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Extension and Generalization of Banach Contraction in Metric and in 

Menger Space 

 

Ajay Kumar Chaudhary1, Chet Raj Bhatta2, and Uday Kumar Karn3* 

1Department of Mathematics, Tri-Chandra Multiple Campus, Tribhuvan University, Kathmandu, Nepal 

2Central Department of Mathematics, Kirtipur, Tribhuvan University, Kathmandu, Nepal 

3Department of Mathematics, Patan Multiple Campus, and PhD Scholar of Tribhuvan University, Kathmandu, Nepal 

*Corresponding e-mail: udaykarn984@gmail.com 

 

Article History: 

Received: 21-07-2024 

Revised: 01-09-2024 

Accepted: 14-09-2024 

Abstract:  

The root of metric fixed point theory is Stefen Banach's contraction mapping, a research 

source for shrinking the distance between two points in space. As a source, many authors 

have introduced many contraction mappings as extensions and generalizations of Banach 

contraction and established fixed point theorems under the property that each such mapping 

in complete metric and Menger space has a unique fixed point. 

This article presents updated results of Banach contraction generalization and extension 

forms in metric and Menger space which helps the comparative and interrelationship study 

in these spaces. 

Keywords: Banach contraction, distribution function, Triangular norm and Menger space. 

 

1. Introduction 

In the nineteenth century, existence theorems emerged in analysis when basic mathematical facts were 

considered critically. The first mathematician to demonstrate the existence of a theorem for differential 

equation systems with analytic right-hand sides was A. L. Cauchy [9]. E. Picard [43] suggested the 

method of successive approximations to prove the existence of the theorems. In 1922, Birkhoff and 

Kellogg[4] gave proof of the classical existence theorem for the equation 
𝑑𝑦

𝑑𝑥
 =  𝑓(𝑥, 𝑦) in function 

spaces. However, the most elementary and by far the most fruitful method for proving theorems on the 

existence and uniqueness of solutions is the principle formulated and proved by S. Banach [1]1920 in 

his PhD thesis published in 1922. Although the idea of successive approximations in some concrete 

situations (solving differential and integral equations) appears in some works of E. Picard, R. 

Caccioppoli, et al. [10], it was Banach who placed it in the right abstract setting, making it suitable for 

a wide range of applications. This principle results from the geometric interpretation of Picard's method 

of successive approximations. Several generalizations of this principle have appeared and many 

authors have done their comparative study see references [[35], [43], [44], [48]] in metric space. 

Menger [38] introduced the probabilistic metric space in 1942 to generalize Frechet's [25] metric space 

by replacing the distance function with the distribution function. It helps to solve uncertainty cases 

regarding the distance between two points in space. This probabilistic metric space became active for 

mathematicians after the great contribution in this space from Schweizer and A. Sklar. [49] 



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In 1966, Sehgal [50] introduced the first generalized form of Banach contraction in probabilistic metric 

space and established the fixed point theorem in complete Menger probabilistic metric space [51], and 

Hicks [32] defined another contraction in 1983. After that many authors worked in this space and 

introduced different variants of contraction in this space, see references [[13 – 17], [20], [26], [28], 

[39], [41]]. 

In this paper, we shall update the study of various general contractive results to the Banach contraction 

principle. Each mapping has a property to establish unique fixed points in complete metric and Menger 

space. 

2. Preliminaries: 

Definition 2.1 [25]:  Let  𝑋 be an abstract set and 𝑑  be a distance function from 𝑋 × 𝑋 →  ℝ+. Then, 

an ordered pair  (𝑋, 𝑑) is said to be metric space if 𝑑 satisfies the following conditions for all 

𝑥, 𝑦, 𝑧 ∈ 𝑋: 

(i)   𝑑(𝑥, 𝑦) ≥ 0;  

(ii)  𝑑(𝑥, 𝑦) = 0  if and only if  𝑥 =  𝑦; 

(iii)  𝑑(𝑥, 𝑦) = 𝑑(𝑦, 𝑥); and 

(iv)  𝑑(𝑥, 𝑦) ≤ 𝑑(𝑥, 𝑧) + 𝑑(𝑧, 𝑦)   

Here, (i) and (ii) guarantee that the distance between any two points of 𝑋 is always positive and only 

zero when the points coincide. (iii) assures that the order of measurement of the distance between two 

points is insignificant. (iv) is a statement of the familiar triangular inequality.   

Definition 2.2 [15]: A mapping  F: ℝ → ℝ+ is said to be a distribution function  if it is  

non-decreasing and left continuous with 𝑖𝑛𝑓𝑥∈ℝ 𝐹 (𝑥)  =  0, and  𝑠𝑢𝑝𝑥∈ℝ 𝐹 (𝑥) =  1, where ℝ+  

denotes the set of non-negative real numbers. 

Definition 2.3 [15]: Let  𝑋 be a non-empty set and  𝐹: 𝑋 × 𝑋 →  𝐿 (set of all distribution functions) 

be a distribution function. Then, a pair (𝑋, 𝐹) is said to be a Probabilistic metric space (abbreviated 

as Pm-space) if the distribution function 𝐹 (𝑥, 𝑦), also denoted by 𝐹𝑥,𝑦 satisfies the following 

conditions: 

             (i)        𝐹𝑥,𝑦(𝑡) = 1 for every 𝑡 >  0 if and only if  𝑥 =  𝑦; 

 (ii)       𝐹𝑥,𝑦(0) = 0  for every  𝑥, 𝑦 ∈ 𝑋;  

            (iii)      𝐹𝑥,𝑦(𝑡) = 𝐹𝑦,𝑥(𝑡)   for every 𝑥, 𝑦 ∈ 𝑋; and  

 (iv)      𝐹𝑥,𝑧(𝑝 + 𝑞) = 1 if and only if 𝐹𝑥,𝑦(𝑝) = 1 and 𝐹𝑦,𝑧(𝑞) = 1. 

Example 2.1: Let (𝑋, 𝑑) be metric space where 𝑋 = [0, 5]with usual metric 𝑑(𝑥, 𝑦) = |𝑥 − 𝑦| and 

distribution function 𝐹 defined as: 

                          𝐹𝑥,𝑦(𝑡) =  {𝑒 
|𝑥−𝑦|

𝑡 ,   𝑖𝑓 𝑡 > 0,
0,   𝑖𝑓 𝑡 = 0.

   for all 𝑥, 𝑦 ∈ 𝑋.  Then,  (𝑋, 𝐹) be Pm space. 

Definition 2.4 [28]: A function  𝑇 ∶  [0, 1]  ×  [0, 1]  → [0, 1] is referred to as a Triangular norm   



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             (shortly T-norm) if it satisfies the following conditions: 

            T1:       𝑇 (0, 0) =  0;  

            T2:     𝑇 (𝑎, 1) = 𝑎  for all 𝑎 ∈ [0, 1]; 

            T3:     𝑇 (𝑎, 𝑏) =  𝑡 (𝑏, 𝑎) for all 𝑎, 𝑏 ∈  [0, 1]; 

            T4:        𝑖𝑓 𝑎 ≤  𝑐, 𝑏 ≤  𝑑 then 𝑇 (𝑎, 𝑏)  ≤  𝑇(𝑐, 𝑑) and  

           T5: 𝑇 (𝑡 (𝑎, 𝑏), 𝑐) =  𝑇 (𝑎, 𝑡 (𝑏, 𝑐)), where 𝑎, 𝑏, 𝑐, 𝑑 ∈ [0, 1]. 

Definition 2.5. [17] A triplet (𝑋, 𝐹, 𝑇) is said to be Menger space, where 𝑋 is a non-empty set, 𝐹 be a 

distribution function, and 𝑇 is a 𝑡-norm such that the following are satisfied 

 for every 𝑡, 𝑠 >  0 & 𝑥, 𝑦, 𝑧 ∈ 𝑋:   

             (i)        𝐹𝑥,𝑦(𝑡) = 1 for every 𝑥 >  0 if and only if  𝑥 =  𝑦, 

            (ii)       𝐹𝑥,𝑦(0) = 0  ; 

           (iii)      𝐹𝑥,𝑦(𝑡) = 𝐹𝑦,𝑥(𝑡), and  

           (iv)      𝐹𝑥,𝑧(𝑡 + 𝑠) ≥ 𝑇(𝐹𝑥,𝑦(𝑡), 𝐹𝑦,𝑧(𝑠)).   

Definition 2.6.[36] Let 𝑇: 𝑋 → 𝑋 be a mapping in metric space (𝑋, 𝑑). Given 𝑥 ∈ 𝑋, 𝑂(𝑥) =

{𝑓𝑛: 𝑛 ∈ 𝑁}, �̅�(𝑥) be its closure. A point 𝑥 ∈ 𝑋 is said to be regular for 𝑇 if 𝑑𝑖𝑎𝑚 𝑂(𝑥) < ∞. 

Given, 𝑥, 𝑦 ∈ 𝑋, let  𝑚(𝑥, 𝑦)  =  𝑚𝑎𝑥{𝑑(𝑥, 𝑦), 𝑑(𝑥, 𝑓𝑥), 𝑑(𝑦, 𝑓𝑦), 𝑑(𝑥, 𝑓𝑦), 𝑑(𝑦, 𝑓𝑥)}, 

And        𝛿(𝑥, 𝑦)  =  𝑑𝑖𝑎𝑚{𝑂(𝑥)  ∪  𝑂(𝑦)}. 

Definition 2.7.[1]  Let  (𝑋, 𝑑) be a metric space. Then, a mapping  𝑇: 𝑋 → 𝑋 is said to be contraction 

mapping if there exists a number  λ ∈ [0,1) such that for every 𝑥, 𝑦 ∈ 𝑋, we have 

                                                𝑑(𝑇𝑥, 𝑇𝑦) ≤ λ 𝑑(𝑥, 𝑦).       (Contraction, in 1922) 

Example 2.2 [16]:  Let function  𝑓: [0,2] → [0,2] be defined by  

                          f(x) =   
0 for x ∈ [0,1] 

 1    for x ∈ (1, 2]
  

                       Then, 𝑓2 is a contraction but 𝑓 is not a contraction. 

3. Various generalizations and extensions of Banach contraction are: 

Definition 3.1. A mapping 𝑇: 𝑋 → 𝑋   of a metric space (𝑋, 𝑑) is said to be contractive if        

                          𝑑(𝑇𝑥, 𝑇𝑦) < 𝑑(𝑥, 𝑦), for every 𝑥 ≠ 𝑦 ∈ 𝑋.   (Edelstein in 1962, [24])  

It is extended forms as: 

(i) 𝑑(𝑇𝑥, 𝑇𝑦) < 𝑚(𝑥, 𝑦)                                             (Rhodes in 1977, [48]) 

(ii) If 𝑥 and 𝑦 are regular, 𝑑(𝑇𝑥, 𝑇𝑦)  < 𝛿(𝑥, 𝑦).         ( Park in 1980, [43]) 



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Remark 1. A contractive map is continuous; if such a mapping has a fixed point, then this fixed point 

is unique. While the condition 𝑑(𝑇𝑥, 𝑇𝑦)  <  𝑑(𝑥, 𝑦) is sufficient to assure that 𝑇 has a fixed point but 

it is too weak to guarantee the existence of one as will be seen from the following examples: 

Example 3.1. Let 𝑇: ℝ → ℝ   be defined by 𝑇(𝑥) =  𝑥 +
𝜋

2
−  𝑇𝑎𝑛−1𝑥 

                  Since  𝑇𝑎𝑛−1𝑥 <
𝜋

2
 for every 𝑥, the operator 𝑇 has no fixed point although 𝑇 is a 

contractive map for 𝑇′(𝑥) = 1 −  
1

1+𝑥2
< 1. 

Example 3.2. Let 𝑇: ℝ → ℝ    be defined by 𝑇(𝑥) = ln(1 + 𝑒𝑥), 

Differentiating, we obtain 

                                                                 𝑇′(𝑥) =  
𝑒𝑥

1+𝑒𝑥 < 1. 

i.e. 𝑇 is a Contractive mapping, and it is easy to see that 𝑇 has no fixed point. 

Definition 3.2.  Let 𝑇: 𝑋 → 𝑋  be a mapping of a metric space  (𝑋, 𝑑). And 𝛼(= 𝛼[ 𝑜𝑟 (𝑥, 𝑦) =

𝛼𝑑(𝑥, 𝑦)]: [0, ∞) → [0,1] be a monotone decreasing function. Then, 𝑇 is a  Rakotch contractive 

mapping if, for every 𝑥 ≠ 𝑦 ∈ 𝑋, we have 𝑑(𝑇𝑥, 𝑇𝑦) ≤ 𝛼𝑑(𝑥, 𝑦).    ( Rakotch in 1962, [45]) 

Definition 3.3.   A mapping 𝑇: 𝑋 → 𝑋  in metric space (𝑋, 𝑑) is said to be Caristi contractions if there 

is  ∅: 𝑋 → [0, ∞) lower semi-continuous function such that 𝑑(𝑥, 𝑇𝑥) ≤ ∅(𝑥) − 𝑇(𝑥).  

                                                                                                             (Caristi in 1966, [8]) 

Note that every Banach contraction 𝑇 satisfies Caristi contractions if for some λ ∈ [0,1), 

∅(𝑥) =
𝑑(𝑥, 𝑇𝑥)

1 − λ
. 

Definition 3.4.  Let 𝑇: 𝑋 → 𝑋  be a contractive mapping of a metric space  (𝑋, 𝑑). Then, 𝑇 is said to 

be Kanan contraction if there exists 𝛼 ∈ (0,
1

2
) such that, we have  

𝑑(𝑇𝑥, 𝑇𝑦) ≤ 𝛼[𝑑(𝑥, 𝑇𝑥) + (𝑦, 𝑇𝑦), for every 𝑥, 𝑦 ∈ 𝑋    ( Kanan in 1968, [37]) 

The following example shows that 𝑇 is not continuous but 𝑇 is a Kanan contraction at 𝛼 =
1

5
 : 

Example 3.3: Let 𝑋 =  ℝ be a usual metric and mapping 𝑇: 𝑋 → 𝑋  be defined by  

                         𝑇(𝑥) = {
0,    𝑓𝑜𝑟  𝑥 ∈ (−∞,   2]
1

2
,   𝑓𝑜𝑟 𝑥 ∈ (2, +∞).

 

Bianchini extends the Kanan contractions as: 

Definition 3.5. A mapping 𝑇: 𝑋 → 𝑋  in metric space (𝑋, 𝑑) is said to be Bianchini contractions if 

𝑑(𝑇𝑥, 𝑇𝑦) ≤ 𝑟 𝑚𝑎𝑥 {(𝑑(𝑥,   𝑇𝑥), 𝑑(𝑦,   𝑇𝑦) } for all 𝑥, 𝑦 ∈ 𝑋, and 𝑟 ∈ [0,1). 

                                                                                                        (Bianchini in 1972, [6]) 



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Definition 3.6. A mapping 𝑇: 𝑋 → 𝑋  in metric space (𝑋, 𝑑) is said to be Reich contractions if 

𝑑(𝑇𝑥, 𝑇𝑦) ≤ 𝑎 𝑑(𝑥, 𝑇𝑥) + 𝑏 𝑑(𝑦,   𝑇𝑦) + 𝑐 𝑑(𝑥, 𝑦), where 𝑎, 𝑏, 𝑐 are nonnegative and satisfy a + b + 

c < 1  and for all 𝑥, 𝑦 ∈ 𝑋.             (Reich in 1971, [46]) 

Note that 𝑎 = 𝑏 = 0 yields Banach's fixed point theorem, while 𝑎 = 𝑏, 𝑐 = 0 yields Kannan's theorem. 

Definition 3.7.  Let 𝑇: 𝑋 → 𝑋  be a contractive mapping of a metric space  (𝑋, 𝑑). Then, 𝑇 is said to 

be Chatterjee contraction if there exists 𝛼 ∈ (0,
1

2
) such that, we have  

𝑑(𝑇𝑥, 𝑇𝑦) ≤ 𝛼[𝑑(𝑥, 𝑇𝑦) + (𝑦, 𝑇𝑥), for every 𝑥, 𝑦 ∈ 𝑋    (Chatterjee in 1972, [12]) 

Definition 3.8.  Let 𝑇: 𝑋 → 𝑋  be a contractive mapping of a metric space  (𝑋, 𝑑). Then, 𝑇 is said to 

be T. Zamfirescu contraction if there exist real numbers 𝑎, 𝑏, 𝑐 satisfying 0 ≤ 𝑎 < 1, 0 ≤ 𝑏, 𝑐 <
1

2
, such 

that for every 𝑥, 𝑦 ∈ 𝑋, at least one of the following is true:                     (Zamfirescu in 1972, [57]) 

(i) 𝑑(𝑇𝑥, 𝑇𝑦) ≤ 𝑎[𝑑(𝑥, 𝑦); 

(ii)  𝑑(𝑇𝑥, 𝑇𝑦) ≤ 𝑏[𝑑(𝑥, 𝑇𝑥) + (𝑦, 𝑇𝑦)];                                                       

(iii) 𝑑(𝑇𝑥, 𝑇𝑦) ≤ 𝑐[𝑑(𝑥, 𝑇𝑦) + (𝑦, 𝑇𝑥).      

Definition 3.9.   A mapping 𝑇: 𝑋 → 𝑋  in metric space (𝑋, 𝑑) is said to be Sehgal contractions if, for 

every 𝑥, 𝑦 ∈ 𝑋, 𝑥 ≠ 𝑦, we have  

      𝑑(𝑇𝑥, 𝑇𝑦) < max {𝑑(𝑥, 𝑇𝑥), 𝑑(𝑦, 𝑇𝑦), 𝑑(𝑥, 𝑦)}     (Sehgal in 1972, [52])  

Definition 3.10.   A mapping 𝑇: 𝑋 → 𝑋  in metric space (𝑋, 𝑑) is said to be Rhodes contractions if, for 

each 𝑥, 𝑦 ∈ 𝑋, 𝑥 ≠ 𝑦,    

(i)  𝑑(𝑇𝑥, 𝑇𝑦) <  𝑚𝑎𝑥 {(𝑑(𝑥,   𝑇𝑦), 𝑑(𝑦,   𝑇𝑥) }.  (Rhodes in 1977, [48]) 

(ii) 𝑑(𝑇𝑥, 𝑇𝑦) ≤  𝑚𝑎𝑥 {(𝑑(𝑥,   𝑇𝑦), 𝑑(𝑦,   𝑇𝑥), 𝑑(𝑥, 𝑦) }.          (Rhodes in 1977, [48]) 

(iii) 𝑑(𝑇𝑥, 𝑇𝑦) ≤  𝑚𝑎𝑥 {(𝑑(𝑥,   𝑦), 𝑑(𝑥,   𝑇𝑥), 𝑑(𝑦, 𝑇𝑦), 𝑑(𝑥,   𝑇𝑦), 𝑑(𝑦,   𝑇𝑥)} 

                                                                                              (Rhodes in 1977, [48])  

Definition 3.11.   A mapping 𝑇: 𝑋 → 𝑋  in metric space (𝑋, 𝑑) is said to be Ciric contractions if there 

exists a non-negative number 𝑞(𝑥, 𝑦), 𝑟(𝑥, 𝑦), 𝑠(𝑥, 𝑦), 𝑎𝑛𝑑 𝑡(𝑥, 𝑦) such that  

sup
𝑥,𝑦∈𝑋

{𝑞(𝑥, 𝑦) +  𝑟(𝑥, 𝑦), +𝑠(𝑥, 𝑦) + 2 𝑡(𝑥, 𝑦)} < 1 

And 𝑑(𝑇𝑥, 𝑇𝑦) ≤ q(x, y) d(x, y) + r(x, y) d(x, Tx) + s(x, y) d(y, Ty) + t(x, y) d(x, Ty) + 𝑑(𝑦, 𝑇𝑥)      

 for every 𝑥, 𝑦 ∈ 𝑋.                                                                     (Ciric in 1971, [19])    

Definition 3.12. A mapping 𝑇: 𝑋 → 𝑋  in metric space (𝑋, 𝑑) is said to be Hardy and Roger 

contractions if there exists 𝑎, 𝑏, 𝑐, 𝑒, 𝑓 monotonically decreasing function from [0, ∞) 𝑡𝑜 (0,1), and 

 𝑎 + 𝑏 + 𝑐 + 𝑒 + 𝑓 < 1 such that 

 𝑑(𝑇𝑥, 𝑇𝑦) ≤a d(x, Tx)+b d(y, Ty)+c d(x, Ty)+𝑒𝑑(𝑦, 𝑇𝑥) + 𝑓𝑑(𝑥, 𝑦), for every 𝑥, 𝑦 ∈ 𝑋,with 

𝑎 = 𝑎(𝑑(𝑥, 𝑦)), 𝑏 = 𝑏(𝑑(𝑥, 𝑦)), 𝑐 = 𝑐(𝑑(𝑥, 𝑦)), 𝑒 = 𝑒(𝑑(𝑥, 𝑦)), 𝑓 = 𝑓(𝑑(𝑥, 𝑦))                                                                           

                                                                                              (Hardy and Roger in 1973, [31]) 



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The Browder contraction [7] was extended as follows: 

Definition 3.13.   Let  (𝑋, 𝑑)  be a metric space and 𝑇: 𝑋 → 𝑋 be a mapping. 

There exists a non-decreasing right continuous function 𝜙 ∶  [0, ∞)  →  [0, ∞) such that 

 𝜙(𝑡)  <  𝑡 𝑓𝑜𝑟 𝑡 >  0 and, for any 𝑥, 𝑦 ∈  𝑋, 

(i)  𝑑(𝑇𝑥, 𝑇𝑦)  ≤  𝜙(𝑑(𝑥, 𝑦)) (Browder [7]) 

(ii)  𝑑(𝑇𝑥, 𝑇𝑦)  ≤  𝜙(𝑚(𝑥, 𝑦)) (Danes [21]) 

(iii) 𝑑(𝑇𝑥, 𝑇𝑦)  ≤  𝜙(𝛿(𝑥, 𝑦)) if 𝑥, 𝑦 are regular (Kasahara [36]) 

Definition 3.14. Let (𝑋, 𝑑) be a metric space, and let 𝑇: 𝑋 → 𝑋  be a mapping. Then, T is said to be 

Meir Keeler's contraction if for any 휀 > 0, ∃𝛿 > 0, such that 

  휀 ≤ 𝑑(𝑥, 𝑦) < 휀 + 𝛿 implies 𝑑(𝑇𝑥, 𝑇𝑦) ≤ 휀 for all 𝑥, 𝑦 ∈ 𝑋 𝑤𝑖𝑡ℎ 𝑥 ≠ 𝑦.  

                                                                                           (Meir Keeler in 1969, [40]) 

Its extended forms are as: 

(i) 휀 ≤ 𝑚(𝑥, 𝑦) < 휀 + 𝛿 𝑖𝑚𝑝𝑙𝑖𝑒𝑠 𝑑(𝑇𝑥, 𝑇𝑦) < 휀;   

(ii) 휀 ≤ 𝛿(𝑥, 𝑦) < 휀 + 𝛿 𝑖𝑚𝑝𝑙𝑖𝑒𝑠 𝑑(𝑇𝑥, 𝑇𝑦) < 휀.              (Park in 1980, [43]) 

Definition 3.15. Let (𝑋, 𝑑) be a metric space, and let 𝑇: 𝑋 → 𝑋  be a mapping. Then, T is said to be 

Park contraction if for any 𝜖 > 0, ∃𝜖0 < 𝜖, and 𝛿0 > 0 such that for any 𝑥, 𝑦 ∈ 𝑋, 

(i)  𝜖 ≤ 𝑑(𝑥, 𝑦) < 𝜖 + 𝛿0 implies 𝑑(𝑇𝑥, 𝑇𝑦) ≤ 𝜖0. (Park in 1980, [43]) 

(ii)   𝜖 ≤ 𝑚(𝑥, 𝑦) < 𝜖 + 𝛿0 implies 𝑑(𝑇𝑥, 𝑇𝑦) ≤ 𝜖0. (Park in 1980, [43]) 

(iii)   𝜖 ≤ 𝛿(𝑥, 𝑦) < 𝜖 + 𝛿0 implies 𝑑(𝑇𝑥, 𝑇𝑦) ≤ 𝜖0.  (Hegedlis-Szilagyi [29]). 

 Definition 3.16.   Let  (𝑋, 𝑑)  be a metric space. Then a mapping 𝑇: 𝑋 → 𝑋 is said to be contractive if 

there exists 𝛼, 𝛽 >  0 with 𝛼 +  𝛽 <  1 satisfying  

𝑑(𝑇𝑥, 𝑇𝑦) ≤  𝛼
𝑑(𝑦,𝑇𝑦)(1 + 𝑑(𝑥,𝑇𝑥))

1 + 𝑑(𝑥,𝑦)
+  𝛽𝑑(𝑥, 𝑦), ∀𝑥, 𝑦 ∈ 𝑋. (Das and Gupta in1975, [22]) 

 Definition 3.17.   A mapping 𝑇: 𝑋 → 𝑋 is said to be contractive in metric space (𝑋, 𝑑) if for all 𝑥, 𝑦 ∈

 𝑋, 𝑥 ≠ 𝑦 and some 𝛼, 𝛽 ∈  [0, 1) with 𝛼 +  𝛽 <  1 satisfying  

 𝑑(𝑇𝑥, 𝑇𝑦) ≤  𝛼
   𝑑(𝑥,𝑇𝑥)𝑑(𝑦,𝑇𝑦)

 𝑑(𝑥,𝑦)
+  𝛽𝑑(𝑥, 𝑦)   (Jaggi in 1977, [33])  

Definition 3.18.  Consider a right upper semi-continuous function 𝜓: �̅�  →  [0, ∞) satisfying 

 𝜓(𝑡) <  𝑡 𝑓𝑜𝑟 𝑎𝑙𝑙 𝑡 ∈   �̅� \ {0},  where P is the range of d. A mapping 𝑇: 𝑋 → 𝑋  in metric space 

(𝑋, 𝑑) is said to be Boyd-Wong contractions if 𝑑(𝑇𝑥, 𝑇𝑦) ≤ 𝜓 (𝑑(𝑥, 𝑦)) for every 𝑥, 𝑦 ∈ 𝑋. (Boyd-

Wong in 1969, [5]) 

Definition 3.19.   A mapping 𝑇: 𝑋 → 𝑋  in metric space (𝑋, 𝑑) is said to be Singh contractions if there 

exists a positive number 𝑚 and a number 𝛼 ∈ (0,
1

2
), for each  𝑥, 𝑦 ∈ 𝑋, we have  

𝑑(𝑓𝑚(𝑥), 𝑓𝑚(𝑦)) ≤ 𝛼[𝑑(𝑥, 𝑓𝑚(𝑥)) +  𝑑(𝑦, 𝑓𝑚(𝑦)).     (Singh in 1969, [53]) 



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Definition 3.20.   A mapping 𝑇: 𝑋 → 𝑋  in metric space (𝑋, 𝑑) is said to be Yen contractions if there 

exists a positive integer 𝑚 and 𝑛 and a  number 𝛼 ∈ (0,1),  such that for each  𝑥, 𝑦 ∈ 𝑋, we have  

𝑑(𝑓𝑚(𝑥), 𝑓𝑛(𝑦)) ≤ 𝛼𝑑(𝑥, 𝑦).     (Yen in 1972, [56]) 

Definition 3.21.   A mapping 𝑇: 𝑋 → 𝑋  in metric space (𝑋, 𝑑) is said to be Guseman contractions if 

there exists a positive integer 𝑚 and a  number 𝛼 ∈ (0,1),  such that for each  𝑥, 𝑦 ∈ 𝑋, we have  

𝑑(𝑓𝑚(𝑥), 𝑓𝑚(𝑦)) ≤ 𝛼𝑑(𝑥, 𝑦).     (Guseman in 1970, [27]) 

Definition 3.22.   A mapping 𝑇: 𝑋 → 𝑋  in metric space (𝑋, 𝑑) is said to be Bailey contractions if there 

exists a positive integer 𝑚 and 0 < 𝑑(𝑥, 𝑦),  such that for each  𝑥, 𝑦 ∈ 𝑋, we have  

𝑑(𝑓𝑚(𝑥), 𝑓𝑚(𝑦)) ≤ 𝑑(𝑥, 𝑦).     (Bailey in 1966, [2]) 

Definition 3.23.   A mapping 𝑇: 𝑋 → 𝑋  in metric space (𝑋, 𝑑) is said to be Berinde contractions if 

there exists  𝛼 ∈ [0,1), and λ ≥ 0  such that for each  𝑥, 𝑦 ∈ 𝑋, we have  

𝑑(𝑇𝑥, 𝑇𝑦) ≤  𝛼𝑑(𝑥, 𝑦) + λd(y, Tx)  (Berinde weak contractions in 2004, [3]) 

Definition 3.24.   A mapping 𝑇: 𝑋 → 𝑋  in metric space (𝑋, 𝑑) is said to be a contractive mapping for 

every 𝑥, 𝑦 ∈ 𝑋, 𝜑(𝑑(𝑇𝑥, 𝑇𝑦)) ≤ 𝜑(𝑑(𝑥, 𝑦)) − ∅(𝑑(𝑥, 𝑦)), where 𝜑, ∅: [0, ∞) → [0, ∞) are 

continuous, non-decreasing, and 𝜑−1({0}) =  ∅−1({0}) = {0}.    (Dutta and Choudhury in 2008, 

[23]) 

Definition 3.35.   Let  (𝑋, 𝑑)  be a metric space and 𝑇: 𝑋 → 𝑋 be a mapping. Define a non-increasing 

function 𝜃: [0, 1) → (
1

2
, 1] by   

                         𝜃(𝑟) = {
1,    if 0 ≤  r ≤

 √5− 1

2
,

(1 −  r)r−2 if
 √5− 1

2
≤  r ≤  2−

1

2

 

                                         (1 +  r)−1 if 2−
1

2  ≤  r <  1. 

Assume that there exists 𝑟 ∈  [0, 1) such that 𝜃(𝑟) 𝑑(𝑥, 𝑇 𝑥)  ≤  𝑑(𝑥, 𝑦) implies 

 𝑑(𝑇 𝑥, 𝑇 𝑦)  ≤  𝑟 𝑑(𝑥, 𝑦) for all 𝑥, 𝑦 ∈  𝑋.                  (Suzuki contraction in 2008, [54]) 

Definition 3.36.   A mapping 𝑇: 𝑋 → 𝑋  in metric space (𝑋, 𝑑) is said to be 𝐹 contractions if there 

exist 𝜏 > 0 such that for every 𝑥, 𝑦 ∈ 𝑋,   𝑑(𝑇𝑥, 𝑇𝑦) > 0 𝑖𝑚𝑝𝑙𝑖𝑒𝑠 𝜏 +  F (𝑑(𝑇𝑥, 𝑇𝑦)) ≤ 𝐹𝑑(𝑥, 𝑦), 

Where 𝐹: ℝ+ → ℝ be a mapping satisfying: 

(i) 𝐹 is strictly increasing i.e. for all 𝛼, 𝛽 ∈ ℝ+ such that  𝛼 <  𝛽, 𝐹(𝛼) < 𝐹(𝛽); 

(ii) For each sequence {𝑥𝑛}, 𝑛 ∈ 𝑁 of positive numbers 𝑙𝑖𝑚 
𝑛→∞

𝑥𝑛 = 0 if and only if 

  𝑙𝑖𝑚 
𝑛→∞

𝐹(𝑥𝑛) = −∞; 

(iii) There exists 𝑘 ∈ (0,1) such that  𝑙𝑖𝑚 
𝛼→0+

𝛼𝑘𝐹(𝛼) = 0. 

(F Contractions Wardowaski in 2012, [55])     



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Definition 3.37.   A mapping 𝑇: 𝑋 → 𝑋  in metric space (𝑋, 𝑑) is said to be θ-contraction if there exist 

𝑘 ∈ (0,1) such that for every 𝑥, 𝑦 ∈ 𝑋,   𝑑(𝑇𝑥, 𝑇𝑦) ≠ 0 ⇒    θ(𝑑(𝑇𝑥, 𝑇𝑦)) ≤ [θ(𝑑(𝑥, 𝑦))]𝑘, 

Where 𝜃: (0, ∞) → (1, ∞) is a function satisfying the following conditions: 

(i) θ is non-decreasing; 

(ii) for each sequence,{𝑡𝑛} ⊂ (0, ∞), 𝑙𝑖𝑚 
𝑛→∞

θ(𝑡𝑛) = 1if and only if 𝑙𝑖𝑚 
𝑛→∞

 𝑡𝑛 = 0+;  

(iii) θ is continuous on (0, ∞); 

(iv) there exist 𝑟 ∈ (0,1), and 𝑙 ∈ (0, ∞]such that  𝑙𝑖𝑚 
𝑡→0+

𝜃(𝑡)−1

𝑡𝑟 = 𝑙. 

( Jleli and Samet in 2014, [34])   

Definition 3.38. Let (𝑋, 𝑑) be a metric space, and 𝑇: 𝑋 →  𝑋 be a mapping. Then for all x, y ∈ X, we 

denote 

𝑚(𝑇𝑥, 𝑇𝑦)  =  𝑎𝑑(𝑥, 𝑦)  +  𝑏 𝑚𝑎𝑥{𝑑(𝑥, 𝑇𝑥), 𝑑(𝑦, 𝑇𝑦)}  +  𝑐[𝑑(𝑥, 𝑇𝑦)  +  𝑑(𝑦, 𝑇𝑥)],  

where 𝑎, 𝑏, and 𝑐 are non-negative reals such that 𝑎 + 𝑏 + 2𝑐 =  𝑟 𝑤𝑖𝑡ℎ 𝑟 ∈  [0, 1).  

Now, we consider the following generalized contractive condition: 

𝜃(𝑟) 𝑚𝑖𝑛{𝑑(𝑥, 𝑇𝑥), 𝑑(𝑥, 𝑇𝑦)}  ≤  𝑑(𝑥, 𝑦) 𝑖𝑚𝑝𝑙𝑖𝑒𝑠 𝑑(𝑇𝑥, 𝑇𝑦)  ≤  𝑚(𝑇𝑥, 𝑇𝑦).   

                                                                                    (Chandra, Joshi, and Joshi in 2022, ]11])  

4. Generalization of contraction mapping in Menger Space: 

 Sehgal [50] first defined probabilistic contraction in his PhD dissertation in 1966 as: 

Definition 4.1: Let (𝑋, 𝐹) be a probabilistic metric space. A mapping 𝑇: 𝑋 → 𝑋 is a probabilistic 

contraction or Sehgal contraction if there exists  𝑘 ∈ (0,1) such that   𝐹𝑇𝑝,𝑇𝑞(𝑘𝑡) ≥ 𝐹𝑝,𝑞(𝑡) 

for all 𝑝, 𝑞 ∈ 𝑋, and 𝑡 > 0.  

Hicks [32] defined another contraction mapping in probabilistic metric space in 1983:  

 Definition 4.2: A mapping 𝑇: 𝑋 → 𝑋 in probabilistic metric space (𝑋, 𝐹) is said to be Hicks 

contraction or C-contraction if there exists 𝑘 ∈ (0,1) such that for every 𝑝, 𝑞 ∈ 𝑋, and every 𝑡 > 0: 

𝐹𝑝,𝑞(𝑡) > 1 − 𝑡 ⇒ 𝐹𝑇𝑝,   𝑇𝑞(𝑘𝑡) > 1 − 𝑘𝑡. 

 A weaker form of Hicks contraction was introduced by D. Mihet [39] in 2005 as: 

 Definition 4.3: A mapping 𝑇: 𝑋 → 𝑋 is said to be weak - Hicks contraction (w-H contraction) if 

there exists 𝑘 ∈ (0,1) such that, for all 𝑝, 𝑞 ∈ 𝑆. 

(𝑤 − 𝐻): 𝑡 ∈ (0,1),  𝐹𝑝,𝑞(𝑡) > 1 − 𝑡 ⇒ 𝐹𝑇𝑝,   𝑇𝑞(𝑘𝑡) > 1 − 𝑘𝑡. 

Example 4.1: Let 𝑋 = [0, ∞), and 𝐹𝑝,𝑞(𝑡) =
min (𝑝,𝑞)

max (𝑝,𝑞)
, ∀ 𝑝, 𝑞 ∈ 𝑋, 𝑝 ≠ 𝑞. Then, (𝑋, 𝐹, 𝑇) be a 

complete Menger space under triangular norm 𝑇 = 𝑇𝑃 >  𝑇𝐿.  It can be seen that the mapping 𝑔: 𝑋 →

𝑋, 𝑔(𝑥) =  {  1,𝑖𝑓 𝑥>0
0,𝑖𝑓 𝑥=0

 

is a 𝑤 − 𝐻 contraction for 𝑘 ∈ (0,1). 



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4.1 Generalized form of probabilistic contraction: 

A probabilistic (m, k) contraction is a generalization of Sehgal contraction, where 𝑚 ≥ 1 and 𝑘 ∈

(0,1) and is defined as: 

Definition 4.1.1:  [16] If (𝑋, 𝐹) is a PM - space, 𝑚 ≥ 1 and 𝑘 ∈ (0,1), a function 𝑓: 𝑋 → 𝑋 is called 

probabilistic (m, k)-contraction if for any 𝑝, 𝑞 ∈ 𝑋 there is an 𝑖 with 1 < 𝑖 < 𝑚 such that for every 

𝑡 > 0, 

𝐹𝑓𝑖 𝑝,𝑓𝑖𝑞(𝑘𝑖𝑡) ≥ 𝐹𝑝,𝑞(𝑡). 

If 𝑚 = 1 and 𝑘 ∈ (0,1) then a probabilistic (1, 𝑘)-Sehgal contraction, 𝑓 is a probabilistic                     

Sehgal contraction. 

 Following is a generalization of Hicks C-contraction: 

Definition 4.1.2: [16] If (𝑋, 𝜑) is a PM - space, 𝑚 ≥ 1 and 𝑘 ∈ (0,1), a function 𝑓: 𝑋 → 𝑋 is called a 

(m, k)-C-contraction if for any 𝑝, 𝑞 ∈ 𝑋 there is an 𝑖 with 𝑙 < 𝑖 < 𝑚 such that for every 𝑡 > 0. 

𝐹𝑝,𝑞(𝑡) > 1 − 𝑡 ⇒ 𝐹𝑓𝑖 𝑝,𝑓𝑖𝑞(𝑘𝑖𝑡) > 1 − 𝑘𝑖𝑡. 

If 𝑚 = 1 and 𝑘 ∈ (0,1) then a probabilistic (1, 𝑘)-C-contraction 𝑓 is a probabilistic C-contraction. 

Definition 4.1.3: [26] Let 𝑓, 𝑔 be two mappings defined on a Menger space (𝑋, 𝐹, 𝑇) with values into 

itself, and let us suppose that 𝑔 is bijective. The mapping 𝑓 is called a probabilistic g-contraction 

with a constant 𝑘 ∈ (0,1) if 

𝑡 > 0 and 𝐹𝑔(𝑥),𝑔(𝑦)(𝑡) > 1 − 𝑡 impies 𝐹𝑓(𝑥),𝑓(𝑦)(𝑘𝑡) > 1 − 𝑘𝑡. 

Conclusion: This study discusses only Banach contraction's extended and generalized form in 

complete metric space and Menger space so that each mapping establishes unique fixed point 

theorems. It helps in comparative studies and may solve many related open problems. 

Acknowledgment: Authors are grateful to the anonymous referees for their precise remarks, and 

suggestions which led to the improvement of the paper. And the corresponding author is highly grateful 

for their financial support for researchers. 

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