Communications on Applied Nonlinear Analysis 

ISSN: 1074-133X 

Vol 32 No. 6s (2025) 

 

454 
https://internationalpubls.com 

 Some Fixed Point Theorems Using 𝚽𝐩 Operator 

 

K Dinesh 1* , Kastriot Zoto 2 , Adriana Topi 3 , B Shoba 4 , Doaa Rizk 5 
1 Department of Mathematics, K.Ramakrishnan College of Engineering, Tiruchirappalli, Tamilnadu, India. 

Email id: dinesh.skksv93@gmail.com 

2 Department of Mathematics, Informatics and Physics, Faculty of Natural Sciences,University of Gjirokastra, Albania. 

Email id: kzoto@uogj.edu.al 

3 Department of Informatics and Technology, Faculty of Engineering, Informatics, and Architecture, European 

University of Tirana, Tirana, 1000, Albania Email id: ardiana.topi@uet.edu.al 

4 Department of Mathematics, St Joseph's College of Engineering, OMR Chennai - 600 019, India 

Email id shobabalasubramaniyam@gmail.com 

5 Department of Mathematics, College of Science, Qassim University, Buraydah, 51452, Saudi Arabia. 

Email id: d.hussien@qu.edu.sa 

 

Article History: 

Received: 22-10-2024 

Revised: 06-12-2024 

Accepted: 13-12-2024 

Abstract:  

This article's goal is to use the Φp operator to prove a few fixed point theorems in 

complete metric space. Furthermore, we investigate whether fixed points for self 

mappings that meet the requirements of rational expression exist and are unique in a 

complete metric space. Our findings expand upon and generalize a great deal of 

previously published research. 

Keywords: Rational Expression, Complete Metric Spaces, Fixed Point, Φp operator, 

Self mapping 

1. Introduction 

In 1889, H. Poincare the French mathematician,  introduced the fixed points in different version The 

origin of fixed point theory, in the 19th century was notorious by mathematicians like Cauchy, 

Fredholm, Caristi, Liouville, Lipschitz, Peano and Picard. Banach’s contribution to metric fixed point 

theory was not recognized until F. Brouwer’s work  and contribution to the development of the non-

linear functional analysis as an active and vital branch of mathematics. In this paper, we investigate 

some Fixed point theorems of Complete Metric Sapces using Φp operator.  

Definition 1.1:   Let 𝑋 be a none-empty set, a function 𝑑: 𝑋 × 𝑋 → 𝑅 is called a metric on 𝑋, if it 

satisfies the following conditions, 

(i) 𝑑(𝜛, 휁) ≥ 0 and 𝑑(𝜛, 휁) = 0 if and only if 𝜛 = 휁, ∀𝜛, 휁 ∈ 𝑋 

(ii) 𝑑(𝜛, 휁) = 𝑑(휁, 𝜛)∀ 𝜛, 휁 ∈ 𝑋  

(iii) 𝑑(𝜛, 휁) ≤ 𝑑(𝜛, 𝑧) + 𝑑(𝑧, 휁) ∀ 𝜛, 휁, 𝑧 ∈ 𝑋 

Then (𝑋, 𝑑) is called metric space. 

Definition 1.2:   A sequence {𝜛𝑛} is said to be a Cauchy sequence if give  휀 > 0, there exists a positive 

integer 𝑚 such that |𝜛𝑛 − 𝜛𝑚| < 휀 whenever 𝑛 ≥ 𝑚.  

mailto:dinesh.skksv93@gmail.com
mailto:kzoto@uogj.edu.al
mailto:ardiana.topi@uet.edu.al
mailto:shobabalasubramaniyam@gmail.com


Communications on Applied Nonlinear Analysis 

ISSN: 1074-133X 

Vol 32 No. 6s (2025) 

 

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Definition 1.3:  A metric space (𝑋, 𝑑) is said to be complete if every Cauchy sequence in 𝑋 is 

convergent in 𝑋 

Definition 1.4:  Let (𝑋, 𝑑) be a complete metric space Φp: 𝑋 → 𝑋 is an increasing and positive 

mapping.  

If 𝑋 = 𝑅, then Φp: 𝑅 → 𝑅 is a p-Lapalcian operator,  

Φp(𝜛) = |𝜛|𝑝−2𝜛 for some 𝑝 > 1  

Lemma 1.5: Show that the operator Φp: 𝑋 → 𝑋 holds the following properties  

(i) If 𝜛 ≤ 휁 then, Φp(𝜛) ≤ Φp(휁)  ∀𝜛, 휁 ∈ 𝑋 

(ii) Φp is continuous bijection and its inverse mapping is also continuous.  

      That is Φp is homeomorphishm  

(iii) Φp(𝜛휁) = Φp(𝜛)Φp(휁) ∀ 𝜛, 휁 ∈ 𝑋 

(iv) Φp (𝜛 + 휁) ≤ Φp(𝜛) + Φp(휁)  ∀ 𝜛, 휁 ∈ 𝑋 

2. Main result 

Theorem 2.1: Let 𝑇 be continuous self map, defined on a complete metric space 𝑋 and  

Φp: 𝑋 → 𝑋 Further 𝑇 satisfies the following conditions 

 Φp(𝑑(𝑇𝜛, 𝑇휁)) ≤ 𝜆1Φp(𝑑(𝜛, 휁)) + 𝜆2Φp[𝑑(𝑇𝜛, 𝜛) + 𝑑(𝑇휁, 휁)] 

                              +𝜆3Φp[𝑑(𝑇휁, 𝜛) + 𝑑(𝑇𝜛, 휁)] +𝜆4Φp [
𝑑(𝜛,𝑇𝜛)𝑑(𝜁,𝑇𝜁)

𝑑(𝜛,𝜁)
] 

                             +𝜆5Φp [
𝑑(𝜛,𝑇𝜁)𝑑(𝜁,𝑇𝜛)

𝑑(𝜛,𝜁)
] 

                            +𝜆6Φp [
𝑑(𝜛,𝑇𝜛)𝑑(𝜁,𝑇𝜁)+𝑑(𝜛,𝑇𝜁)𝑑(𝜁,𝑇𝜛)

𝑑(𝜛,𝜁)
] + 𝜆7Φp [

𝑑(𝜛,𝑇𝜛)𝑑(𝜁,𝑇𝜁)+𝑑(𝜛,𝑇𝜁)𝑑(𝜁,𝑇𝜛)

𝑑(𝜛,𝜁)
] 

                           +𝜆8Φp [
𝑑(𝜛,𝑇𝜁)[𝑑(𝜛,𝑇𝜛)+𝑑(𝜁,𝑇𝜁)+𝑑(𝜛,𝑇𝜁)+𝑑(𝜁,𝑇𝜛)]

𝑑(𝜛,𝑇𝜛)+𝑑(𝜁,𝑇𝜁)+𝑑(𝜛,𝑇𝜁)+𝑑(𝜁,𝑇𝜛)
] 

For all 𝜛, 휁 ∈ 𝑋, 𝜛 ≠ 휁 and 

 Φp(𝜆1) + 2Φp(𝜆2) + 2Φp(𝜆3) + Φp(𝜆4) + Φp(𝜆6) + Φp(𝜆7) < 1 then 𝑇 has unique fixed point in 𝑇. 

Proof: 

Let 𝜛0 be an arbitrary point in 𝑋, and we define a sequence {𝜛𝑛} by means of iterates of 𝑇.  

By setting 𝑇𝑛𝜛0 = 𝜛𝑛, where 𝑛 is a positive integers. If 𝜛𝑛 = 𝜛𝑛+1, for  some 𝑛, then we have 𝑇𝜛𝑛 =

𝜛𝑛, then 𝜛𝑛 is a fixed point of 𝑇 taking 𝜛𝑛 ≠ 𝜛𝑛+1 for all 𝑛 

Φp(𝑑(𝜛𝑛+1, 𝜛𝑛)) = Φp(𝑑(𝑇𝜛𝑛, 𝑇𝜛𝑛−1)) 

 Φp(𝑑(𝑇𝜛𝑛, 𝑇𝜛𝑛−1)) ≤ 𝜆1Φp(𝑑(𝜛𝑛, 𝜛𝑛−1)) + 𝜆2Φp([𝑑(𝑇𝜛𝑛, 𝜛𝑛) + 𝑑(𝑇𝜛𝑛−1, 𝜛𝑛−1)]) 



Communications on Applied Nonlinear Analysis 

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                                        +𝜆3Φp([𝑑(𝑇𝜛𝑛−1, 𝜛𝑛) + 𝑑(𝑇𝜛𝑛, 𝜛𝑛−1)]) 

                                       +𝜆4Φp ([
𝑑(𝜛𝑛,𝑇𝜛𝑛)𝑑(𝜛𝑛−1,𝑇𝜛𝑛−1)

𝑑(𝜛𝑛,𝜛𝑛−1)
]) 

                                     +𝜆5Φp ([
𝑑(𝜛𝑛,𝑇𝜛𝑛−1)𝑑(𝜛𝑛−1,𝑇𝜛𝑛)

𝑑(𝜛𝑛,𝜛𝑛−1)
]) 

                                     +𝜆6Φp ([
𝑑(𝜛𝑛,𝑇𝜛𝑛)𝑑(𝜛𝑛−1,𝑇𝜛𝑛−1)+𝑑(𝜛𝑛,𝑇𝜛𝑛−1)𝑑(𝜛𝑛−1,𝑇𝜛𝑛)

𝑑(𝜛𝑛,𝜛𝑛−1)
]) 

                                     +𝜆7Φp ([
𝑑(𝜛𝑛,𝑇𝜛𝑛)𝑑(𝜛𝑛−1,𝑇𝜛𝑛−1)+𝑑(𝜛𝑛,𝑇𝜛𝑛−1)𝑑(𝜛𝑛−1,𝑇𝜛𝑛)

𝑑(𝜛𝑛,𝜛𝑛−1)
])  

                                     +𝜆8Φp ([
𝑑(𝜛𝑛,𝑇𝜛𝑛−1)[𝑑(𝜛𝑛,𝑇𝜛𝑛)+𝑑(𝜛𝑛−1,𝑇𝜛𝑛−1)+𝑑(𝜛𝑛,𝑇𝜛𝑛−1)+𝑑(𝜛𝑛−1,𝑇𝜛𝑛)]

𝑑(𝜛𝑛,𝑇𝜛𝑛)+𝑑(𝜛𝑛−1,𝑇𝜛𝑛−1)+𝑑(𝜛𝑛,𝑇𝜛𝑛−1)+𝑑(𝜛𝑛−1,𝑇𝜛𝑛)
]) 

                            ≤ 𝜆1Φp(𝑑(𝜛𝑛, 𝜛𝑛−1) + 𝜆2[𝑑(𝜛𝑛+1, 𝜛𝑛) + 𝑑(𝜛𝑛, 𝜛𝑛−1)] 

                              +𝜆3Φp[𝑑(𝜛𝑛, 𝜛𝑛) + 𝑑(𝜛𝑛+1, 𝜛𝑛−1)] + 𝜆4Φp [
𝑑(𝜛𝑛,𝜛𝑛+1)𝑑(𝜛𝑛−1,𝜛𝑛)

𝑑(𝜛𝑛,𝜛𝑛−1)
] 

                              +𝜆5Φp [
𝑑(𝜛𝑛,𝜛𝑛)𝑑(𝜛𝑛−1,𝜛𝑛+1)

𝑑(𝜛𝑛,𝜛𝑛−1)
] 

                              +𝜆6Φp [
𝑑(𝜛𝑛,𝜛𝑛+1)𝑑(𝜛𝑛−1,𝜛𝑛)+𝑑(𝜛𝑛,𝜛𝑛)𝑑(𝜛𝑛−1,𝜛𝑛+1)

𝑑(𝜛𝑛,𝜛𝑛−1)
] 

                              +𝜆7Φp [
𝑑(𝜛𝑛,𝜛𝑛+1)𝑑(𝜛𝑛−1,𝜛𝑛)+𝑑(𝜛𝑛,𝜛𝑛)𝑑(𝜛𝑛−1,𝜛𝑛+1)

𝑑(𝜛𝑛,𝜛𝑛−1)
] 

                              +𝜆8Φp [
𝑑(𝜛𝑛,𝜛𝑛)[𝑑(𝜛𝑛,𝜛𝑛+1)+𝑑(𝜛𝑛−1,𝜛𝑛)+𝑑(𝜛𝑛,𝜛𝑛)+𝑑(𝜛𝑛−1,𝜛𝑛+1)]

𝑑(𝜛𝑛,𝜛𝑛+1)+𝑑(𝜛𝑛−1,𝜛𝑛)+𝑑(𝜛𝑛,𝜛𝑛)+𝑑(𝜛𝑛−1,𝜛𝑛+1)
] 

From the property of Φp  Operator,   

          Φp(𝑑(𝜛𝑛+1, 𝜛𝑛)) ≤ 𝜆1Φp(𝑑(𝜛𝑛, 𝜛𝑛−1)) + 𝜆2Φp(𝑑(𝜛𝑛+1, 𝜛𝑛)) + 𝜆2Φp(𝑑(𝜛𝑛, 𝜛𝑛−1)) 

                                         +𝜆3Φp[𝑑(𝜛𝑛+1, 𝜛𝑛) + 𝑑(𝜛𝑛, 𝜛𝑛−1)] + 𝜆4Φp(𝑑(𝜛𝑛, 𝜛𝑛+1)) 

                                        +𝜆6Φp(𝑑(𝜛𝑛, 𝜛𝑛+1)) + 𝜆7Φp(𝑑(𝜛𝑛, 𝜛𝑛+1)) 

     Φp(𝑑(𝜛𝑛+1, 𝜛𝑛)) − 𝜆2Φp(𝑑(𝜛𝑛+1, 𝜛𝑛)) − 𝜆3Φp(𝑑(𝜛𝑛+1, 𝜛𝑛)) − 𝜆4Φp(𝑑(𝜛𝑛, 𝜛𝑛+1)) 

−𝜆6Φp(𝑑(𝜛𝑛, 𝜛𝑛+1)) − 𝜆7Φp(𝑑(𝜛𝑛, 𝜛𝑛+1)) 

≤ 𝜆1Φp(𝑑(𝜛𝑛, 𝜛𝑛−1)) + 𝜆2Φp(𝑑(𝜛𝑛, 𝜛𝑛−1)) + 𝜆3Φp(𝑑(𝜛𝑛, 𝜛𝑛−1)) 

                                                   

𝑑(𝜛𝑛+1, 𝜛𝑛) ≤ (
Φp(𝜆1) + Φp(𝜆2) + Φp(𝜆3)

1 − Φp(𝜆2) − Φp(𝜆3) − Φp(𝜆4) − Φp(𝜆6) − Φp(𝜆7)
) 𝑑(𝜛𝑛, 𝜛𝑛−1) 

On applying the same process, we get 

𝑑(𝜛𝑛+1, 𝜛𝑛) ≤ (
Φp(𝜆1) + Φp(𝜆2) + Φp(𝜆3)

1 − Φp(𝜆2) − Φp(𝜆3) − Φp(𝜆4) − Φp(𝜆6) − Φp(𝜆7)
)

𝑛

𝑑(𝜛1, 𝜛0) 

𝑑(𝜛𝑛+1, 𝜛𝑛) ≤ 𝛿𝑛 𝑑(𝜛1, 𝜛0) where  𝛿 = (
Φp(𝜆1)+Φp(𝜆2)+Φp(𝜆3)

1−Φp(𝜆2)−Φp(𝜆3)−Φp(𝜆4)−Φp(𝜆6)−Φp(𝜆7)
) < 1 



Communications on Applied Nonlinear Analysis 

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By triangular inequality, we have for 𝑚 > 𝑛 

𝑑(𝜛𝑛, 𝜛𝑚) ≤ 𝑑(𝜛𝑛, 𝜛𝑛+1) + 𝑑(𝜛𝑛+1, 𝜛𝑛+2) + 𝑑(𝜛𝑛+2, 𝜛𝑛+3) + ⋯ + 𝑑(𝜛𝑚−1, 𝜛𝑛) 

≤ (𝛿𝑛 + 𝛿𝑛+1 + ⋯ + 𝛿𝑚−1)𝑑(𝜛1, 𝜛0) 

Therefore, 𝑑(𝜛𝑛, 𝜛𝑚) ≤
𝛿𝑛

1−𝛿
𝑑(𝜛0, 𝑇𝜛0) → 0 as 𝑚, 𝑛 → ∞ 

So, {𝜛𝑛} is Cauchy sequence in 𝜛, so by completeness of 𝑋, there is a point 𝑢 ∈ 𝑋, such that 𝜛𝑛 →

𝑢 as 𝑛 → ∞. Further, the continuity of 𝑇 in 𝑋 implies 

𝑇(𝑢) = 𝑇 ( lim
𝑛→∞

𝜛𝑛) = lim
𝑛→∞

𝑇𝜛𝑛 = lim
𝑛→∞

𝜛𝑛+1 = 𝑢 

Therefore, 𝑢 is a fixed point of 𝑇 in 𝜛. 

Suppose if there is any other 𝜛1 ≠ 𝜛2 in 𝑋 such that 𝑇(𝜛2) = 𝜛2, then 𝑑(𝜛1, 𝜛2) = 𝑑(𝑇𝜛1, 𝑇𝜛2) 

    Φp(𝑑(𝜛1, 𝜛2)) ≤ 𝜆1Φp(𝑑(𝜛1, 𝜛2)) + 𝜆2Φp([𝑑(𝑇𝜛1, 𝜛2) + 𝑑(𝑇𝜛2, 𝜛2)]) 

                                 +𝜆3Φp([𝑑(𝑇𝜛2, 𝜛1) + 𝑑(𝑇𝜛1, 𝜛2)]) + 𝜆4Φp ([
𝑑(𝜛1,𝑇𝜛2)𝑑(𝜛2,𝑇𝜛2)

𝑑(𝜛1,𝜛2)
]) 

                                 +𝜆5Φp ([
𝑑(𝜛1,𝑇𝜛2)𝑑(𝜛2,𝑇𝜛1)

𝑑(𝜛1,𝜛2)
]) 

                                +𝜆6Φp ([
𝑑(𝜛1,𝑇𝜛1)𝑑(𝜛2,𝑇𝜛2)+𝑑(𝜛1,𝑇𝜛2)𝑑(𝜛2,𝑇𝜛1)

𝑑(𝜛1,𝜛2)
]) 

                                +𝜆7Φp ([
𝑑(𝜛1,𝑇𝜛1)𝑑(𝜛2,𝑇𝜛2)+𝑑(𝜛1,𝑇𝜛2)𝑑(𝜛2,𝑇𝜛1)

𝑑(𝜛1,𝑣)
]) 

                               +𝜆8Φp ([
𝑑(𝜛1,𝑇𝜛2)[𝑑(𝜛1,𝑇𝜛1)+𝑑(𝜛2,𝑇𝜛2)+𝑑(𝜛1,𝑇𝜛2)+𝑑(𝜛2,𝑇𝜛1)]

𝑑(𝜛1,𝑇𝜛2)+𝑑(𝜛2,𝑇𝜛2)+𝑑(𝜛1,𝑇𝜛2)+𝑑(𝜛2,𝑇𝜛1)
]) 

   Φp(𝑑(𝜛1, 𝜛2)) ≤ 𝜆1Φp(𝑑(𝜛1, 𝜛2)) + 𝜆2Φp([𝑑(𝜛1, 𝜛1) + 𝑑(𝜛2, 𝜛2)]) 

                               +𝜆3Φp([𝑑(𝜛2, 𝜛1) + 𝑑(𝜛1, 𝜛2)]) + 𝜆4Φp ([
𝑑(𝜛1,𝜛1)𝑑(𝜛2,𝜛2)

𝑑(𝜛1,𝜛2)
]) 

                             +𝜆5Φp ([
𝑑(𝜛1,𝜛2)𝑑(𝜛2,𝜛1)

𝑑(𝜛1,𝜛2)
]) + 𝜆6Φp ([

𝑑(𝜛1,𝜛1)𝑑(𝜛2,𝜛2)+𝑑(𝜛1,𝜛2)𝑑(𝜛2,𝜛1)

𝑑(𝜛1,𝜛2)
])  

                            +𝜆7Φp ([
𝑑(𝜛1,𝜛1)𝑑(𝜛2,𝜛2)+𝑑(𝜛1,𝜛2)𝑑(𝜛2,𝜛1)

𝑑(𝜛1,𝜛2)
]) 

                           +𝜆8Φp ([
𝑑(𝜛1,𝜛2)[𝑑(𝜛1,𝜛1)+𝑑(𝜛2,𝜛2)+𝑑(𝜛1,𝜛2)+𝑑(𝜛2,𝜛1)]

𝑑(𝜛1,𝜛2)+𝑑(𝜛2,𝜛2)+𝑑(𝜛1,𝜛2)+𝑑(𝜛2,𝜛1)
]) 

 Φp(𝑑(𝜛1, 𝜛2)) ≤ 𝜆1Φp(𝑑(𝜛1, 𝜛2)) + 2𝜆3Φp(𝑑(𝜛1, 𝜛2)) + 𝜆5Φp(𝑑(𝜛1, 𝜛2)) 

+𝜆6(𝑑(𝜛1, 𝜛2)) + 𝜆7Φp(𝑑(𝜛1, 𝜛2)) + 𝜆8Φp(𝑑(𝜛1, 𝜛2)) 

    𝑑(𝜛1, 𝜛2) ≤ (Φp(𝜆1) + 2Φp(𝜆3) + Φp(𝜆5) + Φp(𝜆6) + Φp(𝜆7) + Φp(𝜆8))𝑑(𝜛1, 𝜛2) 

Which is contradiction. Hence 𝜛1 is a fixed point 

 



Communications on Applied Nonlinear Analysis 

ISSN: 1074-133X 

Vol 32 No. 6s (2025) 

 

458 
https://internationalpubls.com 

Corollary 2.2:  Let 𝑇 be continues self map, defined on a complete metric space 𝑋. Further 𝑇 

satisfies the following conditions 

 Φp(𝑑(𝑇𝜛, 𝑇휁)) ≤ 𝜆1Φp(𝑑(𝜛, 휁)) + 𝜆2([𝑑(𝑇𝜛, 𝜛) + 𝑑(𝑇휁, 휁)]) 

                               +𝜆3Φp([𝑑(𝑇휁, 𝜛) + 𝑑(𝑇𝜛, 휁)]) 

                              +𝜆4Φp ([
𝑑(𝜛,𝑇𝜛)𝑑(𝑦,𝑇𝜁)

𝑑(𝜛,𝜁)
]) + 𝜆5Φp ([

𝑑(𝜛,𝑇𝜁)𝑑(𝜁,𝑇𝜛)

𝑑(𝜛,𝜁)
]) 

For all 𝜛, 휁 ∈ 𝑋, 𝜛 ≠ 휁 and Φp(𝜆1) + 2Φp(𝜆2) + 2Φp(𝜆3) +  Φp(𝜆4) < 1 then 𝑇 has unique fixed 

point in 𝑇. 

Proof: The proof is comes instead of 𝜆6 = 𝜆7 = 𝜆8 = 0 instead of above theorem. 

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