EUROPEAN JOURNAL OF PURE AND APPLIED MATHEMATICS
2025, Vol. 18, Issue 2, Article Number 5958
ISSN 1307-5543 – ejpam.com
Published by New York Business Global

Fixed Points for Generalized Contractions in b-Gauge
Spaces and Applications

Khadidja Nisse1, Haitham Qawaqneh2,∗, Gawhara Al-Musannef3, Habes Alsamir4,
Said Beloul5

1 Laboratory of Operators Theory and PDEs: Foundations and Applications,
Department of Mathematics, Faculty of Exact Sciences, University of El Oued,
P.O.Box 789, El Oued 39000, Algeria
2 Al-Zaytoonah University of Jordan, Amman 11733, Jordan
3 Faculty of Business Studies, Arab Open University, Jeddah, Saudi Arabia
4 Finance and Banking Department, Business Administration College, Dar Aluloom
University, Riyadh, Saudi Arabia
5 Laboratory of Operators Theory and PDE LABTHOP, Department of Mathematics,
Faculty of Exact Sciences, University of El Oued, P.O.Box 789, El Oued 39000, Algeria.

Abstract. In this work, we extend and generalize, the concept of α-Ψ contraction mappings in
the setting of b-gauge spaces, where a new aspect of extension has been added . Subsequently, we
give some related fixed point results that generalize many existing ones in the literature on this
topic. Some of their applications to nonlinear integral equations on unbounded domains, including
fractional differential equations with maxima, are also presented.

2020 Mathematics Subject Classifications: 47H10, 54H25

Key Words and Phrases: α-Ψ contraction, fixed point theorem, b-gauge spaces

1. Introduction

Banach’s contraction principle, is one of the most important and significant results
in the fixed point theory. Due to its effective applications in various areas of pure and
applied mathematics, it has attracted a wide research interest in this theory. Indeed, the
related existing literature is fulled with different results extending Banach’s principle in
two main directions: in the sense of the contraction mappings or (and) in the frame of
generalized spaces. The metric space has been generalized in many different directions.
One of the most main generalizations directly related to this work, is the gauge space

∗Corresponding author.
DOI: https://doi.org/10.29020/nybg.ejpam.v18i2.5958

Email addresses: nisse-khadidja@univ-eloued.dz (K. Nisse),
h.alqawaqneh@zuj.edu.jo (H. Qawaqneh), G.almusannef@arabou.edu.sa (J.M. Al-musannef),
habes@dau.edu.sa (H. Alsamir), beloulsaid@gmail.com (S. Beloul)

https://www.ejpam.com 1 Copyright: © 2025 The Author(s). (CC BY-NC 4.0)



K. Nisse et al. / Eur. J. Pure Appl. Math, 18 (2) (2025), 5958 2 of 22

which come back to Dugundji [1]. Briefly, a gauge space is a topological space whose
topology is generated by a separating family of pseudo-metrics. It is distinguished from
the metric space by the fact that the distance between two distinct points may be zero.
For more details on the gauge spaces and related fixed point results, we refer to [1–5]. An
other generalization of the metric space which relaxes the triangular inequality’s axiom,
is known in the fixed point theory as b-metric space. To be more precise, we state the
following definition.

Definition 1.1. [6] Let X be a nonempty set and let s ≥ 1 be a given real number. A
mapping d : X ×X −→ R+ is said to be a b-metric, if for all u, v, w ∈ X, the following
conditions hold true

(b1) d(u, v) = 0 if and only if u = v;

(b2) d(u, v) = d(v, u);

(b3) d(u,w) ≤ s [d(u, v) + d(v, w)] .

In this case, (X, d) is called a b-metric space with constant s.

It should be noted that this structure is found in the literature under other names
such as quasi-metric space [7] and metric type space [8]. For more information on the
concept and origins of b-metric spaces, we reefer to the recent survey [9]. Recently, Ali et
al. [10] extended gauge spaces in the setting of b-pseudo metrics and introduced the so
called b-gauge spaces and proved some fixed point results for multi-valued mappings in
this new space. Further generalizations of the metric structure, such as generalized metric
space (known as Branciari metric space), rectangular b-metric space (known as Branciari
b-metric space) and extended b-metric space and other generalized metric spaces can be
found in [11–26].

The following generalized contraction condition called an α-ψ contraction in a met-
ric space (X, d) is introduced and fixed point results for such type of contractions are
established by Samet et al. [27]

α(x, y)d(Fx, Fy) ≤ ψ(d(x, y)), ∀x, y ∈ X,

where α and ψ are auxiliary functions satisfying some conditions. Many other results in
this direction have been obtained later in the setting of b-metric spaces and gauge spaces
with applications, see e.g. [5, 28–35] and the references therein. While so far in the ex-
isting literature, there are not enough contributions on this or even other trends in the
frame of b-gauge spaces, expect in a few papers such as [36–38].

Motivated by the last observation and inspired by [27, 33, 39], we aim through this
work to extend and generalize the concept of α-Ψ contraction mappings in the setting of
b-gauge spaces, where a new aspect of extension has been added. Subsequently, we give
some related fixed point results that generalize many existing ones in the literature on this
topic. Some of their applications to nonlinear integral equations on unbounded domains,
including fractional differential equations with maxima, are also presented.



K. Nisse et al. / Eur. J. Pure Appl. Math, 18 (2) (2025), 5958 3 of 22

2. Preliminaries

We start by recollecting some definitions from [10] to define b-gauge spaces introduced
therein.

Definition 2.1. [10]
Let E be a non-empty set and let s ≥ 1 be a given real number. A mapping d :

E × E −→ R+ is said to be a b-pseudo metric on E, if for all u, v, w ∈ E, the following
conditions hold true

1. d(u, u) = 0;

2. d(u, v) = d(v, u);

3. d(u,w) ≤ s [d(u, v) + d(v, w)].

The d-ball of radius ϵ > 0 centred at u ∈ E is the set:

B(u, d, ϵ) = {v ∈ E : d(u, v) < ϵ} .

Definition 2.2. [10] A family D = {dν}ν∈N of b-pseudo metrics on E is said to be
separating if for every two distinct points u and v, there exists dν ∈ D such that dν(u, v) ̸=
0.

Definition 2.3. [10] Let E be a nonempty set and D = {dν}ν∈N a family of b-pseudo
metrics on E. The topology generated by the family D and denoted by T (D), is the
topology whose subbase B(T ) is the family of all balls dν(u, ϵ), namely:

B(T ) = {dν(u, ϵ) : u ∈ E, ϵ > 0, ν ∈ N} .

The pair (E,B(T )) is called a b-gauge space and is Hausdorff if D is separating.

The notions of convergent sequences, Cauchy sequences and completeness in b-gauge
spaces, are similar to those in metric spaces. For more details on these notions and further
properties and examples on b-gauge spaces, we refer to [10].

In the aim of generalizing the contraction conditions, various families of auxiliary
functions are introduced in the existing literature. In this regard, we introduce now one
of such families.

For s ≥ 1, let Ψs be the family of functions ψ : R+ −→ R+ satisfying the following
conditions, where ψi denotes the ith iteration of ψ.

(Ψs
1) : ψ is non-decreasing;

(Ψs
2) : ψ(st) = sψ(t), ∀ t > 0;

(Ψs
3) :

∞∑
i=1

siψi(t) < +∞ for each t > 0;

(Ψs
4) : ψ(t1) + ψ(t2) ≤ ψ(t1 + t2), ∀ t1, t2 > 0.



K. Nisse et al. / Eur. J. Pure Appl. Math, 18 (2) (2025), 5958 4 of 22

Example 1.

(i) Let ψ : R+ −→ R+ be the function defined by: ψ(t) = c t.

Then, ψ ∈ Ψs, for all s ≥ 1 such that sc < 1.

(ii) Let ψ : R+ −→ R+ be the function defined by: ψ(t) =

{
t2

2 : 0 ≤ t < 1
t
2 : t ≥ 1

Then ψ ∈ Ψ1.

Lemma 2.4.
For every ψ ∈ Ψs, the following properties are satisfied:

(i) ψ(t) ≤ ψ(st) < t, ∀t > 0;

(ii) lim
t→0+

ψ(t) = 0.

Proof.
We begin by demonstrating the following statement:

ψ(st) < t, ∀t > 0. (2.1)

To this end, we proceed by contradiction.
Let us suppose that ψ(st0) ≥ t0 for some t0 > 0. From (Ψs

1) and (Ψs
2), we get:

s2ψ2(t0) = sψ(ψ(st0)) ≥ sψ(t0) = ψ(st0) ≥ t0.

Similarly it can be easily deduced by induction that:

∀i ≥ 1 : siψi(t0) ≥ t0.

Consequently:
lim
i→∞

siψi(t0) ≥ t0 > 0,

which is a contradiction with (Ψs
3). Hence, (2.1) is proved. The first inequality in the

statement (1) follows directly from (Ψs
1) (recall that s ≥ 1).

Note that from (1), we have:

0 ≤ lim
t→0+

ψ(t) ≤ lim
t→0+

t = 0.

Hence, (2) is proved.

Remark 2.5.
Note that if ψ is a function satisfying (Ψs

1), (Ψ
s
2) and (Ψs

4) such that ψ(st) < t, then

to conclude that ψ ∈ Ψs, it is sufficient to show that
ψ(s .)

.
is non-decreasing. Indeed, we

have:
si+1ψi+1(t)

siψi(t)
=
sψi+1(t)

ψi(t)
=
ψ
(
sψi(t)

)
ψi(t)

.



K. Nisse et al. / Eur. J. Pure Appl. Math, 18 (2) (2025), 5958 5 of 22

On the other hand, from the statement (1) in Lemma 2.4, we deduce by induction that:

∀i ≥ 1 : ψi(t) < t.

Hence, from the fact that
ψ(s .)

.
is non-decreasing, we obtain:

si+1ψi+1(t)

siψi(t)
≤ ψ(st)

t
<
t

t
= 1,

which is a sufficient condition leading to (Ψs
3).

For a mapping α : E × E −→ R+ and a non-decreasing function ψ : R+ −→ R+,

such that
∞∑
i=1

ψi(t) < +∞ for all t > 0, the concepts of α-admissible mappings and α-Ψ

contraction mappings in a metric space (E, d), were introduced for the first time by Samet
et al. [27].

Definition 2.6.
A map F : E −→ E is said to be

• α-admissible, if for all x, y ∈ E : α(x, y) ≥ 1 implies α(Fx, Fy) ≥ 1

• α-Ψ contraction mapping, if

α(x, y)d(Fx, Fy) ≤ ψ(d(x, y)), ∀x, y ∈ E.

Later, other contraction conditions of such type have been considered by many authors
to extend the Banach’s principle. In these results, the following condition for α-admissible
mappings F , is often imposed

∃x0 ∈ E, such that α(x0, Fx0) ≥ 1. (2.2)

In [33], the authors introduced the following relaxed condition

∃N ∈ N∗, ∃ (xp)Np=0 ⊂ E, with xN = Fx0, such that α(xp−1, xp) ≥ 1, ∀p = 1, .., N.
(2.3)

Where N∗ = N\{0}. For α-admissible mapping F , it is clear that if (2.2) is satisfied, then
(2.3) is satisfied too with N = 1. But the following simple example illustrates that the
converse is not necessarily true.

Example 2.
Let X = {0, 1, 2, 3},
F : X −→ X

0 7→ 1
1 7→ 0
2 7→ 3
3 7→ 2

α : X ×X −→ {0, 1}

and α(x, y) =

{
0 : (x, y) ∈ {(0, 1), (1, 0), (2, 3), (3, 2)}
1 : otherwise

.

It can be easily seen that F is α-admissible and (2.2) is not satisfied.
Whereas, there exist x0 = 2, x1 = 1 and x2 = Fx0 = 3, such that α(x0, x1) =

α(x1, x2) = 1 ≥ 1. That is (2.3) is satisfied with N = 2.



K. Nisse et al. / Eur. J. Pure Appl. Math, 18 (2) (2025), 5958 6 of 22

3. Main results

Throughout the sequel, E is a non-empty set endowed with a separating complete
b-gauge structure D = {dν}ν∈N , where N is an index set.

Inspired by [27, 39], we give in what follows generalized concepts of α-admissibility and
α-Ψ contractivity in the setting of b-gauge spaces. To this end, we start by introducing
the following auxiliary family and mapping.

We denote by αααν , the following family: αααν = {αν : E×E −→ R+}ν∈N .
w : N −→ N is a mapping from the index set N into itself, such that:

∀ν ∈ N , ∀u, v ∈ E : dν(u, v) ≤ dw(ν)(u, v). (3.1)

Definition 3.1.
A mapping F : E −→ E is said to be αααν-admissible, if

∀ν ∈ N , ∀u, v ∈ E : αν(u, v) ≥ 1 implies αν(Fu, Fv) ≥ 1.

Definition 3.2.
Let F : E −→ E be a given mapping and {ψν}ν∈N ⊂ Ψs.
F is said to be a generalized (αααν ,Ψ

s,w) contraction if

αν(u, v) dν(Fu, Fv) ≤ ψν

(
dw(ν)(u, v)

)
, ∀u, v ∈ E, ∀ν ∈ N . (3.2)

Remark 3.3. It should be noted that many α-Ψ contractive type mappings in the literature
are generalized by that given in (3.2) in two distinct aspects. The introduction of a family
of mappings αααν = {αν}ν∈N instead of only one mapping α is the clear first aspect of
generalization. While the introduction of the mapping w is the second one. Indeed, since
some α-Ψ contraction conditions introduced in similar studies in this direction correspond
to w = IN [4, 10, 33, 36], then in view of (3.1), our contraction condition (3.2) is weaker
than those mentioned above.

Example 3. Let X be the space of all real sequences:

X = {u = (u1, u2, ..., un, ...) : un ∈ R, n ∈ N∗} .

For each n ∈ N∗, let πn : X −→ R be the mapping defined by πn(u) = un.
Let {dn}n∈N∗ be the family of b-pseudo-metrics with constant s = 2 defined on X by

dn(u, v) = |πn(u)− πn(v)|2

Let w : N∗ −→ N∗ be the mapping defined by w(n) = n+ 1.
Consider the map F : X −→ X defined as follows:

Fu =


((1− 1

2)(2− u2), (1− 2
3)(2− u3), ..., (1− n

n+1)(2− un+1), ...), ∃n ∈ N∗ : un ≤ 2;

(2u2 − 2, 2u3 − 2, ..., 2un+1 − 2, ...), otherwise.



K. Nisse et al. / Eur. J. Pure Appl. Math, 18 (2) (2025), 5958 7 of 22

Let αααn = {α}, where α : X ×X −→ R+ is the function given by:

α(u, v) =

{
1 : un, vn ≤ 2 for some n ∈ N∗

0 : otherwise

Now, let Ψ2 be the family of the functions ψn defined for each n ∈ N∗ by:

ψn(t) =
1

(n+ 1)2
t

Let u, v ∈ X, we distinguish two cases:
Case 1: There exists n ∈ N∗ such that un, vn ≤ 2. Then:

α(u, v)dn(Fu, Fv) = dn(Fu, Fv) =
∣∣∣(1− n

n+1)(2− un+1)− (1− n
n+1)(2− vn+1)

∣∣∣2
= (1− n

n+1)
2 |un+1 − vn+1|2 = 1

(n+1)2
|un+1 − vn+1|2

= ψn (dn+1(u, v)) = ψn

(
dw(n)(u, v)

)
Case 2: For every n ∈ N∗ : un > 2 or vn > 2. Since α(u, v) = 0, clearly we have:

α(u, v)dn(Fu, Fv) = 0 ≤ ψn

(
dw(n)(u, v)

)
Consequently, F is a generalized (αααn,Ψ

2,w) contraction.

We state now our first main result.

Theorem 1.
Let F : E −→ E be a a generalized (αααν ,Ψ

s,w) contraction. Suppose that the following
conditions hold:

(C1) F is αααν-admissible.

(C2) ∃x0 ∈ E, N ∈ N∗ and (ap0)
N
p=0 ⊂ E, with a00 = x0 and aN0 = Fx0, such that:

(i) αν(a
p−1
0 , ap0) ≥ 1, ∀p = 1, .., N, ∀ν ∈ N ;

(ii)

N∑
p=1

spdwi(ν)(a
p−1
0 , ap0) ≤Ms,ν(x

0) < +∞, ∀i ∈ N, ∀ν ∈ N .

(C3) ∀ν ∈ N , ∃ψ̃ν ∈ Ψs : ψwi(ν) ≤ ψ̃ν , ∀i ∈ N

(C4) (i) F is continuous or



K. Nisse et al. / Eur. J. Pure Appl. Math, 18 (2) (2025), 5958 8 of 22

(ii) For every sequence
{
uk
}
k∈N of E, such that for all k ∈ N and the same positive

integer N given in (C2):

∃ (apk)
N
p=0 ⊂ E, s.t. a0k = uk, aNk = uk+1 and αν(a

p−1
k , apk) ≥ 1, ∀p = 1, N,∀ν ∈ N ,

(3.3)
if uk −−−→

k→∞
u, then there exists a sub-sequence

{
ukl
}
l∈N of

{
uk
}
k∈N and l0 ∈ N

such that αν(u
kl , u) ≥ 1 for all l ≥ l0.

Then, F has a fixed point.

Proof.
Note first that according to (C2(i)) and (C1), we deduce by induction that

∀p = 1, ..., N, ∀k ∈ N, ∀ν ∈ N : αν(F
kap−1

0 , F kap0) ≥ 1.

Consequently, using (3.2), the following inequalities hold true:

dν

(
F kap−1

0 , F kap0

)
≤ αν(F

k−1ap−1
0 , F k−1ap0)dν

(
F kap−1

0 , F kap0

)
≤ ψν

(
dw(ν)

(
F k−1ap−1

0 , F k−1ap0

))
,

for all ν ∈ N , k ∈ N and p = 1, ..., N .
Now, since ψν is non-decreasing for each ν ∈ N , repeated application of the previous

inequalities yield:

dν(F
kap−1

0 , F kap0) ≤ ψν

(
ψw(ν)

(
...ψwk−1(ν)

(
dwk(ν)(a

p−1
0 , ap0)

)
...
))

,

for all k ∈ N, ν ∈ N and every p = 1, ..., N .
Hence, by means of (C3), we obtain:

dν(F
kap−1

0 , F kap0) ≤ ψ̃ν
k
(
dwk(ν)(a

p−1
0 , ap0)

)
. (3.4)

Let now x0 be the element introduced in (C2(i)) and let
{
xk
}
k∈N be the sequence in

E defined by xk+1 = Fxk.
Assume that xk+1 ̸= xk for all k ∈ N, since otherwise the result is clear. Recall that

a00 = x0 and aN0 = Fx0, then for all k ∈ N, we have:

dν(F
kx0, F k+1x0) ≤ sdν(F

ka00, F
ka10) + s2dν(F

ka10, F
ka20) + ...+ sNdν(F

kaN−1
0 , F kaN0 ).



K. Nisse et al. / Eur. J. Pure Appl. Math, 18 (2) (2025), 5958 9 of 22

Hence, using (3.4) together with (Ψs
1), (Ψ

s
2) and (Ψs

4), we obtain:

dν(F
kx0, F k+1x0) ≤ sψ̃ν

k
(
dwk(ν)(a

0
0, a

1
0)
)
+ s2 ψ̃ν

k
(
dwk(ν)(a

1
0, a

2
0)
)
+ ...+ sN ψ̃ν

k
(
dwk(ν)(a

N−1
0 , aN0 )

)
= ψ̃ν

k
(
sdwk(ν)(a

0
0, a

1
0)
)
+ ψ̃ν

k
(
s2dwk(ν)(a

1
0, a

2
0)
)
+ ...+ ψ̃ν

k
(
sNdwk(ν)(a

N−1
0 , aN0 )

)

= ψ̃ν

(
ψ̃ν

k−1
(
sdwk(ν)(a

0
0, a

1
0)
))

+ ψ̃ν

(
ψ̃ν

k−1
(
s2dwk(ν)(a

1
0, a

2
0)
))

+ ...

+ ψ̃ν

(
ψ̃ν

k−1
(
sNdwk(ν)(a

N−1
0 , aN0 )

))

≤ ψ̃ν

(
ψ̃ν

k−1
(
s dwk(ν)(a

0
0, a

1
0)
)
+ ψ̃ν

k−1
(
s2 dwk(ν)(a

1
0, a

2
0)
)
+ ...

+ψ̃ν
k−1

(
sN dwk(ν)(a

N−1
0 , aN0 )

))

= ψ̃ν

(
s ψ̃ν

k−1
(
dwk(ν)(a

0
0, a

1
0)
)
+ s2 ψ̃ν

k−1
(
dwk(ν)(a

1
0, a

2
0)
)
+ ...

+sN ψ̃ν
k−1

(
dwk(ν)(a

N−1
0 , aN0 )

))
.

Since ψν is non-decreasing, repeated application of the above inequalities yields:

dν(F
kx0, F k+1x0) ≤ ψ̃ν

k

 N∑
p=1

sp dwk(ν)(a
p−1
0 , ap0)

 .

Consequently, it follows from (C2(ii)):

dν(F
kx0, F k+1x0) ≤ ψ̃ν

k (
Ms,ν

(
x0
))
. (3.5)

We are now ready to prove that
{
xk
}
k∈N is a Cauchy sequence. Indeed, let k ∈ N and

m ∈ N∗. We have:

dν(F
kx0, F k+mx0) ≤ sdν

(
F kx0, F k+1x0

)
+ s2 dν

(
F k+1x0, F k+2x0

)
+ ...

+ sm−1 dν

(
F k+m−2x0, F k+m−1x0

)
+ sm dν

(
F k+m−1x0, F k+mx0

)
.



K. Nisse et al. / Eur. J. Pure Appl. Math, 18 (2) (2025), 5958 10 of 22

It follows so from (3.5), that

dν(F
kx0, F k+mx0) ≤ sψ̃ν

k (
Ms,ν(x

0)
)
+ s2ψ̃ν

k+1 (
Ms,ν(x

0)
)
+ ...+ smψ̃ν

k+m−1 (
Ms,ν(x

0)
)

=
1

sk−1

[
skψ̃ν

i (
Ms,ν(x

0)
)
+ sk+1ψ̃ν

i (
Ms,ν(x

0)
)
+ ...+ sk+m−1ψ̃ν

i (
Ms,ν(x

0)
)]

≤ 1

sk−1

∞∑
i=k

si ψ̃ν
i (
Ms,ν(x

0)
)
.

(3.6)
Since in view of (Ψs

3) together with the second statement of Lemma 2.4, we have:

lim
k→∞

∞∑
i=k

si ψ̃ν
i (
Ms,ν(x

0)
)
= 0,

then we deduce from (3.6) that
{
F kx0 = xk

}
k∈N is a Cauchy sequence in the complete

b-gauge space E and so convergent to some x∗ ∈ E. That is, for all ν ∈ N :

lim
k→∞

dν(x
k, x∗) = 0.

On the other hand, the continuity of F guaranteed by (C4(i)), implies that for all
ν ∈ N :

lim
k→∞

dν(x
k, Fx∗) = dν(Fx

k−1, Fx∗) = 0.

Thus, for all ν ∈ N :

dν(x
∗, Fx∗) ≤ s

(
dν(x

∗, xk) + dν(x
k, Fx∗)

)
−−−→
k→∞

0.

Since the b-gauge structure is separating, we conclude that x∗ = Fx∗.
Suppose now that (C4(ii)) is satisfied. Note first that (C2(i)) means that (3.3) is

satisfied for k = 0. Since F is αααν-admissible, it follows by induction that (3.3) is satisfied
for each k ≥ 1 with apk = Fapk−1, for all p = 0, ..., N . Thus, according to (C4(ii)), there

exists a sub-sequence
{
xkl
}
l∈N and some l0 such that for all l ≥ l0 and ν ∈ N , we have

αν(x
kl , x∗) ≥ 1.

Hence, applying (3.2), we obtain:

dν(Fx
kl , Fx∗) ≤ αν(x

kl , x∗) dν(Fx
kl , Fx∗) ≤ ψν

(
dw(ν)(x

kl , x∗)
)
.

Now, letting l −→ ∞ in the right hand side of the above inequality taking into account
the second statement of Lemma 2.4, we deduce:

lim
l→∞

dν(Fx
kl , Fx∗) = 0, ∀ν ∈ N .



K. Nisse et al. / Eur. J. Pure Appl. Math, 18 (2) (2025), 5958 11 of 22

That is
lim
l→∞

Fxkl = Fx∗.

Noting that
lim
l→∞

xkl+1 = lim
l→∞

Fxkl ,

we deduce that x∗ = Fx∗. The proof is complete.

Remark 3.4. Condition (C2(i)) is an extension of (2.3) in the setting of b-gauge spaces.
Thus, according to Example 2, this condition is weaker than the condition (2.2), frequently
imposed in the existing literature on this topic, like in [37, 38, 40].

Sufficient conditions guaranteeing the uniqueness of the fixed point is given in the
following theorem.

Theorem 2.
Let F : E −→ E be a generalized (αααν ,Ψ

s,w) contraction satisfying conditions (C1),
(C3) and (C4) in Theorem 1. Suppose that the following condition holds

˜(C2) ∀x, y ∈ E with x ̸= y, there exists N = N(x, y) ∈ N∗ and (apx,y)Np=0 ⊂ E such that:

(i) a0x,y = x, aNx,y = y, and αν(a
p−1
x,y , a

p
x,y) ≥ 1, ∀p = 1, ..., N,∀ν ∈ N ;

(ii)
N∑
p=1

spdwi(ν)(a
p−1
x,y , a

p
x,y) ≤Ms,ν(x, y) < +∞, ∀i ∈ N, ∀ν ∈ N .

Then F has a unique fixed point.

Proof.
The existence of a fixed point for F results from Theorem 1. Indeed, let x0 be an

arbitrary element in E.

• If x0 = Fx0, then x0 is a fixed point.

• If x0 ̸= Fx0, then with x = x0 and y = Fx0 condition ˜(C2) reduces to condition
(C2) in Theorem 1, from which follows that F has a fixed point.

Suppose now that x, y are two fixed points of F such that x ̸= y. By means of (C1),
˜(C2), (C3) and in a similar way as that used to get (3.5), we have also the following

inequality:

dν(F
kx, F ky) ≤ ψ̃ν

k
(Ms,ν (x, y)) ,

for all k ∈ N and all ν ∈ N . Hence

dν(x, y) = dν(F
kx, F ky) ≤ ψ̃ν

k
(Ms,ν (x, y)) , (3.7)

for all k ∈ N and all ν ∈ N . Noting that

ψ̃ν
k
(Ms,ν (x, y)) ≤ sk ψ̃ν

k
(Ms,ν (x, y)) ,



K. Nisse et al. / Eur. J. Pure Appl. Math, 18 (2) (2025), 5958 12 of 22

and by letting k −→ ∞ in (3.7) taking into account (Ψs
3), we deduce:

dν(x, y) = 0, ∀ν ∈ N ,

which is a contradiction with x ̸= y, since D is separating. The proof is complete.

Example 4. Let X = R be the complete b-gauge space with constant s = 2, endowed
with the separated family of b-pseudo-metrics D = {dn, n ≥ 1} defined by: dn(x, y) =
n(|x| − |y|)2. Let Fx = x

2 , ψn(t) =
1

n+1 t and w(n) = (n+ 1)3 for all n ≥ 1.

αn(x, y) =

{
n, x ̸= y
0, otherwise.

For x, y ∈ R with x ̸= y we have:

αn(x, y)dn(Fx, Fy) =
n2

4
(|x| − |y|)2

and
ψn(d(n+1)3(x, y)) = (n+ 1)2(|x| − |y|)2

Then, for x ̸= y and n ≥ 1 we have:

αn(x, y)dn(Fx, Fy) ≤ ψn(dw(n)(x, y)) = (n+ 1)2(|x| − |y|)2

Hence, F is a generalized (αααn,Ψ
2,w) contraction. Let us now show that F verifies the

other conditions of Theorem 2. Indeed, for (C1) we have, for all n ≥ 1

αn(x, y) ≥ 1 ⇒ αn(Fx, Fy) = n ≥ 1,

then, F is αααn-admissible.
It can be easily seen that (C3) is satisfied with ψ̃n = ψn, for all n ≥ 1.
Let x, y ∈ R such that x ̸= y. Then, there exists z ∈ R such that x ̸= z and y ̸= z. Hence,
for all n ≥ 1, we have αn(x, z) = n ≥ 1 and αn(x, z) = n ≥ 1. Then ˜(C2)-(i) holds with
N = 1 and ˜(C2)-(ii) follows immediately from the fact that ψi

n ≤ ψn for all i ∈ N.
It is not hard to see that F is continuous and so (C4) is satisfied. Thus, all conditions of
Theorem 2 are fulfilled and consequently F has a unique fixed point, which is 0.

Let us state the following conditions

(PC2) There exists x0 ∈ E such that αν(x
0, Fx0) ≥ 1, ∀ν ∈ N and furthermore:

dwi(ν)(x
0, Fx0) < +∞, ∀i ∈ N, ∀ν ∈ N ;

(P̃C2) ∀x, y ∈ E with x ̸= y, there exists z ∈ E such that αν(x, z) ≥ 1, and αν(y, z) ≥
1, ∀ν ∈ N ;

(PC4) (i) F is continuous, or



K. Nisse et al. / Eur. J. Pure Appl. Math, 18 (2) (2025), 5958 13 of 22

(ii) for every sequence
{
uk
}
k∈N of E, such that αν(u

k−1, uk) ≥ 1, ∀ν ∈ N ,

if uk −−−→
k→∞

u, then there exists a sub-sequence
{
ukl
}
l∈N of

{
uk
}
k∈N and l0 ∈ N

such that αν(u
kl , u) ≥ 1 for all l ≥ l0.

as spacial cases of (C2), (C̃2) and (C4) respectively. The following corollaries follow
immediately from Theorem 1 and Theorem 2.

Corollary 3.5.
Let F : E −→ E be a generalized (αααν ,Ψ

s,w) contraction. Suppose that in addition
of conditions (C1), (C3) and (C4) of Theorem 1, (PC2) holds true. Then, F has a fixed
point.

Corollary 3.6.
Let F : E −→ E be a generalized (αααν ,Ψ

s,w) contraction. Suppose that in addition of
conditions (C1) and (C3) of Theorem 1, conditions (P̃C2) and (PC4) hold. Then, F has a
unique fixed point.

4. Application

In this section, we focus on the existence of solutions of some nonlinear integral equa-
tions as an application to the results proved in the previous section.

Let us consider the following integral equation:

x(t) =


φ(0) +

∫ t

0
G(t, τ)f(τ, x(τ), gx(τ))dτ, t > 0

φ(t), t ≤ 0,

(4.1)

where G : R2
+ −→ R+, f : R+ × R2 −→ R, φ : ] −∞, 0] −→ R are nonlinear continuous

functions and g : C(R) −→ C(R) where C(R) denotes the set of all real continuous functions
on R and gx is a delay function.

Let E = C(R) be the complete b-gauge space with constant s = 2, endowed with the
separated family of b-pseudo-metrics {dK}K∈K defined by:

dK (x, y) = sup
t∈K

{
e−λt |x(t)− y(t)|2

}
,

where λ is a positive real number to be specified later and K is the set of all compact
sub-sets of R.
Note that, for dK defined above, conditions 1. and 2. of Definition 2.1 are clearly satisfied.
Moreover, for all x, y, z ∈ E and for every t ∈ K ∈ K, by means of Young’s inequality, we
get:

|x(t)− y(t)|2 ≤ (|x(t)− z(t)|+ |z(t)− y(t)|)2

≤ 2
(
|x(t)− z(t)|2 + |z(t)− y(t)|2

)



K. Nisse et al. / Eur. J. Pure Appl. Math, 18 (2) (2025), 5958 14 of 22

Consequently:
e−λt|x(t)− y(t)|2 ≤ 2 (dK (x, z) + dK (z, y))

Thus, taking the supremum over K on the left-hand side of the above inequality, we
obtain condition 3. of Definition 2.1.

Let w : K −→ K be the mapping defined by:

w(K) =


K, if K ⊂ R− =]−∞, 0],

[0, K∗] , otherwise,
(4.2)

where K∗ = supK.
Let us now consider the following assumptions:

(B1) f is a positive function, non-decreasing with respect to the second and third argu-
ments, and for some real valued function W defined on R+, the following inequality
holds:

|f (t, x(t), gx(t))− f (t, y(t), gy(t))| ≤
√
dw(K)(x, y) eλtW (t), (4.3)

for all x, y ∈ E, t ∈ K+ and λ ≥ 0.

(B2) There exist p, q > 1 with 1
p +

1
q = 1, µ > 1 such that for all λ ≥ 0, the following hold:

(i) Rµ(λ) :=

∫ +∞

0
e

−pλτ
µ W

p
2 (τ) dτ <∞;

(ii) ∀t > 0, Sµ,t(λ) :=

∫ t

0
Gq(t, τ)e

−λq
2

[
t−

(
µ+2
µ

)
τ
]
dτ <∞;

(iii) Rµ(λ)Sµ,t(λ) −−−→
λ→∞

0, ∀t > 0.

(B3) For every x, y ∈ E such that x(t) = y(t) for t ≤ 0, if x(t) ≤ y(t) for t > 0, then
gx(t) ≤ gy(t).

Theorem 3.
Under assumptions (B1)-(B3), the problem (4.1) has at last one global solution in E.

Proof.
Let F : E −→ E be the mapping defined by:

Fx(t) =


φ(0) +

∫ t

0
G(t, τ)f(τ, x(τ), gx(τ))dτ, t > 0

φ(t), t ≤ 0.

(4.4)



K. Nisse et al. / Eur. J. Pure Appl. Math, 18 (2) (2025), 5958 15 of 22

The solutions of (4.1) are the fixed points of F .
Let α : E×E −→ R+ be the function defined by:

α(x, y) =


1 : x(t) ≤ y(t) : ∀t > 0 and x(t) = y(t) = φ(t) : t ≤ 0

0 : otherwise.

Let us check the generalized (αααν ,Ψ
s,w) contraction condition (3.2), where {αK}K∈K = {α}

and {ψK}K∈K is the family of functions ψK defined by (4.8).
The following obvious fact is necessary for the final conclusion.

∀x, y ∈ E s.t. α(x, y) = 0, α(x, y) dK(Fx, Fy) = 0, ∀K ∈ K (4.5)

Let now x, y ∈ E such that α(x, y) = 1. For K ∈ K and t ∈ K such that t ≤ 0. We have:

|Fx(t)− Fy(t)| = |φ(t)− φ(t)| = 0.

Hence, for all t ∈ K such that t ≤ 0 we have

e−λt |Fx(t)− Fy(t)|2 = 0. (4.6)

Now, for t ∈ K such that t > 0, using (B1) we obtain:

|Fx(t)− Fy(t)| ≤
∫ t

0
G(t, τ) |f (τ, x(τ), gx(τ))− f (τ, y(τ), gy(τ))| dτ

≤
√
dw(K)(x, y)

∫ t

0
G(t, τ)

√
eλτW (τ) dτ.

.

Now, multiplying the above inequality by e−
λt
2 , we get:

e−
λt
2 |Fx(t)− Fy(t)| ≤

√
dw(K)(x, y)

[∫ t

0
e−

λt
2 G(t, τ) e

λτ
2

√
W (τ) dτ

]
=

√
dw(K)(x, y)

[∫ t

0
e−

λt
2 G(t, τ) e

λ(µ+2)τ
2µ e

−λτ
µ

√
W (τ) dτ,

]2
,

where

µ is the constant introduced in (B2).
In view of (B2(i).(ii)), Hölder’s inequality gives:

e−
λt
2 |Fx(t)− Fy(t)| ≤

√
dw(K)(x, y)

(∫ t

0
e

−pλτ
µ W

p
2 (τ) dτ

) 1
p

×
(∫ t

0
Gq(t, τ) e

−λq
2

[
t−

(
µ+2
µ

)
τ
]
dτ

) 1
q

=
√
dw(K)(x, y)R

1
p
µ (λ)S

1
q

µ,t(λ).

In conclusion, for all t ∈ K such that t > 0 we have:

e−λt |Fx(t)− Fy(t)|2 ≤ dw(K)(x, y) R
2
p
µ (λ) S

2
q

µ,t(λ). (4.7)



K. Nisse et al. / Eur. J. Pure Appl. Math, 18 (2) (2025), 5958 16 of 22

Let us now define the function ψK : R+ −→ R+ as follows:

ψK(t) =


R

2
p
µ (λ) S

2
q

µ,K∗(λ) t, K∗ > 0

0, K∗ ≤ 0,

(4.8)

where, thanks to (B2(iii)) λ is fixed such that

2R
2
p
µ (λ) S

2
q

µ,K∗(λ) < 1. (4.9)

It is clear that ψK satisfies (Ψs
1), (Ψ

s
2) and (Ψs

4). Furthermore,
ψK(s .)

.
is constant,

thus non-decreasing and in view of (4.9), it satisfies also ψK(2t) < t. Consequently,
according to Remark 2.5, ψK ∈ Ψs.

Combining (4.6) and (4.7) taking into account (4.8), leads to

∀x, y ∈ E s.t. α(x, y) = 1, α(x, y) dK(Fx, Fy) ≤ ψK

(
dw(K)(x, y)

)
, ∀K ∈ K. (4.10)

Now, (3.2) follows immediately from (4.5) and (4.10). In other means, F is a general-
ized (αααν ,Ψ

s,w) contraction.
Condition (I): Let (x, y) ∈ E × E such that α(x, y) ≥ 1. Then for t > 0, we have

x(t) ≤ y(t), which implies according to (B3) that gx(t) ≤ gy(t). The following inequality
follows so for t > 0, from the fact that f is non-decreasing with respect to the second and
third arguments∫ t

0
G(t, τ)f(τ, x(τ), gx(τ))dτ ≤

∫ t

0
G(t, τ)f(τ, y(τ), gy(τ))dτ,

which clearly leads to Fx(t) ≤ Fy(t) for t > 0. On the other hand, from (4.4), we have
Fx(t) = Fy(t) = φ(t) for t ≤ 0. That is α(Fx, Fy) ≥ 1, and consequently (C1) is satisfied.

Condition (II): Let x0 ∈ E be the function defined by:

x0(t) =


φ(0), if t > 0

φ(t), if t ≤ 0.

Since f is positive, then: ∫ t

0
G(t, τ)f(τ, x0(τ), gx0(τ))dτ ≥ 0.

Hence, for t > 0, we have:

x0(t) = φ(0) ≤ φ(0) +

∫ t

0
G(t, τ)f(τ, x0(τ), gx0(τ))dτ = Fx0(t),



K. Nisse et al. / Eur. J. Pure Appl. Math, 18 (2) (2025), 5958 17 of 22

and for t ≤ 0, Fx0(t) = φ(t) = x0(t). That is α(x0, Fx0) ≥ 1.
Furthermore, we have:

dwi(K)(x
0, Fx0) = dw(K)(x

0, Fx0) = sup
t∈[0,K∗]

e−λt
∣∣x0(t)− Fx0(t)

∣∣2 <∞,

for all i ∈ N. Consequently (PC2) is satisfied.
Note also that:

∀K ∈ K, ∀t > 0 : ψK(t) = ψwi(K)(t),

for all i ∈ N∗, and so (C3) is satisfied with ψ̃K = ψK .
Let {xn}n∈N be a sequence of E such that:

α(xn, xn+1) ≥ 1, ∀n ∈ N.

That is:

xn(t) ≤ xn+1(t), for all t > 0 and xn(t) = xn+1(t) = φ(t) for all t ≤ 0. (4.11)

Suppose now that {xn}n∈N converges to some x ∈ E, that is:

∀K ∈ K, sup
t∈K

{
e−λt |xn(t)− x(t)|2 −−−→

n→∞
0,
}
,

which implies that

∀t ∈ R, {xn(t)}n∈N converges to x(t) in R.

Hence, according to (4.11), {xn(t)}n∈N is a non-decreasing real sequence for t > 0 and
therefore for all n ∈ N:

xn(t) ≤ x(t), ∀t > 0 and xn(t) = x(t) = φ(t), ∀t ≤ 0.

This means that α(xn, x) ≥ 1 for all n ∈ N and consequently (PC4) is satisfied.
Then, all conditions of Corollary 3.6 are fulfilled and the proof is complete.

The following Corollary illustrates the efficiency of Theorem 3 in the study of some
fractional differential equations with ”maxima”, namely:CDδx(t) = f

(
t, x(t), max

σ∈[a(t), b(t)]
x (σ)

)
, t > 0

x(t) = φ(t), t ≤ 0,

(4.12)

where CDδ denotes the Caputo fractional derivative operator of order δ ∈ ]0, 1[, a, b, are
real continuous functions defined on R+ such that a(t) ≤ b(t) ≤ t , f : R+ × R2 −→ R is
a nonlinear continuous function and φ : ]−∞, 0] −→ R is a continuous function.

Corollary 4.1.
Assume that the following conditions hold:



K. Nisse et al. / Eur. J. Pure Appl. Math, 18 (2) (2025), 5958 18 of 22

(H1) f is a positive function and non-decreasing with respect to the second and third
arguments, such that

(i) |f (t, ξ1, η1)− f (t, ξ2, η2)| ≤
√

Υ
(
t, |ξ1 − ξ2|2 , |η1 − η2|2

)
, whenever the left hand

side is defined;

(ii) Υ : R3
+ −→ R+ is a non-decreasing function with respect to the second and

third arguments;

(iii) there exists a real valued function W defined on R+, such that:

∀z ≥ 0 : Υ(., z, z) ≤ zW (.).

(H2) There exists µ > 1 such that:

(i) Rµ(λ) :=

∫ +∞

0
e
− (1+δ)λτ

δµ W
1+δ
2δ (τ) dτ <∞, for all λ > 0.

(ii) Rµ(λ) −−−→
λ→∞

0, ∀t > 0.

Then (4.12) has at least one global solution in E.

Proof.
Using the properties of fractional integral and derivative operators, problem (4.12) is

transformed into the following integral equation, see, e.g. [28, 31, 34, 41–43].

x(t) =


φ(0) +

∫ t

0

(t− τ)δ−1

Γ(δ)
f

(
τ, x(τ), max

σ∈[a(τ), b(τ)]
x (σ)

)
dτ, t > 0

φ(t), t ≤ 0,

(4.13)

which is identified to (4.1), with

G(t, τ) =
(t− τ)δ−1

Γ(δ)
and gx(t) = max

σ∈[a(t), b(t)]
x (σ) .

Therefore, it is sufficient to show that conditions (B1)-(B3) are fulfilled, to deduce then
the result from Theorem 3.

Let x, y ∈ E, K ∈ K and t ∈ K such that t > 0. Using (H1(i), (ii)) we obtain:∣∣∣∣f(t, x(t), max
σ∈[a(t), b(t)]

x (σ))− f(t, y(t), max
σ∈[a(t), b(t)]

y (σ))

∣∣∣∣ ≤
√√√√Υ

(
t, |x(t)− y(t)|2 ,

∣∣∣∣ max
σ∈[a(t), b(t)]

x (σ)− max
σ∈[a(t), b(t)]

y (σ)

∣∣∣∣2
)

≤



K. Nisse et al. / Eur. J. Pure Appl. Math, 18 (2) (2025), 5958 19 of 22√
Υ

(
t, |x(t)− y(t)|2 , max

σ∈[a(t), b(t)]
|x (σ)− y (σ)|2

)
≤
√

Υ
(
t, eλtdw(K)(x, y), eλtdw(K)(x, y)

)
,

which yields to (4.3) thanks to (H1(iii)). Consequently, (B1) is fulfilled. (H2(i)) implies
(B2(i)) with p = 1 + 1

δ . Let us now check (B2(ii)) where q = 1 + δ.
We have:

Sµ,t(λ) = 1
Γq(δ)

∫ t

0
(t− τ)q(δ−1)e

−λq
2

[
t−

(
µ+2
µ

)
τ
]
dτ

≤ 1
Γq(δ)

∫ t

0
(t− τ)q(δ−1)e

−λq
2

(
µ+2
µ

)
(t−τ)

dτ.

Performing the change of variable X = λq
2

(
µ+2
µ

)
(t− τ), we get:

Sµ,t(λ) ≤ 1
Γq(δ)

∫ ∞

0

(
2µ

λq(µ+ 2)

)q(δ−1)

Xq(δ−1)e−X dX

= 1
Γ1+δ(δ)

(
2µ

λq(µ+2)

)δ2
Γ(δ2).

Consequently, (B2(ii)) is satisfied and furthermore Sµ,t(λ) −−−→
λ→∞

0, ∀t > 0.

The last fact, combined with (H2(ii)) implies (B2(iii)).
Let x, y ∈ E, such that x(t) = y(t) for t ≤ 0. If x(t) ≤ y(t) for t > 0, then we have:{

x(σ) ≤ y(σ), σ ∈ [a(t), b(t)]+

x(σ) = y(σ), σ ∈ [a(t), b(t)]−,

where [a(t), b(t)]+ = [a(t), b(t)] ∩ R+ and [a(t), b(t)]− = [a(t), b(t)] ∩ R−.
Then 

sup
σ∈[a(t), b(t)]+

x (σ) ≤ sup
σ∈[a(t), b(t)]+

y (σ)

max
σ∈[a(t), b(t)]−

x (σ) = max
σ∈[a(t), b(t)]−

y (σ) .

Consequently max
σ∈[a(t), b(t)]

x (σ) ≤ max
σ∈[a(t), b(t)]

y (σ), that is (B3) is satisfied.

Then all conditions of Theorem 3 are fulfilled and the proof is complete.

5. Conclusion

In this work, we have introduced a new concept in b-gauge metric spaces called gener-
alized (αααν ,Ψ

s,w) contraction, which extended α-ψ contraction in ordinary metric spaces.
Some related fixed point results were given using such concept, where weaker conditions
have been applied in comparison with existing results. Moreover, applications to delay
integral equations on unbounded domain, including fractional differential equations with
maxima are provided.



K. Nisse et al. / Eur. J. Pure Appl. Math, 18 (2) (2025), 5958 20 of 22

Acknowledgements

We extend their appreciation to Al-Zaytoonah University of Jordan(ZUJ) and to the
Arab Open University, Jeddah, Saudi Arabia for funding this work.

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