𝜇), ∁𝜌+1(𝜇)) + 

                                                             𝐺(∁𝜌+1(𝜇), ∁𝜌+𝑗(𝜇), ∁𝜌+𝑗(𝜇)) 

                                           ⪯ 𝐺(∁𝜌(𝜇), ∁𝜌+1(𝜇), ∁𝜌+1(𝜇)) + 

                                               𝐺(∁𝜌+1(𝜇), ∁𝜌+2(𝜇), ∁𝜌+2(𝜇)) 

                                             +𝐺(∁𝜌+2(𝜇), ∁𝜌+𝑗(𝜇), ∁𝜌+𝑗(𝜇) 

                                                               ⋮ 

  

                                            ⪯ ∑𝑗
𝑖=1 𝐺(∁𝜌+𝑖−1(𝜇), ∁𝜌+𝑖(𝜇), ∁𝜌+𝑖(𝜇)) 



Communications on Applied Nonlinear Analysis 

ISSN: 1074-133X 

Vol 32 No. 9s (2025) 

 

86 
https://internationalpubls.com 

                                            ⪯ ∑𝑗
𝑖=1 𝑞𝜌+𝑖−1𝐺(𝜇, ∁𝜇, ∁𝜇) 

 𝐺(∁𝜌(𝜇), ∁𝜌+𝑗(𝜇), ∁𝜌+𝑗(𝜇)) ⪯
𝑞𝜌

1−𝑞
𝐺(𝜇, ∁𝜇, ∁𝜇) (18) 

This implies {∁𝜌𝜇} is a VCS in G-complete VGMS and so it converges to 𝜇1 ∈ 𝜕. 

Now  

 𝐺(𝜇1, ∁𝜇1, ∁𝜇1) ⪯ 𝐺(𝜇1, ∁𝜌+1𝜇, ∁𝜌+1𝜇) + 𝐺(∁𝜌+1𝜇, ∁𝜇1, ∁𝜇1) 

                              ⪯ 𝐺(𝜇1, ∁𝜌+1𝜇, ∁𝜌+1𝜇) +
2𝜕1+𝜕3+𝜕4

2
[𝐺(∁𝜌𝜇, ∁𝜌+1𝜇, ∁𝜌+1𝜇) 

                                  +𝐺(𝜇1, ∁𝜇1, ∁𝜇1)] +
2𝜕2+𝜕3+𝜕4

2
[𝐺(∁𝜌𝜇, ∁𝜇1, ∁𝜇1) 

                                   +𝐺(𝜇1, ∁𝜌+1𝜇, ∁𝜌+1𝜇)] (19) 

By using  

 𝜇1 = lim
𝜌→∞

∁𝜌𝜇 = lim
𝜌→∞

∁𝜌+1𝜇. 

Then eq(19) becomes  

𝐺(𝜇1, ∁𝜇1, ∁𝜇1) ⪯ 0 +
2𝜕1 + 𝜕3 + 𝜕4

2
[0 + 𝐺(𝜇1, ∁𝜇1, ∁𝜇1)] 

                                 +
2𝜕2 + 𝜕3 + 𝜕4

2
[𝐺(𝜇1, ∁𝜇1, ∁𝜇1) + 0] 

                                      ⪯ ∑4
𝑙=1 𝜕𝑖𝐺(𝜇1, ∁𝜇1, ∁𝜇1) (20) 

Since 0 ≤ ∑4
𝑖=1 𝜕𝑖 < 1, it follows that 𝜇1 = ∁𝜇1. 

Now, we assert that 𝜇1 is unique FP. If 𝜕1 is another FP of ∁, then ∀𝜇1 ∈ 𝜒𝑞 and 𝜕1 ∈ 𝜒𝑞+1, 

𝑗 = 1,2, … 𝑚, we have  

 𝐺(𝜇1, 𝜕1, 𝜕1) = 𝐺(∁𝜇1, ∁𝜕1, ∁𝜕1) 

                         ⪯ {𝜕1[𝐺(𝜇1, ∁𝜇1, ∁𝜇1) + 𝐺(𝜕1, ∁𝜕1, ∁𝜕1)], 𝜕2[𝐺(𝜇1, ∁𝜕1, ∁𝜕1) + 

                            𝐺(𝜕1, ∁𝜇1, ∁𝜇1)], 𝜕3[𝐺(𝜇1, ∁𝜇1, ∁𝜇1) + 𝐺(𝜇1, ∁𝜕1, ∁𝜕1)], 

                                     𝜕4[𝐺(𝜕1, ∁𝜇1, ∁𝜇1) + 𝐺(𝜕1, ∁𝜕1, ∁𝜕1)]} 

                                 ⪯ {𝜕1(0), 𝜕2[𝐺(𝜇1, 𝜕1, 𝜕1) + 𝐺(𝜕1, 𝜇1, 𝜇1)], 𝜕3𝐺(𝜇1, 𝜕1, 𝜕1), 

                               𝜕4𝐺(𝜇1, 𝜕1, 𝜕1)} 

                     ⪯ {2𝜕2𝐺(𝜇1, 𝜕1, 𝜕1), 𝜕3𝐺(𝜇1, 𝜕1, 𝜕1), 𝜕4𝐺(𝜇1, 𝜕1, 𝜕1)} 

                     ⪯ (2𝜕2 + 𝜕3 + 𝜕4)𝐺(𝜇1, 𝜕1, 𝜕1). 

 Since 0 ≤ 2𝜕2 + 𝜕3 + 𝜕4 < 1. So ∁ has FP in 𝜇 which is unique. 



Communications on Applied Nonlinear Analysis 

ISSN: 1074-133X 

Vol 32 No. 9s (2025) 

 

87 
https://internationalpubls.com 

3  Conclusion 

Within this manuscript, we illustrate FPT for self-transformations in V-complete VGMS by 

utilizing cyclic contraction. These outcomes are expected to inspire researchers to explore 

problem-solving opportunities in diverse areas such as differential equations and functional 

analysis. 

Refrences 

[1] B. C. Dhage, Bull.Cal. Math. Soc., 84(1992), 329-336.  

[2] B. C. Dhage, Bull. Cal. Math. Soc., 86(1994), 503-508.  

[3] D. W. Boyd, J. S. W. Wong , On nonlinear contractions, Proc. Am. Math. Soc., 

20(1969), 458-464.  

[4] H. Aydi, W. Shatanawi, C. Vetro, On generalized weakly G-contraction mapping in 

G-metric Spaces, Comput. Math. Appl., 62(2011),  4222-4229. 

[5] E. Karapinar, I. M. Erhan, Cyclic contraction and fixed point theorems, Filomat, 

26(2012),  777-782.  

[6] H. Aydi, Generalized cyclic contraction in G-metric spaces, J. Nonlinear Sci. Appl. In 

press.  

[7] L. K. Dey, Fixed Point Theorems in Vector Metric Spaces, Int. J. Math. Anal., 

4(2010),  2395-2400.  

[8] Luxemburg, Zannen, Riesz Spaces, North-Holland Pub. Co., Amsterdam, 1971.  

[9] M. A. Alghamdi, A. Petrusel, N. Shahzad, A fixed point theorems for cyclic 

generalized contraction in metric spaces, Fixed Point Theory Appl., 122(2012). 

[10] N. Bilgili, E. Karapınar, Cyclic contractions via auxiliary functions on G-metric 

spaces, Fixed Point Theory Appl., 49 (2013).  

[11] P. Yadav, M. Kamra, Common Fixed Point Theorems for Four Transformations in 

Vector S-Metric Spaces, AIP Conf. Proc., 3081(2024),  020006-1-020006-11. 

doi.org/10.1063/5.0196129.  

[12] P. Yadav, M. Kamra, Some fixed point results on a vector valued S-metric space, J. 

Tianjin Univ. Sci. Technol., 55(2022),  209-228. DOI 10.17605/OSF.IO/XUHGK.  

[13] P. Yadav, M. Kamra, Rajpal, Common fixed point theorems for self mappings on a 

complete vector S-metric space, Spec Ugdym., 43(2022), 6439-6450.  

[14] R. P. Pathak, R. Bhardwaj, Fixed point and common fixed point theorems in vector 

metric spaces, Math. Theory Model., 3(2013).  

[15] R. Saadati, S. M. Vaezpour, P. Vetro, B.E. Rhoades, Fixed point theorems in 

generalized partially ordered G-metric spaces, Math. Compat. Modelling,52(2010), 

797-801.  

[16] W. Shatanawi, Some fixed point theorems in ordered G-metric spaces and 

applications, Abst. Appl. Anal., 2011(2011), Article ID:126205. 

[17] W. A. Kirk, P. S. Srinavasan, P. Veeramani, Fixed point for mappings satisfying 

cyclical contraction condition, Fixed point theory, 4(2003),  79-89.   

[18] Z. Mustafa, B. I. Sims, A new approach to generalized metric spaces, J. Nonlinear 

Convex Anal., 7(2006),  289-297.  



Communications on Applied Nonlinear Analysis 

ISSN: 1074-133X 

Vol 32 No. 9s (2025) 

 

88 
https://internationalpubls.com 

[19] Z. Mustafa, B. I. Sims, Fixed point theorems for contractive mappings in complete 𝐺-

metric spaces, J. Fixed Point Theory Appl., 2009. Article ID 917175.