Int. J. Anal. Appl. (2023), 21:55

Existence of Solutions via C-Class Functions in Ab-Metric Spaces With Applications

N. Mangapathi1,3,∗, B. Srinuvasa Rao2, K.R.K. Rao3, M.I. Pasha1,3

1Department of Mathematics, B V Raju Institute of Technology, Narsapur,

Medak-502313,Telangana, India
2Department of Mathematics, Dr.B.R.Ambedkar University, Srikakulam, Etcherla-532410, Andhra

Pradesh, India
3Department of Mathematics, Gitam School of Science, GITAM Deemed to be University,

Hyderabad, Rudraram-502329, Telangana, India

∗Corresponding author: nmp.maths@gmail.com

Abstract. Using C-class functions, we demonstrate a few popular common coupled fixed point theo-

rems on Ab-metric spaces and discuss some implications of the main findings. Additionally, we provide

examples to illustrate the findings and their applications to both homotopy theory and integral equa-

tions.

1. Introduction

Fixed point theory plays a significant role in many parts of the development of nonlinear analysis.

It has been applied to various fields of engineering and research. This research was inspired by recent

work on the extension of the Banach contraction principle on Ab metric spaces, which was carried

out by M. Ughade et al. [1] and studied various fixed point results on these spaces. In the further, N.

Mlaiki et al. [2] and K. Ravibabu et al. [3], [4] and P. Naresh et al. [5] succeeded in deriving unique

coupled common fixed point theorems in Ab-metric spaces.

Sessa [6] began researching common fixed point theorems for weakly commuting pair of mappings

in 1982. Later, in 1986, Jungck [7] expanded the idea of weakly commuting mappings to compatible

mappings in metric spaces and proved compatible pair mappings commute on the sets of coincidence

point of the involved mappings. When they commute at their coincidence sites, Jungck and Rhoades [8]

Received: Mar. 23, 2023.

2020 Mathematics Subject Classification. 54H25, 47H10, 54E50.

Key words and phrases. C-class function; ω-compatible mapping; Ab-completeness; coupled fixed points.

https://doi.org/10.28924/2291-8639-21-2023-55
ISSN: 2291-8639

© 2023 the author(s).

https://doi.org/10.28924/2291-8639-21-2023-55


2 Int. J. Anal. Appl. (2023), 21:55

introduced the idea of weak compatibility in 1998 and demonstrated that compatible mappings are

weakly compatible but the reverse is not true.

However, Khan et al. [9] first proposed the idea of modifying distance function, which is a control

function that modifies the distance between two locations in a metric space. Ansari [10] presented the

idea of C-class functions in 2014 and proved the unique fixed point theorems for certain contractive

mappings with regard to the C-class functions, which started a lot of work in this field (See. [11],

[12], [13], [14], [15], [16], [17])

The idea of coupled fixed point was first developed by Guo and Lakshmikantham [18] in 1987 .

Later, employing a weak contractivity type assumption, Bhaskar and Lakshmikantham [19] developed

a novel fixed point theorem for a mixed monotone mapping in a metric space driven with partial

ordering. See study results in ( [20], [21], [22], [23], [24]) and related references for additional results

on coupled fixed point outcomes.

In the framework of Ab-metric spaces, the goal of the current study is to establish original common

coupled fixed point theorems using C-class functions. Additionally, we may provide relevant applications

for homotopy, integral equations, and appropriate examples.

First we recall some basic results.

2. Preliminaries

Definition 2.1. ( [1]) Let = be a non-empty set and b ≥ 1 be given real number. A mapping
Ab : =n → [0,∞) is called an Ab-metric on = if and only if for all Υi,a ∈ = i = 1, 2, 3, ..n; the
following conditions hold :

(Ab1) Ab(Υ1, Υ2, ........, Υn−1, Υn) ≥ 0,
(Ab2) Ab(Υ1, Υ2, ........, Υn−1, Υn) = 0 ⇔ Υ1 = Υ2 = · · · · · · = Υn−1 = Υn,

(Ab3) Ab(Υ1, Υ2, ........, Υn−1, Υn) ≤ b




Ab (Υ1, Υ1, ........, (Υ1)n−1,a)

+Ab (Υ2, Υ2, ........, (Υ2)n−1,a)

+ · · · · · · + Ab (Υn−1, Υn−1, ........, (Υn−1)n−1,a)
+Ab (Υn, Υn, ........, (Υn)n−1,a)




Then the pair (=,Ab) is called an Ab-metric space.

Remark 2.1. ( [1]) The class of Ab-metric spaces is actually larger than that of A-metric spaces, it

should be emphasised. Each A-metric space is, in fact, a Ab-metric space with b = 1. The opposite

isn’t always true, though. A n-dimensional Sb-metric space is also a Ab-metric space. As a result, a

Ab-metric with n = 3 is a particular instance of a Sb-metric.

The example below demonstrates that an Ab-metric on = need not be an A-metric on =.

Example 2.1. ( [1]) Let = = [0, +∞), define Ab : =n → [0, +∞)
as Ab (Υ1, Υ2, ........, Υn−1, Υn) =

∑n
i=1

∑
i<j |Υi − Υj|

2 for all Υi ∈ =, i = 1, 2, · · · . Then (=,Ab)
is an Ab-metric space with b = 2 > 1.



Int. J. Anal. Appl. (2023), 21:55 3

Definition 2.2. ( [1]) Let (=,Ab) be a Ab-metric space. Then, for Υ ∈=, r > 0 we defined the open
ball BAb (Υ, r) and closed ball BAb [Υ, r] with center Υ and radius r as follows respectively:

BAb (Υ, r) = {f∈= : Ab(f,f, · · · , (f)n−1, Υ) < r},

and

BAb [Υ, r] = {f∈= : Ab(f,f, · · · , (f)n−1, Υ) ≤ r}.

Lemma 2.1. ( [1]) In a Ab-metric space, we have

(1) Ab(Υ, Υ, · · · , (Υ)n−1,f) ≤ bAb(f,f, · · · , (f)n−1, Υ);
(2) Ab(Υ, Υ, · · · , (Υ)n−1,ð) ≤ b(n− 1)Ab(Υ, Υ, · · · , (Υ)n−1,f) + b2Ab(f,f, · · · , (f)n−1,ð).

Definition 2.3. ( [1]) If (=,Ab) be a Ab-metric space. A sequence {Υk} in = is said to be:

(1) Ab-Cauchy sequence if, for each � > 0, there exists n0 ∈ N such that
Ab(Υk, Υk, · · · · · ·(Υk)n−1, Υm) < � for each m,k ≥ n0.

(2) Ab-convergent to a point Υ ∈= if, for each � > 0, there exists a positive integer n0 such that
Ab(Υk, Υk, · · · · · ·(Υk)n−1, Υ) < � for all n ≥ n0 and we denote by lim

k→∞
Υk = Υ.

(3) If every Ab-Cauchy sequence is Ab-convergent in =, then the Ab-metric space (=,Ab) is said
to be complete.

Definition 2.4. [10] A continuous mapping Γ : [0, +∞) × [0. + ∞) → R is called a C-class function
if for all s∗,t∗ ∈ [0,∞),

(a) Γ(s∗,t∗) ≤ s;
(b) Γ(s∗,t∗) = s∗ implies that either s∗ = 0 or t∗ = 0.

The family of all C-class functions is denoted by C.

Example 2.2. [10] Each of the functions Γ : [0, +∞) × [0. + ∞) → R defined below are elements of
C.

(a) Γ(s∗,t∗) = s? − t?;
(b) Γ(s∗,t∗) = ms∗ where m ∈ (0, 1).
(c) Γ(s∗,t∗) = s

∗

(1+t?)r
where r ∈ (0,∞).

(d) Γ(s∗,t∗) = s?η(s?) where η : [0,∞) → [0,∞) is continuous function.
(e) Γ(s∗,t∗) = s? −ϕ(s?) for all s∗,t∗ ∈ [0, +∞) where, the continuous function

ϕ : [0,∞) → [0,∞) such that ϕ(s?) = 0 ⇔ s? = 0.
(f ) Γ(s∗,t∗) = sΩ(s?,t?) for all s∗,t∗ ∈ [0, +∞) where, the continuous function

Ω : [0,∞)2 → [0,∞) such that Ω(s?,t?) < 1.

Definition 2.5. [9] A function θ? : [0,∞) → [0,∞) is called an altering distance function if the
following properties are satisfied:

(a) θ? is nondecreasing and continuous;



4 Int. J. Anal. Appl. (2023), 21:55

(b) θ?(t) = 0 if and only if t = 0.

Here Θ represents the family of all altering distance functions.

We must take the following into consideration in order to get our outcomes.

3. Main Results

Definition 3.1. Let (=,Ab) be a Ab-metric spaces and suppose Ω : =2 → = be a mapping. If
Ω (℘,$) = ℘, Ω ($,℘) = $ for ℘,$ ∈= then (℘,$) is called a coupled fixed point of Ω.

Definition 3.2. Let (=,Ab) be a Ab-metric spaces and suppose Ω : =2 → = and Λ : = → = be two
mappings. An element (℘,$) is said to be a coupled coincident point of Ω and Λ if

F (℘,$) = Λ℘, Ω ($,℘) = Λ$

Definition 3.3. Let (=,Ab) be a Ab-metric spaces and suppose Ω : =2 → =, Λ : = → = be two
mappings. An element (℘,$) is said to be a coupled common point of Ω and Λ if

Ω (℘,$) = Λ℘ = ℘, Ω ($,℘) = Λ$ = $,

Definition 3.4. Let (=,Ab) be a Ab-metric space.

(a) A pair (Ω, Λ) is called weakly compatible if Λ(Ω(℘,$)) = Ω(Λ℘, Λ$) whenever for all

℘,$ ∈= such that F (℘,$) = Λ℘, Ω ($,℘) = Λ$
(b) A pair (Ω, Λ) is called compatible if

lim
p→∞

Ab(ΛΩ(ıp, p), ΛΩ(ıp, p) · · · , Ω(Λıp, Λp)) = lim
p→∞

Ab(ΛΩ(p, ıp), ΛΩ(p, ıp) · · · , Ω(Λp, Λıp)) = 0

wherever {ıp},{p} are sequences in =, such that

lim
p→∞

Ω(ıp, p) = Λıp = ı and lim
p→∞

Ω(p, ıp) = Λp = .

Lemma 3.1. If the pair (Ω, Λ) of mappings on the Ab-metric space (=,Ab) is compatible, then it is
weakly compatible. The converse does not hold.

Proof. Let Ω(i, j) = Λi and Ω(j, i) = Λj for some i, j ∈=. we have to prove that ΛΩ(i, j) = Ω(Λi, Λj)
and ΛΩ(j, i) = Ω(Λj, Λi). Put ıp = i and p = j for every p ∈ N. we have Ω(ıp, p) = Λıp → Λi and
Ω(p, ıp) = Λp → Λj. Since (Ω, Λ) is compatible

lim
p→∞

Ab(ΛΩ(ıp, p), ΛΩ(ıp, p) · · · , Ω(Λıp, Λp)) = lim
p→∞

Ab(ΛΩ(p, ıp), ΛΩ(p, ıp) · · · , Ω(Λp, Λıp)) = 0

Therefore, ΛΩ(i, j) = Ω(Λi, Λj) and ΛΩ(j, i) = Ω(Λj, Λi) and hence the pair (Ω, Λ) is weakly compat-

ible. However, the opposite need not be the case.

For example, Let = = [0, +∞), define Ab : =n → [0, +∞) as
Ab (Υ1, Υ2, ........, Υn−1, Υn) =

∑n
i=1

∑
i<j |Υi − Υj|

2 for all Υi ∈ =, i = 1, 2, · · · . Then (=,Ab) is
an Ab-metric space with b = 2 > 1.



Int. J. Anal. Appl. (2023), 21:55 5

Define two mappings Ω : =2 → = by Ω(i, j) =

{
6i−3j+6n−3

6n
for i, j ∈ [0, 1

2
)

n
4

for i, j ∈ [1
2
,∞)

and Λ : = → =

by Λ(i) =

{
9i+6n−6
6n

for i ∈ [0, 1
2

)

n
4

for i ∈ [1
2
,∞)

Now we define the two sequences {ıp},{p} as ıp = 1p and p = 1 +
1
p
, then Ω(ıp, p) =

3
p
+6n−6
6n

→ n−1
n

as p → ∞ and Λ(ıp) =
9
p
+6n−6
6n

→ n−1
n

as p → ∞, also Ω(p, ıp) =
3
p
+6n+3

6n
→ 2n+1

2n
as p → ∞ and

Λ(p) =
9
p
+6n+3

6n
→ 2n+1

2n
as p →∞. But

lim
p→∞

Ab(ΛΩ(ıp, p), ΛΩ(ıp, p) · · · , Ω(Λıp, Λp))

= lim
p→∞

Ab(

27
p

+ 36n2 + 18n− 54
36n2

,

27
p

+ 36n2 + 18n− 54
36n2

· · · ,
27
p

+ 36n2 − 45
36n2

)

= (n− 1)|
2n− 1

4n2
|2 6= 0, if n = 2.

and

lim
p→∞

Ab(ΛΩ(p, ıp), ΛΩ(p, ıp) · · · , Ω(Λp, Λıp))

= lim
p→∞

Ab(

27
p

+ 36n2 + 18n + 27

36n2
,

27
p

+ 36n2 + 18n + 27

36n2
· · · ,

27
p

+ 36n2 + 36

36n2
)

= (n− 1)|
2n− 1

4n2
|2 6= 0, if n = 2.

Thus the pair (Ω, Λ) is not compatible. Also,Then for any i, j ∈ [1
2
,∞), (n

4
, n
4

)is a coupled coincidence

point of Ω and Λ it is namely that i = j = n
4
, Ω(i, j) = 7

8
= Λi and Ω(j, i) = 7

8
= Λj for n = 2 and

λΩ(i, j) = Ω(Λi, Λj), λΩ(j, i) = Ω(Λj, Λi), showing that Ω, Λ are weakly compatible maps on =.
�

Theorem 3.1. Let (=,Ab) be a complete Ab-metric space. Suppose T : =2 →= and f : =→= be
a two mappings satisfying the following:

η?
(

2b2Ab(T (ı, ),T (ı, ), · · · ,T (ð,f))
)
≤ Γ (η? (Ab(f ı, f ı, · · · , fð)) ,θ? (Ab(f , f , · · · , ff))) (3.1)

for all ı, ,ð,f∈=, where η?,θ? ∈ Θ and Γ ∈ C
a) T (=2) ⊆ f (=);
b) pair (T,f ) is compatible;

c) f is continuous.

Then there is a unique common coupled fixed point of T and f in =.

Proof. Let ı0, 0 ∈= be arbitrary, and from (a), we construct the sequences {ıp} ,{p} , in = as

T (ıp, p) = f ıp+1 = ℵp, T (p, ıp) = f p+1 = Υp, where p = 0, 1, 2, . . . .



6 Int. J. Anal. Appl. (2023), 21:55

Now from (3.1), we have

η?
(

2b2Ab(ℵ1,ℵ1, · · · ,ℵ2)
)

= η?
(

2b2Ab(T (ı1, 1),T (ı1, 1), · · · ,T (ı2, 2))
)

≤ Γ (η? (Ab(f ı1, f ı1, · · · , f ı2)) ,θ? (Ab(f 1, f 1, · · · , f 2)))

≤ η? (Ab(f ı1, f ı1, · · · , f ı2))

≤ η? (Ab(ℵ0,ℵ0, · · · ,ℵ1))

By the definition of η?, we have that

Ab(ℵ1,ℵ1, · · · ,ℵ2) ≤
1

2b2
Ab(ℵ0,ℵ0, · · · ,ℵ1).

Also

η?
(

2b2Ab(ℵ2,ℵ2, · · · ,ℵ3)
)

= η?
(

2b2Ab(T (ı2, 2),T (2, 2), · · · ,T (ı3, 3))
)

≤ Γ (η? (Ab(f ı2, f ı2, · · · , f ı3)) ,θ? (Ab(f 2, f 2, · · · , f 3)))

≤ η? (Ab(f ı2, f ı2, · · · , f ı3))

≤ η? (Ab(ℵ1,ℵ1, · · · ,ℵ2))

By the definition of η?, we have that

Ab(ℵ2,ℵ2, · · · ,ℵ3) ≤
1

2b2
Ab(ℵ1,ℵ1, · · · ,ℵ2)

≤
1

(4b2)2
Ab(ℵ0,ℵ0, · · · ,ℵ1)

Continuing this process, we can conclude that

Ab(ℵp,ℵp, · · · ,ℵp+1) ≤
1

(2b2)p
Ab(ℵ0,ℵ0, · · · ,ℵ1) → 0 as p →∞.

that is

lim
p→∞

Ab(ℵp,ℵp, · · · ,ℵp+1) = 0.

Similarly, we can prove that

lim
p→∞

Ab(Υp, Υp, · · · , Υp+1) = 0.

Now for q > p, by use of (Ab3), we have

Ab (ℵp,ℵp, · · · ,ℵn−1,ℵq) ≤ b




Ab (ℵp,ℵp, ........, (ℵp)n−1,ℵp+1)
+Ab (ℵp,ℵp, ........, (ℵp)n−1,ℵp+1)

+ · · · · · · + Ab (ℵp,ℵp, ........, (ℵp)n−1,ℵp+1)
+Ab (ℵq,ℵq, ........, (ℵq)n−1,ℵp+1)






Int. J. Anal. Appl. (2023), 21:55 7

≤ b(n− 1)Ab (ℵp,ℵp, ........, (ℵp)n−1,ℵp+1)

+bAb (ℵq,ℵq, ........, (ℵq)n−1,ℵp+1)

≤ b(n− 1)Ab (ℵp,ℵp, ........, (ℵp)n−1,ℵp+1)

+b2Ab (ℵp+1,ℵp+1, ........, (ℵp+1)n−1,ℵq)

≤ b(n− 1)Ab (ℵp,ℵp, ........, (ℵp)n−1,ℵp+1)

+b3(n− 1)Ab (ℵp+1,ℵp+1, ........, (ℵp+1)n−1,ℵp+2)

+b4Ab (ℵp+2,ℵp+2, ........, (ℵp+2)n−1,ℵq)

≤ b(n− 1)Ab (ℵp,ℵp, ........, (ℵp)n−1,ℵp+1)

+b3(n− 1)Ab (ℵp+1,ℵp+1, ........, (ℵp+1)n−1,ℵp+2)

+b5(n− 1)Ab (ℵp+2,ℵp+2, ........, (ℵp+2)n−1,ℵp+3)

+b7(n− 1)Ab (ℵp+3,ℵp+3, ........, (ℵp+3)n−1,ℵp+4)

+ . . . + b2q−2p−2(n− 1)Ab (ℵq−2,ℵq−2, ........, (ℵq−2)n−1,ℵq−1)

+b2q−2p−3Ab (ℵq−1,ℵq−1, ........, (ℵq−1)n−1,ℵq)

≤ (n− 1)
(
b

1

(2b2)p
+ b3

1

(2b2)p+1
+ b5

1

(2b2)p+2
+ . . . + b2q−2p−2

1

(2b2)q−2

)
Ab (ℵ0,ℵ0, · · · ,ℵ1)

+b2q−2p−3
1

(2b2)q−1
Ab (ℵ0,ℵ0, · · · ,ℵ1)

≤ (n− 1)b
1

(2b2)p

(
1 + b2

1

2b2
+ b4(

1

2b2
)2 + . . . + b2q−2p−4(

1

2b2
)q−p−2

)
Ab (ℵ0,ℵ0, · · · ,ℵ1)

+b2q−2p−3(
1

2b2
)q−p−1Ab (ℵ0,ℵ0, · · · ,ℵ1)

≤ (n− 1)b
1

(2b2)p

(
1 +

1

2
+

1

22
+ . . . +

1

2q−p−2
+ · · ·

)
Ab (ℵ0,ℵ0, · · · ,ℵ1)

≤ 2(n− 1)b
1

(2b2)p
Ab (ℵ0,ℵ0, · · · ,ℵ1) → 0 as p,q →∞.

Hence {ℵp} is a Cauchy sequence in = . We can also demonstrate that {Υp}, is Cauchy sequence
in =. Therefore,

lim
p,q→∞

Ab(ℵp,ℵp, · · · ,ℵq) = 0, and lim
p,q→∞

Ab(Υp, Υp, · · · , Υq) = 0.

Since (=,Ab) is complete, there exist ℵ, Υ ∈= such that

lim
p→∞

ℵp = lim
p→∞

T (ıp, p) = lim
p→∞

f ıp+1 = ℵ lim
p→∞

Υp = lim
p→∞

T (p, ıp) = lim
p→∞

f p+1 = Υ.

Since f : =→= is continuous

lim
p→∞

f 2ıp+1 = fℵ and lim
p→∞

f T (ıp, p) = fℵ



8 Int. J. Anal. Appl. (2023), 21:55

lim
p→∞

f 2p+1 = f Υ and lim
p→∞

f T (p, ıp) = f Υ

Since {T,f} is compatible, we have F (f ıp, f p) → fℵ and F (f p, f ıp) → f Υ

lim
p→∞

Ab(T (f ıp, f p),T (f ıp, f p) · · · , f T (ıp, p)) = 0. (3.2)

lim
p→∞

Ab(T (f p, f ıp),T (f p, f ıp) · · · , f T (p, ıp)) = 0. (3.3)

Now, we prove that fℵ = T (ℵ, Υ) and f Υ = T (Υ,ℵ).
For all p ≥ 0, we have

Ab (fℵ, fℵ, · · · , (fℵ)n−1,T (f ıp, f p))

≤ b

(
(n− 1)Ab (fℵ, fℵ, ........, (fℵ)n−1, f T (ıp, p))

+Ab (f T (ıp, p), f T (ıp, p), ........, (f T (ıp, p))n−1,T (f ıp, f p))

)
≤ (n− 1)bAb (fℵ, fℵ, ........, (fℵ)n−1, f T (ıp, p))

+b2Ab (T (f ıp, f p),T (f ıp, f p), ........, (T (f ıp, f p))n−1, f T (ıp, p))

On taking limits as p →∞ and from (3.2 ) we get

Ab (fℵ, fℵ, · · · , (fℵ)n−1,T (ℵ, Υ)) = 0.

Similarly it is easy to see that Ab (f Υ, f Υ, · · · , (f Υ)n−1,T (Υ,ℵ)) = 0.
Thus, T (ℵ, Υ) = fℵ and T (Υ,ℵ) = f Υ. Hence (ℵ, Υ) is a coupled coincidence point of T and f .
Now we prove that fℵ = ℵ and f Υ = Υ. Now consider

η?
(

2b2Ab (fℵ, fℵ, · · · , (fℵ)n−1,ℵp)
)

= η?
(

2b2Ab (T (ℵ, Υ),T (ℵ, Υ), · · · , (T (ℵ, Υ))n−1,T (ıp, p))
)

≤ Γ (η? (Ab(fℵ, fℵ, · · · , f ıp)) ,θ? (Ab(f Υ, f Υ, · · · , f )))

≤ η? (Ab(fℵ, fℵ, · · · , f ıp))

By the definition of η?, we have

Ab (fℵ, fℵ, · · · , (fℵ)n−1,ℵp) ≤
1

2b2
Ab(fℵ, fℵ, · · · , f ıp)

Letting p → ∞ , we get Ab (fℵ, fℵ, · · · , (fℵ)n−1,ℵ) ≤ 12b2Ab(fℵ, fℵ, · · · ,ℵ) which implies that
fℵ = ℵ. Similarly, we can prove f Υ = Υ. Therefore, T (ℵ, Υ) = fℵ = ℵ and T (Υ,ℵ) = f Υ = Υ.
Thus,(ℵ, Υ) is a common coupled point of T and f . In order to demonstrate uniqueness, we first
assume that (ℵ?, Υ?) is a another coupled common fixed point of T and f .

Now

η?
(

2b2Ab (ℵ,ℵ, · · · , (ℵ)n−1,ℵ?)
)

= η?
(

2b2Ab (T (ℵ, Υ),T (ℵ, Υ), · · · , (T (ℵ, Υ))n−1,T (ℵ?, Υ?))
)

≤ Γ (η? (Ab(fℵ, fℵ, · · · , fℵ?)) ,θ? (Ab(f Υ, f Υ, · · · , f Υ?)))

≤ η? (Ab(fℵ, fℵ, · · · , fℵ?))

≤ η? (Ab(ℵ,ℵ, · · · ,ℵ?))



Int. J. Anal. Appl. (2023), 21:55 9

By the definition of η?, we have Ab(ℵ,ℵ, · · · ,ℵ?) ≤ 12b2Ab(ℵ,ℵ, · · · ,ℵ
?)

Therefore, Ab(ℵ,ℵ, · · · ,ℵ?) = 0 implies ℵ = ℵ?. Similarly, we can shows that Υ = Υ?. Thus,(ℵ, Υ)
is a unique common coupled point of T and f . Finally, we will show ℵ = Υ.

η?
(

2b2Ab (ℵ,ℵ, · · · , (ℵ)n−1, Υ)
)

= η?
(

2b2Ab (T (ℵ, Υ),T (ℵ, Υ), · · · , (T (ℵ, Υ))n−1,T (Υ,ℵ))
)

≤ Γ (η? (Ab(fℵ, fℵ, · · · , f Υ)) ,θ? (Ab(f Υ, f Υ, · · · , fℵ)))

≤ η? (Ab(fℵ, fℵ, · · · , f Υ))

≤ η? (Ab(ℵ,ℵ, · · · , Υ))

By the definition of η?, we have Ab(ℵ,ℵ, · · · , Υ) ≤ 12b2Ab(ℵ,ℵ, · · · , Υ).
Therefore, Ab(ℵ,ℵ, · · · , Υ) = 0 implies ℵ = Υ. Thus,(ℵ,ℵ) is a common fixed point of T and f .

�

Theorem 3.2. Let (=,Ab) be a complete Ab-metric space. Suppose T : =2 →= and f : =→= be
a two mappings satisfying the following:

η?
(

2b2Ab(T (ı, ),T (ı, ), · · · ,T (ð,f))
)
≤ Γ (η? (Ab(f ı, f ı, · · · , fð)) ,θ? (Ab(f , f , · · · , ff))) (3.4)

for all ı, ,ð,f∈=, where η?,θ? ∈ Θ and Γ ∈ C
a) T (=2) ⊆ f (=);
b) pair (T,f ) is weakly compatible;

c) f (=) is closed in =.
Then there is a unique common coupled fixed point of T and f in =.

Proof. Let ı0, 0 ∈= and from Theorem 3.1, we construct the sequences {ℵp} ,{Υp} in = are Cauchy
sequences. Since (=,Ab) is complete, {ℵp} ,{Υp} are convergent as follows

lim
p→∞

ℵp = lim
p→∞

T (ıp, p) = lim
p→∞

f ıp+1 = ℵ lim
p→∞

Υp = lim
p→∞

T (p, ıp) = lim
p→∞

f p+1 = Υ.

Since f (=) is closed in (=,Ab), so {ℵp} ,{Υp} ⊆ f (=) are converges in the complete Ab- metric
spaces (=,Ab), therefore, there exist ℵ, Υ ∈ f (=) with

lim
p→∞

ℵp+1 = ℵ lim
p→∞

Υp+1 = Υ

Since f : = → = and ℵ, Υ ∈ f (=), there exist f,℘ ∈ = such that ff = ℵ and f ℘ = Υ. We claim
that T (f,℘) = ℵ and T (℘,f) = Υ. By using (3.4), we have

η?
(

2b2Ab (T (ıp, p),T (ıp, p), · · · , (T (ıp, p))n−1,T (f,℘))
)

≤ Γ (η? (Ab(f ıp, f ıp, · · · , ff)) ,θ? (Ab(f p, f p, · · · , f ℘)))

≤ η? (Ab(f ıp, f ıp, · · · , ff))

By the definition of η?,

Ab (T (ıp, p),T (ıp, p), · · · , (T (ıp, p))n−1,T (f,℘)) ≤ 12b2Ab(f ıp, f ıp, · · · , ff)



10 Int. J. Anal. Appl. (2023), 21:55

Letting p →∞, it yields that

lim
p→∞

Ab (T (ıp, p),T (ıp, p), · · · , (T (ıp, p))n−1,T (f,℘)) ≤ lim
p→∞

1

2b2
Ab(f ıp, f ıp, · · · , ff) = 0.

It follows that Ab (ℵ,ℵ, · · · , (ℵ)n−1,T (f,℘)) = 0 implies that T (f,℘) = ℵ. Similarly, we can
prove T (℘,f) = Υ.
Hence, T (f,℘) = ℵ = ff and T (℘,f) = Υ = f ℘.

Since {T,f} is weakly compatible pair, we have T (ℵ, Υ) = fℵ and T (Υ,ℵ) = f Υ. Now we shall
prove that fℵ = ℵ and f Υ = Υ. By using (3.4), we have

η?
(

2b2Ab (fℵ, fℵ, · · · , (fℵ)n−1,ℵp)
)

= η?
(

2b2Ab (T (ℵ, Υ),T (ℵ, Υ), · · · , (T (ℵ, Υ))n−1,T (ıp, p))
)

≤ Γ (η? (Ab(fℵ, fℵ, · · · , f ıp)) ,θ? (Ab(f Υ, f Υ, · · · , f p)))

≤ η? (Ab(fℵ, fℵ, · · · , f ıp))

By the definition of η?, Ab (fℵ, fℵ, · · · , (fℵ)n−1,ℵp) ≤ 12b2Ab(fℵ, fℵ, · · · , f ıp)
Letting p → ∞, it yields that Ab (fℵ, fℵ, · · · , (fℵ)n−1,ℵ) ≤ 12b2Ab(fℵ, fℵ, · · · ,ℵ), which possible
holds only, Ab (fℵ, fℵ, · · · , (fℵ)n−1,ℵ) = 0 implies that fℵ = ℵ. Similarly, we shall show f Υ = Υ. It
follows that T (ℵ, Υ) = fℵ = ℵ
and T (Υ,ℵ) = f Υ = Υ. Therefore, the common coupled fixed point of T and f is (ℵ, Υ).
It is simple to demonstrate the connected fixed point’s uniqueness and the common fixed point’s

uniqueness of T and f , just like in the proof of Theorem 3.1. �

Corollary 3.1. Let (=,Ab) be a complete Ab-metric space. Suppose a mapping T : =2 → = be
satisfies:

η?
(

2b2Ab(T (ı, ),T (ı, ), · · · ,T (ð,f))
)
≤ Γ (η? (Ab(ı, ı, · · · ,ð)) ,θ? (Ab(, , · · · ,f)))

for all ı, ,ð,f∈=, where η?,θ? ∈ Θ and Γ ∈ C. Then T has a unique coupled fixed point in =.

Example 3.1. Let = = [0, +∞), define Ab : =n → [0, +∞)
as Ab (℘1,℘2, ........,℘n−1,℘n) =

∑n
i=1

∑
i<j |℘i −℘j|

2 for all ℘i ∈ =, i = 1, 2, · · · . Then (=,Ab) is
an complete Ab-metric space with b = 2.

Let T : =2 →= and f : =→= be given by T (℘,$) = sin
(
4℘−2$+32n−2

32n

)
and f (℘) = 2℘+4n−2

4n
and

Γ(s?,t?) = s
?

1+2s?
∀ s?,t? ∈ [0,∞). Let η?(t?) = t

?

4
and θ?(t?) = t?

2 Then obviously, T (=2) ⊆ f (=)
and the pair (T,f ) are ω-compatible and clearly for all ı, ,ð,f∈=, where η?,θ? ∈ Θ and Γ ∈ C, we
have

Γ (η? (Ab(f ı, f ı, · · · , fð)) ,θ? (Ab(f , f , · · · , ff))) −η?
(

2b2Ab(T (ı, ),T (ı, ), · · · ,T (ð,f))
)

=
η? (Ab(f ı, f ı, · · · , fð))

1 + 2η? (Ab(f ı, f ı, · · · , fð))
− 2b2

Ab(T (ı, ),T (ı, ), · · · ,T (ð,f))
4

=
1
4
Ab(f ı, f ı, · · · , fð)

1 + 1
2
Ab(f ı, f ı, · · · , fð)

−
b2

2
(n− 1)|sin(

4ı − 2 + 32n− 2
32n

) − sin(
4ð− 2f + 32n− 2

32n
)|2



Int. J. Anal. Appl. (2023), 21:55 11

=
1
4
|2ı−2ð
4n
|2

1 + 1
2
|2ı−2ð
4n
|2
− 2b2(n− 1)|cos(

4ı − 2 + 64n− 4 + 4ð− 2f
64n

) sin(
4ı − 2 − 4ð + 2f

64n
)|2

≥
|2ı−2ð
4n
|2

2
(

2 + |2ı−2ð
4n
|2
) ≥ 0.

Then we have

η?
(

2b2Ab(T (ı, ),T (ı, ), · · · ,T (ð,f))
)
≤ Γ (η? (Ab(f ı, f ı, · · · , fð)) ,θ? (Ab(f , f , · · · , ff))). Thus,

all assumptions of Theorem 3.1 are satisfied and (1, 1) is unique common coupled fixed point of T

and f .

4. APPLICATION TO INTEGRAL EQUATIONS

In this part, we examine the existence of a singular initial value solution with reference to Corollary

3.1

Theorem 4.1. Consider the initial value problem

℘1(t) = T (t, (℘,$)(t)), t ∈ I = [0, 1], (℘,$)(0) = (℘0,$0) (4.1)

where T : I × [℘0
4
,∞) → [℘0

4
,∞) and ℘0 ∈R Then there exists unique solution in C

(
I, [

℘0
4
,∞)

)
for

the initial value problem (4.1).

Proof. The integral equation for the initial value problem (4.1) is

℘(t) = ℘0 + 2b
2
√
n− 1

t∫
0

T (s, (℘,$)(s))ds.

Let = = C
(
I, [

℘0
4
,∞)

)
and Ab (℘1,℘2, ........,℘n−1,℘n) =

∑n
i=1

∑
i<j |℘i −℘j|

2 for all ℘i ∈=,
i = 1, 2, · · · , define η?,θ? : [0,∞) → [0,∞) as η?(t) = t, θ?(t) = 3t5 and Γ : [0,∞) → [0,∞) as
Γ(s?,t?) = ms? where m ∈ (0, 1) R : =2 →= by

R(℘,$)(t) =
℘0

2b2
√

(n− 1)
+

t∫
0

T (s, (℘,$)(s))ds.

Now for all ℘,$ ∈=, we have

η?
(

2b2Ab (R(℘,$)(t),R(℘,$)(t), · · · ,R(ð,f)(t))
)

= 2b2(n− 1)|R(℘,$)(t) −R(ð,f)(t)|2

= 2b2(n− 1)|
℘0

2b2
√

(n− 1)
+

t∫
0

T (s, (℘,$)(s))ds −
ð0

2b2
√

(n− 1)
+

t∫
0

T (s, (ð,f)(s))ds|2

=
1

2b2
|℘(t) −ð(t)|2 ≤

1

2
(n− 1)|℘(t) −ð(t)|2 ≤

1

2
Ab(℘,℘, · · · ,ð)

≤ Γ (η? (Ab(ı, ı, · · · ,ð)) ,θ? (Ab(, , · · · ,f)))

It follows from Corollary 3.1, we conclude that R has a unique solution in =. �



12 Int. J. Anal. Appl. (2023), 21:55

5. Application to Homotopy

In this part, we examine the possibility that homotopy theory has a single solution.

Theorem 5.1. Let (=,Ab) be complete Ab-metric space, V and V be an open and closed subset of
= such that V ⊆ V . Suppose Hb : V

2 × [0, 1] → = be an operator with following conditions are
satisfying,

τ0) ℘ 6= Hb(℘,$,s), $ 6= Hb($,℘,s), for each ℘,$ ∈ ΥV and s ∈ [0, 1] (Here ΥV is boundary of
V in =);
τ1) for all ℘,$,ı,  ∈ V , s ∈ [0, 1] and η?,θ? ∈ Θ, Γ ∈ C such that

η?
(

2b2Ab(Hb(℘,$,s),Hb(℘,$,s), · · · ,Hb(ı, ,s))
)
≤ Γ (η? (Ab(℘,℘, · · · , ı)) ,θ? (Ab($,$, · · · , ))) .

τ2) ∃ M ≥ 0 3 Ab(Hb(℘,$,s),Hb(℘,$,s), · · · ,Hb(℘,$,t)) ≤ M|s − t|
for every ℘,$ ∈ V and s,t ∈ [0, 1].
Then Hb(., 0) has a coupled fixed point ⇐⇒ Hb(., 1) has a coupled fixed point.

Proof. Let the set

= =
{
s ∈ [0, 1] : Hb(℘,$,s) = ℘,Hb($,℘,s) = $ for some ℘,$ ∈ V

}
.

We have that (0, 0) in =2 if Hb(., 0) has a coupled fixed point in V 2. Therefore, the set = is not
empty. We now demonstrate that = is both closed and open in [0, 1] and that = = [0, 1] is the result
of this connectivity.

As a result, V 2 has a coupled fixed point for Hb(., 1). We begin by demonstrating that = closed in
[0, 1]. Observing this, Let {sp}∞p=1 ⊆ = with sp → κ ∈ [0, 1] as p → ∞. We must show that s ∈ =.
Since sp ∈ = for p = 0, 1, 2, 3, · · · , there exists sequences {℘p} ,{$p} with ℘p = Hb(℘p,$p,sp),
$p = Hb($p,℘p,sp).
Consider

Ab(℘p,℘p, · · · , (℘p)n−1,℘p+1)

= Ab (Hb(℘p,$p,sp),Hb(℘p,$p,sp), · · · , (Hb(℘p,$p,sp))n−1,Hb(℘p+1,$p+1,sp+1))

≤
b(n− 1)Ab

(
Hb(℘p,$p,sp),Hb(℘p,$p,sp), · · · ,Hb(℘p+1,$p+1,sp)

)
+b2Ab

(
Hb(℘p+1,$p+1,sp),Hb(℘p+1,$p+1,sp), · · · ,Hb(℘p+1,$p+1,sp+1)

)
≤

b(n− 1)Ab
(
Hb(℘p,$p,sp),Hb(℘p,$p,sp), · · · ,Hb(℘p+1,$p+1,sp)

)
+b2M|sp − sp+1|

Letting p →∞, we get

lim
p→∞

Ab(℘p,℘p, · · · , (℘p)n−1,℘p+1)

≤ lim
p→∞

b(n− 1)Ab (Hb(℘p,$p,sp),Hb(℘p,$p,sp), · · · ,Hb(℘p+1,$p+1,sp)) + 0



Int. J. Anal. Appl. (2023), 21:55 13

Since η?,θ? are continuous and non-decreasing, we obtain

lim
p→∞

η?

(
2b

n− 1
Ab(℘p,℘p, · · · , (℘p)n−1,℘p+1)

)
≤ lim

p→∞
η?
(

2b2Ab (Hb(℘p,$p,sp),Hb(℘p,$p,sp), · · · ,Hb(℘p+1,$p+1,sp))
)

≤ lim
p→∞

Γ (η? (Ab(℘p,℘p, · · · ,℘p+1)) ,θ? (Ab($p,$p, · · · ,$p+1)))

≤ lim
p→∞

η? (Ab(℘p,℘p, · · · ,℘p+1)) .

By the definition of η?, it follows that

lim
p→∞

(
2b

n− 1
− 1)Ab(℘p,℘p, · · · , (℘p)n−1,℘p+1) ≤ 0

So that

lim
p→∞

Ab(℘p,℘p, · · · , (℘p)n−1,℘p+1) = 0

Now for q > p, by use of (Ab3), we have

Ab (℘p,℘p, · · · ,℘n−1,℘q) ≤ b(n− 1)Ab (℘p,℘p, ........, (℘p)n−1,℘p+1)

+b2Ab (℘p+1,℘p+1, ........, (℘p+1)n−1,℘q)

≤ b(n− 1)Ab (℘p,℘p, ........, (℘p)n−1,℘p+1)

+b3(n− 1)Ab (℘p+1,℘p+1, ........, (℘p+1)n−1,℘p+2)

+b4Ab (℘p+2,℘p+2, ........, (℘p+2)n−1,℘q)

≤ b(n− 1)Ab (℘p,℘p, ........, (℘p)n−1,℘p+1)

+b3(n− 1)Ab (℘p+1,℘p+1, ........, (℘p+1)n−1,℘p+2)

+b5(n− 1)Ab (℘p+2,℘p+2, ........, (℘p+2)n−1,℘p+3)

+b7(n− 1)Ab (℘p+3,℘p+3, ........, (℘p+3)n−1,℘p+4)

+ . . . + b2q−2p−2(n− 1)Ab (℘q−2,℘q−2, ........, (℘q−2)n−1,℘q−1)

+b2q−2p−3Ab (℘q−1,℘q−1, ........, (℘q−1)n−1,℘q) → 0 as p,q →∞.

Hence {℘p} is a Cauchy sequence in Ab metric spaces (=,Ab).Similarly, we may demonstrate that
the Cauchy sequence in (=,Ab) is {$p} and by the fact that (=,Ab) is complete, there exist u,v ∈=
with

lim
p→∞

℘p+1 = u lim
p→∞

℘p lim
p→∞

$p+1 = v = lim
p→∞

$p

We have

η?
(

2b2Ab (u,u, · · · ,Hb(u,v,s))
)

= lim
p→∞

η?
(

2b2Ab (Hb(℘p,$p,s),Hb(℘p,$p,s), · · · ,Hb(u,v,s))
)

≤ lim
n→∞

Γ (η? (Ab(℘p,℘p, · · · ,u)) ,θ? (Ab($p,$p, · · · ,v)))

≤ lim
n→∞

η? (Ab(℘p,℘p, · · · ,u)) = 0



14 Int. J. Anal. Appl. (2023), 21:55

It follows thatHb(u,v,s) = u. Similarly, we can proveHb(v,u,s) = v. Thus s ∈=. Hence= is closed
in [0, 1]. Let s0 ∈ =, then there exist ℘0,$0 ∈ V with ℘0 = Hb(℘0,$0,s0), $0 = Hb($0,℘0,s0).
Since V is open, then there exist r > 0 such that BAb (℘0, r) ⊆ V . Choose s ∈ (s0−�,s0+�) such that
|s − s0| ≤ 1Mp <

�
2
, then for ℘ ∈ BAb (℘0, r) = {℘ ∈=/Ab(℘,℘, · · · ,℘0) ≤ r + Ab(℘0,℘0, · · · ,℘0)}.

Now we have

Ab (Hb(℘,$,s),Hb(℘,$,s), · · · ,℘0)

= Ab (Hb(℘,$,s),Hb(℘,$,s), · · · ,Hb(℘0,$0,s0))

≤ (n− 1)bAb (Hb(℘,$,s),Hb(℘,$,s), · · · ,Hb(℘,$,s0))

+b2Ab (Hb(℘,$,s0),Hb(℘,$,s0), · · · ,Hb(℘0,$0,s0))

≤ b(n− 1)M|s − s0| + b2Ab (Hb(℘,$,s0),Hb(℘,$,s0), · · · ,Hb(℘0,$0,s0))

≤ b(n− 1)
1

Mp−1
+ b2Ab (Hb(℘,$,s0),Hb(℘,$,s0), · · · ,Hb(℘0,$0,s0))

Letting p →∞, we obtain

Ab (Hb(℘,$,s),Hb(℘,$,s), · · · ,℘0) ≤ b2Ab (Hb(℘,$,s0),Hb(℘,$,s0), · · · ,Hb(℘0,$0,s0))

Since η?,θ? are continuous and non-decreasing, we obtain

η? (Ab (Hb(℘,$,s),Hb(℘,$,s), · · · ,℘0))

≤ η?
(

2b2Ab (Hb(℘,$,s0),Hb(℘,$,s0), · · · ,Hb(℘0,$0,s0))
)

≤ Γ (η?(Ab(℘,℘, · · · ,℘0)),θ?(Ab($,$, · · · ,$0)))

≤ η?(Ab(℘,℘, · · · ,℘0))

Since η? is non-decreasing, we have

Ab (Hb(℘,$,s),Hb(℘,$,s), · · · ,℘0) ≤ Ab(℘,℘, · · · ,℘0)

≤ r + Ab(℘0,℘0, · · · ,℘0).

Similarly, we can prove,

Ab (Hb($,℘,s),Hb($,℘,s), · · · ,$0) ≤ r + Ab($0,$0, · · · ,$0).

Thus for each fixed s ∈ (s0 − �,s0 + �), Hb(.,s) : BAb (℘0, r) → BAb (℘0, r),
Hb(.,s) : BAb ($0, r) → BAb ($0, r). All of the Theorem 5.1’s requirements are then met. Accord-
ingly, we deduce that Hb(.,s) has a coupled fixed point in V

2
. But it has to be in V 2. Since (τ0) is

true. The result is that s ∈ = for any s ∈ (s0 − �,s0 + �). Because of this, (s0 − �,s0 + �) ⊆ =. In
[0, 1], = is obviously open. We follow the same approach for the opposite inference. �

Conclusion: This paper wraps up a few applications to integral equations and homotopy theory using

coupled fixed point theorems for C-class functions in the framework of Ab-metric spaces.



Int. J. Anal. Appl. (2023), 21:55 15

Conflicts of Interest: The authors declare that there are no conflicts of interest regarding the publi-

cation of this paper.

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	1. Introduction
	2. Preliminaries
	3. Main Results
	4. Application to Integral Equations
	5. Application to Homotopy
	References