Communications on Applied Nonlinear Analysis 

ISSN: 1074-133X 

Vol 32 No. 10s (2025) 

  

2612 
 

https://internationalpubls.com 

 Some Results on Partial Cone Metric Spaces with an Application 

 Heeramani Tiwari1,*, Anil Mishra2 ,  Padmavati3  

1,2,3Government V.Y.T. Autonomous P.G. College, Durg, Chhattisgarh, India 

toravi.tiwari@gmail.com, mshranil@gmail.com, padmavati.sudha62@gmail.com 

 

 

Article History: 

Received: 12-01-2025 

Revised: 15-02-2025 

Accepted: 01-03-2025 

Abstract:  

 The objective of this paper is to determine some fixed point theorems 

for generalized 𝛼 −  𝜓 contractive mappings in the framework of 

partial cone metric spaces. In addition, we prove a unique fixed point 

theorem using a rational contractive condition. Our findings align with 

previous research in this area. We also show that our result can be 

applied to the problem of determining the existence of solutions to 

second-order differential equations. 

Keywords:  𝛼 − 𝜓  contractive mappings, partial cone metric spaces, α admissible 

Mappings. 

 

1. Introduction 

The Banach contraction principle [1] was a foundation for a development of metric fixed point 

theory which has been generalized by utilizing various contractive conditions in various contexts. In 

1906, Frechet [2] introduced the notion of metric spaces. In 2007, Huang and Zhang [7] introduced 

the concept of cone metric space which is a generalization of metric space. Another generalization of 

metric spaces is partial metric spaces which was introduced by Matthews [3, 4] in which the self 

distance need not be equal to zero and proved the partial metric version of Banach fixed point 

theorem. Partial cone metric spaces have been investigated by Mahlotra et al. [10] and Sonmez. They 

proved some fixed point theorems in this space. Recently many papers on cone metric spaces and 

partial cone metric spaces have been appeared e.g. see . [8, 9, 13, 14, 15, 16, 17, 18]. On the other 

hand, Samet et al. [5] extended and generalized the Banach contraction principle by introducing a 

new class of contractive type mappings known as 𝛼 −  𝜓 contractive type mappings. Karapinar and 

Samet [6] generalized the 𝛼 −  𝜓 contractive type mappings and established various fixed point 

theorems. 

To begin, we will define partial metric spaces , cone metric spaces, and partial cone metric spaces as 

well as their properties:  

Definition 1.1. (Partial metric space) A partial metric on a non-empty set X is a function 

 𝜌: 𝑋 × 𝑋 → ℝ+ such that for all 𝑥, 𝑦, 𝑧 ∈ 𝑋 the following hold  

1. 𝑥 = 𝑦 ⇔ 𝜌(𝑥, 𝑥) = 𝜌(𝑦, 𝑦) = 𝜌(𝑥, 𝑦);  

mailto:toravi.tiwari@gmail.com
mailto:mshranil@gmail.com
mailto:padmavati.sudha62@gmail.com


Communications on Applied Nonlinear Analysis 

ISSN: 1074-133X 

Vol 32 No. 10s (2025) 

  

2613 
 

https://internationalpubls.com 

2. 𝜌(𝑥, 𝑥) ≤ 𝜌(𝑥, 𝑦); 

3. 𝜌(𝑥, 𝑦) = 𝜌(𝑦, 𝑥); 

4. 𝜌(𝑥, 𝑦) ≤ 𝜌(𝑥, 𝑧) + 𝜌(𝑧, 𝑦) − 𝜌(𝑧, 𝑧). For all 𝑥, 𝑦, 𝑧 ∈ 𝑋. 

Then the pair (X, ρ) is called a partial metric space. It is clear that if ρ(x, y) = 0, then (1) and (2) 

imply that x = y. But if x = y,  ρ(x, y) may not be 0. A basic example of partial metric space is the 

pair (ℝ+, 𝜌) where  𝜌(𝑥, 𝑦)  =  𝑚𝑎𝑥{𝑥, 𝑦} for all 𝑥, 𝑦 ∈ ℝ+.  

Let E be a real Banach space and P a subset of E. P is called a cone if it satisfies the following. 

(1) P is closed, non-empty, and 𝑃 ≠ 0, 

(2) 𝑎𝑥 +  𝑏𝑦 ∈ 𝑃 for all 𝑥, 𝑦 ∈ 𝑃 and non-negative real numbers 𝑎, 𝑏 ∈  ℝ, 

(3) 𝑃 ∩  (−𝑃 )  =  {0}. 

  For a specified cone 𝑃 ⊂  𝐸, we can establish a partial ordering ≤ on E in relation to P by defining 

𝑥 ≤  𝑦 if and only if 𝑦 −  𝑥 ∈  𝑃 . The notation 𝑥 <  𝑦 is used to signify that 𝑥 ≤  𝑦 and 𝑥 ≠  𝑦, 

while 𝑥 ≪  𝑦 indicates that 𝑦 −  𝑥 ∈  𝑖𝑛𝑡𝑃 , with 𝑖𝑛𝑡𝑃 representing the interior of P . The cone P is 

termed normal if there exists a constant K >  0  such that for all x, y ∈  E where 0 ≤  x ≤  y, it 

follows that ||x||  ≤  K||y||. The smallest positive value that satisfies this condition is referred to as 

the normal constant of P.  

Let E be a Banach space, P a cone in E with 𝑖𝑛𝑡𝑃 ≠  𝜙 and ≤ is partial ordering with respect to P. 

Definition 1.2 (Cone metric space) Let X be a non empty set. The mapping 𝑑𝑐 ∶  𝑋 ×  𝑋 →  𝐸 is 

said to be a cone metric on X if for all 𝑥, 𝑦, 𝑧 ∈  𝑋. The followings hold: 

(1)  0 ≤  𝑑𝑐(𝑥, 𝑦) and 𝑑𝑐(𝑥, 𝑦)  =  0 if and only if 𝑥 =  𝑦, 

(2)  𝑑𝑐(𝑥, 𝑦)  =  𝑑𝑐(𝑦, 𝑥), 

(3) 𝑑𝑐(𝑥, 𝑦)  ≤  𝑑𝑐(𝑥, 𝑧)  + 𝑑𝑐(𝑦, 𝑧). 

and (X, dc) is called a cone metric space. 

Mahlotra et al. [10] and Sonmez [11] introduced the notion of partial cone metric space and its 

topological characterization. We now state the definition of partial cone metric space. 

Definition 1.3 (Partial cone metric space)  A partial cone metric on a non-empty 

set X is a function 𝜌𝑐: 𝑋 ×  𝑋 →  𝐸 such that for all 𝑥, 𝑦, 𝑧 ∈  𝑋 

(1) 0 ≤  𝜌𝑐(𝑥, 𝑥)  ≤  𝜌𝑐(𝑥, 𝑦), 

(2) 𝑥 =  𝑦  if and only if  𝜌𝑐(𝑥, 𝑥)  =  𝜌𝑐(𝑥, 𝑦)  =  𝜌𝑐(𝑦, 𝑦), 



Communications on Applied Nonlinear Analysis 

ISSN: 1074-133X 

Vol 32 No. 10s (2025) 

  

2614 
 

https://internationalpubls.com 

(3) 𝜌𝑐(𝑥, 𝑦)  =  𝜌𝑐(𝑦, 𝑥), 

(4) 𝜌𝑐(𝑥, 𝑦)  ≤  𝜌𝑐(𝑥, 𝑧)  +  𝜌𝑐(𝑧, 𝑦)  − 𝜌𝑐(𝑧, 𝑧). 

A partial cone metric space is a pair (𝑋, 𝜌𝑐) such that X is a non-empty set and 𝜌𝑐 is a partial cone 

metric on X . It is clear that, if 𝜌𝑐(𝑥, 𝑦)  =  0, then (1) and (2) imply that x = y. But the converse is 

not true in general. A cone metric space is a partial cone metric space, but there exist partial cone 

metric spaces which are not cone metric spaces. we give the following example from [11] 

Example 1.4 Consider a Banach space 𝐸 =  ℝ2, 𝑃 =  {(𝑥, 𝑦)  ∈  𝐸 ∶  𝑥, 𝑦 ≥  0} 

and 𝑋 =  ℝ+ and 𝜌𝑐 ∶  𝑋 ×  𝑋 →  𝐸 defined 𝑏𝑦 𝜌𝑐(𝑥, 𝑦)  =  (𝑚𝑎𝑥{𝑥, 𝑦}, 𝑘𝑚𝑎𝑥{𝑥, 𝑦}) 

where 𝑘 ≥  0 is a constant. Then (𝑋, 𝜌𝑐) is a partial cone metric space which is not a cone metric 

space.  

Remark 1.5  Suppose (𝑋, 𝜌𝑐)  is a partial cone metric space, then 

𝑑𝑐(𝑥, 𝑦)  =  2𝜌𝑐(𝑥, 𝑦) – 𝜌𝑐(𝑥, 𝑥) – 𝜌𝑐(𝑦, 𝑦) 

For all 𝑥, 𝑦, 𝑧 ∈  𝑋 defines a cone metric on X. 

Theorem1.6  Every partial cone metric space (𝑋, 𝜌𝑐)  is a topological space. 

Following, We give some properties of partial cone metric spaces, for more details see [11]. 

Definition 1.7 Let (𝑋, 𝜌𝑐) be a partial cone metric space. Let {𝑥𝑛} be a sequence in X  

and 𝑥 ∈  𝑋 

(1) {𝑥𝑛} is said to be convergent to x and x is called a limit of {𝑥𝑛} if 

lim
𝑛→∞

𝜌𝑐(𝑥𝑛, 𝑥) = lim
𝑛→∞

𝜌𝑐(𝑥𝑛, 𝑥𝑛) = 𝜌𝑐(𝑥, 𝑥) 

 

(2) {𝑥𝑛}  is Cauchy sequence if there is 𝑥 ∈  𝑃 such that for every ϵ >  0 there 

is ℕ such that for all 𝑛, 𝑚 > ℕ,  ||𝜌𝑐(𝑥𝑛, 𝑥𝑚)  −  𝑥||  <  𝜖. 

(3) (𝑋, 𝜌𝑐)  is said to be complete if every Cauchy sequence in (𝑋, 𝜌𝑐) is convergent in (𝑋, 𝜌𝑐). 

 

In 2012, Samet et al. [5] introduced α-admissible mapping as follows: 

 

Definition 1.8  [5] Let 𝑇 ∶  𝑋 →  𝑋 and 𝛼 ∶  𝑋 ×  𝑋 →  [0, ∞). T is said to α-admissible if 

𝛼(𝑥, 𝑦)  ≥  1 ⇒  𝛼(𝑇 𝑥, 𝑇 𝑦)  ≥  1 

for all 𝑥, 𝑦 ∈  𝑋. 

 

 



Communications on Applied Nonlinear Analysis 

ISSN: 1074-133X 

Vol 32 No. 10s (2025) 

  

2615 
 

https://internationalpubls.com 

2. Main Results 

[12] Let 𝛹 be the family of non-decreasing function 𝜓 ∶  [0, ∞)  →  [0, ∞) such that ∑ 𝜓𝑛(𝑡) <∞
𝑛=1

∞ for each 𝑡 >  0 where 𝜓𝑛 is nth iterate of ψ. 

Lemma 2.1  [12] For every function  𝜓 ∶  [0, ∞)  →  [0, ∞) the following holds: 

If ψ is non decreasing, then for each 𝑡 >  0,  lim
𝑛→∞

𝜓𝑛(𝑡) = 0 implies 𝜓(𝑡)  <  𝑡 

and 𝜓(0)  =  0. 

Definition 2.2  Let (𝑋, 𝜌𝑐) be a partial cone metric space P is a normal cone with constant K.  

Let 𝑇 ∶  𝑋 →  𝑋 be a self mapping. Then T is said to be generalized 𝛼 −  𝜓 contractive mapping if 

there exists two functions 𝛼 ∶  𝑋 ×  𝑋 →  [0, ∞) and 𝜓 ∈  𝛹 for all  𝑥, 𝑦 ∈  𝑋 we have 

                       𝛼(𝑥, 𝑦)𝜌𝑐(𝑇 𝑥, 𝑇 𝑦)  ≤  𝜓(𝑀 (𝑥, 𝑦))                                                         (2.1) 

where 

            𝑀 (𝑥, 𝑦)  =  𝑚𝑎𝑥{𝜌𝑐(𝑥, 𝑦), 𝜌𝑐(𝑥, 𝑇 𝑥), 𝜌𝑐(𝑦, 𝑇 𝑦)}                                                   (2.2) 

Theorem 2.3 Let (𝑋, 𝜌𝑐 ) be a complete partial cone metric space and 𝑇 ∶  𝑋 → 𝑋 be self mapping. 

Suppose 𝛼 ∶  𝑋 ×  𝑋 →  [0, ∞) be the mappings satisfying the conditions: 

(i) T is α admissible; 

(ii) T is generalized 𝛼 −  𝜓 contractive mapping; 

(iii) there exists 𝑥0  ∈  𝑋 such that 𝛼(𝑥0, 𝑇 𝑥0)  ≥  1; 

(iv) T is continuous or if {𝑥𝑛} be a sequence in X such that 𝛼(𝑥𝑛, 𝑥𝑛+1)  ≥  1 for 

all n and 𝑥𝑛  →  𝑥 as 𝑛 →  ∞ then 𝛼(𝑥𝑛, 𝑥)  ≥  1 for all n. 

Then T has a fixed point in X. 

Proof: Let 𝑥0 be an arbitrary point such that 𝛼(𝑥0, 𝑇 𝑥0)  ≥  1. Suppose we have a sequence {𝑥𝑛} in 

X such that 𝑥𝑛+1  =  𝑇 𝑥𝑛 for all 𝑛 ∈  ℕ. If 𝑥𝑛  =  𝑥𝑛+1  for some 𝑛 ∈  ℕ, then 𝑥𝑛 is a fixed point of 

T and the existence part of the proof is finished. Suppose 𝑥𝑛 ≠ 𝑥𝑛+1for every 𝑛 ∈  ℕ Now, since T is 

α-admissible, so 

𝛼(𝑇 𝑥0, 𝑇 𝑥1)  =  𝛼(𝑥1, 𝑥2)  ≥  1 

𝛼(𝑇 𝑥1, 𝑇 𝑥2)  =  𝛼(𝑥2, 𝑥3)  ≥  1 

and using induction we have 𝛼(𝑥𝑛, 𝑥𝑛+1)  ≥  1 for all 𝑛 ∈  ℕ. 

Now, from (2.1) we have 

                            𝜌𝑐(𝑥𝑛, 𝑥𝑛+1)  =  𝜌𝑐(𝑇 𝑥𝑛−1, 𝑇 𝑥𝑛)                                                         (2.3) 

≤  𝛼(𝑥𝑛−1, 𝑥𝑛)𝜌𝑐(𝑇 𝑥𝑛−1, 𝑇 𝑥𝑛) 

                                                       ≤  𝜓(𝑀 (𝑥𝑛−1, 𝑥𝑛))                                                    (2.4)    

where 

𝑀 (𝑥𝑛−1, 𝑥𝑛)  =  𝑚𝑎𝑥{𝜌𝑐(𝑥𝑛−1, 𝑥𝑛), 𝜌𝑐(𝑥𝑛−1, 𝑇 𝑥𝑛−1), 𝜌𝑐(𝑥𝑛, 𝑇 𝑥𝑛)} 



Communications on Applied Nonlinear Analysis 

ISSN: 1074-133X 

Vol 32 No. 10s (2025) 

  

2616 
 

https://internationalpubls.com 

                                            = 𝑚𝑎𝑥{𝜌𝑐(𝑥𝑛−1, 𝑥𝑛), 𝜌𝑐(𝑥𝑛−1, 𝑥𝑛), 𝜌𝑐(𝑥𝑛, 𝑥𝑛+1)}                (2.5) 

Now, if 𝜌𝑐(𝑥𝑛, 𝑥𝑛+1)  >  𝜌𝑐(𝑥𝑛−1, 𝑥𝑛). Then 

||𝜌𝑐(𝑥𝑛, 𝑥𝑛+1))|| ≤  𝜓(||𝜌𝑐(𝑥𝑛, 𝑥𝑛+1))||)  <  ||𝜌𝑐(𝑥𝑛, 𝑥𝑛+1))||                                       (2.6) 

This is a contradiction. Thus for all 𝑛 ≥  1 we have 

𝑚𝑎𝑥{𝜌𝑐(𝑥𝑛−1, 𝑥𝑛), 𝜌𝑐(𝑥𝑛, 𝑥𝑛+1)}  =  𝜌𝑐(𝑥𝑛−1, 𝑥𝑛)                                                          (2.7) 

                𝜌𝑐(𝑥𝑛, 𝑥𝑛+1))  ≤  𝜓(𝜌𝑐(𝑥𝑛−1, 𝑥𝑛))                                                                   (2.8) 

Continuing this process inductively, we obtain 

𝜌𝑐(𝑥𝑛, 𝑥𝑛+1))  ≤ 𝜓𝑛  (𝜌𝑐(𝑥0, 𝑥1))                                                                                    (2.9) 

Now for  𝑚 > 𝑛, using (2.9) and triangular inequality , we obtain 

𝜌𝑐(𝑥𝑚, 𝑥𝑛)  ≤  𝜌𝑐(𝑥𝑚, 𝑥𝑚−1)  + 𝜌𝑐(𝑥𝑚−1, 𝑥𝑚−2). . . . . . . . 𝜌𝑐(𝑥𝑛+1, 𝑥𝑛) – ∑ 𝜌𝑐(𝑥𝑚−𝑘, 𝑥𝑚−𝑘)

𝑚−𝑛−1

𝑘=1

 

                   ≤  𝜌𝑐(𝑥𝑚 , 𝑥𝑚−1)  + 𝜌𝑐(𝑥𝑚−1, 𝑥𝑚−2). . . . . . . . 𝜌𝑐(𝑥𝑛+1, 𝑥𝑛) 

                   ≤  (𝜓𝑚−1  +  𝜓𝑚−2 + . . . . . . . 𝜓𝑛)𝜌𝑐(𝑥0, 𝑥1) 

                    =
𝜓𝑛

1−𝜓
𝜌𝑐(𝑥0, 𝑥1)                                                                                (2.10) 

Since P is normal cone with normal constant K, we find that 

              ||𝜌𝑐(𝑥𝑚, 𝑥𝑛))|| ≤  𝐾||
𝜓𝑛

1−𝜓
 𝜌𝑐(𝑥0, 𝑥1)||                                                    (2.11) 

Which implies that 𝜌𝑐(𝑥𝑚, 𝑥𝑛)  →  0 as 𝑛, 𝑚 →  ∞. Hence {𝑥𝑛} a Cauchy sequence 

in partial cone metric space which is complete hence it must be convergent in X, 

let lim
𝑛→∞

𝑥𝑛  =  𝑧 therefore 

𝜌𝑐(𝑧, 𝑧)  =  lim
𝑛→∞

 𝜌𝑐(𝑥𝑛, 𝑧)  = lim
𝑛→∞

 𝜌𝑐(𝑥𝑛, 𝑥𝑛)  =  0 

Case 1. T is continuous, then we have 𝑥𝑛+1  =  𝑇 𝑥𝑛  →  𝑇 𝑧 as 𝑛 →  ∞. By uniqueness of limit 

𝑇 𝑧 =  𝑧. Hence z is a fixed point of T . 

Case 2 If {𝑥𝑛} is a sequence in X such that 𝛼(𝑥𝑛, 𝑥𝑛+1)  ≥  1 for all n and 𝑥𝑛  →  𝑧 as 𝑛 →  ∞. Then 

𝛼(𝑥𝑛, 𝑧)  ≥  1 for all n. Now we show that ||𝜌𝑐(𝑇 𝑧, 𝑧)|| ≥  0, On contrary, assume ||𝜌𝑐(𝑇 𝑧, 𝑧)|| >

 0 we have  

𝜌𝑐(𝑇 𝑧, 𝑧)  ≤  𝜌𝑐(𝑇 𝑧, 𝑇 𝑥𝑛)  + 𝜌𝑐(𝑇 𝑥𝑛, 𝑧) – 𝜌𝑐(𝑇 𝑥𝑛, 𝑇 𝑥𝑛) 

                                                   ≤ 𝛼(𝑥𝑛, 𝑧)𝜌𝑐(𝑇 𝑧, 𝑇 𝑥𝑛)  + 𝜌𝑐(𝑇 𝑥𝑛, 𝑧) – 𝜌𝑐(𝑇 𝑥𝑛, 𝑇 𝑥𝑛) 

                                                    ≤  𝜓(𝑀 (𝑥𝑛, 𝑧))  +  𝜌𝑐(𝑥𝑛+1, 𝑧)                         (2.12) 

Since P is normal cone with normal constant K, we have 

            ||𝜌𝑐(𝑇 𝑧, 𝑧)|| ≤  𝐾||𝜓(𝑀 (𝑥𝑛, 𝑧))  + 𝜌𝑐(𝑥𝑛+1, 𝑧)||                                    (2.13) 



Communications on Applied Nonlinear Analysis 

ISSN: 1074-133X 

Vol 32 No. 10s (2025) 

  

2617 
 

https://internationalpubls.com 

where 

𝑀 (𝑥𝑛 , 𝑧)  =  𝑚𝑎𝑥{𝜌𝑐(𝑥𝑛, 𝑧), 𝜌𝑐(𝑥𝑛, 𝑇 𝑥𝑛), 𝜌𝑐(𝑧, 𝑇 𝑧)} 

                                                       = 𝑚𝑎𝑥{𝜌𝑐(𝑥𝑛, 𝑧), 𝜌𝑐(𝑥𝑛, 𝑥𝑛+1), 𝜌𝑐(𝑧, 𝑇 𝑧)}        (2.14) 

Taking 𝑛 →  ∞ we get 

                                      𝑀 (𝑥𝑛, 𝑧)  =  𝜌𝑐(𝑧, 𝑇 𝑧)                                                        (2.15) 

 

Now, Taking 𝑛 →  ∞ in (2.12) we get that 

  ||𝜌𝑐(𝑇 𝑧, 𝑧)|| ≤  𝐾||𝜓(𝜌𝑐(𝑧, 𝑇 𝑧)||≤  𝐾||𝜌𝑐(𝑧, 𝑇 𝑧)||                                               (2.16) 

which is not true for all 𝐾 >  0. So we get a contradiction. 𝑇ℎ𝑒𝑟𝑒𝑓𝑜𝑟𝑒 ||𝜌𝑐(𝑇 𝑧, 𝑧)|| → 0 as 

 𝑛 →  ∞. It implies that 𝑇 𝑧 =  𝑧 and hence z is a fixed point of T .This completes the proof. 

Example 2.4  𝐿𝑒𝑡 𝑋 =  [0, ∞) and 𝜌𝑐(𝑥, 𝑦)  =  (𝑚𝑎𝑥{𝑥, 𝑦}, 𝑘 𝑚𝑎𝑥{𝑥, 𝑦}). Then (𝑋, 𝜌𝑐) is a 

complete partial cone metric space. Consider the mapping 𝑇 ∶  𝑋 →  𝑋defined by 

𝑇(𝑥) = {
𝑥 –

2

3
𝑥 > 1

𝑥

3
0 ≤ 𝑥 ≤ 1

                                                          (2.17) 

and let  𝜓 ∶  [0, ∞)  →  [0, ∞) be such that 𝜓(𝑡) =  
𝑡

2
 for all 𝑡 ≥  0. If we define the functions 𝛼, 𝛽 ∶

 𝑋 ×  𝑋 →  [0, ∞) as 

𝛼(𝑥, 𝑦) = {
3

2
𝑥, 𝑦 ∈ [0,1]

0 𝑜𝑡ℎ𝑒𝑟𝑤𝑖𝑠𝑒
                                                         (2.18) 

We show that contractive condition of Theorem 2.3 is satisfied. Without loss of generality we 

assume that 𝑥 ≥  𝑦. Then for 𝑥, 𝑦 ∈  [0, 1] we get 

𝛼(𝑥, 𝑦)𝜌𝑐(𝑇 𝑥, 𝑇 𝑦) =  𝛼(𝑥, 𝑦)𝜌𝑐 (
𝑥

3
 ,

𝑦

3
 ) =

3

2
(
𝑥

3
 ,

𝑘𝑥

3
 ) 

=
1

4
(𝑥, 𝑘𝑥) 

≤
1

2
(𝑥, 𝑘𝑥) 

                                                                        =
1

2
 𝑚𝑎𝑥{(𝑥, 𝑘𝑥), (𝑥, 𝑘𝑥), (𝑦, 𝑘𝑦)} 

                                                                       =
1

2
 𝑚𝑎𝑥{𝜌𝑐(𝑥, 𝑦), 𝜌𝑐(𝑥, 𝑇 𝑥), 𝜌𝑐(𝑦, 𝑇 𝑦)} 

                                                                       =  𝜓(𝑚𝑎𝑥{𝜌𝑐(𝑥, 𝑦), 𝜌𝑐(𝑥, 𝑇 𝑥), 𝜌𝑐(𝑦, 𝑇 𝑦)})  (2.19) 

Theorem 2.5  Let (𝑋, 𝜌𝑐) be a complete partial cone metric space P is a normal cone with constant 

K. Suppose the mapping 𝑇 ∶  𝑋 →  𝑋 satisfies the contractive condition 



Communications on Applied Nonlinear Analysis 

ISSN: 1074-133X 

Vol 32 No. 10s (2025) 

  

2618 
 

https://internationalpubls.com 

𝜌𝑐(𝑇 𝑥, 𝑇 𝑦))  ≤ 𝑎1 𝜌𝑐(𝑥, 𝑦)  + 𝑎2  
𝜌𝑐(𝑥,𝑇 𝑥)𝜌𝑐(𝑦,𝑇 𝑦)

𝜌𝑐(𝑥,𝑦)+  𝜌𝑐(𝑦,𝑇 𝑥)+  𝜌𝑐(𝑥,𝑇 𝑦)
  + 𝑎3  

𝜌𝑐(𝑦,𝑇 𝑦) 𝜌𝑐(𝑥,𝑇 𝑥)

𝜌𝑐(𝑥,𝑦)
  +

                                𝑎4 𝜌𝑐(𝑥, 𝑇 𝑦)  + 𝑎5 𝜌𝑐(𝑦, 𝑇 𝑥)                                                       (2.20) 

where 𝑎1, 𝑎2, 𝑎3, 𝑎4, 𝑎5  ≥  0 are constants such that 𝑎1  +  𝑎2  + 𝑎3  +  2𝑎4  + 𝑎5  <  1. 

Then T has a unique fixed point in X. 

Proof  Choose 𝑥0  ∈  𝑋 such that 𝑇 𝑥0  =  𝑥1, 𝑇 𝑥1  =  𝑇2𝑥0  =  𝑥2. . . 𝑥𝑛  =  𝑇 𝑥𝑛−1 = 𝑇𝑛𝑥0. Then 

𝜌𝑐(𝑥𝑛, 𝑥𝑛+1)  =  𝜌𝑐(𝑇 𝑥𝑛−1, 𝑇 𝑥𝑛) 

 ≤ 𝑎1 𝜌𝑐(𝑥𝑛−1, 𝑥𝑛)  +  𝑎2
𝜌𝑐(𝑥𝑛−1,𝑇 𝑥𝑛−1)𝜌𝑐(𝑥𝑛,𝑇 𝑥𝑛)

𝜌𝑐(𝑥𝑛−1,𝑥𝑛)+ 𝜌𝑐(𝑥𝑛,𝑇 𝑥𝑛−1) + 𝜌𝑐(𝑥𝑛−1,𝑇 𝑥𝑛)
 + 𝑎3

𝜌𝑐(𝑥𝑛,𝑇 𝑥𝑛)𝜌𝑐(𝑥𝑛−1,𝑇 𝑥𝑛−1)

𝜌𝑐(𝑥𝑛−1,𝑥𝑛)
 +                     

𝑎4𝜌𝑐(𝑥𝑛−1, 𝑇 𝑥𝑛)  +  𝑎5𝜌𝑐(𝑥𝑛, 𝑇 𝑥𝑛−1) 

≤ 𝑎1 𝜌𝑐(𝑥𝑛−1, 𝑥𝑛)  +  𝑎2
𝜌𝑐(𝑥𝑛−1,𝑥𝑛)𝜌𝑐(𝑥𝑛,𝑥𝑛+1)

𝜌𝑐(𝑥𝑛−1,𝑥𝑛)+ 𝜌𝑐(𝑥𝑛,𝑥𝑛) + 𝜌𝑐(𝑥𝑛−1,𝑥𝑛+1)
 + 𝑎3

𝜌𝑐(𝑥𝑛,𝑥𝑛+1)𝜌𝑐(𝑥𝑛−1,𝑥𝑛)

𝜌𝑐(𝑥𝑛−1,𝑥𝑛)
 +                     

𝑎4𝜌𝑐(𝑥𝑛−1, 𝑥𝑛+1)  +  𝑎5𝜌𝑐(𝑥𝑛, 𝑥𝑛) 

≤  𝑎1𝜌𝑐(𝑥𝑛−1, 𝑥𝑛)  + 𝑎2𝜌𝑐(𝑥𝑛−1, 𝑥𝑛)  + 𝑎3𝜌𝑐(𝑥𝑛, 𝑥𝑛+1)  +  𝑎4𝜌𝑐(𝑥𝑛−1, 𝑥𝑛+1)  + 𝑎5𝜌𝑐(𝑥𝑛, 𝑥𝑛) 

≤  𝑎1𝜌𝑐(𝑥𝑛−1, 𝑥𝑛)  + 𝑎2𝜌𝑐(𝑥𝑛−1, 𝑥𝑛)  + 𝑎3𝜌𝑐(𝑥𝑛, 𝑥𝑛+1)  +  𝑎4𝜌𝑐(𝑥𝑛−1, 𝑥𝑛)  +  𝑎4𝜌𝑐(𝑥𝑛, 𝑥𝑛+1)

−  𝑎4𝜌𝑐(𝑥𝑛, 𝑥𝑛)  +  𝑎5𝜌𝑐(𝑥𝑛, 𝑥𝑛) 

=  (𝑎1  + 𝑎2  +  𝑎4)𝜌𝑐(𝑥𝑛−1, 𝑥𝑛)  + (𝑎3  + 𝑎4)𝜌𝑐(𝑥𝑛, 𝑥𝑛+1)  + (𝑎5 – 𝑎4)𝜌𝑐(𝑥𝑛, 𝑥𝑛) 

=  (𝑎1  + 𝑎2  +  𝑎4)𝜌𝑐(𝑥𝑛−1, 𝑥𝑛)  + (𝑎3  + 𝑎4  +  𝑎5)𝜌𝑐(𝑥𝑛, 𝑥𝑛+1) 

                           ≤  
(𝑎1 + 𝑎2 + 𝑎4)

1 − (𝑎3 + 𝑎4 + 𝑎5)
𝜌𝑐(𝑥𝑛−1, 𝑥𝑛)                                                         (2.21) 

 

𝐿𝑒𝑡 𝜆 =
(𝑎1  + 𝑎2  +  𝑎4)

1 − (𝑎3  + 𝑎4  +  𝑎5)
 

 Since 𝑎1  + 𝑎2  +  𝑎3  +  2𝑎4  + 𝑎5  <  1 and 𝑎3  +  𝑎4  + 𝑎5  <  1 implies 

that 𝜆 <  1. Hence 

                              𝜌𝑐(𝑥𝑛, 𝑥𝑛+1)  ≤  𝜆𝜌𝑐(𝑥𝑛−1, 𝑥𝑛)                                                        (2.22) 

for all 𝑛 ∈  ℕ. For any 𝑚 >  𝑛 where 𝑚, 𝑛 ∈  ℕ we have 

𝜌𝑐(𝑥𝑚, 𝑥𝑛)  ≤  𝜌𝑐(𝑥𝑚, 𝑥𝑚−1)  + 𝜌𝑐(𝑥𝑚−1, 𝑥𝑚−2). . . . . . . . 𝜌𝑐(𝑥𝑛+1, 𝑥𝑛)  − ∑ 𝜌𝑐(𝑥𝑚−𝑘, 𝑥𝑚−𝑘)

𝑚−𝑛−1

𝑘

 

≤  𝜌𝑐(𝑥𝑚, 𝑥𝑚−1) + 𝜌𝑐(𝑥𝑚−1, 𝑥𝑚−2). . . . . . . . 𝜌𝑐 (𝑥𝑛+1, 𝑥𝑛) 

≤  (𝜆𝑚−1  + 𝜆𝑚−1 + . . . . . . . 𝜆𝑛)𝜌𝑐(𝑥0, 𝑥1) 

                                     =  
𝜆𝑛

1 − 𝜆
𝜌𝑐(𝑥0, 𝑥1)                                                                   (2.23) 

Since P is normal cone with normal constant K, we have 



Communications on Applied Nonlinear Analysis 

ISSN: 1074-133X 

Vol 32 No. 10s (2025) 

  

2619 
 

https://internationalpubls.com 

                    ||𝜌𝑐(𝑥𝑚, 𝑥𝑛))|| ≤  𝐾||
𝜆𝑛

1 − 𝜆
𝜌𝑐(𝑥0, 𝑥1)||                                                  (2.24) 

Now since 𝜆 <  1, ||𝜌𝑐(𝑥𝑚, 𝑥𝑛))|| ≤  𝐾||
𝜆𝑛

1−𝜆
  𝜌𝑐(𝑥0, 𝑥1)|| →  0 as 𝑛 →  ∞ 

Hence {𝑥𝑛} is a Cauchy sequence in a partial cone metric space which is complete hence it must be 

convergent in X, let  lim
𝑛→∞

𝑥𝑛  =  𝑧 therefore  

𝜌𝑐(𝑧, 𝑧)  =  lim
𝑛→∞

𝜌𝑐(𝑥𝑛, 𝑧)   = lim
𝑛→∞

𝜌𝑐(𝑥𝑛, 𝑥𝑛)  =  0 

 

Now we show that ||𝜌𝑐(𝑇 𝑧, 𝑧)|| ≥  0, On contrary, assume ||𝜌𝑐(𝑇 𝑧, 𝑧)|| >  0 we have 

𝜌𝑐(𝑇 𝑧, 𝑧)  ≤  𝜌𝑐(𝑇 𝑧, 𝑇 𝑥𝑛)  +  𝜌𝑐(𝑇 𝑥𝑛 , 𝑧) – 𝜌𝑐(𝑇 𝑥𝑛, 𝑇 𝑥𝑛) 

≤𝑎1𝜌𝑐(𝑧, 𝑥𝑛) + 𝑎2
𝜌𝑐(𝑧,𝑇 𝑧)𝜌𝑐(𝑥𝑛,𝑇 𝑥𝑛)

𝜌𝑐(𝑧,𝑥𝑛)+ 𝜌𝑐(𝑥𝑛,𝑇 𝑧)+ 𝜌𝑐(𝑧,𝑇 𝑥𝑛)
 +  𝑎3

𝜌𝑐(𝑥𝑛,𝑇 𝑥𝑛)𝜌𝑐(𝑧,𝑇 𝑧)

𝜌𝑐(𝑧,𝑥𝑛)
+𝑎4𝜌𝑐(𝑧, 𝑇 𝑥𝑛) +  

𝑎5𝜌𝑐(𝑥𝑛, 𝑇 𝑧)  + 𝜌𝑐(𝑇 𝑥𝑛 , 𝑧) – 𝜌𝑐(𝑇 𝑥𝑛, 𝑇 𝑥𝑛) 

≤𝑎1𝜌𝑐(𝑧, 𝑥𝑛) + 𝑎2
𝜌𝑐(𝑧,𝑇 𝑧)𝜌𝑐(𝑥𝑛,𝑥𝑛+1)

𝜌𝑐(𝑧,𝑥𝑛)+ 𝜌𝑐(𝑥𝑛,𝑇 𝑧)+ 𝜌𝑐(𝑧,𝑥𝑛+1)
 +  𝑎3

𝜌𝑐(𝑥𝑛,𝑥𝑛+1)𝜌𝑐(𝑧,𝑇 𝑧)

𝜌𝑐(𝑧,𝑥𝑛)
+𝑎4𝜌𝑐(𝑧, 𝑇 𝑥𝑛) +  

𝑎5𝜌𝑐(𝑥𝑛, 𝑇 𝑧)  +  𝜌𝑐( 𝑥𝑛+1, 𝑧)  

≤ 𝑎1𝜌𝑐(𝑧, 𝑥𝑛)  + +𝑎4𝜌𝑐(𝑧, 𝑥𝑛+1)  + 𝑎5𝜌𝑐(𝑥𝑛, 𝑇 𝑧)  +  𝜌𝑐(𝑥𝑛+1, 𝑧) 

≤ 𝑎1𝜌𝑐(𝑧, 𝑥𝑛) + +𝑎4 𝜌𝑐(𝑧, 𝑥𝑛+1)  + 𝑎5𝜌𝑐(𝑇 𝑧, 𝑧)  + 𝑎5𝜌𝑐(𝑧, 𝑥𝑛) – 𝑎5𝜌𝑐(𝑧, 𝑧)  + 𝜌𝑐(𝑥𝑛+1, 𝑧) 

≤  (𝑎1  +  𝑎5)𝜌𝑐(𝑧, 𝑥𝑛)  + +[𝑎4𝜌𝑐(𝑥𝑛+1, 𝑧)  + 𝜌𝑐(𝑥𝑛+1, 𝑧)]  +  𝑎5𝜌𝑐(𝑇 𝑧, 𝑧)                  (2.25) 

So using (2.25) we have 

𝜌𝑐(𝑇 𝑧, 𝑧)  ≤  
(𝑎1 + 𝑎5)

1 – 𝑎5
𝜌𝑐(𝑧, 𝑥𝑛)  + 

𝑎4

1 – 𝑎5
𝜌𝑐(𝑥𝑛+1, 𝑧)  + 

1

1 – 𝑎5
𝜌𝑐(𝑥𝑛+1, 𝑧)                        (2.26) 

Hence 

||𝜌𝑐(𝑇 𝑧, 𝑧)||  ≤  
(𝑎1 + 𝑎5)

1 – 𝑎5
𝐾||𝜌𝑐(𝑧, 𝑥𝑛)|| +  

𝑎4

1 – 𝑎5
𝐾||𝜌𝑐(𝑥𝑛+1, 𝑧)|| + 

1

1 – 𝑎5
𝐾||𝜌𝑐(𝑥𝑛+1, 𝑧)|| →  0           

(2.27)      

So we have ||𝜌𝑐(𝑇 𝑧, 𝑧)|| =  0 therefore 𝜌𝑐(𝑇 𝑧, 𝑧)  =  0 or T z = z. 

Uniqueness  If 𝑧1 is s another Fixed Point of T , Then 𝑇 𝑧1  =  𝑧1 replacing x by z 

and y by 𝑧1 in (2.1) we get 

𝜌𝑐(𝑧, 𝑧1) =  𝜌𝑐(𝑇 𝑧, 𝑇 𝑧1)

≤ 𝑎1𝜌𝑐(𝑧, 𝑧1) +  𝑎2

𝜌𝑐(𝑧, 𝑇 𝑧)𝜌𝑐(𝑧1, 𝑇 𝑧1)

𝜌𝑐(𝑧, 𝑧1)  + 𝜌𝑐(𝑧1, 𝑇 𝑧)  +  𝜌𝑐(𝑧, 𝑇 𝑧1)
 

+ 𝑎3

𝜌𝑐(𝑧1, 𝑇 𝑧1)𝜌𝑐(𝑧, 𝑇 𝑧)

𝜌𝑐(𝑧, 𝑧1)
 +  𝑎4𝜌𝑐(𝑧, 𝑇 𝑧1)  +  𝑎5𝜌𝑐(𝑧1, 𝑇 𝑧) 



Communications on Applied Nonlinear Analysis 

ISSN: 1074-133X 

Vol 32 No. 10s (2025) 

  

2620 
 

https://internationalpubls.com 

                     ≤ 𝑎1𝜌𝑐(𝑧, 𝑧1) +  𝑎2
𝜌𝑐(𝑧,𝑧)𝜌𝑐(𝑧1,𝑧1)

𝜌𝑐(𝑧,𝑧1) + 𝜌𝑐(𝑧1,𝑧) + 𝜌𝑐(𝑧,𝑧1)
 +  𝑎3

𝜌𝑐(𝑧1,𝑧1)𝜌𝑐(𝑧,𝑧)

𝜌𝑐(𝑧,𝑧1)
 +  𝑎4𝜌𝑐(𝑧, 𝑧1) +

                               𝑎5𝜌𝑐(𝑧1, 𝑧)                                                                                                 (2.28) 

Therefore 𝜌𝑐(𝑧, 𝑧1)  =  0 or 𝑧 =  𝑧1 .  

3. Application  

This section is influenced by the findings presented in the papers [19, 20] which aims to offer an 

application of Theorem 2.3 to the solution of second order differential equation of the form 

                                                                 𝑥′′(𝑡)  =  −𝑓 (𝑡, 𝑥(𝑡)), 𝑡 ∈  𝐼                                     

                                                                 𝑥(0)  =  𝑥(1)  =  0                                            (3.1) 

𝑤ℎ𝑒𝑟𝑒 𝐼 =  [0, 1], 𝑓 ∶  𝐼 ×  ℝ → ℝ is a continuous function.Consider the space X = C(I) of 

continuous function defined on I. It is well-known that the Problem (3.1) is equivalent to the integral 

equation 

                                                    𝑥(𝑡)  = ∫ 𝑔(𝑡, 𝑠)𝑓 (𝑠, 𝑥(𝑠)) 𝑑𝑠
1

0
                                       (3.2) 

for all 𝑡 ∈  [0, 1] where g is the Green function defined by 

 

𝑔(𝑡, 𝑠) = {
(1 −  𝑠)𝑡 0 ≤ t ≤ s ≤ 1

(1 −  𝑡)𝑠 0 ≤ s ≤ t ≤ 1
                                                                         (3.3) 

Then solving problem (3.1) is equivalent to finding fixed point of T in C(I). 

Theorem 3.1 Let 𝑋 =  𝐶(𝐼) and 𝑇 ∶  𝑋 →  𝑋 be an operator given by 

                      𝑇 𝑥(𝑡)  = ∫ 𝑔(𝑡, 𝑠)𝑓 (𝑠, 𝑥(𝑠)) 𝑑𝑠
1

0
                                                                (3.4) 

for all 𝑥 ∈  𝑋 and 𝑡 ∈  𝐼 =  [0, 1]. Suppose the following conditions hold: 

(i) For all 𝑡 ∈  𝐼, for all 𝑎, 𝑏 ∈  ℝ 𝑤𝑖𝑡ℎ ||𝑎||, ||𝑏|| ≤  1, we have 

|𝑓 (𝑡, 𝑎)  −  𝑓 (𝑡, 𝑏)|  ≤  8𝜇(|𝑎 −  𝑏|)                                                              (3.5) 

 

(ii) there exists 𝑥0  ∈  𝐶(𝐼) such that ||𝑥0||
∞

≤  1, 

(iii) for all 𝑥 ∈  𝐶(𝐼) 

|𝑥||∞ ≤  1 → || ∫ 𝑔(𝑡, 𝑠)𝑓 (𝑠, 𝑥(𝑠))𝑑𝑠||
1

0 ∞
≤  1                                            ( 3.6) 

Then the second order differential equation (3.1) has a solution. 

Proof Consider C(I) endowed with the partial metric given by 

𝜌𝑐(𝑥, 𝑦) = {
||𝑥 −  𝑦||

∞
||𝑥|| ≤  1, ||𝑦|| ≤  1

||𝑥 −  𝑦||
∞

+ 𝜏 𝑜𝑡ℎ𝑒𝑟𝑤𝑖𝑠𝑒
                                                         (3.7) 

where 𝜏 >  0. Then (𝐶(𝐼), 𝜌𝑐) is a partial metric space. Now we define partial 

cone metric as 



Communications on Applied Nonlinear Analysis 

ISSN: 1074-133X 

Vol 32 No. 10s (2025) 

  

2621 
 

https://internationalpubls.com 

                    𝜌𝑐(𝑥, 𝑦)  =  (𝜌𝑐(𝑥, 𝑦), 𝛽𝜌𝑐(𝑥, 𝑦))                                              (3.8) 

where 𝛽 ≥  0. Now, let 𝑥, 𝑦 ∈  𝐶(𝐼) such that ||𝑥|| ≤  1, ||𝑦|| ≤  1, then we  have 

𝜌𝑐(𝑇 𝑥, 𝑇 𝑦)  =  (||𝑇 𝑥 −  𝑇 𝑦||
∞

, 𝛽||𝑇 𝑥 −  𝑇 𝑦||
∞

) 

= ( sup
𝑡∈[0,1]

∫ 𝑔(𝑡, 𝑠)|𝑓 (𝑠, 𝑥(𝑠)) − 𝑓 (𝑠, 𝑦(𝑠))|𝑑𝑠
1

0

, 𝛽 sup
𝑡∈[0,1]

∫ 𝑔(𝑡, 𝑠)|𝑓 (𝑠, 𝑥(𝑠)) − 𝑓 (𝑠, 𝑦(𝑠))|𝑑𝑠
1

0

)  

 

= ( sup
𝑡∈[0,1]

∫ 𝑔(𝑡, 𝑠)8 𝜇|𝑥(𝑠) − 𝑦(𝑠)|𝑑𝑠
1

0

, 𝛽 sup
𝑡∈[0,1]

∫ 𝑔(𝑡, 𝑠)8 𝜇|𝑥(𝑠) − 𝑦(𝑠)|𝑑𝑠
1

0

) 

= ( sup
𝑡∈[0,1]

∫ 𝑔(𝑡, 𝑠) × (8 𝜇||𝑥 − 𝑦||
∞

)𝑑𝑠
1

0
, 𝛽 sup

𝑡∈[0,1]
∫ 𝑔(𝑡, 𝑠) × (8 𝜇||𝑥 − 𝑦||

∞
)𝑑𝑠

1

0
)   (3.9) 

Now, as we know    sup
𝑡∈[0,1]

∫ 𝑔(𝑡, 𝑠)𝑑𝑠
1

0
=

1

8
   and taking 𝜓(𝑡)  =  𝜇𝑡 

𝜌𝑐(𝑇 𝑥, 𝑇 𝑦)  ≤  𝜇(𝜌𝑐(𝑥, 𝑦)), 𝛽(𝜌𝑐(𝑥, 𝑦))) 

≤  𝜇𝜌𝑐(𝑥, 𝑦) 

                    ≤  𝜓(𝜌𝑐(𝑥, 𝑦))  ≤  𝜓(𝑚𝑎𝑥{𝜌𝑐(𝑥, 𝑦), 𝜌𝑐(𝑥, 𝑇 𝑥), 𝜌𝑐(𝑦, 𝑇 𝑦)})             (3.10)    

Define the function 𝛼 ∶  𝐶(𝐼)  ×  𝐶(𝐼)  →  [0, ∞) as       

𝛼(𝑥, 𝑦) = {
1 ||𝑥|| ≤  1, ||𝑦|| ≤  1
0 𝑜𝑡ℎ𝑒𝑟𝑤𝑖𝑠𝑒

                                                                         (3.11) 

 

For all 𝑥, 𝑦 ∈  𝐶(𝐼)  

𝛼(𝑥, 𝑦)𝜌𝑐(𝑇 𝑥, 𝑇 𝑦)  ≤  𝜓(𝑚𝑎𝑥{𝜌𝑐(𝑥, 𝑦), 𝜌𝑐(𝑥, 𝑇 𝑥), 𝜌𝑐(𝑦, 𝑇 𝑦)})                         (3.12) 

Clearly, all the conditions of Theorem 2.3 are satisfied and so Γ has a fixed point. Thus the system of 

integral equations (3.2) has a solution. 

References 

[1] S. Banach, Sur les operations dans les ensembles abstraits et leur application aux equations 

integrals, Fund. Math., 3 (1922), 133-181. 

[2] M. Frechet, Sur quelques points du calcul fonctionnel, Rendic. Circ. Mat. Palermo 22 (1906) 

1-74. 

[3] S. G. Matthews,Partial metric topology, Research report 212, Department of Computer 

Science, University of Warwick, (1992). 

[4] S. G. Matthews, Partial metric topology, Proceedings of the 8th summer conference on 

topology and its applications, Annals of the New York Academy of Sciences, 728 (1994), 

183-197. 

[5] B. Samet, C. Vetro and P. Vetro, Fixed point theorem for 𝛼 − 𝜓 contractive type mappings, 

Nonlinear Anal., 75 (2012), 2154-2165. 



Communications on Applied Nonlinear Analysis 

ISSN: 1074-133X 

Vol 32 No. 10s (2025) 

  

2622 
 

https://internationalpubls.com 

[6] E. Karapinar and B. Samet, Generalized α − ψ -contractive type mappings and related fixed 

point theorems with applications, Abstract and Applied Analysis 2012 (2012), Article ID 

793486. 

[7] L. G. Huang, X. Zhang, Cone metric spaces and Fixed point theorems of Contractive 

mappings, J. Math. Anal. App. 332 (2007), 1468-1476. 

[8] M. Abbas, G. Jungck, Common fixed point results for non commuting mappings without 

continuity in cone metric spaces, J. Math. Anal. Appl., 341 (2008), 416–420. 

[9] M. Abbas and B. E. Rhoades, Fixed and periodic point results in cone metric space, Appl. 

Math. Lett., 22(4) (2009), 511–515. 

[10] S. K. Mahlotra, S. Shukla, R. Sen, N. Verma, Fixed point theorems in partial cone metric 

spaces, Inter. J. Math. Arch., 2(4) (2011), 610–616. 

[11] A. Sonmez, Fixed point theorems in partial cone metric spaces, arXiv:1101.2741v1 

[math.GN]. 

[12] M. Verma, N. Hooda, Fixed points for generalized α − ψ contractive mappings in cone metric 

spaces, Adv. Fixed Point Theory, 6(3) (2016), 241-253. 

[13] S. K. Malhotra, S. Shukla, R. Sen and N. Verma, Generalization of fixed point theorems in 

partial cone metric spaces, Int. J. Math. Arch. 2(4), 2011. 

[14] S. Moshokoa, On partial metric spaces and partial cone metric spaces, Hacettepe Journal of 

Mathematics and Statistics Volume 46 (6) (2017), 1069 – 1075. 

[15] J. Fernandez, Some Fixed-Point Results of α-Admissible Mappings in Partial Cone Metric 

Spaces over Banach Algebra, Proceedings of the 1st International Conference on Artificial 

Intelligence for Internet of Things: Accelerating Innovation in Industry and Consumer 

Electronics (AI4IoT 2023), pages 192-194. 

[16] J. Fernandez, K. Saxena and N. Malviya, Fixed points of expansive maps in partial cone 

metric spaces, GU J Sci 27(4):1085-1091 (2014). 

[17] J. Fernandez, S. Saelee, K. Saxena, N. Malviya and P. Kumam, The A-cone metric space over 

Banach algebra with applications, , Cogent Mathematics, 4:1, 1282690. 

[18] D. Turkoglu, M. Abuloha, Cone metric spaces and fixed point theorems in diametrically 

contractive mappings, Acta Math. Sin. Engl. Series, 26(3) (2010), 489–496. 

[19] R.A. Rashwan, H.A. Hammad, M. Gamal, S. Omran, , M. De la Sen, Fixed Point Method- 

ologies for ψ-contraction mappings in cone metric spaces over Banach algebra with sup- 

portive application, Int. J. Anal. Appl. (2024), 22:120. 

[20] P. Kumam, C. Vetro and F. Vetro, Fixed points for Weak ψ-contractions in partial metric 

spaces, Abstract and Applied Analysis Volume 2013, Article ID 986028.