EUROPEAN JOURNAL OF PURE AND APPLIED MATHEMATICS
Vol. 17, No. 4, 2024, 2492-2504
ISSN 1307-5543 – ejpam.com
Published by New York Business Global

Common Fixed Point of Generalized Berinde Type
Contraction and an Application

Habes Alsamir1,∗, Haitham Qawaqneh2, Gawhara Al-Musannef3, Roshdi Khalil4

1 Finance and Banking Department, Business Administration College, Dar Aluloom University,
Riyadh, Saudi Arabia
2 Department of Mathematics,Faculty of Science and Information Technology,
Al-Zaytoonah University of Jordan, Amman 11733, Jordan.
3 Faculty of Business Studies, Arab Open University, Jeddah, Saudi Arabia
4 Department of Mathematics, Faculty of Science, The University of Jordan, Amman, 11942,
Jordan

Abstract. In this paper, we introduce λ(s,φ,ψ,L)-generalized Berinde type contraction and obtain some
common fixed point results for such class of contractions the setting of triangular α-admissible mappings
with respect to η in the framework of b-metric spaces. Our results generalize and extend some theorems
in the literature. An example is given to support our result.

2020 Mathematics Subject Classifications: 47H10, 54H25

Key Words and Phrases: Triangular α-admissible mappings with respect to η, common fixed point,
b−metric spaces

1. Introduction and preliminaries

The most important tools in fixed point theory is Banach contraction principle. A lot of
authors have extended or generalized this contraction and proved the existence of fixed and
common fixed point theorems for single valued and multi-valued mappings and some applica-
tion (see [3–6, 11, 14–18, 21–23]). The concept of the b-metric space was introduced by Czerwik
[12] and he also obtained some fixed-point theorems of contractive mappings in b-metric space.
Since then, this notion has been used by many authors to obtain various fixed point theorems.
Roshan et al. in [18] used the notion of almost generalized contractive mappings in ordered
complete b-metric spaces and established some fixed and common fixed point results.

The main goal of this section is to present some definitions and properties of b-metric spaces:

Definition 1.1. ([12]) Let 𭟋 be a nonempty set. A mapping Λb : 𭟋×𭟋 → [0,+∞) is said to
be a b-metric if the following three conditions hold for all u, v ∈ 𭟋 :
(Λ1) Λ(u, v) = 0 ⇒ u = v;
(Λ2) Λ(u, v) = Λ(v, u);
(Λ3) Λ(u, v) ≤ s[Λ(u,w) + Λ(w, v)].
In this case, the pair (𭟋,Λb) is called a b-metric space.

Example 1.2. Let (𭟋,Λb) be a metric space and let β > 1, ϱ ≥ 0 and µ > 0. For u, v ∈ 𭟋, set
Λb(u, v) = ϱΛb(u, v)+µΛb(u, v)

β. Then (𭟋,Λb) is a b-metric space with the parameter s = 2β−1
and not a metric space on 𭟋.

∗Corresponding author.
DOI: https://doi.org/10.29020/nybg.ejpam.v17i4.5388

Email addresses: habes@dau.edu.sa (H. Alsamir), h.alqawaqneh@zuj.edu.jo (H. Qawaqneh),
G.almusannef@arabou.edu.sa (J.M. Al-musannef), roshdi@ju.edu.jo (R. Khalil)

https://www.ejpam.com 2492 Copyright: © 2024 The Author(s). (CC BY-NC 4.0)



H. Alsamir et al. / Eur. J. Pure Appl. Math, 17 (4) (2024), 2492-2504 2493

Example 1.3. Let 𭟋 be the set of Lebesgue measurable functions on [0,1] such that
∫ 1
0 | p(u) |2<

+∞. Define

Λb(u, v) =

∫ 1

0
| p(u)− q(u) |2 d(u).

Then Λb satisfies the following properties:

(i) Λb(u, v) = 0 ⇔ u = v

(ii) Λb(u, v) = Λb(v, u), for all u, v ∈ 𭟋

(iii) Λb(u, v) ≤ 2[Λb(u,w) + Λb(w, v)], for all u,w, v ∈ 𭟋.

Definition 1.4. ([20]) Let (𭟋,Λb) be a b-metric space. Then a sequence {un} in 𭟋 is called:
(i) b-convergent if and only if there exists v ∈ 𭟋 such that Λb(un, u) → 0, as n → +∞. In this
case, we write limn→+∞ un = u.
(2) b-Cauchy if and only if Λb(un, um) = 0 as n,m→ ∞.

Proposition 1.5. ([11]) In b-metric space (𭟋,Λb) the following assertions holds:
(1) A b-convergent sequence has a unique limit,
(2) Each b-convergent is b-Cauchy,
(3) In general, a b-metric is not continuous.

Proposition 1.6. ([11]) The b-metric space (𭟋,Λb) is complete if every Cauchy sequence in 𭟋
b-converges.

Qawagneh et al. [19] introduced the notion of triangular α-admissible with respect to η for
p and q on a set 𭟋 as the following:

Definition 1.7. ([20])Let p, q : 𭟋 → 𭟋 be two mappings and α, η : 𭟋×𭟋 → R be two functions
such that the following assertions hold:

(i) if α(u, v) ≥ η(u, v), then α(pu, qv) ≥ η(pu, qv), and α(pqu, qpv) ≥ η(pqu, qpv),

(ii) if α(u, h) ≥ η(u, h), and α(h, v) ≥ η(h, v), then α(u, v) ≥ η(u, v),

Lemma 1.8. ([22]) Let p, q : 𭟋 → 𭟋 be two mappings and α, η : 𭟋×𭟋 → R be two functions
such that the pair (p, q) is triangular α-admissible with respect to η. Assume that there exist
u0 ∈ 𭟋 such that α(u0, pu0) ≥ η(u0, pu0). Define a sequence {un} in 𭟋 by pu2n = u2n+1 and
qu2n+1 = u2n+2. Then α(un, um) ≥ η(un, um) for all m,n ∈ N with n < m.

Berinde [[6],[7],[8],[9],[10]] presented many interesting fixed-point results for various types of
contraction mappings. In [8] and [9], he defined the almost contraction map as follows.

Definition 1.9. Let (𭟋,Λ) be a metric space. A map p : 𭟋 → 𭟋 is called an almost contraction
if there exist a constant λ ∈ [0, 1) and some L ≥ 0 such that:

Λ(pu, pv) ≤ λΛ(u, v) + LΛ(v, pu)

for all u, v ∈ 𭟋.

Let Φ the set of all increasing and continuous functions φ : [0,+∞) → [0,+∞) and let ∆
be the set of all lower semi-continuous functions ψ : [0,+∞) → [0,+∞) with ψ(b) = b if and
only if b = 0.



H. Alsamir et al. / Eur. J. Pure Appl. Math, 17 (4) (2024), 2492-2504 2494

2. An λ(s,φ,ψ,L)- generalized Berinde type contraction mapping

Now, we will present λ(s,φ,ϕ,L)- generalized Berinde type contraction mapping prove our
main result for such class of contractions in the framework of b-metric spaces.

Definition 2.1. Let (𭟋,Λb) be a b-metric space with parameter s ≥ 1 and p, q : 𭟋 → 𭟋 be a two
mappings. Then we consider that the pair (p, q) is λ(s,φ,ϕ,L)-generalized Berinde type contraction
mapping if there exists α, η : 𭟋×𭟋 → R be two mappings, φ ∈ Ω, ϕ ∈ Φ, λ ∈ [0, 1), L ≥ 0 such
that

φ
(
s2Λb(pu, qv)

)
≤ λ [φ (MΛb

(u, v))− ϕ (MΛb
(u, v)) + LNΛb

(u, v)] , (2.1)

holds for all u, v ∈ 𭟋, where

MΛb
(u, v) = max

{
Λb(u, v),Λb(u, pu),Λb(v, qv),

Λb(u, qv) + Λb(pu, v)

2s[1 + Λb(pu, v)]

}
,

and
NΛb

(u, v) = min {Λb(u, v),Λb(u, pu),Λb(v, qv),Λb(v, pu)} .

Now we begin with our first result.

Theorem 2.2. Let (𭟋,Λb) be a complete b-metric space with the constant s ≥ 1, and (p, q) be
two self-mappings on 𭟋. Suppose that α, η : 𭟋 × 𭟋 → R are two functions. Assume that the
following conditions hold:
(i) λ(s,φ,ϕ,L)-Berinde type contraction mapping;
(ii) the pair (p, q) is triangular α-admissible with respect to η;
(iii) there exists u0 ∈ 𭟋 such that α(u0, pu0) ≥ η(u0, pu0),
(iv) p and q are continuous mappings.
Then, p and q have a common fixed point in 𭟋.

Proof. Let u0 ∈ 𭟋 such that α(u0, pu0) ≥ η(u0, pu0). We define a sequence {un} ⊂ 𭟋 such
that u2n+1 = pu2n and u2n+2 = qu2n+1 for all n ∈ N. If ∃ an n∗ such that un∗+1 = un∗ for some
n∗ ∈ N, then it is very easy to show that p and q have a common fixed point, which completes
the proof. Since the pair (p, q) is triangular α-admissible with respect to η, then

α(u1, u2) = α(pu0, qu1) ≥ η(pu0, qu1) = η(u1, u2)

and
α(u2, u1) = α(pu1, qu0) ≥ η(pu1, qu0) = η(u2, u1).

One more time by using triangular α-admissible with respect to η, we get

α(u2, u3) = α(pu1, qu2) ≥ η(pu1, qu2) = η(u2, u3)

and
α(u3, u2) = α(pu2, qu1) ≥ η(pu2, qu1) = η(u3, u2).

By repeating the above steps for n−times, we obtain the following α(un, un+1) ≥ η(un, un+1)
and α(un+1, un) ≥ η(un+1, un). By Lemma 1.8, we have α(u2n, u2n+1) ≥ η(u2n, u2n+1) for all
n ∈ N and since (p, q) is λ(s,φ,ψ,L)-generalized Berinde type contraction mapping, we get

φ(Λb(u2n+1, u2n+2)) ≤ φ(s2Λb(pu2n, qu2n+1)

≤ λ[φ(MΛb
(u2n, u2n+1))− ϕ(MΛb

(u2n, u2n+1)) + LNΛb
(u2n, u2n+1)](2.2)



H. Alsamir et al. / Eur. J. Pure Appl. Math, 17 (4) (2024), 2492-2504 2495

for all n ∈ N, where

MΛb
(u2n, u2n+1) = max {Λb(u2n, u2n+1),Λb(u2n, pu2n),Λb(u2n+1, qu2n+1),

Λb(u2n, qu2n+1) + Λb(pu2n, u2n+1)

2s(1 + Λb(pu2n, u2n+1))

}
= max {Λb(u2n, u2n+1),Λb(u2n, u2n+1),Λb(u2n+1, u2n+2),

Λb(u2n, u2n+2) + Λb(u2n+1, u2n+1)

2s(1 + Λb(u2n+1, u2n+1))

}
= max

{
Λb(u2n, u2n+1),Λb(u2n+1, u2n+2),

Λb(u2n, u2n+2)

2s

}
and

NΛb
(u2n, u2n+1) = min{Λb(u2n, u2n+1),Λb(u2n, pu2n),Λb(u2n+1, qu2n+1),Λb(u2n+1, pu2n)}

i.e.,
NΛb

(u2n, u2n+1) = 0 (2.3)

Since

Λb(u2n, u2n+2)

2s
≤ s[Λb(u2n, u2n+1) + Λb(u2n+1, u2n+2)]

2s

≤ Λb(u2n, u2n+1) + Λb(u2n+1, u2n+2)

2
≤ max{Λb(u2n, u2n+1),Λb(u2n+1, u2n+2)},

we get
MΛb

(u2n, u2n+1)) ≤ max{Λb(u2n, u2n+1),Λb(u2n+1, u2n+2)}. (2.4)

Taking (2.3) and (2.4) into account,(2.2) yields

φ(Λb(u2n+1, u2n+2)) ≤ λ [φ (max{Λb(u2n, u2n+1),Λb(u2n+1, u2n+2)})
−λϕ (max{Λb(u2n, u2n+1),Λb(u2n+1, u2n+2)})]

< φ (max{Λb(u2n, u2n+1),Λb(u2n+1, u2n+2)})
− ϕ (max{Λb(u2n, u2n+1),Λb(u2n+1, u2n+2)}) .

Now, we will show that Λb(u2n+1, u2n+2) ≤ Λb(u2n, u2n+1). Arguing by contradiction, we
assume Λb(u2n+1, u2n+2) > Λb(u2n, u2n+1). Therefore, we have two cases.
Case 1: MΛb

(u2n, u2n+1) = Λb(u2n, u2n+1). Then

φ(Λb(u2n+1, u2n+2) < φ(Λb(u2n, u2n+1))− ϕ(Λb(u2n, u2n+1)) < φ(Λb(u2n, u2n+1))

Since φ is increasing, we have Λb(u2n+1, u2n+2) < Λb(u2n, u2n+1). which is a contradiction.
Case 2: MΛb

(u2n, u2n+1) = Λb(u2n+1, u2n+2). Then

φ(Λb(u2n+1, u2n+2) < φ(Λb(u2n+1, u2n+2))− ϕ(Λb(u2n+1, u2n+2)) < φ(Λb(u2n+1, u2n+2))

Since φ is increasing, we have Λb(u2n+1, u2n+2) < Λb(u2n+1, u2n+2). Which is a impossible.
Hence from the above we have Λb(u2n+1, u2n+2) ≤ Λb(u2n, u2n+1)
By similar way, we can prove that Λb(u2n, u2n+1) ≤ Λb(u2n−1, u2n). So, we conclude that
Λb(un, un+1) ≤ Λb(un−1, un). that is, the sequence Λb(un+1, un+1) is a decreasing sequence
and bounded below for all n ∈ N. Therefore there ∃ ω ≥ 0 such that

lim
n→∞

Λb(un, un+1) = ω.



H. Alsamir et al. / Eur. J. Pure Appl. Math, 17 (4) (2024), 2492-2504 2496

We want to prove that ω = 0. Now, we have

φ(ω) ≤ λ[φ(ω)− ϕ(ω)] < φ(ω)− ϕ(ω) < φ(ω)

which is a contradiction. Hence

lim
n→∞

Λb(un, un+1) = 0. (2.5)

Now, we want to prove that {un} is a Cauchy sequence by Lemma 1.8, ∃ ε > 0 and two
subsequences {umi} and {uni} of {un} with mi > ni > i such that

Λb(uni , umi) ≥ ε

Λb(uni−1, umi) < ε.

By using the triangular inequality, we have

ε ≤ Λb(uni , umi) ≤ Λb(uni , uni−1) + sΛb(uni−1, umi)

< s[Λb(uni , uni−1) + ε] (2.6)

Letting i→ +∞ on both sides of (2.6) and using (2.5), we obtain

ε ≤ lim
n→+∞

Λb(uni , umi) < sε. (2.7)

From triangular inequality, we have

Λb(uni , umi) ≤ s[Λb(uni , uni+1) + Λb(uni+1, umi)], (2.8)

and
Λb(uni+1, umi) ≤ s[Λb(uni+1, uni) + Λb(uni , umi)]. (2.9)

By taking upper limit as i→ +∞ in (2.8) and applying (2.5) , (2.7) , we get

ε ≤ lim sup
i→+∞

Λb(uni , umi) ≤ s

(
lim sup
+i→+∞

Λb(uni+1, umi)

)
.

Again, by letting the upper limit as i→ +∞ in (2.9), we have

lim sup
i→+∞

Λb(uni+1, umi) ≤ s

(
lim sup
i→+∞

Λb(uni , umi)

)
≤ s.sε = s2ε.

Thus
ε

s
≤ lim sup

i→+∞
Λb(uni+1, umi) ≤ s2ε. (2.10)

Similarly,
ε

s
≤ lim sup

i→+∞
Λb(uni , umi+1) ≤ s2ε. (2.11)

By using the triangular inequality, we get

Λb(uni+1, umi) ≤ s[Λb(uni+1, umi+1) + Λb(umi+1, umi)]. (2.12)

On letting i→ +∞ in (2.12) and using the inequalities (2.5) , (2.10) , we get

ε

s2
≤ lim sup

i→+∞
Λb(uni+1, umi+1). (2.13)



H. Alsamir et al. / Eur. J. Pure Appl. Math, 17 (4) (2024), 2492-2504 2497

By following the above methods, we find

lim sup
i→+∞

Λb(uni+1, umi+1) ≤ s3ε. (2.14)

From (2.13) and (2.14), we obtain

ε

s
≤ lim sup

i→+∞
Λb(uni+1, umi+1) ≤ s3ε. (2.15)

By Lemma 1.8, we have
α(uni+1, umi+1) ≥ η(uni+1, umi+1).

Thus, we have

φ(Λb(uni+1, umi+1)) ≤ φ(s2Λb(uni+1, umi+1))

≤ λ [φ (MΛb
(uni , umi))− ϕ (MΛb

(uni , umi)) + L (NΛb
(uni , umi))]

= [λφ (MΛb
(uni , umi))− λϕ (MΛb

(uni , umi)) + λL (NΛb
(uni , umi))] ,

where

MΛb
(uni , umi) = max{Λb(uni , umi),Λb(uni , puni),Λb(umi , qumi),

Λb(uni , qumi) + Λb(puni , umi)

2s(1 + Λb(puni , umi))
}.

NΛb
(uni , umi) = min {Λb(uni , umi),Λb(uni , puni),Λb(umi , qumi),Λb(umi , puni)}

= min {Λb(uni , umi),Λb(uni , uni+1),Λb(umi , umi+1),Λb(umi , uni+1)}(2.16)

Taking the limit as i → +∞ in the above two expressions and using (2.5),(2.7) ,(2.10) and
(2.11), we obtain

ε = max{ε,
ε
s +

ε
s

2s
} ≤ lim sup

i→+∞
Λb(uni , umi) ≤ max{sε, s

2ε+ s2ε

2s
} = sε.

lim sup
i→+∞

NΛb
(uni , umi) = 0.

From (2.13), we obtain

φ(sε) ≤ φ(s2
ε

s2
) ≤ φ(s2 lim sup

i→+∞
φ(Λb(uni+1, umi+1))

≤ λ[φ(lim sup
i→+∞

MΛb
(uni , umi)− ϕ(lim inf

i→+∞
MΛb

(uni , umi)

≤ λ[φ(sε)− ϕ(sε)]

≤ λ(φ(sε))− λ(ϕ(sε))

< λφ(sε)

which leads to a contradiction. Thus {un} is a Cauchy sequence. Since 𭟋 is an complete b-
metric space and α(uni+1, umi+1) ≥ η(uni+1, umi+1) for all n ∈ N0, there exists θ such that
limn→+∞ un = θ. If p is continuous, we have pθ = limn→+∞ pu2n = limn→+∞ u2n+1 = θ. From
Condition (2.2), we have:

φ(Λb(θ, qθ)) ≤ φ(s2Λb(θ, θ))

≤ λ[(φ(MΛb
(θ, θ))− ϕ(MΛb

(θ, θ)) + LNΛb
(θ, θ)]



H. Alsamir et al. / Eur. J. Pure Appl. Math, 17 (4) (2024), 2492-2504 2498

for all n ∈ N, where

MΛb
(θ, θ) = max{Λb(θ, θ),Λb(θ, pθ),Λb(θ, qθ),

Λb(θ, qθ) + Λb(pθ, θ)

2s(1 + Λb(pθ, θ))
}

= Λb(θ, qθ)

and

NΛb
(θ, θ) = min{Λb(θ, θ),Λb(θ, pθ),Λb(θ, qθ),Λb(θ, qθ)} = 0.

By using the properties of φ and ϕ, we have

φ(Λb(θ, qθ)) = φ(s2Λb(pθ, qθ))

≤ λ[(φ(MΛb
(θ, qθ))− ϕ(MΛb

(θ, θ))]

= λ[(φ(Λb(θ, qθ))− ϕ(Λb(θ, qθ))]

< λ(φ(Λb(θ, qθ))).

Hence, θ = qθ is θ is the common fixed of p and q. If q is continuous, then, by a similar way of
the above, we can prove that p and q have a common fixed point.

Theorem 2.3. Let (𭟋,Λb) be a complete b-metric space with the constant s ≥ 1, and (p, q) be
two self-mappings on 𭟋. Suppose that α, η : 𭟋 × 𭟋 → R are two functions. Assume that the
following conditions hold:
(i) λ(s,φ,ϕ,L)-Berinde type contraction mapping;
(ii) the pair (p, q) is triangular α-admissible with respect to η;
(iii) If ∃ u0 ∈ 𭟋 such that α(u0, pu0) ≥ η(u0, pu0),
(iv) if {un} is a sequence in 𭟋 such that α(un, un+1) ≥ η(un, un+1), for all n ∈ N and un → θ
as n→ ∞, then ∃ a subsequence {uni} of {un} such that α(uni , u∗) ≥ η(uni , u∗), for all i ∈ N.
Then, p and q have a common fixed point in 𭟋.

Proof. Following similar arguments as in the proof of Theorem 2.2, we obtain a sequence
{un} is defined by u2n+1 = pu2n and u2n+2 = pu2n+1 for all n ∈ N converging to u∗ ∈ 𭟋 such
that α(u2n, u2n+1) ≥ η(u2n, u2n+1) for all n ∈ N. By (iv), there exist a subsequence {uni} of
{un} such that α(uni , u∗) ≥ η(uni , u∗), for all i ∈ N. Therefore

φ(Λb(u2ni+1, qu∗)) ≤ φ(s2Λb(pu2ni , qu∗)

≤ λ[(φ(MΛb
(u2ni , u∗))− ϕ(MΛb

(u2ni , u∗)) + LNΛb
(u2ni , u∗)]

(2.17)

for all n ∈ N, where

MΛb
(u2ni , u∗)) = max{Λb(u2ni , u∗),Λb(u2ni , pu2ni),Λb(u∗, qu∗),

Λb(u2ni , qu∗) + Λb(pu2ni , u∗)

2s(1 + Λb(pu2ni , u∗))
}

= max{Λb(u2n, u∗),Λb(u2n, u2ni+1),Λb(u∗, qu∗),
Λb(u2ni , qu∗) + Λb(u2ni+1, u∗)

2s(1 + Λb(u2ni+1, u∗))
}

and

NΛb
(u2ni , u∗) = min{Λb(u2ni , u∗),Λb(u2ni , pu2ni),Λb(u∗, qu∗),Λb(u∗, pu2ni)}

= min{Λb(u2ni , u∗),Λb(u2ni , u2ni+1),Λb(u∗, qu∗),Λb(u∗, u2ni+1)}.

Since

lim sup
i→∞

Λb(u2ni , qu∗) + Λb(u2ni+1, u∗)

2s(1 + Λb(u2ni+1, u∗))
≤ Λb(u∗, qu∗)

2
.



H. Alsamir et al. / Eur. J. Pure Appl. Math, 17 (4) (2024), 2492-2504 2499

By taking i→ ∞ in (2.18) and (2.18) using (2.5), we deduce that

lim sup
i→∞

MΛb
(u2ni , u∗)) = Λb(u∗, qu∗)

and
lim sup
i→∞

NΛb
(u2ni , u∗)) = 0.

From (2.17)and taking in account () and (), we have

φ(Λb(u∗, qu∗)) ≤ λ[φ(Λb(u∗, qu∗))− ϕ(Λb(u∗, qu∗)))] (2.18)

< λφ(Λb(u∗, qu∗))− λϕ(Λb(u∗, qu∗)). (2.19)

By definition of φ and ϕ, we have a contradiction. Hence Λb(u∗, qu∗) = 0, i.e.,

qu∗ = u∗.

By the same way we can prove that pu∗ = u∗.

Definition 2.4. Let (𭟋,Λb) be a b-metric space with parameter s ≥ 1, p, q : 𭟋 → 𭟋 and
α, η : 𭟋 × 𭟋 → R be two functions. Let φ ∈ Ω, ϕ ∈ Φ and λ ∈ [0, 1). Then the pair (p, q) is
called λ(s, φ, ϕ)-contraction mapping of type (B) if α(u, v) ≥ η(u, v), then

φ
(
s2Λb(pu, qv)

)
≤ λ [φ (MΛb

(u, v))− ϕ (MΛb
(u, v))] , (2.20)

where λ ∈ [0, 1) φ ∈ Ω, ϕ ∈ Φ and

MΛb
(u, v) = max

{
Λb(u, v),Λb(u, pu),Λb(v, qv),

Λb(u, qv) + Λb(pu, v)

2s[1 + Λb(pu, v)]

}
.

.

The proof of the followings two theorems follows from Theorem 2.2 and Theorem 2.3 by
putting L = 0.

Theorem 2.5. Let (𭟋,Λb) be a complete b-metric space with the constant s ≥ 1, and (p, q) be
two self-mappings on 𭟋. Suppose that α, η : 𭟋 × 𭟋 → R are two functions. Assume that the
following conditions hold:
(i) λ(s, φ, ϕ)- contraction type (B) mapping;
(ii) the pair (p, q) is triangular α-admissible with respect to η;
(iii) There exists u0 ∈ 𭟋 such that α(u0, pu0) ≥ η(u0, pu0),
(iv) p and q are continuous mappings.
Then, p and q have a common fixed point in 𭟋.

Theorem 2.6. Let (𭟋,Λb) be a complete b-metric space with the constant s ≥ 1, and (p, q) be
two self-mappings on 𭟋. Suppose that α, η : 𭟋 × 𭟋 → R are two functions. Assume that the
following conditions hold:
(i) λ(s, φ, ϕ)-contraction mapping type (B);
(ii) the pair (p, q) is triangular α-admissible with respect to η;
(iii) If ∃ u0 ∈ 𭟋 such that α(u0, pu0) ≥ η(u0, pu0),
(iv) if {un} is a sequence in 𭟋 such that α(un, un+1) ≥ η(un, un+1), for all n ∈ N and un → θ
as n→ ∞, then there exist a subsequence {uni of {un} such that α(uni , u∗) ≥ η(uni , u∗), for all
i ∈ N.
Then, p and q have a common fixed point in 𭟋.

The following corollaries are consequences of Theorem 2.2 and Theorem 2.3.



H. Alsamir et al. / Eur. J. Pure Appl. Math, 17 (4) (2024), 2492-2504 2500

Corollary 2.7. Let (𭟋,Λb) be a complete b-metric space with the constant s ≥ 1, and p be a
self-mapping on 𭟋. Suppose that α, η : 𭟋×𭟋 → R are two functions. Suppose that the following
conditions hold:
(i)

If α(u, v) ≥ η(u, v) ⇒ φ
(
s2Λb(pu, pv)

)
≤ λ [φ (MΛb

(u, v))− ϕ (MΛb
(u, v)) + (NΛb

(u, v))] ,
(2.21)

(ii)p is triangular α-admissible with respect to η;
(iii) If ∃ u0 ∈ 𭟋 such that α(u0, pu0) ≥ η(u0, pu0),
(iv) p is a continuous mappings.
Then, p has a fixed point in 𭟋.

Proof. The conclusion follows from Theorem 2.2 by taking q = p.

Corollary 2.8. Let (𭟋,Λb) be a complete b-metric space with the constant s ≥ 1, and p be a
self-mapping on 𭟋. Suppose that α : 𭟋×𭟋 → R are two functions. Assume that the following
conditions hold:
(i)

If α(u, v) ≥ 1 ⇒ φ
(
s2Λb(pu, pv)

)
≤ λ [φ (MΛb

(u, v))− ϕ (MΛb
(u, v)) + (NΛb

(u, v))] , (2.22)

(ii)p is triangular α-admissible with respect to η;
(iii) There exists u0 ∈ 𭟋 such that α(u0, pu0) ≥ 1,
(iv) p is a continuous mappings.
Then, p has a fixed point in 𭟋.

Proof. The proof follows Corollary 2.7 by defining η : 𭟋×𭟋 → R via η(u, v) = 1.

Remark 2.9. Since a b-metric space is a metric space when s = 1, so our Theorems can be seen
as a generalizations and extensions of several comparable results in metric spaces and b-metric
spaces.

The following example illustrates the above result.

Example 2.10. Let 𭟋 = {1, 2, 3, 4}. Define Λb : 𭟋×𭟋 → [0,+∞) as follows:

Λb(u, v) = Λb(v, u) = 0 if u ̸= v, u = v

Λb(u, v) = Λb(v, u) = 2 if u = 1, v = 2

Λb(u, v) = Λb(v, u) = 1 if u = 1, v = 3

Λb(u, v) = Λb(v, u) = 10 if u, v = 1, 2, 3, v = 4

Define φ(t) = et, ϕ(t) = et

2+et , λ = 1
2 , L = 2 and define the mappings p, q : 𭟋 → 𭟋 by

p1 = p2 = p3 = 1, p4 = 3

q1 = 2, q2 = q3 = q4 = 1.

It is obvious that (𭟋,Λb) is a complete b-metric space with the constant s = 2. We show that
the condition (2.1) is true. We put

φ
(
s2Λb(pu, qv)

)
= A,φ(MΛb

(u, v)) = B,ϕ(MΛb
(u, v)) = C and NΛb

(u, v) = D.

Then we have the following cases:



H. Alsamir et al. / Eur. J. Pure Appl. Math, 17 (4) (2024), 2492-2504 2501

Table 1: The possible values of u, v

Λb(u, v) A λ[B − C +D] A ≤ λ[B − C +D] ✓

Λb(1, 1) ≈ 2.72 ≈ 3.30 2.72 < 3.30 ✓
Λb(2, 2) 1 ≈ 3.30 1 < 3.30 ✓
Λb(3, 3) 1 ≈ 1.07 1 < 1.07 ✓
Λb(4, 4) ≈ 54.60 ≈ 11012.73 54.60 < 11012.73 ✓
Λb(1, 2) 1 ≈ 3.30 1 < 3.30 ✓
Λb(1, 3) 1 ≈ 1.07 1 < 1.07 ✓
Λb(1, 4) 1 ≈ 11012.73 1 < 11012.73 ✓
Λb(2, 3) 1 ≈ 3.30 1 < 3.30 ✓
Λb(2, 4) 1 ≈ 11012.73 1 < 11012.73 ✓
Λb(3, 4) 1 ≈ 11012.73 1 < 11012.73 ✓

λ[B − C +D]

A
A = λ[B − C +D]

A ≤ λ[B − C +D]

Λb(1, 1)

Λb(2, 2)Λb(3, 3) Λb(4, 4)

Figure 1. Satisfing the enquality A ≤ λ[B − C +D]

Thus, all the conditions of Theorem 2.1 are satisfied and hence p and q have a common fixed
point. Indeed, 1 is a common fixed point of p and q.

3. Application

Fixed point theorem has numerous applications, such as fractional differential equations ([1],
[2], [13]), the significance of these types of equations is their utilization in modeling in many
subjects. In this section, we utilize our results to demonstrate the existence and uniqueness of
the Fredholm type integral equation.
Now, Consider the set 𭟋 = C([0, 1], (−∞,∞)) and the following Fredholm type integral equa-
tion:

ṕ(t) =

∫ 1

0
S(t, s, ṕ(t)) ds, for t, s ∈ [0, 1], (3.1)

where S(t, s, ṕ(t)) is a continuous function on [0, 1]× [0, 1] → (−∞,∞).
Now, define Λb : 𭟋×𭟋 → C and (p, q) 7→| ṕ(t)− q(t) | .

Note that (𭟋,Λb) is a complete b-metric space, where the parameter s = 2.

Theorem 3.1. Suppose that for all p, q ∈ 𭟋
(1) | S(t, s, ṕ(t))− S(t, s, q(t)) |≤ |ṕ(t)−q(t)|

2 .



REFERENCES 2502

(2) | S(t, s,
∫ 1
0 S(t, s, ṕ(t)) ds) − S(t, s, q

∫ 1
0 S(t, s, q(t)) ds) |≤| S(t, s, ṕ(t)) − S(t, s, q(t)) | for all

t, s.
Then the integral equation 3.1 has a unique solution.

Proof. Let ṕ(t) : 𭟋 → 𭟋 defined by ṕ(t) =
∫ 1
0 S(t, s, ṕ(t)) ds, then

Λb(ṕ, q) =| ṕ(t)− q(t) | . Now we have

Λb(ṕ(t), q(t)) = | ṕ(t)− q(t) |

= | S(t, s,
∫ 1

0
S(t, s, ṕ(t)) ds)− S(t, s, q

∫ 1

0
S(t, s, q(t)) ds) |

≤ | S(t, s, ṕ(t))− S(t, s, q(t)) |

≤ | ṕ(t)− q(t) |
2

≤ 1

2
Λb(ṕ(t), q(t))

= λ [φ (MΛb
(ṕ(t), q(t)))− ϕ (MΛb

(ṕ(t), q(t)))] ,

where φ(t) = t and ϕ(t) = t
2 . Also the parameter s < 3.

Hence, all the hypotheses of Theorem 2.2, are fulfilled and then the equation 3.1 has a unique
solution.

4. Conclusion

We have demonstrated the existence and uniqueness of a fixed point for self-mapping in b-
metric spaces under diverse nonlinear mappings with continuous control functions. Also, we
show an application of our results to Fredholm-type integral equations. Additionally, we would
like to bring the researchers consideration to the following question.

4.1. Question

Under what conditions we will get the same results for self-mapping in partial b-metric
spaces?

Acknowledgements

We extend their appreciation to the Deanship of Post Graduate and Scientific Research at
Dar Al Uloom University for funding this work.

References

[1] I M Batiha; J Oudetallah; A Ouannas; A A Al-Nana and I H Jebril. Tuning the fractional-
order pid-controller for blood glucose level of diabetic patients. International Journal of
Advances in Soft Computing and its Applications, 13(2):1–10, 2021.

[2] H Qawaqneh; J Manafian; M Alharthi and Y Alrashed. Stability analysis, modulation
instability, and beta-time fractional exact soliton solutions to the van der waals equation.
Mathematics, 12(14: 2257), 2024.

[3] H Akhadkulov; M S M Noorani; A B Saaban; F M Alipiah and H Alsamir. Notes on
multidimensional fixed-point theorems. Demonstratio Mathematica, 50(1):360–374, 2017.



REFERENCES 2503

[4] W Shatanawi; M S M Noorani; J Ahmad; H Alsamir and M Kutbi. Some common fixed
points of multivalued mappings on complex-valued metric spaces with homotopy result. J.
Nonlinear Sci. Appl., 10:3381–796, 2017.

[5] H Aydi and A Felhi. Best proximity points for cyclic kannan-chatterjea- ciric type contrac-
tions on metric-like spaces. J. Nonlinear Sci. Appl., 9:5202–5218, 2016.

[6] V Berinde. Approximating fixed points of weak contractions using the picard iteration.
Carpathian Journal of Mathematics, pages 7–22, 2003.

[7] V Berinde. Approximating fixed points of weak φ-contractions using the picard iteration.
Fixed Point Theory, 2(2):7–22, 2003.

[8] V Berinde. Approximating fixed points of weak contractions using the picard iteration.
Nonlinear Anal. Forum, 9:43–53, 2004.

[9] V Berinde. General constructive fixed point theorems for C̀iric̀-type almost contractions
in metric spaces. Carpathian Journal of Mathematics, pages 10–19, 2008.

[10] V Berinde. Some remarks on a fixed point theorem for C̀iric̀-type almost contractions.
carpathian journal of mathematics. Carpathian Journal of Mathematics, pages 157–162,
2009.

[11] M Boriceanu; M Bota and A Petrusel. Multivalued fractals in b-metric spaces. Cent. Eur.
J. Math., 8(2):367–377, 2010.

[12] S Czerwik. Nonlinear set-valued contraction mappings in b-metric spaces. atti semin.
International Journal of Advances in Soft Computing and its Applications, 46(2):263–276,
1998.

[13] I M Batiha; S A Njadat; R M Batyha; A Zraiqat; A Dababneh and Sh Momani. Design
fractional-order pid controllers for single-joint robot arm model. International Journal of
Advances in Soft Computing and its Applications, 14(2):96–114, 2022.

[14] A F Roldan-López de-Hierro; E Karapinar; C Roldán-López-de Hierro and J Mart́ıinez-
Moreno. Coincidence point theorems on metric spaces via simulation functions. J. Comput.
Appl. Math., 275:345–75, 2015.

[15] M A Khamsi and N Hussain. Kkm mappings in metric type spaces. Nonlinear Anal.,
73(9):3123–3129, 2010.

[16] H Alsamir; M S M Noorani and W Shatanawi. On new fixed point theorems for three types
of (α, β)−(ψ, θ, ϕ)-multivalued contractive mappings in metric spaces. Cogent Mathematics,
3(1):1257473, 2016.

[17] H Qawagneh; M S M Noorani and W Shatanawi. Fixed point theorems for (α, k, θ)-
contractive multi-valued mapping in b-metric space and applications. International Journal
of Mathematics and Computer Science, 14(1):263–283, 2019.

[18] J Roshan; V Parvaneh and I Altun. Some coincidence point results in ordered b-metric
spaces and applications in a system of integral equations. Appl. Math. Comput., 226:725–
737, 2014.

[19] H Qawaqneh. New contraction embedded with simulation function and cyclic
(α, β)−admissible in metric-like spaces. International Journal of Mathematics and Com-
puter Science, 15(1):1029–1044, 2020.



REFERENCES 2504

[20] H Qawaqneh. Fractional analytic solutions and fixed point results with some applications.
Adv. Fixed Point Theory, 14(1), 2024.

[21] H Alsamir; H Aydi; M S M Noorani; W Shatanawi; H Akhadkulov; H Qawaqneh and
K Alanazi. Fixed point results in metric-like spaces via σ-simulation functions. European
Journal of Pure and Applied Mathematics, 12(1):88–100, 2019.

[22] A H Ansari; S Chandok X L Liu and S Radenovicl. On some results in metric spaces using
auxiliary simulation functions via new functions. J. Comput. Anal. Appl., 24(6):1103–1114,
2018.

[23] H Qawaqneh; M S M Noorani; H Aydi; A Zraiqat and A H Ansari. On fixed pointresults
in partial b-metric spaces. Journal of Function Spaces, 2021, 2021.