Communications on Applied Nonlinear Analysis 

ISSN: 1074-133X 

Vol 31 No. 3s (2024) 

 

367 
https://internationalpubls.com 

Some New Types of 𝜶𝜷−𝟏-admissible Mappings and Fixed Point Results 

 

Dev Raj Joshi, Piyush Kumar Tripathi, Alok Kumar Agrawal and Ashutosh Mishra 

Department of Mathematics, ASAS, Amity University Uttar Pradesh Lucknow campus, Lucknow-226028, India 

Email:dev.joshi@s.amity.edu, pktripathi@lko.amity.edu, akagraw@lko.amity.edu and ashutosh.mishra8@s.amity.edu 

 

 

Article History: 

Received: 05-04-2024 

Revised: 30-05-2024 

Accepted: 14-06-2024 

Abstract:  

In this paper fixed point and common fixed point theorems are proved for 𝛼 − 𝜑 

contractions under the 𝛼𝛽−1-admissible and  𝜇 − 𝛼𝛽−1   admissible conditions. These 

results generalize the 𝛼 –admissible condition of self-mappings as explained by many 

researchers in the literature herein. In general, many research reports introduced the 𝛼–

admissible condition on the basis of 𝛼(𝑝, 𝑞) > 1. However, this condition is not 

applicable for finding the fixed points of self-mappings in many cases. We propose 𝛼–

admissible on the basis of 𝛼(𝑝, 𝑞) > 𝛽−1 which generalize the case we mentioned in 

the literature. To support our new concepts, we present two examples at the end of this 

paper. 

Keywords: 𝛼𝛽−1-admissible mappings, 𝜇 − 𝛼𝛽−1admissible mapping, 𝜇 − 𝛼, 𝜑 

contraction. 

Mathematics Subject Classification: 47H10, 54H25. 

 

1. Introduction 

In the field of fixed point theory, Banach contraction principle [1] is one of the fundamental tools to 

explain existence of fixed point in complete metric spaces. After this principle, many researchers 

have suggested different generalizations of contraction principle and their applications to solve many 

mathematical problems such as solution of differential equations, integral equations and non-linear 

analysis.  

Among the generalizations of Banach contraction principle, we discuss 𝛼 − 𝜑 contraction here. It 

was initiated by Samet et al. [2] in which they  introduced 𝛼 − 𝜑 contraction type mappings and 

proved fixed point theorems in complete metric spaces. This concept was further improved and 

modified by B. Samet [3] alone. After the paper of Samet et al. the 𝛼 − 𝜑  contraction is used by 

many researchers in different metric spaces. Karapinar et al. [4] used this contraction in b-metric 

space with 𝛼-orbital admissible condition. Li. and Guan [5] introduced 𝛼𝑠𝑝  and 𝜌 − 𝛼𝑠𝑝 admissible 

condition in b-metric space. Likewise, Hussain et al. [6] introduced 𝛼 –admissible mapping with 

respect to 𝜂 . Zoto et al. [7] introduced 𝛼𝑞𝑠𝑝 admissible mapping and prove the fixed point theorem 

in b-metric space. Almost researchers in this direction have used 𝛼-admissible condition. For 

example see [8],[9],[10], [11],[12],[13],[14],[15]. 

As we motivated and inspired by the papers [2],[3], [4], [5] and [7], our aim is to introduce 𝛼𝛽−1 –

admissible mapping to prove some fixed point theorems by using 𝛼 − 𝜑 contraction in complete 



Communications on Applied Nonlinear Analysis 

ISSN: 1074-133X 

Vol 31 No. 3s (2024) 

 

368 
https://internationalpubls.com 

metric spaces in this paper.  Additionally, we introduce 𝜇 − 𝛼𝛽−1admissible mappings to prove the 

common fixed point theorem by using (𝜇 − 𝛼, 𝜑)-contraction in complete metric space.  

2. Preliminaries   

In this section, we present some basic concepts and notations related to 𝛼 − 𝜑 contractions and 𝛼 -

admissible conditions. 

Definition 2.1: [16] A mapping 𝜑: [0, ∞) → [0, ∞) is called comparison function if satisfy following 

assumptions:   

a) 𝜑 is non-decreasing. 

b) 𝜑(𝑝) < 𝑝  ∀ 𝑝 > 0. 

c) 𝜑𝑚(𝑝) → 0 as 𝑚 → ∞   ∀ 𝑝 > 0 

d) 𝜑(0) = 0 

The set of all comparison functions is denoted by ∅. 

Definition 2.2: [2] Let 𝑋 be a non-empty set. The mapping 𝜇: 𝑋 → 𝑋 is called 𝛼-admissible, if there 

exists a function 𝛼: 𝑋 × 𝑋 → [0, ∞) such that 𝛼(𝑎, 𝑏) ≥ 1 ⇒ 𝛼(𝜇(𝑎), 𝜇(𝑏)) ≥ 1 for 𝑎, 𝑏 ∈ 𝑋. 

Definition 2.3: [2] Let 𝜇 be a self-map on a metric space (𝑋, 𝑑). We say 𝜇 is 𝛼 − 𝜑 contraction if 

there exist 𝛼: 𝑋 × 𝑋 → [0, ∞) and a comparison function 𝜑: [0, ∞) → [0, ∞) such that, 

𝛼(𝑎, 𝑏)𝑑(𝜇(𝑎), 𝜇(𝑏)) ≤ 𝜑(𝑑(𝑎, 𝑏))     ∀  𝑎, 𝑏 ∈ 𝑋  

Definition 2.4: [3] For a given 𝜑 ∈ ∅, Let us define a set 𝑅𝜑 = {𝛾 ∈ [0, ∞): 𝛾𝜑 ∈ ∅}  

The main theorem in [2] is re stated here. 

Theorem 2.1: Let  (𝑋, 𝑑) be complete metric space and 𝜇  be a given continuous self- map. Suppose 

that there exist two functions 𝛼: 𝑋 × 𝑋 → [0, ∞) and 𝜑 ∈ ∅  such that, 

a) 𝜇 is 𝛼 − 𝜑 contraction.   

b)  𝜇 is 𝛼-admissible. 

c)  There exists 𝑎0 ∈ 𝑋 such that 𝛼(𝑎0, 𝜇(𝑎0) ≥ 1  

d)  𝜇 is continuous or  

e)  For every {𝑎𝑛} ⊂ 𝑋 such that 𝑎𝑛 → 𝑎 ∈ 𝑋  and 𝛼(𝑎𝑛, 𝑎𝑛+1) ≥ 1 for 𝑛 ∈ 𝑁, we have  

𝛼(𝑎𝑛, 𝑎) ≥ 1 for 𝑛 ∈ 𝑁 .  

Then 𝜇 has a fixed point. Moreover, if in addition we suppose that for every pair 𝑢, 𝑣 ∈ 𝑋 there 

exists 𝑤 ∈ 𝑋 such that  𝛼(𝑢, 𝑤) ≥ 1 and 𝛼(𝑣, 𝑤) ≥ 1 , we have a unique fixed point. 

 Following are the main results; we obtained in Samt [3]    

Theorem 2.2: Let  (𝑋, 𝑑) be complete metric space and 𝜇  be a given continuous self- map. Suppose 

that there exist two functions 𝛼: 𝑋 × 𝑋 → [0, ∞) and 𝜑 ∈ ∅  such that 𝜇 is 𝛼 − 𝜑 contraction. Also 

let there exist 𝛾 ∈ 𝑅𝜑 and a finite sequence {𝜏𝑖} ⊂ 𝑋, 𝑖 = 1,2 … . 𝑝 such that 

 𝜏0 = 𝑥0, 𝜏𝑝 = 𝜇(𝑥0), 𝛼(𝜇𝑛(𝜏𝑖), 𝜇𝑛(𝜏𝑖+1)) ≥ 𝛾−1  ∀𝑛 ∈ 𝑁, 𝑖 = 1,2. . 𝑝 − 1……….  (2. 1) 

Then there exist a fixed point of 𝜇. 



Communications on Applied Nonlinear Analysis 

ISSN: 1074-133X 

Vol 31 No. 3s (2024) 

 

369 
https://internationalpubls.com 

The uniqueness of fixed point is guaranteed by following conditions 

Theorem 2.3: Let  (𝑋, 𝑑) be complete metric space and 𝜇  be a given self- map. Suppose that there 

exist two functions 𝛼: 𝑋 × 𝑋 → [0, ∞) and 𝜑 ∈ ∅  such that 𝜇 is 𝛼 − 𝜑 contraction. Suppose also 

that  

a) The fixed point set 𝐹𝑖𝑥(𝜇) is not empty.  

b) For all 𝑢, 𝑣 ∈ 𝐹𝑖𝑥(𝜇) with 𝑢 ≠ 𝑣,if 𝛼(𝑢, 𝑣) < 1 then there exist 𝛾 ∈ 𝑅𝜑 and for some 𝑞 ∈ 𝑁 

There is a finite sequence {𝛿𝑖} ⊂ 𝑋, 𝑖 = 1,2 … . 𝑞 such that 𝛿0 = 𝑢  , 𝛿𝑞 =

𝑣, 𝛼(𝜇𝑛(𝛿𝑖), 𝜇𝑛(𝛿𝑖+1)) ≥ 𝛾−1   ∀ 𝑛 ∈ 𝑁 𝑎𝑛𝑑 𝑖 = 0,1,2 … 𝑞 − 1.  

Then 𝜇 has a unique fixed point. 

Definition 2.5: [17] Let 𝜇 𝑎𝑛𝑑 𝜌 be two self-mappings of a complete metric space (𝑋, 𝑑).We say 

they are weakly compatible if they commute at their coincident point. That is if 𝜇(𝑠) = 𝜌(𝑠) for 

some 𝑠 ∈ 𝑋 implies 𝜌𝜇(𝑠) = 𝜇𝜌(𝑠) 

Here, we restate the proposition by M.Abbas and G.Jungck [17] 

Proposition 2.1: Let (𝑋, 𝑑) be a complete metric space and  𝜇, 𝜌 are weakly compatible self- 

mappings on 𝑋.If there is unique coincident point 𝜇(𝑧) = 𝜌(𝑧) = 𝑢 then 𝑢  is the unique common 

fixed point of 𝜇 𝑎𝑛𝑑 𝜌.  

The definition 2.2 was generalized by H.Aydi[18] as follows. 

Definition 2.6: Let 𝜇 𝑎𝑛𝑑 𝜌 be two self-mappings on a non-empty set 𝑋.We say the pair of 

mappings (𝜇, 𝜌)is 𝛼-admissible if there exist a mapping 𝛼: 𝑋 × 𝑋 → [0, ∞) such that 

 𝛼(𝑎, 𝑏) ≥ 1 ⇒ 𝛼(𝜇(𝑎), 𝜌(𝑏)) ≥ 1  𝑎𝑛𝑑   𝛼(𝜌(𝑏), 𝜇(𝑎)) ≥ 1. 

 

Here, it is noted that there is no any mathematical support to find the finite sequence satisfying 

(2.1).Actually we feel, it is most serious task to get such sequence.  Similarly the hypothesis (b) in 

theorem 2.3 is not suitable for the aim of paper [3]. Because of these reasons we introduce general 

cases to extend and improve theorems 2.1, 2.2 and 2.3 without using condition defined in definition 

2.2. 

3. Main Results 

In this section we present some new concepts, fixed point theorems and common fixed point 

theorems.  

From definition2.4, let us take the set 𝑅𝜑+ = {𝜆 ∈ 𝑅𝜑 ∶  𝜑(𝑝) < 𝜆𝜑(𝑝) < 𝑝     ∀ 𝑝 > 0}……… (3.1) 

It is clear that this set (3.1) non-empty because 𝜑(𝑝) < 𝑝  ∀ 𝑝 > 0 and by properties of real numbers 

there are infinite real numbers between 𝜑(𝑝)and p. Now we can take maximum value from (3.1), say 

 𝛽 = 𝑚𝑎𝑥. {𝜆 ∈ 𝑅𝜑+}.Obviously,  𝛽 > 1 and it is maximum value for which  𝛽𝜑 ∈ ∅ .Thus 𝛽−1 < 1 

is minimum value for which 𝛼(𝑎, 𝑏) > 𝛽−1 for 𝑎, 𝑏 ∈ 𝑋. 

Now we can define 𝛼-admissible mapping on the basis of 𝛽 as follows:  



Communications on Applied Nonlinear Analysis 

ISSN: 1074-133X 

Vol 31 No. 3s (2024) 

 

370 
https://internationalpubls.com 

Definition 3.1: Let  (𝑋, 𝑑) be a metric space and 𝜇  be a given self- map. We say 𝜇 is 𝛼𝛽−1 – 

admissible, if there exist 𝛼: 𝑋 × 𝑋 → [0, ∞) such that, 𝛼(𝑎, 𝑏) > 𝛽−1 ⇒ 𝛼(𝜇(𝑎), 𝜇(𝑏)) > 𝛽−1 for  

𝑎, 𝑏 ∈ 𝑋.  

Now we prove following fixed point theorems as our main findings. 

 Theorem 3.1: Let (𝑋, 𝑑) be a complete metric space.   Let 𝛽 is maximum of all member of 𝑅𝜑+. If a 

self -mapping  𝜇 satisfies following conditions, 

a) 𝜇  is 𝛼 − 𝜑 contraction on X. 

b) 𝜇 is 𝛼𝛽−1- admissible  

c) There exist 𝑥0 ∈ 𝑋 such that 𝛼(𝑥0, 𝜇(𝑥0)) > 𝛽−1 

d) 𝜇 is continuous in 𝑋 

  Then there exist a fixed point of 𝜇 in 𝑋.  

Proof: -.Let there exist 𝑥0 ∈ 𝑋 such that 𝛼(𝑥0, 𝜇(𝑥0)) > 𝛽−1. Being 𝜇 is 𝛼 𝛽−1 - admissible, so by 

induction we can get, 

 𝛼(𝜇𝑛(𝑥0), 𝜇𝑛+1(𝑥0)) > 𝛽−1, ∀ 𝑛 ∈ 𝑁…………….. (3.2) 

Now we have from condition (a), 

𝛽−1𝑑(𝜇(𝑥0), 𝜇2(𝑥0) ≤   𝛼 (𝑥0,𝜇(𝑥0)) 𝑑(𝜇(𝑥0), 𝜇2(𝑥0)) ≤  𝜑(𝑑(𝑥0,𝜇(𝑥0)))   

⇒ 𝑑(𝜇(𝑥0), 𝜇2(𝑥0) ≤ 𝛽𝜑(𝑑(𝑥0,𝜇(𝑥0)))   

Let 𝛽𝜑 = 𝜎 ∈ ∅, then 

 𝑑(𝜇(𝑥0), 𝜇2(𝑥0) ≤ 𝜎(𝑑(𝑥0,𝜇(𝑥0))) …………. (3.3) 

Again  

 𝛽−1𝑑(𝜇2(𝑥0), 𝜇3(𝑥0) ≤   𝛼(𝑓(𝑥0), 𝜇2(𝑥0))𝑑(𝜇2(𝑥0), 𝜇3(𝑥0)) ≤  𝜑(𝑑(𝜇(𝑥0), 𝜇2(𝑥0))) 

 ⇒ 𝑑(𝜇2(𝑥0), 𝜇3(𝑥0) ≤  𝜎(𝑑(𝜇(𝑥0), 𝜇2(𝑥0))) , from (3.3) 

 𝑑(𝜇2(𝑥0), 𝜇3(𝑥0) ≤ 𝜎2(𝑑(𝑥0,𝜇(𝑥0))) 

Continuing this process, we have  

 𝑑(𝜇𝑛(𝑥0), 𝜇𝑛+1(𝑥0) ≤ 𝜎𝑛(𝑑(𝑥0,𝜇(𝑥0)))…….. (3.4) 

Now for 𝑚, 𝑛 ∈ 𝑁 with 𝑚 < 𝑛, 

𝑑(𝜇𝑚(𝑥0), 𝜇𝑛(𝑥0))

≤ 𝑑(𝜇𝑚(𝑥0), 𝜇𝑚+1(𝑥0 )) + 𝑑(𝜇𝑚+1(𝑥0), 𝜇𝑚+2(𝑥0)) + ⋯ + 𝑑(𝜇𝑛−1(𝑥0), 𝜇𝑛(𝑥0)) 

 From (3.4) 

𝑑(𝜇𝑚(𝑥0), 𝜇𝑛(𝑥0)) ≤ 𝜎𝑚 (𝑑 (𝑥0,𝜇(𝑥0))) + 𝜎𝑚+1 (𝑑 (𝑥0,𝜇(𝑥0))) + ⋯ … … + 𝜎𝑛−1(𝑑(𝑥0,𝜇(𝑥0))) 



Communications on Applied Nonlinear Analysis 

ISSN: 1074-133X 

Vol 31 No. 3s (2024) 

 

371 
https://internationalpubls.com 

Taking 𝑚 → ∞ and by property of comparison mapping, 

 𝜎𝑚 (𝑑 (𝑥0,𝜇(𝑥0))) = 0 .  

Thus we have,    

𝑑(𝜇𝑚(𝑥0), 𝜇𝑛(𝑥0)) = 0  

This implies {𝜇𝑛(𝑥0)} is Cauchy sequence in (𝑋, 𝑑)  .Since (𝑋, 𝑑)  is complete, there exist  𝑥𝜇 ∈ 𝑋 

such that lim
𝑛→∞

𝜇𝑛(𝑥0) =  𝑥𝜇. 

The continuity of 𝜇 implies that, 

 lim
𝑛→∞

𝜇𝑛+1(𝑥0) =  𝜇(𝑥𝜇) . The uniqueness of limiting value, gives 𝜇(𝑥𝜇) = 𝑥𝜇. □  

It is clear that our suggested hypothesis, “there exist 𝑥0 ∈ 𝑋 such that 𝛼(𝑥0, 𝜇(𝑥0)) > 𝛽−1” is more 

general than the hypothesis given in theorem 2.1 in preliminaries section. 

Next we show the continuity assumption of 𝜇 does not require to exist a fixed point of 𝜇. 

Theorem 3.2: Let (𝑋, 𝑑) be a complete metric space.   Let 𝛽 is maximum of all member of 𝑅𝜑+. If 

the self -mapping  𝜇 satisfies following conditions, 

a) 𝜇 is 𝛼 − 𝜑 contraction on X.  

b) 𝜇 is 𝛼𝛽−1 -admissible. 

c) There exist 𝑥0 ∈ 𝑋 such that 𝛼(𝑥0, 𝜇(𝑥0)) > 𝛽−1 

d) If 𝛼(𝜇𝑛(𝑥0), 𝜇𝑛+1(𝑥0)) > 𝛽−1, ∀ 𝑛 ∈ 𝑁 and lim
𝑛→∞

𝑑(𝜇𝑛(𝑥0), 𝑧) = 0 implies 𝛼(𝜇𝑛(𝑥0), 𝑧) >

𝛽−1, ∀ 𝑛 ∈ 𝑁  

  Then there exist a fixed point of 𝜇 in 𝑋. 

Proof: From theorem 3.1 the sequence {𝜇𝑛(𝑥0)} is Cauchy sequence in (𝑋, 𝑑)  .Since (𝑋, 𝑑)  is 

complete, there exist  𝑥𝜇 ∈ 𝑋 such that lim
𝑛→∞

𝜇𝑛(𝑥0) =  𝑥𝜇.Thus lim
𝑛→∞

𝑑(𝜇𝑛(𝑥0), 𝑥𝜇) = 0. 

Now from given hypothesis (d) and from (3.2), we can get, 

 𝛼(𝜇𝑛(𝑥0), 𝑥𝑓) > 𝛽−1, ∀ 𝑛 ∈ 𝑁…………… (3.5) 

Now, 

𝛽−1𝑑( 𝜇𝑛+1(𝑥0), 𝜇(𝑥𝜇)  ≤ 𝛼( 𝜇𝑛(𝑥0), 𝑥𝜇 )𝑑( 𝜇𝑛+1(𝑥0), 𝜇(𝑥𝜇) ≤ 𝜑(𝑑(𝜇𝑛(𝑥0), 𝑥𝜇))  

⇒ 𝑑( 𝜇𝑛+1(𝑥0), 𝜇(𝑥𝜇) ≤ 𝛽𝜑(𝑑(𝜇𝑛(𝑥0), 𝑥𝜇))   

⇒ 𝑑( 𝜇𝑛+1(𝑥0), 𝜇(𝑥𝜇) ≤ 𝜎(𝑑(𝜇𝑛(𝑥0), 𝑥𝜇))  < 𝑑(𝜇𝑛(𝑥0), 𝑥𝜇)  

Taking 𝑛 → ∞ , we have  

𝑑( 𝑥𝜇, 𝜇(𝑥𝜇) ≤ 𝑑( 𝑥𝜇 , 𝑥𝜇)  

⇒ 𝑑( 𝑥𝜇, 𝑓(𝑥𝜇) = 0  



Communications on Applied Nonlinear Analysis 

ISSN: 1074-133X 

Vol 31 No. 3s (2024) 

 

372 
https://internationalpubls.com 

∴  𝜇(𝑥𝜇) = 𝑥𝜇   □   

The uniqueness of fixed point is given by following theorem. 

Theorem 3.3: By adding the condition:  “if for every pair 𝑢, 𝑣 ∈ 𝑋 there exist 𝑤 ∈ 𝑋  such that 

𝛼(𝑢, 𝑤) > 𝛽−1 and 𝛼(𝑣, 𝑤) > 𝛽−1"  in the theorem 3.1 and 3.2, then there exist unique fixed point 

of 𝜇.  

Proof: If possible suppose there are two fixed points  𝑥𝑓 and 𝑦𝑓 . By our hypothesis there exist 𝑤 ∈

𝑋 such that 𝛼(𝑥𝜇, 𝑤) > 𝛽−1 and 𝛼(𝑦𝜇, 𝑤) > 𝛽−1. 

Being  𝜇 is 𝛼-admissible, 𝛼(𝑥𝜇, 𝑤) > 𝛽−1 implies 𝛼(𝑥𝜇 , 𝜇𝑛(𝑤)) > 𝛽−1. 

Now we have, 

𝛽−1𝑑 (𝑥𝜇, 𝜇𝑛(𝑤)) = 𝛽−1𝑑(𝜇 (𝑥𝜇), 𝜇𝑛(𝑤) ≤  𝛼(𝑥𝜇,𝜇
𝑛−1(𝑤))𝑑(𝜇(𝑥𝜇), 𝜇𝑛(𝑤)) ≤

 𝜑(𝑑(𝑥𝜇,𝜇
𝑛−1(𝑤))  

⇒ 𝑑(𝑥𝜇 , 𝜇𝑛(𝑤)) ≤ 𝜎(𝑑(𝑥𝜇,𝜇
𝑛−1(𝑤)). 

 Since 𝜎  is non-decreasing, then by induction we can have, 

 𝑑(𝑥𝜇, 𝜇𝑛(𝑤)) ≤ 𝜎𝑛(𝑑(𝑥𝜇,𝑤)). 

But 

 lim
𝑛→∞

𝜎𝑛(𝑑(𝑥𝜇,𝑤)) = 0.Hence, lim
𝑛→∞

𝑑(𝑥𝜇, 𝜇𝑛(𝑤)) = 0 

Similarly we can show that lim
𝑛→∞

𝑑(𝑦𝜇, 𝜇𝑛(𝑤)) = 0.  

By triangular property of metric 

𝑑(𝑥𝜇, 𝑦𝜇) ≤  𝑑 (𝑥𝜇, 𝜇𝑛(𝑤)) + 𝑑(𝑦𝜇, 𝜇𝑛(𝑤)). Now, taking 𝑛 → ∞ 

𝑑(𝑥𝜇, 𝑦𝜇) = 0. Hence fixed point is unique. □ 

Next, we introduce some new types of 𝛼-admissible mappings and contraction conditions to prove 

common fixed point theorem. 

First of all let us denote the set of all coincident points of pair of mappings (𝜇, 𝜌) by 𝐶(𝜇, 𝜌). 

Definition 3.2: Let (𝑋, 𝑑) be a complete metric space and  𝜇, 𝜌 are self- mappings on 𝑋 with 𝜌(𝑋) ⊂

𝜇(𝑋). We say 𝜌 is 𝜇 − 𝛼, 𝜑 -contraction if there exist 𝛼: 𝑋 × 𝑋 → [0, ∞) and a comparison function 

𝜑: [0, ∞) → [0, ∞) such that,  

 𝛼(𝜇(𝑎), 𝜇(𝑏))𝑑(𝜌(𝑎), 𝜌(𝑏)) ≤ 𝜑(𝑑(𝜇(𝑎), 𝜇(𝑏)))   ∀  𝑎, 𝑏 ∈ 𝑋. 

Definition3.3: Let (𝑋, 𝑑) be a metric space and  𝜇, 𝜌  self- mappings on 𝑋 with 𝜌(𝑋) ⊂ 𝜇(𝑋).We say 

𝜌 is 𝜇 − 𝛼𝛽−1 -admissible if  there exists a function 𝛼: 𝑋 × 𝑋 → [0, ∞) such that, 

 𝛼(𝜇(𝑎), 𝜇(𝑏)) > 𝛽−1 ⇒ 𝛼(𝜌(𝑎), 𝜌(𝑏)) > 𝛽−1 For   𝑎, 𝑏 ∈ 𝑋 



Communications on Applied Nonlinear Analysis 

ISSN: 1074-133X 

Vol 31 No. 3s (2024) 

 

373 
https://internationalpubls.com 

Now we apply above results to prove the common fixed point in complete metric spaces. 

Theorem 3.4:  Let (𝑋, 𝑑) be a complete metric space and  𝜇, 𝜌 are weakly compatible self- mappings 

on 𝑋 with 𝜌(𝑋) ⊂ 𝜇(𝑋).Also let 𝜇(𝑋) is closed subset of 𝑋.There exist  𝛼: 𝑋 × 𝑋 → [0, ∞),    𝜑 ∈ ∅ 

and 𝛽 is maximum of all member of 𝑅𝜑+  such that 𝜇 𝑎𝑛𝑑 𝜌 satisfy following conditions: 

a) 𝜌 is 𝜇 − 𝛼, 𝜑 contraction. 

b) 𝜌 is 𝜇 − 𝛼𝛽−1 admissible. 

c) There exist 𝑥0 ∈ 𝑋 such that 𝛼(𝜇(𝑥0), 𝜌(𝑥0)) > 𝛽−1 

d) If for a sequence {𝑥𝑛} in 𝑋, the sequence 𝜇(𝑥𝑛) converges to 𝜇(𝑥)  and 𝛼(𝜇(𝑥𝑛), 𝜇(𝑥𝑛+1) >

𝛽−1 then there exist sub-sequence 𝜇(𝑥𝑛𝑘
) of 𝜇(𝑥𝑛) such that 𝛼(𝜇(𝑥𝑛𝑘

), 𝜇(𝑥)) > 𝛽−1. 

e) For all 𝑝, 𝑞 ∈ 𝐶(𝜇, 𝜌)the condition 𝛼(𝜇(𝑝), 𝜇(𝑞)) > 𝛽−1 or 𝛼(𝜇(𝑞), 𝜇(𝑝)) > 𝛽−1 exist. 

Then 𝜇 and 𝜌 have a unique point of coincident. Further this point is unique common fixed 

point.  

Proof: Since 𝜌(𝑋) ⊂ 𝜇(𝑋), for 𝑥0 ∈ 𝑋 there exist 𝑥1 ∈ 𝑋 such that 𝜌(𝑥0) = 𝜇(𝑥1). In general we 

can write 𝑠𝑛 = 𝜌(𝑥𝑛) = 𝜇(𝑥𝑛+1).If 𝑠𝑛 = 𝑠𝑛+1  for some  𝑛 ∈ 𝑁, then 𝑠𝑛 = 𝜌(𝑥𝑛+1) = 𝜇(𝑥𝑛+1).This 

implies there is a point of coincident. Thus let 𝑠𝑛 ≠ 𝑠𝑛+1  ∀𝑛 ∈ 𝑁 . By condition (c), there exist 𝑥0 ∈

𝑋 such that 

 𝛼(𝜇(𝑥0), 𝜌(𝑥0)) > 𝛽−1 

⇒ 𝛼(𝜇(𝑥0), 𝜇(𝑥1)) > 𝛽−1  

Since 𝑔 is 𝜇 − 𝛼 admissible, so 

 𝛼(𝜇(𝑥0), 𝜇(𝑥1)) > 𝛽−1 ⇒ 𝛼(𝜌(𝑥0), 𝜌(𝑥1)) > 𝛽−1 

                                           ⇒ 𝛼(𝜇(𝑥1), 𝜇(𝑥2)) > 𝛽−1 

                                          ⇒ 𝛼(𝜌(𝑥1), 𝜌(𝑥2)) > 𝛽−1 

                                          ⇒ 𝛼(𝜇(𝑥2), 𝜇(𝑥3)) > 𝛽−1 

Continuing this process we have, 

𝛼(𝜇(𝑥𝑛), 𝜇(𝑥𝑛+1)) > 𝛽−1 …… (3.6) 

From condition (a) 

 𝛽−1𝑑(𝜌(𝑥0), 𝜌(𝑥1) ≤ 𝛼(𝜇(𝑥0), 𝜇(𝑥1))𝑑(𝜌(𝑥0), 𝜌(𝑥1) ≤ 𝜑(𝑑(𝜇(𝑥0), 𝜇(𝑥1)) 

⇒ 𝑑(𝜌(𝑥0), 𝜌(𝑥1) ≤ 𝛽𝜑(𝑑(𝜇(𝑥0), 𝜇(𝑥1))            [  ∵ 𝛽𝜑 = 𝜎 ∈ ∅] 

 ⇒ 𝑑(𝜌(𝑥0), 𝜌(𝑥1) ≤ 𝜎(𝑑(𝜇(𝑥0), 𝜇(𝑥1))…………. (3.7) 

 ⇒ 𝑑(𝜇(𝑥1), 𝜇(𝑥2) ≤ 𝜎(𝑑(𝜇(𝑥0), 𝜇(𝑥1))………………. (3.8) 

Similarly we can show 

 𝑑(𝜇(𝑥2), 𝜇(𝑥3) ≤ 𝜎(𝑑(𝜇(𝑥1), 𝜇(𝑥2)) 



Communications on Applied Nonlinear Analysis 

ISSN: 1074-133X 

Vol 31 No. 3s (2024) 

 

374 
https://internationalpubls.com 

From (3.8) and non-decreasing property of 𝜎, we have 

𝑑(𝜇(𝑥2), 𝜇(𝑥3) ≤ 𝜎2(𝑑(𝜇(𝑥0), 𝜇(𝑥1))  

Continuing this process we have, 

 𝑑(𝜇(𝑥𝑛), 𝜇(𝑥𝑛+1) ≤ 𝜎𝑛(𝑑(𝜇(𝑥0), 𝜇(𝑥1))……… (3.9) 

Now for any 𝑚, 𝑛 ∈ 𝑁 with 𝑚 < 𝑛, 

 𝑑(𝜇(𝑥𝑚), 𝜇(𝑥𝑛) ≤  𝑑(𝜇(𝑥𝑚), 𝜇(𝑥𝑚+1) +  𝑑(𝜇(𝑥𝑚+1), 𝜇(𝑥𝑛+2) + ⋯ … +  𝑑(𝜇(𝑥𝑛−1), 𝜇(𝑥𝑛) 

From (3.9) 

𝑑(𝜇(𝑥𝑚), 𝜇(𝑥𝑛) ≤ 𝜎𝑚(𝑑(𝜇(𝑥0), 𝜇(𝑥1)) + 𝜎𝑚+1(𝑑(𝜇(𝑥0), 𝜇(𝑥1)) + ⋯ … . . +𝜎𝑛−1(𝑑(𝜇(𝑥0), 𝜇(𝑥1)) 

 Using the property 𝜎𝑚(𝑡) = 0 ∀  𝑡 > 0  for 𝑚 → ∞ 

𝑑(𝜇(𝑥𝑚), 𝜇(𝑥𝑛) = 0 …………… (3.10) 

This yield { 𝜇(𝑥𝑛)} is Chauchy sequence in 𝑋. Being (𝑋, 𝑑) is complete metric space, there exist 𝑢 ∈

𝑋 such that lim
𝑛→∞

𝜇(𝑥𝑛) = 𝑢. Here 𝜌(𝑥𝑛) = 𝜇(𝑥𝑛+1), implies  lim
𝑛→∞

𝜌(𝑥𝑛) = 𝑢.   

As our hypothesis 𝜇(𝑋) is closed sub-set of 𝑋, so 𝑢 ∈ 𝜇(𝑋)  and there exist 𝑧 ∈ 𝑋 such that 𝜇(𝑧) = 𝑢 

thus  lim
𝑛→∞

𝑑( 𝜇(𝑥𝑛) − 𝜇(𝑧)) = 0………. (3.11) 

Now by hypothesis (d), from (3.6) and (3.11); there exist sub-sequence 𝜇(𝑥𝑛𝑘
) of 𝜇(𝑥𝑛) such 

that 𝛼 (𝜇(𝑥𝑛𝑘
), 𝜇(𝑧)) > 𝛽−1    ∀ 𝑘 ∈ 𝑁  

Now by   (3.7) we have, 

   𝛽−1𝑑(𝜌(𝑥𝑛𝑘
), 𝜌(𝑧)) ≤ 𝛼 (𝜇(𝑥𝑛𝑘

), 𝜇(𝑧)) 𝑑(𝜌(𝑥𝑛𝑘
), 𝜌(𝑧) ≤ 𝜑(𝑑(𝜇(𝑥𝑛𝑘

), 𝜇(𝑧))) 

 𝑑(𝜌(𝑥𝑛𝑘
), 𝜌(𝑧)) ≤ 𝜎(𝑑(𝜇(𝑥𝑛𝑘

), 𝜇(𝑧))) 

Taking 𝑘 → ∞ 

𝑑(𝑢, 𝜌(𝑧)) ≤ 𝜎(𝑑(𝜇(𝑧), 𝜇(𝑧))  

⇒ 𝑑(𝑢, 𝜌(𝑧)) = 0  

∴ 𝜌(𝑧) = 𝑢 = 𝜇(𝑧)  

Therefore 𝑧 is a coincident of 𝜇 and 𝜌. 

We show 𝑧 is unique coincident point. If possible suppose 𝑤 is another coincident point then by 

hypothesis (e) 

𝛼(𝜇(𝑧), 𝜇(𝑤)) > 𝛽−1 …….. (3.12) 

Applying condition (a) and from (3.7) and (3.12) we have, 

𝛽−1𝑑(𝜌(𝑧), 𝜌(𝑤)) ≤ 𝛼(𝜇(𝑧), 𝜇(𝑤))𝑑(𝜌(𝑧), 𝜌(𝑤) ≤ 𝜑(𝑑(𝜇(𝑧), 𝜇(𝑤)) ) 

⇒ 𝑑(𝜌(𝑧), 𝜌(𝑤)) ≤ 𝜎(𝑑(𝜇(𝑧), 𝜇(𝑤)) ≤ 𝑑(𝜇(𝑧), 𝜇(𝑤))   



Communications on Applied Nonlinear Analysis 

ISSN: 1074-133X 

Vol 31 No. 3s (2024) 

 

375 
https://internationalpubls.com 

 ⇒ 𝑑(𝜌(𝑧), 𝜌(𝑤)) ≤ 𝑑(𝜌(𝑧), 𝜌(𝑤)) 

⇒ 𝑑(𝜌(𝑧), 𝜌(𝑤)) = 0  

⇒ 𝜌(𝑧) = 𝜌(𝑤)  

∴ 𝜌(𝑧) = 𝜇(𝑧) = 𝜌(𝑤) = 𝜇(𝑤)  

Hence 𝑧 is unique coincident point and 𝑢 is unique point of coincident. From proposition 2.1, it is 

clear that 𝑢 is unique common fixed point of 𝜇 and 𝜌  □ 

Example 1: Let 𝑋 = [0, 1] and metric on it is 𝑑(𝑥, 𝑦) = ⌊𝑥 − 𝑦⌋.Take the mapping 𝜇: 𝑋 → 𝑋 defined 

by 𝜇(𝑥) =
𝑥3

2
+

7

16
 .It is clear that 𝜇 is continuous but not Banach contraction at 𝑥 = 1 and 𝑦 =

0.9.Actualy it is 𝛼 − 𝜑 contraction. Where, 𝛼: 𝑋 × 𝑋 → [0, ∞) defined by 𝛼(𝑥, 𝑦) =
1

𝑥2+𝑥𝑦+𝑦2 and 

𝜑: 𝑋 × 𝑋 → [0, ∞) defined by 𝜑(𝑡) =
𝑡

2
. Take 𝛽 = 1.999 …. so that 𝛽𝜑 is again a comparison 

function. There exist 𝑥0 = 0.1 such that 𝛼(𝑥0, 𝜇(𝑥0)) > 𝛽−1.By calculation we can easily find that 𝜇  

is 𝛼 𝛽−1-admissible. More over for every pair 𝑢, 𝑣 ∈ 𝑋 there exist 𝑤 ∈ 𝑋 such that 𝛼(𝑢, 𝑤) > 𝛽−1 

and 𝛼(𝑣, 𝑤) > 𝛽−1.Here 𝜇 satisfy all condition of theorem 3.1 and 3.3.Hence there exist unique 

fixed point 𝑥 =
1

2
.   

Example 2: Let 𝑋 = [0, 1] and metric on it is 𝑑(𝑥, 𝑦) = ⌊𝑥 − 𝑦⌋.Take two self-mappings defined 

by 𝜇(𝑥) =
𝑥3

2
+

7

16
   and 𝜌(𝑥) =

𝑥4

3
+

23

48
 . It is clear that 𝜇 and 𝜌 are continuous but neither of them 

are Banach contraction at 𝑥 = 1 and 𝑦 = 0.9.Actualy they satisfy  𝑓 − 𝛼, 𝜑 contraction. Where, 

𝛼: 𝑋 × 𝑋 → [0, ∞) defined by 𝛼(𝑥, 𝑦) =
𝑥2+𝑥𝑦+𝑦2

𝑥3+𝑥𝑦2+𝑥2𝑦+𝑦3 and 𝜑: 𝑋 × 𝑋 → [0, ∞) defined by 𝜑(𝑡) =

2𝑡

3
. Take 𝛽 = 1.499 …. so that 𝛽𝜑 is again a comparison function. There exist 𝑥0 = 0.1 such that 

𝛼(𝜇(𝑥0), 𝜌(𝑥0)) > 𝛽−1.By calculation we can easily find that 𝜌  is 𝜇 − 𝛼 𝛽−1-admissible. More over 

for every pair 𝑢, 𝑣 ∈ 𝐶(𝜇, 𝜌)  such that 𝛼(𝜇(𝑢), 𝜇(𝑣)) > 𝛽−1 or 𝛼(𝜇(𝑣), 𝜇(𝑢)) > 𝛽−1.Here 𝜇 and  𝜌 

commute at only coincident point .Hence there exist unique common fixed point 𝑥 =
1

2
.  

4. Remarks 

a) If 𝛽 = 1 then the theorem 3.1 becomes theorem 2.1. 

b) For 𝛽 = 1 the mapping 𝜇 becomes 𝛼-admissible. 

c) If 𝜇 is identity map and 𝛽 = 1 then the theorem 3.4 becomes theorem 2.1. 

Refrences 

[1] S. Banach, “Sur les opérations dans les ensembles abstraits et leur application aux équations intégrales,” 

Fundamenta mathematicae, vol. 3, no. 1, pp. 133–181, 1922. 

[2] B. Samet, C. Vetro, and P. Vetro, “Fixed point theorems for α–ψ-contractive type mappings,” Nonlinear 

analysis: theory, methods & applications, vol. 75, no. 4, pp. 2154–2165, 2012. 

[3] B. Samet, “FIXED POINTS FOR α-ψ CONTRACTIVE MAPPINGS WITH AN APPLICATION TO 

QUADRATIC INTEGRAL EQUATIONS,” vol. 1914, pp. 1–18, 1914. 



Communications on Applied Nonlinear Analysis 

ISSN: 1074-133X 

Vol 31 No. 3s (2024) 

 

376 
https://internationalpubls.com 

[4] E. Karapinar, S. Czerwik, and H. Aydi, “Research Article (𝛼, 𝜓)-Meir-Keeler Contraction Mappings in 

Generalized 𝑏-Metric Spaces,” 2018, Accessed: May 18, 2024. [Online]. Available: 

https://www.academia.edu/download/72105343/3264620.pdf 

[5] J. Li and H. Guan, “Common Fixed Point Results for Generalized -Contractive Mappings with Applications,” 

Journal of Function Spaces, vol. 2021, p. e5020027, Jun. 2021, doi: 10.1155/2021/5020027. 

[6] N. Hussain, P. Salimi, and A. Latif, “Fixed point results for single and set-valued α-η-ψ-contractive mappings,” 

Fixed Point Theory Appl, vol. 2013, no. 1, p. 212, Dec. 2013, doi: 10.1186/1687-1812-2013-212. 

[7] K. Zoto, B. Rhoades, and S. Radenović, “Some generalizations for (α-ψ, φ)-contractions in b-metric-like spaces 

and an application,” Fixed Point Theory and Applications, vol. 2017, pp. 13663–2017, Nov. 2017, doi: 

10.1186/s13663-017-0620-1. 

[8] B. N. Abagaro, K. K. Tola, and M. A. Mamud, “Fixed point theorems for generalized $(\alpha ,\psi )$-

contraction mappings in rectangular quasi b-metric spaces,” Fixed Point Theory Algorithms Sci Eng, vol. 2022, 

no. 1, p. 13, Dec. 2022, doi: 10.1186/s13663-022-00723-w. 

[9] J. R. Morales and E. M. Rojas, “Common fixed points for $$(\psi -\varphi )$$-weak contractions type in b-

metric spaces,” Arab. J. Math., vol. 10, no. 3, pp. 639–658, Dec. 2021, doi: 10.1007/s40065-021-00347-9. 

[10] P. Manthena and R. Manchala, “Fixed point results for $\backslashmathscr {H} $-contractions in fuzzy metric 

spaces via admissible,” Malaya Journal of Matematik, vol. 6, no. 03, pp. 588–594, 2018. 

[11] P. Salimi, N. Hussain, A. Roldan, and E. Karapınar, “On Modified 𝛼-ϕ-Asymmetric Meir-Keeler Contractive 

Mappings,” Filomat, vol. 28, no. 9, pp. 1855–1869, 2014. 

[12] A. Latif, M. E. Gordji, E. Karapınar, and W. Sintunavarat, “Fixed point results for generalized ( α , ψ ) -Meir-

Keeler contractive mappings and applications,” J Inequal Appl, vol. 2014, no. 1, p. 68, Dec. 2014, doi: 

10.1186/1029-242X-2014-68. 

[13] N. Hussain, E. Karapınar, P. Salimi, and P. Vetro, “Fixed point results for G m -Meir-Keeler contractive and G - 

( α , ψ ) -Meir-Keeler contractive mappings,” Fixed Point Theory Appl, vol. 2013, no. 1, p. 34, Dec. 2013, doi: 

10.1186/1687-1812-2013-34. 

[14] N. Shahzad, M. A. Alghamdi, S. Alshehri, and I. Aranđelović, “Semi-metric spaces and fixed points of α−φ -

contractive maps,” J. Nonlinear Sci. Appl., vol. 09, no. 05, pp. 3147–3156, May 2016, doi: 

10.22436/jnsa.009.05.104. 

[15] A. FELHI, S. SAHMIM, and H. AYDI, “On common fixed points for (α, ψ)-contractions and generalized cyclic 

contractions in b-metric-like spaces and consequences,” 2016, Accessed: May 18, 2024. [Online]. Available: 

https://www.emis.de/journals/TJNSA/includes/files/articles/user_articles/20151029125204_669_On_common_f

ixed_points_for_(alpha,p.pdf 

[16] I. A. Rus, A. Petruşel, and G. Petruşel, Fixed point theory, vol. 2. Cluj University Press Cluj-Napoca, 2008. 

[17] M. Abbas and G. Jungck, “Common fixed point results for noncommuting mappings without continuity in cone 

metric spaces,” Journal of Mathematical Analysis and Applications, vol. 341, no. 1, pp. 416–420, May 2008, 

doi: 10.1016/j.jmaa.2007.09.070. 

[18] H. Aydi, “α-implicit contractive pair of mappings on quasi b-metric spaces and an application to  integral 

equations,” J. Nonlinear Convex Anal, vol. 17, no. 12, pp. 2417–2433, 2016.