Communications on Applied Nonlinear Analysis 

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Some Common Fixed Point Theorems for Weakly Compatible Mappings 

in Rectangular S-Metric Spaces 

T. R. Shimpi1 and S. G. Dapke2 
1Department of Mathematics, GF’s Godavari College of Engineering, Jalgaon, India. 

Email. trushalishimpi@gmail.com  
2Department of Mathematics, Iqra’s H. J. Thim College of Arts and Science, Mehrun, Jalgaon, India. 

Email. sadashivgdapke@gmail.com  

 

Article History: 

Received: 02-01-2025 

Revised: 25-02-2025 

Accepted: 20-03-2025 

Abstract:  

This paper explores the existence of common fixed points for weakly compatible 

mappings in the framework of rectangular S-metric spaces, a generalization of 

traditional S-metric spaces. By introducing suitable contractive conditions, we 

establish new common fixed point theorems that unify and extend various known 

results in the field of fixed point theory. These theorems provide broader 

applicability and deeper insights into the behaviour of such mappings under 

generalized metric conditions. The developed results not only contribute to the 

theoretical understanding but also enhance the scope of fixed point analysis in more 

generalized settings. To illustrate the validity and applicability of the main results, 

we provide several relevant examples. These examples serve to clarify the theoretical 

findings and demonstrate how the established theorems can be effectively applied in 

practice. Overall, the study offers a meaningful advancement in the general theory 

of fixed points within the context of rectangular S-metric spaces. 

Keywords: S-metric space; rectangular S-metric space; Common fixed point. 

 
 

1. Introduction 

Fixed-point theory finds applications in numerous disciplines. Within this context, coincidence points, 

which generalize fixed points, hold significant importance. Jungck [5] was the first to introduce the 

concept of commuting mappings and to define common fixed points in metric spaces. Although 

commuting mappings possess strong structural properties, their stringent conditions restrict their 

applicability. To overcome this limitation, Sessa [15] proposed the concept of weakly commuting 

mappings, offering a more flexible alternative. Expanding on this notion, Jungck [6] introduced 

compatible mappings, which further generalized weak commutatively. He showed that every weakly 

commuting pair is compatible, although the converse is not necessarily true. Later, Jungck [7, 8] 

introduced the concept of weak compatibility, defining a pair of self-maps as weakly compatible if 

they commute at their coincidence points.  

Building upon this foundational work, several researchers have investigated coincidence points for 

various types of mappings in metric spaces, where distances are traditionally measured between pairs 

of points. While effective in many settings, this pairwise approach can be limiting in more complex 

scenarios. To address such limitations, Sedghi et al. [13] introduced the concept of S-metric spaces, 

where distances are defined over triplets of points, thereby broadening the scope of fixed-point theory. 

Abbas and Jungck [2] established results on coincidence and common fixed points under contractive 

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conditions in cone metric spaces, while Abbas and Rhoades [3] further generalized fixed-point 

theorems by eliminating the requirement of commutatively. Saluja [12] extended fixed-point results 

for weak contractions within complete S-metric spaces.  

In this paper, we investigate coincidence and common fixed points for weakly compatible mappings 

satisfying weak contraction conditions in complete S-metric spaces. The theoretical results are 

illustrated with relevant examples. 

2. Methodology 

We begin by introducing the necessary definitions and properties of rectangular S-metric spaces and 

weakly compatible mappings. Using suitable contractive conditions, we formulate and prove new 

common fixed point theorems within this generalized framework. The results are supported by 

illustrative examples that verify the applicability of the proposed theorems. This approach extends 

existing results in fixed point theory to a broader class of metric spaces. 

3. Preliminaries 

In this section, we present the basic definitions and foundational concepts necessary for the 

development of our main results.  

Definition 3.1 [14] Let 𝑋 be a non empty set and S: X3 → ℝ+ be a function satisfying the properties 

given below;   

    • 𝑆(𝑥, 𝑦, 𝑧) = 0 if and only if 𝑥 = 𝑦 = 𝑧  

    • 𝑆(𝑥, 𝑦, 𝑧) ≤ 𝑆(𝑥, 𝑥, 𝑎) + 𝑆(𝑦, 𝑦, 𝑎) + 𝑆(𝑧, 𝑧, 𝑎) for all 𝑎, 𝑥, 𝑦, 𝑧 ∈ 𝑋 (termed as rectangle 

inequality).  

 Then, the pair (𝑋, 𝑆) is named as a S-metric space.  

Definition 3.2 [1] Let X be a non void set and S: X3 → ℝ+ be a function satisfying the conditions listed 

below:   

    • 𝑆(𝑥, 𝑦, 𝑧) = 0 if and only if 𝑥 = 𝑦 = 𝑧  

    • 𝑆(𝑥, 𝑦, 𝑧) ≤ 𝑆(𝑥, 𝑥, 𝑎) + 𝑆(𝑦, 𝑦, 𝑎) + 𝑆(𝑧, 𝑧, 𝑎) for all 𝑥, 𝑦, 𝑧 ∈ 𝑋 and all distinct points 𝑎 ∈
𝑋 − {𝑥, 𝑦, 𝑧}. Then, the pair (𝑋, 𝑆) is referred as a rectangular S-metric space.  

Definition 3.3 [14] Let (𝑋, 𝑆) be an S-metric space and 𝐴 ⊂ 𝑋.   

    • A sequence {𝑥𝑛} in 𝑋 is converge to 𝑥 if 𝑆(𝑥𝑛, 𝑥𝑛, 𝑥) → 0 𝑎𝑠 𝑛 → ∞. In other words, for 

every 𝜖 > 0 there exists 𝑛0 ∈ 𝑁 such that for all 𝑛 ≥ 𝑛0, 𝑆(𝑥𝑛, 𝑥𝑛, 𝑥) < 𝜖. In this case, we write 

𝑙𝑖𝑚𝑛→∞ 𝑥𝑛 = 𝑥 and we say that 𝑥 is the limit of {𝑥𝑛} in 𝑋.  

    • A sequence {𝑥𝑛} in 𝑋 is reffered as Cauchy sequence if for each 𝜖 > 0, there exists 𝑛0 ∈ 𝑁 

such that for each 𝑛, 𝑚 ≥ 𝑛0, 𝑆(𝑥𝑛, 𝑥𝑛, 𝑥𝑚) < 𝜖.  

    • The 𝑆-metric space (𝑋, 𝑆) is said to be complete if every Cauchy sequence in 𝑋 converges 

to a limit in 𝑋.  



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Definition 3.4 [12] Let (𝑋, 𝑆) be a 𝑆-metric space. A mapping 𝑇: 𝑋 → 𝑋 is said to be a weak 

contraction on 𝑋 if there exists a function 𝜓: [0, ∞) → [0, ∞) with 𝜓(𝑡) = 0 if and only if 𝑡 = 0 that 

satisfies  

 𝑆(𝑇𝑥, 𝑇𝑥, 𝑇𝑦) ≤ 𝑆(𝑥, 𝑥, 𝑦) − 𝛿𝜓(𝑆(𝑥, 𝑥, 𝑦)) 

for all 𝑥, 𝑦 ∈ 𝑋, where 0 ≤ 𝛿 < 1.  

Example 3.5  Let X = R and 𝑆: 𝑋 × 𝑋 × 𝑋 → 𝑅+   be a function such that  

   𝑆(𝑥, 𝑥, 𝑦) = {
9𝑥2 + 𝑦2 if𝑥 ≠ 𝑦,
0 if𝑥 = 𝑦,

 (3.1) 

 for all 𝑥, 𝑦 ∈ 𝑋. Then (𝑋, 𝑆) becomes a 𝑆-metric space.  

Let 𝑓: 𝑋 → 𝑋 is defined as 𝑓(𝑥) =
𝑥

9
 and 𝜓(𝑡) = 80𝑡 for all 𝑡 ≥ 0, where 𝜓: [0, ∞) → [0, ∞) is non 

decreasing continuous function. Then  

   𝑆(𝑓𝑥, 𝑓𝑥, 𝑓𝑦) = 𝑆 (
𝑥

9
,

𝑥

9
,

𝑦

9
) 

                           = 9 (
𝑥

9
)

2

+ (
𝑦

9
)

2

 

                           = 9
𝑥2

81
+

𝑦2

81
 

                           = 9𝑥2 + 𝑦2 −
80

81
(9𝑥2 + 𝑦2) 

                           = 𝑆(𝑥, 𝑥, 𝑦) −
1

81
𝜓(𝑆(𝑥, 𝑥, 𝑦)). 

 Thus, we found that 𝑓 is a weak contraction on 𝑋. Since, it fulfills the requirements of Definition (3.4) 

for 𝛿 =
1

81
.  

Proposition 3.6 [2] If weakly compatible functions 𝑓 and 𝑔 have exactly only one point of coincidence 

𝑤 = 𝑓𝑥 = 𝑔𝑥 on 𝑋, then 𝑤 is the unique common fixed point of 𝑓 and 𝑔.  

Lemma 3.7 [14] A condition 𝑆(𝑥, 𝑥, 𝑦) = 𝑆(𝑦, 𝑦, 𝑥) holds ∀𝑥, 𝑦 ∈ 𝑋 in 𝑆-metric space (𝑋, 𝑆).  

Lemma 3.8 [14] Let (𝑋, 𝑆) be an 𝑆-metric space. If {𝑥𝑛} and {𝑦𝑛} are sequences in 𝑋 converging to 𝑥 

and 𝑦 respectively, that is, 𝑥𝑛 → 𝑥 and 𝑦𝑛 → 𝑦 as 𝑛 → ∞, then 𝑆(𝑥𝑛, 𝑥𝑛 , 𝑦𝑛) → 𝑆(𝑥, 𝑥, 𝑦) as 𝑛 → ∞.  

Lemma 3.9 [14] Let (𝑋, 𝑆) be an 𝑆-metric space. If the sequence {𝑥𝑛} in 𝑋 converges to 𝑥, then the 

limit 𝑥 is unique.  

Lemma 3.10 [14] Let (𝑋, 𝑆) be an 𝑆-metric space. If the sequence {𝑥𝑛} in 𝑋 converges to 𝑥, then {𝑥𝑛} 

is a Cauchy sequence.  

4. Main results 

In this section, we present new common fixed point theorems for weakly compatible mappings defined 

on complete rectangular 𝑆-metric spaces. These results are obtained by imposing specific contractive 

conditions that ensure the existence and uniqueness of common fixed points.  



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Theorem 4.1  Let 𝑓 and 𝑔 be a self mapping on complete 𝑆-metric space (𝑋, 𝑆). Assume that 𝑓 and 

𝑔 satisfies the following restrictions,  

 𝜓(𝑆(𝑓𝑥, 𝑓𝑥, 𝑓𝑦), 𝑆(𝑓𝑦, 𝑓𝑦, 𝑓𝑥)) ≤ 𝑞𝜓(𝑆(𝑔𝑥, 𝑔𝑥, 𝑔𝑦), 𝑆(𝑔𝑦, 𝑔𝑦, 𝑔𝑥)), 

 where 0 < 𝑞 < 1 and 𝜓: [0, ∞)2 → [0, ∞)2 is a continuous function on [0, ∞)2 with 𝜓(𝑎, 𝑏) = 0 if 

and only if 𝑎 = 0 = 𝑏. Also,   

    • 𝑓(𝑋) ⊆ 𝑔(𝑋),  

    • If 𝑔(𝑋) is complete.  

 Then 𝑓 and 𝑔 have a unique coincidence point in 𝑋. Moreover, if 𝑓 and 𝑔 are weakly 

compatible, then 𝑓 and 𝑔 have a unique common fixed point in 𝑋.  

Proof. Let 𝑥0 be any point in 𝑋. Since 𝑓𝑥0 ∈ 𝑓(𝑋) and 𝑓(𝑋) ⊆ 𝑔(𝑋), there exists a point 𝑥1 in 𝑋 such 

that 𝑓𝑥0 = 𝑔𝑥1. Clearly 𝑥1 ∈ 𝑋 and again, by inclusion 𝑓𝑥1 ∈ 𝑓(𝑋) there exists 𝑥2 in 𝑋 such that 

𝑓𝑥1 = 𝑔𝑥2. Proceeding in this way, we generate a sequence {𝑥𝑛} in 𝑋, such that each 𝑥𝑛+1 ∈ 𝑋 

satisfies 𝑓𝑥𝑛 = 𝑔𝑥𝑛+1, ∀𝑛. 

Now, consider the following expression:  

 𝜓(𝑆(𝑔𝑥𝑛+1, 𝑔𝑥𝑛+1, 𝑔𝑥𝑛+2), 𝑆(𝑔𝑥𝑛+2, 𝑔𝑥𝑛+2, 𝑔𝑥𝑛+1))  

= 𝜓(𝑆(𝑓𝑥𝑛, 𝑓𝑥𝑛, 𝑓𝑥𝑛+1), 𝑆(𝑓𝑥𝑛+1, 𝑓𝑥𝑛+1, 𝑓𝑥𝑛)) 

                    ≤ 𝑞𝜓(𝑆(𝑔𝑥𝑛, 𝑔𝑥𝑛, 𝑔𝑥𝑛+1), 𝑆(𝑔𝑥𝑛+1, 𝑔𝑥𝑛+1, 𝑔𝑥𝑛)) 

                                  ≤   𝑞2𝜓(𝑆(𝑔𝑥𝑛−1, 𝑔𝑥𝑛−1, 𝑔𝑥𝑛), 𝑆(𝑔𝑥𝑛, 𝑔𝑥𝑛, 𝑔𝑥𝑛−1)). 

 By continuing this process inductively, we obtain: 

𝜓(𝑆(𝑔𝑥𝑛+1, 𝑔𝑥𝑛+1, 𝑔𝑥𝑛+2), 𝑆(𝑔𝑥𝑛+2, 𝑔𝑥𝑛+2, 𝑔𝑥𝑛+1)) ≤ 𝜓(𝑆(𝑔𝑥0, 𝑔𝑥0, 𝑔𝑥1), 𝑆(𝑔𝑥1, 𝑔𝑥1, 𝑔𝑥0)). 

Since 0 < 𝑞 < 1 taking the limit as 𝑛 → ∞, gives  

 𝜓(𝑆(𝑔𝑥𝑛+1, 𝑔𝑥𝑛+1, 𝑔𝑥𝑛+2), 𝑆(𝑔𝑥𝑛+2, 𝑔𝑥𝑛+2, 𝑔𝑥𝑛+1)) → 0. 

Due to continuity of 𝜓, it follows that  

   0 = lim
𝑛→∞

𝜓(𝑆(𝑔𝑥𝑛+1, 𝑔𝑥𝑛+1, 𝑔𝑥𝑛+2), 𝑆(𝑔𝑥𝑛+2, 𝑔𝑥𝑛+2, 𝑔𝑥𝑛+1)) 

   0 = 𝜓( lim
𝑛→∞

𝑆(𝑔𝑥𝑛+1, 𝑔𝑥𝑛+1, 𝑔𝑥𝑛+2), lim
𝑛→∞

𝑆(𝑔𝑥𝑛+2, 𝑔𝑥𝑛+2, 𝑔𝑥𝑛+1)). 

 Thus, by using the property of 𝜓, we conclude  

 lim
𝑛→∞

𝑆(𝑔𝑥𝑛+1, 𝑔𝑥𝑛+1, 𝑔𝑥𝑛+2) = lim
𝑛→∞

𝑆(𝑔𝑥𝑛+2, 𝑔𝑥𝑛+2, 𝑔𝑥𝑛+1) = 0. 

Now, we will prove that the sequence {𝑔𝑥𝑛} is a Cauchy sequence. Assume, for contradiction, that 

{𝑔𝑥_𝑛} is not a Cauchy sequence. Then there is 𝜖 > 0 and subsequences {𝑔𝑥𝑛(𝑘)} and {𝑔𝑥𝑚(𝑘)} such 

that, ∀ 𝑘 ∈ 𝑁. Then, we have 

𝑛(𝑘) > 𝑚(𝑘) > 𝑘, 𝑆(𝑔𝑥𝑛(𝑘), 𝑔𝑥𝑛(𝑘), 𝑔𝑥𝑚(𝑘)) ≥ 𝜖 and 𝑆(𝑔𝑥𝑛(𝑘), 𝑔𝑥𝑛(𝑘), 𝑔𝑥𝑚(𝑘)−1)) < 𝜖. 



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Hence,  

 𝜖 ≤ 𝑆(𝑔𝑥𝑛(𝑘), 𝑔𝑥𝑛(𝑘), 𝑔𝑥𝑚(𝑘)) 

 = 𝑆(𝑔𝑥𝑚(𝑘), 𝑔𝑥𝑚(𝑘), 𝑔𝑥𝑛(𝑘)) 

 ≤ 𝑆(𝑔𝑥𝑚(𝑘), 𝑔𝑥𝑚(𝑘), 𝑔𝑥𝑚(𝑘)−1)) + 𝑆(𝑔𝑥𝑚(𝑘), 𝑔𝑥𝑚(𝑘), 𝑔𝑥𝑚(𝑘)−1)) 

                  +𝑆(𝑔𝑥𝑛(𝑘), 𝑔𝑥𝑛(𝑘), 𝑔𝑥𝑚(𝑘)−1)) 

 < 0 + 0 + 𝜖. 

 Which is leads to a contradiction. So,  

 𝑆(𝑔𝑥𝑛(𝑘), 𝑔𝑥𝑛(𝑘), 𝑔𝑥𝑚(𝑘)) = 𝜖. 

Additionally,  

𝑆(𝑔𝑥𝑛(𝑘)−1, 𝑔𝑥𝑛(𝑘)−1, 𝑔𝑥𝑚(𝑘)−1) ≤ 𝑆(𝑔𝑥𝑛(𝑘)−1, 𝑔𝑥𝑛(𝑘)−1, 𝑔𝑥𝑛(𝑘)) + 𝑆(𝑔𝑥𝑛(𝑘)−1, 𝑔𝑥𝑛(𝑘)−1, 𝑔𝑥𝑛(𝑘)) 

                                   +𝑆(𝑔𝑥𝑚(𝑘)−1, 𝑔𝑥𝑚(𝑘)−1, 𝑔𝑥𝑛(𝑘)) 

                                  = 2𝑆(𝑔𝑥𝑛(𝑘)−1, 𝑔𝑥𝑛(𝑘)−1, 𝑔𝑥𝑛(𝑘)) + 𝑆(𝑔𝑥𝑛(𝑘), 𝑔𝑥𝑛(𝑘), 𝑔𝑥𝑚(𝑘)−1) 

                                 < 2(0) + 𝜖 = 𝜖. 

 Thus, 𝑆(𝑔𝑥𝑛(𝑘)−1, 𝑔𝑥𝑛(𝑘)−1, 𝑔𝑥𝑚(𝑘)−1) < 𝜖. 

Consequently, 𝑆(𝑔𝑥𝑚(𝑘)−1, 𝑔𝑥𝑚(𝑘)−1, 𝑔𝑥𝑛(𝑘)−1) = 𝑆(𝑔𝑥𝑛(𝑘)−1, 𝑔𝑥𝑛(𝑘)−1, 𝑔𝑥𝑚(𝑘)−1) < 𝜖. 

Now,  

 𝜓(𝑆(𝑔𝑥𝑛(𝑘), 𝑔𝑥𝑛(𝑘), 𝑔𝑥𝑚(𝑘)), 𝑆(𝑔𝑥𝑚(𝑘), 𝑔𝑥𝑚(𝑘), 𝑔𝑥𝑛(𝑘))) 

                = 𝜓(𝑆(𝑓𝑥𝑛(𝑘)−1, 𝑓𝑥𝑛(𝑘)−1, 𝑓𝑥𝑚(𝑘)−1), 𝑆(𝑓𝑥𝑚(𝑘)−1, 𝑓𝑥𝑚(𝑘)−1, 𝑓𝑥𝑛(𝑘)−1)) 

 ≤ 𝑞𝜓(𝑆(𝑔𝑥𝑛(𝑘)−1, 𝑔𝑥𝑛(𝑘)−1, 𝑔𝑥𝑚(𝑘)−1), 𝑆(𝑔𝑥𝑚(𝑘)−1, 𝑔𝑥𝑚(𝑘)−1, 𝑔𝑥𝑛(𝑘)−1)) 

 < 𝑞𝜓(𝜖, 𝜖), 

 which gives 𝜓(𝜖, 𝜖) < 𝑞𝜓(𝜖, 𝜖).  

Since 0 < 𝑞 < 1, this inequality is only possible if 𝜖 = 0𝜓(𝜖, 𝜖) = 0. 

By the property of 𝜓 we conclude  

 𝑆(𝑔𝑥𝑛(𝑘), 𝑔𝑥𝑛(𝑘), 𝑔𝑥𝑚(𝑘)) = 0. 

This leads to a contradiction. Thus we obtain, {𝑔𝑥𝑛} is a Cauchy sequence in 𝑔(𝑋). Since 𝑔(𝑋) is 

complete, there exist a point 𝑞 in 𝑔(𝑋) such that 𝑔𝑥𝑛 → 𝑞, as 𝑛 → ∞. Thus, there exists a point 𝑝 in 𝑋 

with 𝑔𝑝 = 𝑞. Hence,  

𝜓(𝑆(𝑔𝑥𝑛+1, 𝑔𝑥𝑛+1, 𝑓𝑝), 𝑆(𝑓𝑝, 𝑓𝑝, 𝑔𝑥𝑛+1)) = 𝜓(𝑆(𝑓𝑥𝑛, 𝑓𝑥𝑛, 𝑓𝑝), 𝑆(𝑓𝑝, 𝑓𝑝, 𝑓𝑥𝑛)) 

                                                                   ≤ 𝑞𝜓(𝑆(𝑔𝑥𝑛, 𝑔𝑥𝑛, 𝑔𝑝), 𝑆(𝑔𝑝, 𝑔𝑝, 𝑔𝑥𝑛)), 



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 taking the limit as 𝑛 → ∞, we get  

 𝜓(𝑆(𝑞, 𝑞, 𝑓𝑝), 𝑆(𝑓𝑝, 𝑓𝑝, 𝑞)) ≤ 𝑞𝜓(𝑆(𝑞, 𝑞, 𝑞), 𝑆(𝑞, 𝑞, 𝑞)) = 𝑞𝜓(0,0). 

 So,  

 𝜓(𝑆(𝑞, 𝑞, 𝑓𝑝), 𝑆(𝑓𝑝, 𝑓𝑝, 𝑞)) = 0𝑓𝑝 = 𝑞. 

 Thus, 𝑓𝑝 = 𝑔𝑝 = 𝑞 showing that 𝑞 is the point of coincidence of 𝑓 and 𝑔.  

To show uniqueness, suppose another point 𝑤 ∈ 𝑋 such that 𝑓𝑤 = 𝑔𝑤 = 𝑤. Now  

 𝜓(𝑆(𝑞, 𝑞, 𝑤), 𝑆(𝑤, 𝑤, 𝑞)) = 𝜓(𝑆(𝑓𝑝, 𝑓𝑝, 𝑓𝑤), 𝑆(𝑓𝑤, 𝑓𝑤, 𝑓𝑝)) 

                                               ≤   𝑞𝜓(𝑆(𝑔𝑝, 𝑔𝑝, 𝑔𝑤), 𝑆(𝑔𝑤, 𝑔𝑤, 𝑔𝑝)) 

                                               =   𝑞𝜓(𝑆(𝑞, 𝑞, 𝑤), 𝑆(𝑤, 𝑤, 𝑞)). 

 Again, since 0 < 𝑞 < 1, this yields;  

 𝜓(𝑆(𝑞, 𝑞, 𝑤), 𝑆(𝑤, 𝑤, 𝑞)) = 0𝑞 = 𝑤. 

Therefore 𝑓 and 𝑔 have a unique point of coincidence. Since, 𝑓 and 𝑔 are weakly compatible, by 

Proposition (3.6), it follows that they have a unique common fixed point in 𝑋. 

Theorem 4.2  Let (𝑋, 𝑆) be a complete rectangular S-metric space and 𝑓, 𝑔: 𝑋 → 𝑋 be two mappings 

such that for all 𝑥, 𝑦 ∈ 𝑋,  

 𝑆(𝑓𝑥, 𝑓𝑥, 𝑓𝑦) ≤ 𝛼𝑆(𝑔𝑥, 𝑔𝑥, 𝑔𝑦) + 𝛽[𝑆(𝑔𝑥, 𝑔𝑥, 𝑓𝑥) + 𝑆(𝑔𝑦, 𝑔𝑦, 𝑓𝑦)], 

 where 𝛼, 𝛽 > 0 and 𝛼 + 2𝛽 < 1.  

Assume the following conditions hold;   

    • 𝑓(𝑋) ⊆ 𝑔(𝑋),  

    • If 𝑔(𝑋) is complete.  

Then 𝑓 and 𝑔 have a unique coincidence point in 𝑋 (i.e. there exists unique 𝑧 ∈ 𝑋 such that 𝑓𝑧 = 𝑔𝑧). 

Moreover, if 𝑓 and 𝑔 are weakly compatible (i.e., they commute at their coincidence point), then 𝑓 

and 𝑔 have a unique common fixed point in 𝑋 (that is, there exists a unique 𝑝 ∈ 𝑋 such that 𝑓𝑝 =

𝑔𝑝 = 𝑝).  

Proof. Let 𝑥0 be any point in 𝑋. Since 𝑓𝑥0 ∈ 𝑓(𝑋) and 𝑓(𝑋) ⊆ 𝑔(𝑋), there exists a point 𝑥1 in 𝑋 such 

that 𝑓𝑥0 = 𝑔𝑥1. As 𝑥1 ∈ 𝑋, we have 𝑓𝑥1 ∈ 𝑓(𝑋), so there exists 𝑥2 in 𝑋 such that 𝑓𝑥1 = 𝑔𝑥2. 

Continuing in this way, we construct a sequence {𝑥𝑛} in 𝑋 such that, 𝑓𝑥𝑛 = 𝑔𝑥𝑛+1 for all n, where 

𝑥𝑛+1 ∈ 𝑋 

Now, consider  

   𝑆(𝑔𝑥𝑛+1, 𝑔𝑥𝑛+1, 𝑔𝑥𝑛) = 𝑆(𝑓𝑥𝑛, 𝑓𝑥𝑛 , 𝑓𝑥𝑛−1) 

 ≤ 𝛼𝑆(𝑔𝑥𝑛, 𝑔𝑥𝑛 , 𝑔𝑥𝑛−1) + 𝛽[𝑆(𝑔𝑥𝑛 , 𝑔𝑥𝑛, 𝑓𝑥𝑛) + 𝑆(𝑔𝑥𝑛−1, 𝑔𝑥𝑛−1, 𝑓𝑥𝑛−1)] 

 = 𝛼𝑆(𝑔𝑥𝑛, 𝑔𝑥𝑛, 𝑔𝑥𝑛−1) + 𝛽[𝑆(𝑔𝑥𝑛, 𝑔𝑥𝑛 , 𝑔𝑥𝑛+1) + 𝑆(𝑔𝑥𝑛−1, 𝑔𝑥𝑛−1, 𝑔𝑥𝑛)] 



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 = 𝛼𝑆(𝑔𝑥𝑛, 𝑔𝑥𝑛, 𝑔𝑥𝑛−1) + 𝛽[𝑆(𝑔𝑥𝑛+1, 𝑔𝑥𝑛+1, 𝑔𝑥𝑛) + 𝑆(𝑔𝑥𝑛, 𝑔𝑥𝑛, 𝑔𝑥𝑛−1)] 

 = (𝛼 + 𝛽)𝑆(𝑔𝑥𝑛, 𝑔𝑥𝑛, 𝑔𝑥𝑛−1) + 𝛽𝑆(𝑔𝑥𝑛+1, 𝑔𝑥𝑛+1, 𝑔𝑥𝑛) 

 Rewriting;  

 (1 − 𝛽)𝑆(𝑔𝑥𝑛+1, 𝑔𝑥𝑛+1, 𝑔𝑥𝑛) ≤ (𝛼 + 𝛽)𝑆(𝑔𝑥𝑛, 𝑔𝑥𝑛, 𝑔𝑥𝑛−1) 

 This yields;  

 𝑆(𝑔𝑥𝑛+1, 𝑔𝑥𝑛+1, 𝑔𝑥𝑛) ≤
𝛼+𝛽

1−𝛽
𝑆(𝑔𝑥𝑛, 𝑔𝑥𝑛, 𝑔𝑥𝑛−1) 

                                          =   ℎ𝑆(𝑔𝑥𝑛, 𝑔𝑥𝑛, 𝑔𝑥𝑛−1) 

                                         ≤   ℎ𝑛𝑆(𝑔𝑥1, 𝑔𝑥1, 𝑔𝑥0), 

 where, ℎ =
𝛼+𝛽

1−𝛽
< 1 by assumption.  

By recursion:  

 𝑆(𝑔𝑥𝑛+1, 𝑔𝑥𝑛+1, 𝑔𝑥𝑛) ≤   ℎ𝑛𝑆(𝑔𝑥1, 𝑔𝑥1, 𝑔𝑥0), 

 Let 𝑧 = 𝑆(𝑔𝑥1, 𝑔𝑥1, 𝑔𝑥0). Then, for 𝑚 < 𝑛, we have  

  𝑆(𝑔𝑥𝑛, 𝑔𝑥𝑛, 𝑔𝑥𝑚) ≤   𝑆(𝑔𝑥𝑛, 𝑔𝑥𝑛, 𝑔𝑥𝑛−1) + 𝑆(𝑔𝑥𝑛, 𝑔𝑥𝑛, 𝑔𝑥𝑛−1) + 𝑆(𝑔𝑥𝑚, 𝑔𝑥𝑚 , 𝑔𝑥𝑛−1) 

        ≤   ℎ𝑛−1𝑆(𝑔𝑥1, 𝑔𝑥1, 𝑔𝑥0) + ℎ𝑛−1𝑆(𝑔𝑥1, 𝑔𝑥1, 𝑔𝑥0) + 𝑆(𝑔𝑥𝑚, 𝑔𝑥𝑚 , 𝑔𝑥𝑛−1) 

        =   2ℎ𝑛−1𝑆(𝑔𝑥1, 𝑔𝑥1, 𝑔𝑥0) + 𝑆(𝑔𝑥𝑚, 𝑔𝑥𝑚 , 𝑔𝑥𝑛−1) 

        ≤   2ℎ𝑛−1𝑧 + [𝑆(𝑔𝑥𝑚, 𝑔𝑥𝑚 , 𝑔𝑥𝑛−2) + 𝑆(𝑔𝑥𝑚 , 𝑔𝑥𝑚, 𝑔𝑥𝑛−2) 

                        + 𝑆(𝑔𝑥𝑛−1, 𝑔𝑥𝑛−1, 𝑔𝑥𝑛−2)] 

       =   2ℎ𝑛−1𝑧 + [2𝑆(𝑔𝑥𝑚 , 𝑔𝑥𝑚, 𝑔𝑥𝑛−2) + 𝑆(𝑔𝑥𝑛−1, 𝑔𝑥𝑛−1, 𝑔𝑥𝑛−2)] 

       ≤   2ℎ𝑛−1𝑧 + [2𝑆(𝑔𝑥𝑚, 𝑔𝑥𝑚, 𝑔𝑥𝑛−2) + ℎ𝑛−2𝑆(𝑔𝑥1, 𝑔𝑥1, 𝑔𝑥0)] 

      =   2ℎ𝑛−1𝑧 + ℎ𝑛−2𝑧 + 2𝑆(𝑔𝑥𝑚, 𝑔𝑥𝑚 , 𝑔𝑥𝑛−2) 

      ≤   2ℎ𝑛−1𝑧 + ℎ𝑛−2𝑧 + 2[2𝑆(𝑔𝑥𝑚 , 𝑔𝑥𝑚, 𝑔𝑥𝑛−3) + 𝑆(𝑔𝑥𝑛−2, 𝑔𝑥𝑛−2, 𝑔𝑥𝑛−3)] 

      ≤   2ℎ𝑛−1𝑧 + ℎ𝑛−2𝑧 + 4𝑆(𝑔𝑥𝑚, 𝑔𝑥𝑚, 𝑔𝑥𝑛−3) + 2ℎ𝑛−3𝑧 

      ≤   2ℎ𝑛−1𝑧 + ℎ𝑛−2𝑧 + 2ℎ𝑛−3𝑧 + 4[2𝑆(𝑔𝑥𝑚, 𝑔𝑥𝑚, 𝑔𝑥𝑛−4) 

                        +𝑆(𝑔𝑥𝑛−3, 𝑔𝑥𝑛−3, 𝑔𝑥𝑛−4)] 

      ≤   2ℎ𝑛−1𝑧 + ℎ𝑛−2𝑧 + 2ℎ𝑛−3𝑧 + 8𝑆(𝑔𝑥𝑚, 𝑔𝑥𝑚, 𝑔𝑥𝑛−4) + 4ℎ𝑛−4𝑧 

      =   2ℎ𝑛−1𝑧 + ℎ𝑛−2𝑧 + 2ℎ𝑛−3𝑧 + 4ℎ𝑛−4𝑧 + ⋯ 

      =   2ℎ𝑛−1𝑧 + ℎ𝑛−2𝑧 (  1 +
2

ℎ
+

4

ℎ2 +
8

ℎ3 + ⋯ ) 



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      =   2ℎ𝑛−1𝑧 + ℎ𝑛−2𝑧 (  1 + (
2

ℎ
)   + (

2

ℎ
)

2

+ (
2

ℎ
)

3

+ ⋯ ) 

       =   2ℎ𝑛−1𝑧 + ℎ𝑛−2𝑧 (  1 −
2

ℎ
)

−1

, 

 Since 0 < ℎ < 1, by taking limit as 𝑛 → ∞, we obtain,  

 lim
𝑛→∞

𝑆(𝑔𝑥𝑛, 𝑔𝑥𝑛, 𝑔𝑥𝑚) = 0. 

Hence, we obtain that the sequence {𝑔𝑥𝑛} is a Cauchy sequence in 𝑔(𝑋). Given that 𝑔(𝑋) is complete, 

there exist a point 𝑞 in 𝑔(𝑋) such that 𝑔𝑥𝑛 → 𝑞, as 𝑛 → ∞. Since, 𝑞 ∈ 𝑔(𝑋) there exists a point 𝑝 in 

𝑋 with 𝑔𝑝 = 𝑞. Now,  

   𝑆(𝑓𝑥𝑛, 𝑓𝑥𝑛, 𝑓𝑝) ≤ 𝛼𝑆(𝑔𝑥𝑛, 𝑔𝑥𝑛, 𝑔𝑝) + 𝛽[𝑆(𝑔𝑥𝑛, 𝑔𝑥𝑛 , 𝑓𝑥𝑛) + 𝑆(𝑔𝑝, 𝑔𝑝, 𝑓𝑝)] 

 = 𝛼𝑆(𝑔𝑥𝑛, 𝑔𝑥𝑛, 𝑔𝑝) + 𝛽[𝑆(𝑔𝑥𝑛, 𝑔𝑥𝑛, 𝑔𝑥𝑛+1) + 𝑆(𝑔𝑝, 𝑔𝑝, 𝑓𝑝)]. 

 Taking the limit as 𝑛 → ∞ and using continuity and convergence  

   𝑆(𝑞, 𝑞, 𝑓𝑝) ≤ 𝛼𝑆(𝑞, 𝑞, 𝑞) + 𝛽[𝑆(𝑞, 𝑞, 𝑞) + 𝑆(𝑞, 𝑞, 𝑓𝑝)] 

 Which implies  

 𝑆(𝑞, 𝑞, 𝑓𝑝) ≤ 𝛽𝑆(𝑞, 𝑞, 𝑓𝑝). 

 Since, 0 < 𝛽 < 1, the only solution is 𝑆(𝑞, 𝑞, 𝑓𝑝) = 0, 𝑓𝑝 = 𝑞. But also, 𝑔𝑝 = 𝑞 so 𝑓𝑝 = 𝑔𝑝 = 𝑞 

showing that, 𝑞 is the point of coincidence of 𝑓 and 𝑔. To prove uniqueness, suppose 𝑤 ∈ 𝑋 is another 

coincidence point, i.e., 𝑓𝑤 = 𝑔𝑤 = 𝑤. Then  

         𝑆(𝑞, 𝑞, 𝑤) = 𝑆(𝑓𝑝, 𝑓𝑝, 𝑓𝑤) 

 ≤ 𝛼𝑆(𝑔𝑝, 𝑔𝑝, 𝑔𝑤) + 𝛽[𝑆(𝑔𝑝, 𝑔𝑝, 𝑓𝑝) + 𝑆(𝑔𝑤, 𝑔𝑤, 𝑓𝑤)] 

 = 𝛼𝑆(𝑞, 𝑞, 𝑤) + 𝛽[𝑆(𝑞, 𝑞, 𝑞) + 𝑆(𝑤, 𝑤, 𝑤)] 

 = 𝛼𝑆(𝑞, 𝑞, 𝑤). 

Since, 0 < 𝛼 < 1, we can conclude that 𝑆(𝑞, 𝑞, 𝑤) = 0𝑞 = 𝑤. Hence, the coincidence point is unique. 

Finally since, 𝑓 and 𝑔 are weakly compatible, by Proposition (3.6), they have a unique common fixed 

point in 𝑋.  

Theorem 4.3  Let (𝑋, 𝑆) be a complete rectangular S-metric space, and let 𝑓, 𝑔: 𝑋 → 𝑋 be two self-

maps satisfying the inequality  

𝑆(𝑓𝑥, 𝑓𝑥, 𝑔𝑦) ≤ 𝛼𝑆(𝑥, 𝑥, 𝑦) + 𝛽[𝑆(𝑥, 𝑥, 𝑓𝑥) + 𝑆(𝑦, 𝑦, 𝑔𝑦)] + 𝛾[𝑆(𝑥, 𝑥, 𝑔𝑦) + 𝑆(𝑦, 𝑦, 𝑓𝑥)] 

 for all 𝑥, 𝑦 ∈ 𝑋, where 𝛼, 𝛽, 𝛾 > 0 and 𝛼 + 2𝛽 + 3𝛾 < 1. 

 Then 𝑓 and 𝑔 have a unique common fixed point in 𝑋. Moreover, every fixed point of 𝑓 is also a fixed 

point of 𝑔, and conversely.  

Proof. Let 𝑥0 be an arbitrary starting point of 𝑋, and define the sequence {𝑥𝑛} by  

 𝑥2𝑛+1 = 𝑓𝑥2𝑛, 𝑥2𝑛+2 = 𝑔𝑥2𝑛+1, 𝑛 = 0,1,2, ⋯ 



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Now,  

   𝑆(𝑥2𝑛+1, 𝑥2𝑛+1, 𝑥2𝑛+2) = 𝑆(𝑓𝑥2𝑛, 𝑓𝑥2𝑛, 𝑔𝑥2𝑛+1) 

 ≤ 𝛼𝑆(𝑥2𝑛, 𝑥2𝑛, 𝑥2𝑛+1) + 𝛽[𝑆(𝑥2𝑛, 𝑥2𝑛, 𝑓𝑥2𝑛) + 𝑆(𝑥2𝑛+1, 𝑥2𝑛+1, 𝑔𝑥2𝑛+1)] 

 +𝛾[𝑆(𝑥2𝑛, 𝑥2𝑛, 𝑔𝑥2𝑛+1) + 𝑆(𝑥2𝑛+1, 𝑥2𝑛+1, 𝑓𝑥2𝑛)] 

 = 𝛼𝑆(𝑥2𝑛, 𝑥2𝑛, 𝑥2𝑛+1) + 𝛽[𝑆(𝑥2𝑛, 𝑥2𝑛, 𝑥2𝑛+1) + 𝑆(𝑥2𝑛+1, 𝑥2𝑛+1, 𝑥2𝑛+2)] 

 +𝛾[𝑆(𝑥2𝑛, 𝑥2𝑛, 𝑥2𝑛+2) + 𝑆(𝑥2𝑛+1, 𝑥2𝑛+1, 𝑥2𝑛+1)] 

 = 𝛼𝑆(𝑥2𝑛, 𝑥2𝑛, 𝑥2𝑛+1) + 𝛽[𝑆(𝑥2𝑛, 𝑥2𝑛, 𝑥2𝑛+1) + 𝑆(𝑥2𝑛+1, 𝑥2𝑛+1, 𝑥2𝑛+2)] 

 +𝛾[𝑆(𝑥2𝑛, 𝑥2𝑛, 𝑥2𝑛+1) + 𝑆(𝑥2𝑛, 𝑥2𝑛, 𝑥2𝑛+1) + 𝑆(𝑥2𝑛+2, 𝑥2𝑛+2, 𝑥2𝑛+1)] 

 = 𝛼𝑆(𝑥2𝑛, 𝑥2𝑛, 𝑥2𝑛+1) + 𝛽[𝑆(𝑥2𝑛, 𝑥2𝑛, 𝑥2𝑛+1) + 𝑆(𝑥2𝑛+1, 𝑥2𝑛+1, 𝑥2𝑛+2)] 

 +𝛾[2𝑆(𝑥2𝑛, 𝑥2𝑛, 𝑥2𝑛+1) + 𝑆(𝑥2𝑛+1, 𝑥2𝑛+1, 𝑥2𝑛+2)] 

 = (𝛼 + 𝛽 + 2𝛾)𝑆(𝑥2𝑛, 𝑥2𝑛, 𝑥2𝑛+1) + (𝛽 + 𝛾)𝑆(𝑥2𝑛+1, 𝑥2𝑛+1, 𝑥2𝑛+2) 

 Rearranging gives  

 (1 − (𝛽 + 𝛾))𝑆(𝑥2𝑛+1, 𝑥2𝑛+1, 𝑥2𝑛+2) ≤ (𝛼 + 𝛽 + 2𝛾)𝑆(𝑥2𝑛, 𝑥2𝑛, 𝑥2𝑛+1) 

 ≤
(𝛼+𝛽+2𝛾)

(1−(𝛽+𝛾))
𝑆(𝑥2𝑛, 𝑥2𝑛, 𝑥2𝑛+1) 

 By taking 𝛿 =
(𝛼+𝛽+2𝛾)

(1−(𝛽+𝛾))
< 1, We get  

 𝑆(𝑥2𝑛+1, 𝑥2𝑛+1, 𝑥2𝑛+2) ≤ 𝛿𝑆(𝑥2𝑛, 𝑥2𝑛, 𝑥2𝑛+1) 

 Applying recursively,  

 𝑆(𝑥2𝑛+1, 𝑥2𝑛+1, 𝑥2𝑛+2) ≤ 𝛿2𝑛+1𝑆(𝑥0, 𝑥0, 𝑥1), 

 For 𝑚, 𝑛 ∈ ℕ with 𝑚 < 𝑛 and some 𝑁 ∈ ℕ, we estimate  

   𝑆(𝑥𝑛, 𝑥𝑛, 𝑥𝑚) ≤   𝑆(𝑥𝑛, 𝑥𝑛 , 𝑥𝑛+1) + 𝑆(𝑥𝑛, 𝑥𝑛, 𝑥𝑛+1) + 𝑆(𝑥𝑚, 𝑥𝑚 , 𝑥𝑛+1) 

 = 2𝑆(𝑥𝑛, 𝑥𝑛, 𝑥𝑛+1) + 𝑆(𝑥𝑚 , 𝑥𝑚, 𝑥𝑛+1) 

 By recursively applying the triangle-like inequality, we proceed as,  

   𝑆(𝑥𝑛, 𝑥𝑛, 𝑥𝑚) ≤ 2𝛿𝑛𝑆(𝑥0, 𝑥0, 𝑥1) + [𝑆(𝑥𝑚, 𝑥𝑚, 𝑥𝑛+2) + 𝑆(𝑥𝑚 , 𝑥𝑚, 𝑥𝑛+2) +

𝑆(𝑥𝑛+1, 𝑥𝑛+1, 𝑥𝑛+2)] 

 By taking 𝑆(𝑔𝑥0, 𝑔𝑥0, 𝑔𝑥1) = 𝑧, this yields  

   𝑆(𝑥𝑛, 𝑥𝑛, 𝑥𝑚) ≤ 2𝛿𝑛𝑧 + [2𝑆(𝑥𝑚, 𝑥𝑚, 𝑥𝑛+2) + 𝑆(𝑥𝑛+1, 𝑥𝑛+1, 𝑥𝑛+2)] 

 ≤ 2𝛿𝑛𝑧 + [2𝑆(𝑥𝑚, 𝑥𝑚, 𝑥𝑛+2) + 𝛿𝑛+1𝑧] 

 ≤ 2𝛿𝑛𝑧 + 𝛿𝑛+1𝑧 + 2[𝑆(𝑥𝑚, 𝑥𝑚 , 𝑥𝑛+3) + 𝑆(𝑥𝑚, 𝑥𝑚 , 𝑥𝑛+3) + 𝑆(𝑥𝑛+2, 𝑥𝑛+2, 𝑥𝑛+3)] 

 ≤ 2𝛿𝑛𝑧 + 𝛿𝑛+1𝑧 + 2[2𝑆(𝑥𝑚, 𝑥𝑚 , 𝑥𝑛+3) + 𝛿𝑛+2𝑧] 



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 ≤ 2𝛿𝑛𝑧 + 𝛿𝑛+1𝑧 + 2𝛿𝑛+2𝑧 + 4[𝑆(𝑥𝑚, 𝑥𝑚, 𝑥𝑛+4) + 𝑆(𝑥𝑚, 𝑥𝑚, 𝑥𝑛+4) 

 ≤ 2𝛿𝑛𝑧 + 𝛿𝑛+1𝑧 + 2𝛿𝑛+2𝑧 + 4[2𝑆(𝑥𝑚, 𝑥𝑚 , 𝑥𝑛+4) + 𝛿𝑛+3𝑧] 

               +𝑆(𝑥𝑛+3, 𝑥𝑛+3, 𝑥𝑛+4)] 

 ≤ 2𝛿𝑛𝑧 + 𝛿𝑛+1𝑧 + 2𝛿𝑛+2𝑧 + 4𝛿𝑛+3𝑧 + 8𝛿𝑛+4𝑧 + ⋯ 

 = 2𝛿𝑛𝑧 + 𝛿𝑛+1𝑧(1 + 2𝛿 + 4𝛿2 + 8𝛿3 + ⋯ ) 

 =   2𝛿𝑛𝑧 + 𝛿𝑛+1𝑧(  1 + (2𝛿)   + (2𝛿)2 + (2𝛿)3 + ⋯ ) 

 =   2𝛿𝑛𝑧 + 𝛿𝑛+1𝑧 (  
1

1−2𝛿
), 

 As 𝛿 < 1, it follows that  

 lim
𝑛→∞

𝑆(𝑥𝑛, 𝑥𝑛, 𝑥𝑚) = 0. 

Thus, the sequence {𝑥𝑛} is a Cauchy sequence in 𝑋. Completeness of 𝑋 implies that there exists a point 

𝑝 in 𝑋 such that 𝑥𝑛 → 𝑝 as 𝑛 → ∞.  

To show that 𝑝 is a fixed point of 𝑔, we consider  

  𝑆(𝑥𝑛, 𝑥𝑛, 𝑔𝑝) ≤   𝑆(𝑥𝑛, 𝑥𝑛, 𝑥𝑛+1) + 𝑆(𝑥𝑛 , 𝑥𝑛, 𝑥𝑛+1) + 𝑆(𝑔𝑝, 𝑔𝑝, 𝑥𝑛+1) 

 = 2𝑆(𝑥𝑛, 𝑥𝑛, 𝑥𝑛+1) + 𝑆(𝑥𝑛+1, 𝑥𝑛+1, 𝑔𝑝) 

 ≤ 2𝛿𝑛𝑧 + [𝑆(𝑥𝑛+1, 𝑥𝑛+1, 𝑥𝑛+2) + 𝑆(𝑥𝑛+1, 𝑥𝑛+1, 𝑥𝑛+2) + 𝑆(𝑔𝑝, 𝑔𝑝, 𝑥𝑛+2)] 

 = 2𝛿𝑛𝑧 + [2𝑆(𝑥𝑛+1, 𝑥𝑛+1, 𝑥𝑛+2) + 𝑆(𝑔𝑝, 𝑔𝑝, 𝑥𝑛+2)] 

 ≤ 2𝛿𝑛𝑧 + [2𝛿𝑛+1𝑧 + 𝑆(𝑥𝑛+2, 𝑥𝑛+2, 𝑔𝑝)] 

 ≤ 2𝛿𝑛𝑧 + 2𝛿𝑛+1𝑧 + [𝑆(𝑥𝑛+2, 𝑥𝑛+2, 𝑥𝑛+3) + 𝑆(𝑥𝑛+2, 𝑥𝑛+2, 𝑥𝑛+3) + 𝑆(𝑔𝑝, 𝑔𝑝, 𝑥𝑛+3)] 

 = 2𝛿𝑛𝑧 + 2𝛿𝑛+1𝑧 + [2𝑆(𝑥𝑛+2, 𝑥𝑛+2, 𝑥𝑛+3) + 𝑆(𝑔𝑝, 𝑔𝑝, 𝑥𝑛+3)] 

 ≤ 2𝛿𝑛𝑧 + 2𝛿𝑛+1𝑧 + 2𝛿𝑛+2𝑧 + 𝑆(𝑥𝑛+3, 𝑥𝑛+3, 𝑔𝑝) 

 ≤ 2𝛿𝑛𝑧 + 2𝛿𝑛+1𝑧 + 2𝛿𝑛+2𝑧 + [𝑆(𝑥𝑛+3, 𝑥𝑛+3, 𝑥𝑛+4) + 𝑆(𝑥𝑛+3, 𝑥𝑛+3, 𝑥𝑛+4) 

                  +𝑆(𝑔𝑝, 𝑔𝑝, 𝑥𝑛+4)] 

 = 2𝛿𝑛𝑧 + 2𝛿𝑛+1𝑧 + 2𝛿𝑛+2𝑧 + [2𝑆(𝑥𝑛+3, 𝑥𝑛+3, 𝑥𝑛+4) + 𝑆(𝑔𝑝, 𝑔𝑝, 𝑥𝑛+4)] 

 ≤ 2𝛿𝑛𝑧 + 2𝛿𝑛+1𝑧 + 2𝛿𝑛+2𝑧 + 2𝛿𝑛+3𝑧 + 𝑆(𝑥𝑛+4, 𝑥𝑛+4, 𝑔𝑝) 

 = 2𝛿𝑛𝑧 + 2𝛿𝑛+1𝑧 + 2𝛿𝑛+2𝑧 + 2𝛿𝑛+3𝑧 + ⋯ 

 = 2𝛿𝑛𝑧(1 + 𝛿 + 𝛿2 + 𝛿3 + ⋯ ) 

Therefore,  𝑆(𝑥𝑛 , 𝑥𝑛, 𝑔𝑝) ≤   2𝛿𝑛𝑧 (  
1

1−𝛿
), 

 where 𝑧 = 𝑆(𝑔𝑥0, 𝑔𝑥0, 𝑔𝑥1). Since 𝛿 < 1, as 𝑛 → ∞. Thus,  lim𝑛→∞𝑆(𝑝, 𝑝, 𝑔𝑝) = 0 and 𝑔𝑝 = 𝑝. 

Also, Now to show 𝑓𝑝 = 𝑝, we proceed as,  



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  𝑆(𝑓𝑝, 𝑓𝑝, 𝑝) = 𝑆(𝑓𝑝, 𝑓𝑝, 𝑔𝑝) 

 ≤ 𝛼𝑆(𝑝, 𝑝, 𝑝) + 𝛽[𝑆(𝑝, 𝑝, 𝑓𝑝) + 𝑆(𝑝, 𝑝, 𝑔𝑝)] + 𝛾[𝑆(𝑝, 𝑝, 𝑔𝑝) + 𝑆(𝑝, 𝑝, 𝑓𝑝)] 

 = 𝛽[𝑆(𝑝, 𝑝, 𝑓𝑝) + 𝑆(𝑝, 𝑝, 𝑝)] + 𝛾[𝑆(𝑝, 𝑝, 𝑝) + 𝑆(𝑝, 𝑝, 𝑓𝑝)] 

 = (𝛽 + 𝛾)𝑆(𝑝, 𝑝, 𝑓𝑝) 

 = (𝛽 + 𝛾)𝑆(𝑓𝑝, 𝑓𝑝, 𝑝). 

 Since, 𝛽, 𝛾 > 0. This is only possible if 𝑆(𝑓𝑝, 𝑓𝑝, 𝑝) = 0 and 𝑓𝑝 = 𝑝. Therefore, we can conclude 

that, 𝑓𝑝 = 𝑔𝑝 = 𝑝. Hence, 𝑓 and 𝑔 have a common fixed point in 𝑝 ∈ 𝑋. 

Uniqueness: Suppose another common fixed point 𝑤 ∈ 𝑋 exists i.e. 𝑓𝑤 = 𝑔𝑤 = 𝑤  

      𝑆(𝑝, 𝑝, 𝑤) = 𝑆(𝑓𝑝, 𝑓𝑝, 𝑔𝑤) 

            ≤ 𝛼𝑆(𝑝, 𝑝, 𝑤) + 𝛽[𝑆(𝑝, 𝑝, 𝑓𝑝) + 𝑆(𝑤, 𝑤, 𝑔𝑤)] + 𝛾[𝑆(𝑝, 𝑝, 𝑔𝑤) + 𝑆(𝑤, 𝑤, 𝑓𝑝)] 

 Substituting 𝑓𝑝 = 𝑔𝑝 = 𝑝 and 𝑓𝑤 = 𝑔𝑤 = 𝑤, we obtain  

         𝑆(𝑝, 𝑝, 𝑤) = 𝛼𝑆(𝑝, 𝑝, 𝑤) + 𝛽[𝑆(𝑝, 𝑝, 𝑝) + 𝑆(𝑤, 𝑤, 𝑤)] + 𝛾[𝑆(𝑝, 𝑝, 𝑤) + 𝑆(𝑤, 𝑤, 𝑝)] 

 = 𝛼𝑆(𝑝, 𝑝, 𝑤) + 𝛾[𝑆(𝑝, 𝑝, 𝑤) + 𝑆(𝑝, 𝑝, 𝑤)] 

 = 𝛼𝑆(𝑝, 𝑝, 𝑤) + 2𝛾𝑆(𝑝, 𝑝, 𝑤) 

 = (𝛼 + 2𝛾)𝑆(𝑝, 𝑝, 𝑤). 

 Since, 𝛼, 𝛾 > 0, the only possibility is 𝑆(𝑝, 𝑝, 𝑤) = 0. Hence, 𝑝 = 𝑤. Hence, the common fixed point 

is unique, and the proof is complete. 

Corollary 4.4  Let (𝑋, 𝑆) be a complete rectangular 𝑆-metric space and 𝑓, 𝑔: 𝑋 → 𝑋 be a self 

mappings such that  

 𝑆(𝑓𝑥, 𝑓𝑥, 𝑓𝑦) ≤ ℎ𝑆(𝑔𝑥, 𝑔𝑥, 𝑔𝑦), where ℎ ∈ [0,1).  

Also, following conditions holds   

    • 𝑓(𝑋) ⊆ 𝑔(𝑋),  

    • If 𝑔(𝑋) is complete.  

 Then 𝑓 and 𝑔 have a unique common fixed point in 𝑋.  

 Proof. The given inequality is a special case of the condition in Theorem (4.2), obtained by choosing 

𝛼 = ℎ and 𝛽 = 0. Hence, the result follows directly from Theorem (4.2). 

Corollary 4.5  Let (𝑋, 𝑆) be a complete rectangular 𝑆-metric space and 𝑓, 𝑔: 𝑋 → 𝑋 be a self 

mapping satisfies the condition,  

 𝑆(𝑓𝑥, 𝑓𝑥, 𝑓𝑦) ≤ 𝑞[𝑆(𝑔𝑥, 𝑔𝑥, 𝑓𝑦) + 𝑆(𝑔𝑦, 𝑔𝑦, 𝑓𝑥)],  where 𝑞 ∈ [0,
1

2
). 

Assume further that,   



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    • 𝑓(𝑋) ⊆ 𝑔(𝑋)  

    • If 𝑔(𝑋) is complete subset of 𝑋.  

Then there exists a unique fixed point in 𝑋 for 𝑓 and 𝑔.  

Corollary 4.6  Let (𝑋, 𝑆) be a complete rectangular 𝑆-metric space and 𝑓, 𝑔: 𝑋 → 𝑋 be a self mapping 

satisfies the condition,  

 𝑆(𝑓𝑥, 𝑓𝑥, 𝑓𝑦) ≤ 𝛽[𝑆(𝑔𝑥, 𝑔𝑥, 𝑓𝑥) + 𝑆(𝑔𝑦, 𝑔𝑦, 𝑓𝑦)], where 𝑞 ∈ [0,
1

2
).  

Suppose further that,   

    • 𝑓(𝑋) ⊆ 𝑔(𝑋),  

    • If 𝑔(𝑋) is complete.  

Then 𝑓 and 𝑔 have a unique common fixed point in 𝑋.  

Proof. This inequality results from Theorem (4.2) by setting 𝛼 = 0. The conclusion then directly 

follows from Theorem(4.2). 

Example 4.7  Let 𝑋 = ℝ and define the function 𝑆: 𝑋 × 𝑋 × 𝑋 → ℝ+ by 𝑆(𝑥, 𝑦, 𝑧) = 𝑥𝑦 + 𝑧2 for all 

𝑥, 𝑦 ∈ ℝ. Then (𝑋, 𝑆) forms a 𝑆-metric space.  

Now, define the mappings 𝑓: 𝑋 → 𝑋 as follows 𝑓(𝑥) =
𝑥

4
 and 𝑔(𝑥) =

𝑥

2
 for all 𝑥 ∈ [0,1]. We then 

compute  

  𝑆(𝑓𝑥, 𝑓𝑥, 𝑓𝑦) = 𝑆 (
𝑥

4
,
𝑥

4
,
𝑦

4
) 

 = (
𝑥

4
)(

𝑥

4
) + (

𝑦

4
)

2

 

 =
𝑥2

16
+

𝑦2

16
 

 =
1

16
(𝑥2 + 𝑦2) 

 =
1

4
(

1

4
(𝑥2 + 𝑦2)) 

 =
1

4
(𝑆(𝑔𝑥, 𝑔𝑥, 𝑔𝑦)) 

 = ℎ(𝑆(𝑔𝑥, 𝑔𝑥, 𝑔𝑦)), where ℎ =
1

4
< 1.  

Thus 𝑓 satisfies all the conditions of Corollary (4.4). Therefore, by applying Corollary(4.4). It follows 

that 𝑓 and 𝑔 have a unique common fixed point in 𝑋. Clearly, 0 ∈ 𝑋 is this unique common fixed 

point.  

5. Conclusion 

The results presented in this paper contribute to the ongoing development of fixed point theory by 

extending classical results to the broader context of rectangular S-metric spaces. By introducing new 

common fixed point theorems for weakly compatible mappings under contractive conditions, we 

provide a more general framework that encompasses several existing results as special cases. The 

illustrative examples confirm the applicability of our theorems and highlight the usefulness of 



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Vol 32 No. 10s (2025) 

  

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rectangular S-metric spaces in analysing complex mapping behaviours. These findings open new 

avenues for further research in generalized metric spaces and their applications. 

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