Journal of Indian Acad. Math.  
ISSN: 0970-5120  
Vol. 48, No. 1 (2026) pp. 121–138.  
FIXED POINT RESULTS FOR (α-ψ)-RATIONAL  
TYPE CONTRACTIVE CONDITIONS IN  
PARTIAL METRIC SPACES  
Thokchom Rosy Devi1, Laishram Shambhu Singh2,  
and Thokchom Chhatrajit Singh3  
Abstract. In this paper, we present the concept of (α, ψ)-rational type contractive  
mapping and provide sufficient condition for the existence and uniqueness of fixed point  
in partial metric space. The concept of partial metric spaces, introduced by Matthews in  
1994, has proven to be a valuable generalization of metric spaces, allowing for a broader  
range of applications in various fields of mathematics and its applications. We also deduce  
several results as consequences and examples are also provided to validate the results  
obtained. Our investigation contributes to the ongoing research on fixed point theory  
in partial metric spaces by introducing and analyzing (α, ψ)-rational type contractive  
conditions. The developed theory not only enriches our understanding of fixed point  
phenomena in non-standard metric spaces but also opens avenues for future research in  
related areas of mathematics and its applications. Our results extend and generalize  
existing fixed point theorems in the literature, providing a deeper understanding of the  
applicability of such contractive conditions in the context of partial metric spaces.  
Keywords: Rational type contraction, partial metric space, fixed point theorem,  
α−admissible mapping.  
2010 AMS Subject Classification: 47H10, 54H25.  
1. Introduction  
The study of fixed points and fixed point theorems have been an active area of research  
for the last five decades for many researchers as fixed points and their related results have  
always been a major theoretical tools in fileds as widely apart as Differential Equation,  
Topology, Economics, Game theory, Dynamics, Optical control, Analysis and Functional  
Analysis. In 1994, the notion of partial metric space was introduced by Matthews to  
study the denotationnal semantics of dataflow nertworks. The partial metric spaces are  
acknowledged for their significant role in the development of models in the theory of  
computation. Matthews[22] also obtained the partial metric version of Banach contraction  
theorem. The concept of α-admissible mappings and (α, ψ)- rational type contraction  
mappings are introduced by samet et. el.[32] and this concept is extended by different  
reseachers (see [5, 6, 7, 8, 9, 10, 11, 13]). In 2015, Hamid et.al [13] used (α, ψ)-rational type  
contraction mappings to establish the fixed point theorem in generalized metric spaces.  
This work examines a further generalization of a metric spaces, known as a partial metric  
spaces, which extends the concept of a normal metric space introduced by Frchet in  
1906. This concept was presented by Mathews[21]. Mathews[21] established the partial  
121  
122  
Thokchom Rosy Devi, Laishram Shambhu Singh, and Thokchom Chhatrajit Singh  
metric contraction theorem, which renders the partial metric function significant in fixed  
point theory. Oltra and Valero [24] extended the results of Mathews in the setting of  
complete partial metric spaces. Subsequently, significant findings in partial metric space  
were developed as enhancements and generalizations of previously established results in  
the literature (see [5, 6, 7, 8, 9, 11, 12, 18, 30, 32]). In this work, we establish some  
theorems for the existence and uniqueness of a fixed point in partial metric space along  
with the concept of (α, ψ)- rational type contractive mapping using auxiliary functions.  
2. Preliminaries  
We start with the following definition. Let X be a nonempty set and ξ : X×X → [0, ∞)  
satisfy the following conditions:  
(i) ξ(u, v) = 0 if and only if u = v for all u, v ∈ X.  
(ii) ξ(u, v) = d(v, u) for all u, v ∈ X.  
(iii) ξ(u, v) ≤ d(u, w) + d(w, v) for all u, v, w ∈ X.  
Then the map ξ is called metric on X and the pair (X, ξ) is called a metric space.  
Definition 2.1. [21] Let X be a nonempty set and let ξ : X × X → [0, ∞) satisfies:  
(i) ξ(u, v) ≥ 0 ∀ u, v ∈ X and ξ(u, v) = ξ(u, u) = ξ(v, v) if and only if u = v.  
(ii) ξ(u, u) ≤ ξ(u, v) ∀ u, v ∈ X.  
(iii) ξ(u, v) = ξ(v, u) ∀ u, v ∈ X.  
(iv) ξ(u, v) ≤ ξ(u, w) + ξ(w, v) − ξ(w, w) ∀ u, v, w ∈ X.  
Consequently, the map ξ is referred to as a partial metric on X, and the pair (X, ξ) is  
designated as a partial metric space.  
Example 2.2. [21] Let X = [0, ∞) and ξ : X × X → [0, ∞) given by  
ξ(u, v) = max{u, v}, ∀ u, v ∈ X.  
Then (X, ξ) is a partial metric space.  
It is reasonable to introduce fundamental topological notions-such as convergence of a  
sequence, the Cauchy sequence criterion, continuity of functions, and the completeness of  
a topological space-within the context of partial metric spaces.  
Definition 2.3. [19, 21, 31, 32] Let (X, ξ) be a partial metric space. We say that  
(i) A sequence {un} converges to the limit u if ξ(u, u) = limn→∞ ξ(u, un).  
(ii) A sequence {un} is Cauchy if limn,m→∞ ξ(un, um) exists and finite.  
(iii) A partial metric space (X, ξ) is complete if each Cauchy sequence {un} converges  
to a point u ∈ X such that ξ(u, u) = limn,m→∞ ξ(un, um).  
(iv) A mapping F : X → X is continuous at a point u0 ∈ X if for each ꢀ > 0, there  
exists δ > 0 such that F(Bξ(u0, δ)) ⊆ Bξ(Fu0, ꢀ).  
For what follows, we shall recall the following lemma that can be derived easily (see  
[21]).  
FIXED POINT RESULTS FOR (α-ψ)-RATIONAL TYPE CONTRACTIVE CONDITIONS  
123  
Lemma 2.4. Let ξ be a partial metric on a non-empty set X and ρξ be the corresponding  
standard metric space on the same set X. Then  
(a) A sequence {un} is fundamental in the framework of a partial metric (X, ξ) if and  
only if it is a fundamental sequence in the setting of the corresponding standard  
metric space (X, ρξ).  
(b) A partial metric space (X, ξ) is complete if and only if the corresponding standard  
metric space (X, ρξ) is complete. Moreover  
lim  
= 0 ⇔ ξ(u, u) = lim ξ(u, un) = lim ξ(un, um)  
n→∞  
n,m→∞  
n→∞ρξ(u,un)  
(c) If un → u as n → ∞ in a partial metric space (X, ξ) with ξ(u, u) = 0, then we  
have  
lim ξ(un, v) = ξ(u, v), for everey v ∈ X  
n→∞  
Definition 2.5. [30] Consider a nonempty set X and let τ : X → X and α : X × X →  
[0, ∞) be two mappings. We define τ as an α-admissible mapping if α(u, v) ≥ 1 implies  
α(τu, τv) ≥ 1 for all u, v ∈ X.  
Definition 2.6. [13] Let (X, ξ) be a partial metric space and α : X × X → [0, ∞).X is  
defined as a α-regular partial metric space if, for a sequence {un} in X where {un → u}  
and α(un, un+1) ≥ 1, there exists a subsequence {un } of {un} such that α(un , x) ≥ 1 for  
k
k
all k ∈ N.  
3. Fixed point results  
In this section, we employ (α, ψ)-rational type contraction mappings to establish a  
unique fixed point theorem in the context of partial metric spaces. The current study  
shall analyze the auxiliary function ψ as defined by Alsulami et al. [3].  
Consider Ψ denote the family of functions ψ : [0, ∞) → [0, ∞) that satisfy the following  
properties:  
(i) ψ is both upper semi-continuous and strictly increasing.  
(ii) Any value of t > 0 will lead to [ψn(t)]n∈N coverges to 0 as n → ∞.  
(iii) For all t > 0, ψ(t) < t.  
Definition 3.1. Let (X, ξ) be a partial metric space and α : X × X → [0, ∞). A self-  
mapping τ : X → X, is characterized as (α, ψ)-rational type-I contractive mapping, is  
defined by the existence of a function ψ ∈ Ψ such that for all u, v ∈ X, the following  
condition holds:  
α(u, v)ξ(τu, τv) ≤ ψ(Ω(u, v))  
124  
Thokchom Rosy Devi, Laishram Shambhu Singh, and Thokchom Chhatrajit Singh  
where  
ꢀ
Ω(u, v) = max ξ(u, v), ξ(u, τu), ξ(v, Tv),  
ξ(u, v) + p(u, τu) ξ(u, τu) + ξ(v, τv)  
,
2
2
ꢁ
ξ(u, τu)ξ(v, τv) ξ(u, τu)ξ(v, τv)  
,
(1)  
1 + ξ(u, v)  
1 + ξ(τu, τv)  
Theorem 3.2. Consider a complete partial metric space (X, ξ)and let τ : X → X be a  
self mapping and  
α : X × X → [0, ∞)  
be a given function. Assume that each of the following conditions holds:  
(i) The mapping τ is α-admissible.  
(ii) τ is an (α, ψ)-rational type-I contractive mapping.  
(iii) there exists u0 ∈ X such that α(u0, τu0) ≥ 1, α(u0, τ2u0) ≥ 1.  
(iv) Either X exhibits α-regular or τ is continuous.  
Consequently, τ possesses a fixed point u∗ ∈ X, and the sequence {τnu0} converges to u∗.  
Moreover, if for any u, v ∈ F(τ), it holds that α(u, v) ≥ 1. Then, τ possesses a unique  
fixed point in X.  
Proof. Let u0 ∈ X such that α(u0, τu0) ≥ 1 and α(u0, τ2u0) ≥ 1. We formulate the  
sequence, {un} ∈ X as un = τnu0 = τun−1 for all n ∈ N.  
By definition, un is a fixed point of τ only if un = un +1 for every n0 ∈ N. Therefore,  
0
0
we will assume that for every n ∈ N, un = un+1  
.
Given that T is α-admissible  
α(u0, τu0) = α(u0, u1) ≥ 1  
α(τu0, τu1) = α(u1, u2) ≥ 1  
α(τu1, τu2) = α(u2, u3) ≥ 1  
implies  
therefore,  
.
.
.
Thus, by induction, we obtain  
α(un, un+1) ≥ 1  
for all n ≥ 0.  
Using analogous reasoning, as  
α(u0, τ2u0) ≥ 1,  
we have  
α(u0, u2) = α(u0, τ2u0) ≥ 1,  
α(τu0, τu2) = α(u1, u3) ≥ 1  
FIXED POINT RESULTS FOR (α-ψ)-RATIONAL TYPE CONTRACTIVE CONDITIONS  
Using induction, we obtain  
125  
α(un, un+2) ≥ 1  
for all n ≥ 0. Let u = un and v = un+1 in (1). We obtain  
ξ(un+1, un+2) = ξ(τun, τun+1) ≤ α(un, un+1)ξ(τun, τun+1) ≤ ψ(Ω(un, un+1  
)
in which,  
ꢀ
Ω(un, un+1) = max ξ(un, un+1), ξ(un, τun), ξ(un+1, τun+1),  
ξ(un, un+1) + ξ(un, τun) ξ(un, τun) + ξ(un+1, τun+1  
)
,
,
2
2
ꢁ
ξ(un, τun)ξ(un+1, τun+1) ξ(un, τun)ξ(un+1, τun+1  
)
,
1 + ξ(un, un+1  
)
1 + ξ(τun, τun+1)  
ꢀ
= max ξ(un, un+1), ξ(un, un+1), ξ(un+1, un+2),  
ξ(un, un+1) + ξ(un, un+1) ξ(un, un+1) + ξ(un+1, un+2  
)
,
,
2
2
ꢁ
ξ(un, un+1)ξ(un+1, un+2) ξ(un, un+1)p(un+1, un+2  
)
,
1 + ξ(un, un+1  
)
1 + ξ(un+1, un+2)  
ꢀ
= max ξ(un, un+1), ξ(un+1, un+2),  
ꢁ
ξ(un, un+1) + ξ(un+1, un+2  
)
(2)  
2
Given that,  
and  
ξ(un, un+1)ξ(un+1, un+2  
1 + ξ(un, un+1  
)
≤ ξ(un+1, un+2  
)
)
ξ(un, un+1)ξ(un+1, un+2  
1 + ξ(un+1, un+2  
)
≤ ξ(un, un+1  
)
)
If for a given n, we have  
Ω(un, un+1) = ξ(un+1, un+2  
)
then  
ξ(un+1, un+2) ≤ ψ(Ω(un, un+1))  
= ψ(ξ(un+1, un+2))  
(3)  
< ξ(un+1, un+2))  
this is impossible.  
If for a given n, we get  
ꢀ
ꢁ
Ω(un, un+1) ≤ max ξ(un, un+1), ξ(un+1, un+2  
)
If  
ꢀ
ꢁ
max ξ(un, un+1), ξ(un+1, un+2) = ξ(un+1, un+2  
)
126  
Thokchom Rosy Devi, Laishram Shambhu Singh, and Thokchom Chhatrajit Singh  
Then,  
Ω(un, un+1) ≤ ξ(un+1, un+2  
)
ξ(un+1, un+2) ≤ ψ(Ω(un, un+1))  
≤ ψ(ξ(un+1, un+2))  
≤ ξ(un+1, un+2  
)
which is impossible.  
Therefore,  
Ω(un, un+1) = ξ(un, un+1),  
for all n ∈ N  
ξ(un+1, un+2) ≤ ψ(Ω(un, un+1  
)
= ψ(ξ(un, un+1)).  
We deduce that  
ξ(un+1, un+2) < ξ(un, un+1), ∀ n ∈ N  
(4)  
for all n ∈ N. From the two inequalities shown above, we can derive that  
ξ(un+1, un+2) ≤ ψn(ξ(u0, u1)),  
∀ n ∈ N. It clearly shows that  
lim ξ(un+1, un+2) = 0  
(5)  
n→∞  
Take into account (1) with u = un−1 and v = vn+1. We obtain  
ξ(un, un+2) = ξ(τun−1, τun+1  
)
≤ α(un−1, un+1)ξ(τun−1, τun+1  
≤ ψ(Ω(un−1, un+1))  
)
(6)  
FIXED POINT RESULTS FOR (α-ψ)-RATIONAL TYPE CONTRACTIVE CONDITIONS  
127  
in which,  
ꢀ
Ω(un−1, un+1) = max ξ(un−1, un+1), ξ(un−1, τun−1), ξ(un+1, τun+1),  
ξ(un−1, un+1) + ξ(un−1, τun−1  
)
,
2
ξ(un−1, τun−1) + ξ(un+1, τun+1  
)
,
2
ξ(un−1, τun−1)ξ(un+1, τun+1  
)
)
,
1 + ξ(un−1, un+1  
)
ꢁ
ξ(un−1, τun−1)ξ(un+1, τun+1  
1 + ξ(Tun−1, τun+1  
)
ꢀ
= max ξ(un−1, un+1), ξ(un−1, un), ξ(un+1, un+2),  
ξ(un−1, un+1) + ξ(un−1, un) ξ(un−1, un) + ξ(un+1, un+2  
)
,
,
2
2
ξ(un−1, un)ξ(un+1, un+2  
)
)
,
1 + ξ(un−1, un+1  
ξ(un−1, un)p(un+1, un+2  
1 + ξ(un, un+2  
)
ꢁ
(7)  
)
From (4), it is evident that  
ξ(un+1, un+2) < ξ(un−1, un)  
Define  
Then,  
sn = ξ(un, un+2), tn = ξ(un, un+1  
)
ꢀ
sn−1 + sn−1  
Ω(un−1, un+1) = max sn−1, tn−1, tn+1  
,
,
2
ꢁ
tn−1 + tn+1 tn−1 n+1  
t
tn−1 n+1  
t
,
,
2
1 + an−1  
1 + an  
If  
tn−1 + tn+1  
tn−1 n+1  
1 + sn−1  
t
tn−1 n+1  
1 + an  
b
Ω(un−1, un+1) = tn−1 or tn+1 or  
or  
or  
,
2
then, by utilizing (6) and the upper semi-continuity of ψ, we can obtain the following:  
lim sup as n → ∞ in (5)  
0 ≤  
lim sup sn ≤ lim sup ψ(Ω(un−1, un+1))  
n→∞  
n→∞  
= ψ( lim sup Ω(un−1, un+1))  
n→∞  
= ψ(0) = 0.  
Consequently,  
lim un = lim ξ(un, un+2) = 0  
n→∞  
128  
If  
Thokchom Rosy Devi, Laishram Shambhu Singh, and Thokchom Chhatrajit Singh  
1
Ω(un−1, un+1) =  
(sn−1 + tn−1  
)
2
1
=
[ξ(un−1, un+1) + ξ(un−1, un)]  
2
< ξ(un−1, un+1) + ξ(un−1, un)  
≤ ξ(un−1, un) + ξ(un, un+1) − ξ(un, un) + p(un−1, un)  
≤ ξ(un−1, un) + ξ(un, un+1) + ξ(un−1, un)  
Again, using the upper semi-continuity of ψ and (6), we can obtain the following:  
lim sup = n → ∞  
in (5)  
0 ≤ lim sup sn ≤ lim sup ψ(Ω(un−1, un+1))  
n→∞  
n→∞  
= ψ( lim sup Ω(un−1, un+1))  
n→∞  
= ψ(0) = 0.  
Hence  
lim sn = lim σ(un, un+2) = 0  
n→∞  
n→∞  
If  
Ω(un−1, un+1) = sn−1  
the equation (6) follows  
sn ≤ ψ(sn−1) < sn−1  
.
The sequence {sn} is characterized as positive monotone decreasing and as a result, it  
converges to a point q. Suppose that q > 0. Subsequently, we obtain the following:  
q = lim sup sn ≤ lim sup ψ(sn−1  
)
n→∞  
n→∞  
= ψ( lim sup sn−1  
)
n→∞  
= ψ(q) < q.  
which is a contradiction. Consequently,  
lim sn = lim ξ(un, un+2) = 0.  
n→∞ n→∞  
(8)  
We will now show that un = um for all n = m. The converse is also true; suppose that  
for any m, n ∈ N where n = m, xn = xm. We may assume that m > n + 1 without losing  
generality since p(uk, uk+1) > 0 for every k ∈ N.  
Again, substitute  
u = un = um  
and  
v = un+1 = um+1  
FIXED POINT RESULTS FOR (α-ψ)-RATIONAL TYPE CONTRACTIVE CONDITIONS  
in (1) which results in  
129  
ξ(un, un+1) = ξ(un, τun) = ξ(um, τum) = ξ(τum−1, τum)  
(9)  
≤ α(um−1, um)ξ(τum−1, τum) ≤ ψ(Ω(um−1, um))  
where,  
ꢀ
Ω(um−1, um) = max ξ(um−1, um), ξ(um−1, τum−1), ξ(um, τum),  
ξ(um−1, um) + ξ(um−1, τum−1) ξ(um−1, τum−1) + ξ(um, Tum)  
,
,
2
2
ξ(um−1, τum−1)ξ(um, Tum)  
,
1 + ξ(um−1, um)  
ξ(um−1, τum−1)ξ(um, τum)  
ꢁ
1 + ξ(τum−1, τum)  
ꢀ
= max ξ(um−1, um), ξ(um−1, um), ξ(um, um+1),  
ξ(um−1, um) + ξ(um−1, um) ξ(um−1, um) + ξ(um, um+1  
)
,
,
2
2
ξ(um−1, um)ξ(um, um+1  
1 + ξ(um−1, um)  
ξ(um−1, um)ξ(um, um+1  
)
)
,
ꢁ
1 + ξ(um, um+1  
)
ꢀ
ꢁ
ξ(um−1, um) + ξ(um, um+1  
)
= max ξ(um−1, um), ξ(um, um+1),  
2
If  
Ω(um−1, um) = p(um−1, um)  
then, (9) implies  
ξ(un, un+1) ≤ ψ(ξ(um−1, um)) ≤ ψm−n(ξ(un, un+1))  
(10)  
(11)  
If  
Ω(um−1, um) = p(um, um+1  
)
from (9), we get  
ξ(un, un+1) ≤ ψ(ξ(um, um+1) ≤ ψm−n+1(ξ(un, un+1))  
If  
Ω(um−1, um) < max{ξ(um−1, um), ξ(um, um+1)}  
If  
max{ξ(um−1, um), ξ(um, um+1)} = ξ(um−1, um)  
(9) implies  
If  
ξ(un, un+1) ≤ ψ(ξ(um−1, um)) ≤ ψm−n(ξ(un, un+1)).  
(12)  
max{ξ(um−1, um), ξ(um, um+1)} = ξ(um, um+1  
)
130  
Thokchom Rosy Devi, Laishram Shambhu Singh, and Thokchom Chhatrajit Singh  
(9) implies  
ξ(un, un+1) ≤ ψ(ξ(um, um+1)) ≤ ψm−n+1(ξ(un, un+1))  
(13)  
The inequalities (10), (11), (12), (13) imply and using the property (iii) of ψ  
ξ(xn, xn+1) < ξ(xn, xn+1),  
which is impossible.  
We are going to show that {xn} is a Cauchy sequence,  
lim ξ(un, un+k) = 0,  
n→∞  
for all k ∈ N. In (5) and (8), we have already established the cases for k = 1 and k = 2,  
respectively.  
Let k be an arbitrary integer such that k ≥ 3. We discuss two cases:  
Case 1: Let k = 2m + 1, where m ≥ 1. Utilizing characteristic (iv) of partial metric  
space, we obtain  
ξ(un, un+k) = ξ(un, un+2m+1) ≤ ξ(un, un+1  
)
+ξ(un+1, un+2m+1) − ξ(un+1, un+1  
)
)
≤ ξ(un, un+1) + ξ(un+1, un+2m+1  
)
≤ ξ(un, un+1) + ξ(un+1, un+2  
)
+ξ(un+2, un+2m+1) − ξ(un+2, un+2  
≤ ξ(un, un+1) + ξ(un+1, un+2) + ξ(un+2, un+2m+1  
)
.
.
.
≤ ξ(un, un+1) + ξ(un+1, un+2) + · · · + ξ(un+2m, un+2m+1  
)
ꢂ
n+2m  
≤
lim ξ(uq, uq+1)  
q=n  
ꢂ
n+2m  
lim ψq(ξ(u0, u1)  
≤
≤
q=n  
ꢂ
∞
lim ψq(ξ(u0, q1) → 0  
q=n  
as n → ∞.  
FIXED POINT RESULTS FOR (α-ψ)-RATIONAL TYPE CONTRACTIVE CONDITIONS  
131  
Case 2: Let k = 2m, where m ≥ 2. By applying the property (iv) of partial metric  
space, we get  
ξ(un, un+k) = ξ(un, un+2m) ≤ ξ(un, un+2  
)
+ ξ(un+2, un+2m) − ξ(un+2, un+2  
)
≤ ξ(un, un+2) + ξ(un+2, un+2m  
)
≤ ξ(un, un+2) + ξ(un+2, un+3  
)
+ ξ(un+3, un+2m) − ξ(un+3, un+3  
)
≤ ξ(un, un+2) + ξ(un+2, un+3) + ξ(un+3, un+2m  
)
.
.
.
≤ ξ(un, un+2) + ξ(un+2, un+3  
)
+ · · · + ξ(un+2m−1, un+2m  
)
ꢂ
n+2m−1  
≤ ξ(un, un+2) +  
lim ψq(ξ(u0, u1))  
q=n+2  
ꢂ
∞
≤ ξ(un, un+2) +  
lim ψq(ξ(u0, u1)) → 0  
q=n+2  
as n → ∞. Since,  
limn→∞ξ(un, un+2) = 0  
because of (8). In the two examples given above, we can see that  
lim ξ(un, un+k) = 0  
n→∞  
for all k ≥ 3. So, we may say that {un} is a Cauchy sequence in (X, ξ). Since (X, ξ) is  
complete, there is a point u∗ ∈ X that  
lim p(un, u∗) = 0.  
(14)  
n→∞  
Next, we shall show that the fixed point τ is the limit u∗ of the sequence {un}. Allow us  
to begin by assuming that τ is a curve. Then,  
lim ξ(τun, τu∗) = lim ξ(un+1, τu∗) = 0.  
n→∞  
n→∞  
We conclude that  
u∗ = Tu∗.  
In other words, u∗ is a fixed point in τ.  
Let us assume that u is α-regular. After that, there is a subsequence {un } of {un}  
k
such that  
α(un −1, u∗) ≥ 1  
k
∀ k ∈ N. From inequality (1), where u = un and v = u∗, we get  
k
ξ(un +1, τu∗) = ξ(τun , τu∗)  
k
k
≤ α(un , u∗)ξ(τun , τu∗)  
(15)  
k
k
≤ ψ(Ω(un , u∗))  
k
132  
Thokchom Rosy Devi, Laishram Shambhu Singh, and Thokchom Chhatrajit Singh  
where,  
ꢀ
Ω(un , u∗) = max ξ(un , u∗), ξ(un , τun ), p(u∗, τu∗),  
k
k
k
k
ξ(un , u∗) + ξ(un , τun )  
k
k
k
,
2
ξ(un , τun ), ξ(u∗, τu∗)  
k
k
,
2
ξ(un , τun )ξ(u∗, τu∗)  
k
k
,
1 + ξ(un , u∗)  
k
ꢁ
ξ(un , τun )ξ(u∗, Tx∗)  
k
k
1 + ξ(τun , τu∗)  
k
ꢀ
= max ξ(un , u∗), ξ(un , un +1), ξ(u∗, τu∗),  
k
k
k
ξ(un , u∗), +ξ(un , un +1  
)
k
k
k
,
2
ξ(un , un +1), ξ(u∗, Tu∗)  
k
k
,
2
ξ(un , un +1)ξ(u∗, τu∗)  
k
k
,
1 + ξ(un , u∗)  
k
ꢁ
ξ(un , un +1)ξ(u∗, Tu∗)  
k
k
(16)  
1 + ξ(un +1, τu∗)  
k
Let k → ∞ in (16), we get  
Ω(un , u∗) = ξ(u∗, Tu∗).  
k
Consequently, by taking the limit as k → ∞ in inequality (16), we get  
ξ(u∗, τu∗) ≤ ψ(ξ(u∗, τu∗)) < ξ(u∗, τu∗),  
which implies that  
u∗ = τu∗,  
that is, u∗ is a fixed point of τ.  
Let us consider two distinct fixed points of τ, denoted as u∗ and v∗, where u∗ = v∗.  
According to the hypothesis, α(u∗, v∗) ≥ 1. Therefore, substituting u = u∗ and v = v∗  
into (1) yields  
ξ(u∗, v∗) = ξ(τu∗, τv∗) < α(u∗, v∗)ξ(σu∗, σv∗)  
(17)  
< ψ(Ω(u∗, v∗))  
where  
ꢀ
ξ(u∗, v∗) + ξ(u∗, τv∗)  
Ω(u∗, v∗) = max ξ(u∗, v∗), ξ(v∗, τv∗), ξ(u∗, τu∗),  
,
2
ξ(u∗, τu∗) + ξ(v∗, τv∗) ξ(u∗, τu∗)ξ(v∗, τv∗)  
,
,
2
1 + ξ(u∗, v∗)  
ꢁ
ξ(u∗, τu∗)ξ(v∗, τv∗)  
1 + ξ(τu∗, τv∗)  
= ξ(u∗, v∗).  
(18)  
FIXED POINT RESULTS FOR (α-ψ)-RATIONAL TYPE CONTRACTIVE CONDITIONS  
Therefore, we get  
133  
ξ(u∗, v∗) ≤ ψ(ξ(u∗, v∗)) < ξ(u∗, v∗),  
which is possible only if σ(u∗, v∗) = 0, that is u∗ = v∗. Thus, τ possesses a unique fixed  
point.  
□
Example 3.3. Assume that X = {1, 2, 3} is a finite set. Define  
ξ : X × X → [0, ∞)  
as:  
ξ(1, 1) = ξ(2, 2) = ξ(3, 3) = 0  
ξ(1, 2) = ξ(2, 1) = 3  
ξ(2, 3) = ξ(3, 2) = ξ(1, 3) = ξ(3, 1) = 8  
In this case, ξ is a partial metric on X, and (X, ξ) is a complete partial metric space.  
Define τ : X → X as  
τ3 = 2, τ1 = τ2 = 1,  
t
Define α(u, v) as α(u, v) = 1, ψ(t) = .  
2
Finally, for u = 1, 2 and v = 1, 2, we get  
α(u, v)ξ(τu, τv) = 0 ≤ ψ(Ω(u, v)) = 0  
Conversely, for u = 1, 2 and v = 3  
α(u, 3)ξ(τu, τ3) = ξ(1, 2) = 3  
ꢀ
ξ(u, 3) + ξ(u, τu)  
Ω(u, 3) = max ξ(u, 3), ξ(u, τu), ξ(3, τ3),  
,
2
ꢁ
ξ(u, τu) + ξ(3, τ3) ξ(u, τu)ξ(3, τ3) ξ(u, τu)ξ(3, τ3)  
,
,
2
1 + ξ(u, 3)  
1 + ξ(τu, τ3)  
ꢀ
ꢁ
8
2
11 8  
11 (0 or 3) × 8 (0 or 3) × 8  
= max 8, 0 or 3, 8,  
or  
11  
,
or  
,
,
2 2  
2
1 + 8  
1 + 8  
ꢀ
ꢃ
24  
9
= max 8, 0 or 3, 4 or  
, 0 or  
2
= 8.  
and hence  
α(u, 3)ξ(τu, τ3) = ξ(1, 2) = 3 ≤ ψ(8) = 4  
The contraction condition is evident when u = 3 and v = 3. It is evident that τ satisfies  
the conditions of theorem (3.2) and has a unique fixed point at u = 1.  
134  
Thokchom Rosy Devi, Laishram Shambhu Singh, and Thokchom Chhatrajit Singh  
4. Some consequences  
Definition 4.1. Consider a partially ordered set (X, ≤). For any pair of elements u, v ∈  
X, where u ≤ v implies τu ≤ Tv, we say that the mapping τ : X → X is non-decreasing  
with respect to ≤.  
Definition 4.2. Consider a partially ordered metric space (X, ξ, ≤). The regular partial  
metric space X is defined as the set of all sequences in X such that for every k ∈ N, there  
exists a subsequence {un } of {un} such that un ≤ u, ∀ u, v ∈ X.  
k
k
Theorem 4.3. Let (X, ξ, ≤) represent a partially ordered complete partial metric space,  
and let τ : X → X denote a non-decreasing self-mapping. Assuming the following condi-  
tions are satisfied:  
(i) ∃ ψ ∈ Ψ for which ξ(τu, τv) ≤ ψ(Ω(u, v))  
where  
ꢄ
ξ(u, v) + ξ(u, τu)  
Ω(u, v) = max ξ(u, v), ξ(u, τu), ξ(v, τv),  
,
2
ξ(u, τu) + ξ(v, τv) ξ(u, τu)ξ(v, τv)  
,
,
2
1 + ξ(u, v)  
ꢃ
ξ(u, τu)ξ(v, τv)  
1 + ξ(τu, τv)  
(ii) ∃ u0 ∈ X such that u0 ≤ τu0 and u0 ≤ τ2u0 .  
(iii) Either X is regular or τ is continuous. Consequently, τ possesses a fixed point  
u∗ ∈ X and {τnu0} converges to u∗.  
Proof. Let α : X × X → [0, ∞) be a mapping defined by:  
ꢅ
1
if u ≤ v or v ≤ u;  
α(u, v) =  
0
otherwise .  
Now that we have shown that Theorem 3.2 holds, we can say that τ has a fixed point,  
which is the limit of the sequence {τnu0}.  
□
Theorem 4.4. Assume (X, ξ, ≤) is a partially ordered complete Partial metric space, and  
τ : X → X is a non-decreasing self mapping. Assume the following criteria are met:  
(i) ∃ ψ ∈ Ψ for which ξ(τu, τv) ≤ ψ(Ω(u, v)) where  
ξ(u, v) + ξ(u, τu)  
Ω(u, v) = max{ξ(u, v), ξ(u, τu), ξ(v, τv),  
,
2
ξ(u, τu) + ξ(v, τv)  
ξ(u, τu)ξ(v, τv)  
,
,
2
1 + ξ(u, v) + ξ(u, τv) + ξ(v, τu)  
ξ(u, τv)ξ(u, v)  
}
1 + ξ(u, τu) + ξ(v, τu) + ξ(v, τv)  
(ii) ∃ u0 ∈ X such that u0 ≤ τu0 and u0 ≤ τ2u0.  
(iii) Either X is regular or τ is continuous. Consequently, τ possesses a fixed point  
u∗ ∈ X and {τnu0} converges to u∗.  
FIXED POINT RESULTS FOR (α-ψ)-RATIONAL TYPE CONTRACTIVE CONDITIONS  
135  
Proof. Consider a mapping α : X × X → [0, ∞) defined by:  
ꢅ
1
if u ≤ v or v ≤ u;  
α(u, v) =  
0
otherwise .  
The existence conditions of Theorem 3.2 show that τ has a fixed point, which is the limit  
of the sequence {τnu0}.  
Several specific cases can be inferred from the above results.  
□
Corollary 4.5. Consider a complete partial metric space (X, σ) and let τ : X → X,  
α : X × X → R. Assume the following criteria are satisfied:  
(i) τ is an α-admissible mapping.  
(ii) τ holds ξ(τu, τv) ≤ k0Ω(u, v)  
where  
ξ(u, v) + ξ(u, τu)  
Ω(u, v) = max{ξ(u, v), ξ(u, τv), ξ(v, τv),  
,
2
ξ(u, τu) + ξ(v, τv)  
ξ(u, τu)ξ(v, τv)  
,
,
2
1 + ξ(u, v) + ξ(u, τv) + ξ(v, τu)  
ξ(u, τv)ξ(u, v)  
}
1 + ξ(u, τu) + ξ(v, τu) + ξ(v, τv)  
for some k0 ∈ [0, 1).  
(iii) ∃ u0 ∈ X such that α(u0, τu0) ≥ 1 andα(u0, τ2u0) ≥ 1 .  
(iii) Either X is α-regular or τ is continuous.  
As a result, {τnu0} converges to u∗ and τ has a fixed point u∗ ∈ X. Additionally, if  
α(u, v) ≥ 1  
for every u, v ∈ F(τ), consequently, τ possesses a unique fixed point in X.  
Proof. Define a mapping ψ(t) = kt. Evidently ψ ∈ ψ . There is a unique fixed point for  
T according to theorem 3.2.  
□
Corollary 4.6. Consider a complete partially ordered Partial metric space (X, ξ, ≤) and  
let τ : X → X be a non-decreasing self mapping. Assume that the following criteria are  
met:  
(i)  
ξ(τu, τv) ≤ k0(Ω(u, v))  
where  
ξ(u, v) + ξ(u, τu)  
Ω(u, v) = max{ξ(u, v), ξ(u, τu), ξ(v, τv),  
,
2
ξ(u, τu) + ξ(v, τv)  
ξ(u, τu)ξ(v, τv)  
,
,
2
1 + ξ(u, v) + ξ(u, τv) + ξ(v, τu)  
ξ(u, τv)ξ(u, v)  
}
1 + ξ(u, τu) + ξ(v, τu) + ξ(v, τv)  
for all u, v ∈ X with u ≤ v and for some k0 ∈ [0, 1).  
136  
Thokchom Rosy Devi, Laishram Shambhu Singh, and Thokchom Chhatrajit Singh  
(ii) ∃ u0 ∈ X such that u0 ≤ τu0 and u0 ≤ τ2u0.  
(iii) Either X is regular or τ is continuous.  
Consequently, τ possesses a fixed point u∗ ∈ X and {τnu0} converges to u∗.  
Proof. Let α : X × X → [0, ∞) be a mapping defined by:  
ꢅ
1
if u ≤ v or v ≤ u;  
α(u, v) =  
0
otherwise .  
It follows from Corollary 4.5 that τ possesses a fixed point.  
□
5. Conclusion  
Our exploration of fixed point results for (α, ψ)-rational type contractive conditions in  
partial metric spaces has yielded significant insights and contributions to the field of fixed  
point theory. Through our investigation, we have established novel theoretical frameworks  
and provided practical tools for analyzing the existence and uniqueness of fixed points in  
diverse settings. Our findings have extended the classical notion of contractive mappings  
to a more flexible and adaptable form, encompassing a broader class of mappings and  
partial metric spaces. This study opens up new avenues for further research in fixed  
point theory, with opportunities to explore more intricate contractive conditions and  
their applications in both theoretical and practical contexts. We anticipate that our  
contributions will inspire future advancements in the understanding and utilization of  
fixed point concepts in partial metric spaces and beyond.  
Conflict of Interest  
The authors have no conflicts of interest to declare. The study was conducted indepen-  
dently without any external funding, and the authors did not have any involvement from  
commercial entities that could have influenced the study design, data collection, analysis,  
interpretation, or publication.  
Acknowledgements  
The authors like to express sincere gratitude to the referees for their helpful comments  
and suggestion to improve the quality of the manuscript.  
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(Received, November 01, 2024)  
(Revised, May 31, 2025)  
1Dhanamanjuri University,  
Department of Mathematics, Manipur,  
India-795001  
Email: rosythokchom 1@gmail.com  
2Dhanamanjuri University,  
Department of Mathematics,  
Manipur, India-795001  
E-mail: lshambhu1162@gmail.com  
3Manipur Technical University,  
Department of Mathematics,  
Takyelpat Imphal, Manipur,  
India-795004.