EUROPEAN JOURNAL OF PURE AND APPLIED MATHEMATICS
2025, Vol. 18, Issue 2, Article Number 6015
ISSN 1307-5543 – ejpam.com
Published by New York Business Global

On Multivalued Contractions via θ-Hyperbolic Sine
Distance Functions

Hassen Aydi1,2,∗, Abdelbasset Felhi3, Irshad Ayoob4, Nabil Mlaiki4

1 Institut Supérieur d’Informatique et des Techniques de Communication, Université de
Sousse, H. Sousse 4000, Tunisia
2 Department of Mathematics and Applied Mathematics, Sefako Makgatho Health Sciences
University, Ga-Rankuwa, South Africa
3 Department of Mathematics and Physics, Preparatory Institute for Engineering Studies,
Carthage University, Bizerte, Tunisia
4 Department of Mathematics and Sciences, Prince Sultan University, Riyadh 11586,
Saudi Arabia

Abstract. Very recently, the concept of θ-hyperbolic sine distance functions has been introduced
by Jleli and Samet in [1]. In this work, we prove some related fixed points results for several classes
of multivalued mappings including manageable functions on metric spaces.

2020 Mathematics Subject Classifications: 54E50; 54E25; 47H10; 33B10

Key Words and Phrases: hyperbolic function, θ-hyperbolic sine distance function, multivalued
mapping, metric space, fixed point

1. Introduction

In 1906, Fréchet [2] defined the concept of a metric space (MS).

Definition 1. [2] Let X be any nonempty set. A function d : X ×X → [0,+∞) is said
to be a distance function or metric on X if for any ϖ, ς, s ∈ X,

(i) d(ϖ, ς) = 0 iff ς = ϖ,

(ii) d(ϖ, ς) = d(ς,ϖ),

(iii) d(ϖ, ς) ≤ d(ς, s) + d(s,ϖ) (triangle inequality).

∗Corresponding author.
DOI: https://doi.org/10.29020/nybg.ejpam.v18i2.6015

Email addresses: hassen.aydi@isima.rnu.tn (H. Aydi), abdelbassetfelhi@gmail.com (A. Felhi),
iayoub@psu.edu.sa (I. Ayoob), nmlaiki@psu.edu.sa (N. Mlaiki), nmlaiki2012@gmail.com (N. Mlaiki)

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



H. Aydi et al. / Eur. J. Pure Appl. Math, 18 (2) (2025), 6015 2 of 14

Banach fixed point (FP) theorem [3] is an fundamental tool in the theory of MSs. It
affirms the existence of a unique FP of contraction maps on complete MSs. Later, it has
been generalized and extended in several directions, either by weakening the topology of
the metric, or by generalizing the contraction itself. Several works arise in this sense, like
[4–9].

On a MS (X, d), CB(X) is the set of nonempty bounded and closed subsets of X. For
Π,Ξ ∈ CB(X), the Hausdorff-Pompieu metric induced by d is

H(Π,Ξ) = max

{
sup
a∈Π

δ(a,Ξ), sup
b∈Ξ

δ(b,Π)

}
,

where δ(ς,Π) = inf{d(ς, a) | a ∈ Π} is the distance from ς to the set Π.

Definition 2. Let X be any nonempty set. An element ς ∈ X is said to be a FP of
a multivalued mapping T : X → 2X if ς ∈ T (ς), where 2X denotes the collection of all
nonempty subsets of X.

Nadler [10] studied the existence of FPs for multivalued contractions.

Theorem 1. [10] Let (X, d) be a complete MS and T : X → CB(X) be a contraction,
i.e.,

H(Tς, Tϖ) ≤ kd(ϖ, ς),

for all ϖ, ς ∈ X, where k ∈ [0, 1). Then, there is a FP of T .

After the work of Nadler [10], many FP results for multivalued mappings appeared in
literature. For more details, see [11–13].

Motivated by the fact that hyperbolic functions have variant applications in many
fields, like physics, mathematics, engineering, etc, recently, Jleli and Samet [1] introduced
the notion of θ-hyperbolic sine distance functions associated to a certain metric and ob-
tained some nice FPs results. Following this direction, we aim to establish some FPs results
for some classes of contractive multivalued mappings on MSs involving the θ-hyperbolic
sine distance function.

For τ > 0, let Θτ be the collection of functions θ : [0,+∞) → [0,+∞) so that

θ(t) ≥ ctτ , (1)

for all t ≥ 0, where c > 0 is a constant.

Definition 3. Let (X, d) be a MS. For all τ > 0 and θ ∈ Θτ , consider dθ : X
2 → [0,+∞)

as
dθ(ϖ, ς) = θ(sinh(d(ϖ, ς))), ∀ϖ, ς ∈ X,

where sinh the hyperbolic sine function is given as

sinh t =
et − e−t

2
, t ∈ R.

The mapping dθ is called the θ-hyperbolic sine distance function associated to the metric
d.



H. Aydi et al. / Eur. J. Pure Appl. Math, 18 (2) (2025), 6015 3 of 14

Some properties of the θ-hyperbolic sine distance function are provided below.

Proposition 1. [1] Let (X, d) be a MS and θ ∈ Θτ for some τ > 0. Then, for all
ϖ, ς ∈ X, we have

(i) dθ(ϖ, ς) = 0 =⇒ ς = ϖ.

(ii) If θ(0) = 0, then dθ(ς, ς) = 0.

(iii) dθ(ϖ, ς) = dθ(ς,ϖ).

Notice that a θ-hyperbolic sine distance function is not necessarily a metric, even if
θ(0) = 0. The next example shows this fact.

Example 1. Let X = R and d(ϖ, ς) = |ς − ϖ| for all ϖ, ς ∈ X. Let θ(t) =
√
t for all

t ≥ 0. Then, θ ∈ Θ 1
2
. The θ-hyperbolic sine distance function associated to the metric d

is defined by
dθ(ϖ, ς) = θ(sinh(|ς −ϖ|))

for all ϖ, ς ∈ X. On the other hand, we have

dθ(1, 5)

dθ(1, 3) + dθ(3, 5)
=

θ(sinh(4))

θ(sinh(2)) + θ(sinh(2))

=

√
sinh(4)

2
√
sinh 2

=
1

2

√
e2 + e−2 > 1,

which shows that dθ does not verify the triangle inequality. Consequently, dθ is not a
metric on X.

Proposition 2. [1] Let (X, d) be a MS.

(i) Let dθ be the θ-hyperbolic sine distance function associated to d, where θ ∈ Θτ for
some τ > 0. Then, for all ι > 0, we have

ιdθ = dθι ,

where θι = ιθ.

(ii) Let θ1, θ2 ∈ Θτ for some τ > 0. Then,

dθ1 + dθ2 = dθ,

where θ = θ1 + θ2.

Proposition 3. [1] Let θ ∈ Θτ for some τ > 0. Assume that:

(i) θ(0) = 0;

(ii) There exists r∗ > 0 such that
θ(sinh r∗) = r∗.

Then, for every nonempty set X, there exists a metric d on X such that the θ-hyperbolic
sine distance function associated to d coincides with d, i.e., dθ = d.



H. Aydi et al. / Eur. J. Pure Appl. Math, 18 (2) (2025), 6015 4 of 14

2. The Hausdorff θ-hyperbolic sine distance function

Our work is concerned with multivalued mappings T : X → 2X . For this, let (X, d) be
a MS. Let θ ∈ Θτ for some τ > 0. For two bounded and closed subsets Π,Ξ in X, consider

Hθ(Π,Ξ) = max{∆θ(Π,Ξ),∆θ(Ξ,Π)},

where

∆θ(Π,Ξ) = sup{θ(sinh(δ(a,Ξ))) : a ∈ Π} = sup
a∈Π

inf
b∈Ξ

{θ(sinh(d(a, b)))}.

The mapping Hθ is called the Hausdorff θ-hyperbolic sine distance function associated to
the metric d. Some properties of Hθ are provided below.

Proposition 4. Let (X, d) be a MS and θ ∈ Θτ for some τ > 0. Then, for every
Π,Ξ ∈ CB(X),

(i) Hθ(Π,Ξ) = 0 =⇒ Π = Ξ.

(ii) If θ(0) = 0, then Hθ(Π,Π) = 0.

(iii) Hθ(Π,Ξ) = Hθ(Ξ,Π).

Proof. (i)Hθ(Π,Ξ) = 0 =⇒ ∆θ(Π,Ξ) = ∆θ(Ξ,Π) = 0. In the case ∆θ(Π,Ξ) = 0, we
get

sup
a∈Π

{θ(sinh(δ(a,Ξ)))} = 0, i.e θ(sinh(δ(a,Ξ))) = 0 ∀a ∈ Π.

Then for all a ∈ Π,

∃(bn) ⊂ Ξ so that lim
n→+∞

θ(sinh(d(a, bn))) = 0.

By (1), we obtain
lim

n→+∞
(sinh(d(a, bn)))

τ = 0, for some τ > 0,

which implies
lim

n→+∞
sinh(d(a, bn)) = 0.

Thus, for all a ∈ Π,
lim

n→+∞
δ(a, bn) = 0, i.e a ∈ Ξ = Ξ.

So, Π ⊂ Ξ. Similarly, as ∆θ(Ξ,Π) = 0, we have Ξ ⊂ Π. Finally, we obtain Π = Ξ.
(ii) If θ(0) = 0, then

Hθ(Π,Π) = ∆θ(Π,Π)

= sup
a∈Π

inf
b∈Ξ

θ(sinh(d(a, b)))

≤ sup
a∈Π

θ(sinh(d(a, a)))

= sup
a∈Π

θ(sinh(0)) = sup
a∈Π

θ(0) = sup
a∈Π

(0) = 0.



H. Aydi et al. / Eur. J. Pure Appl. Math, 18 (2) (2025), 6015 5 of 14

Thus, Hθ(Π,Π) = 0.
(iii) It is obvious.

Notice that a Hausdorff θ-hyperbolic sine distance function is not necessarily a Haus-
dorff metric, even θ(0) = 0. The following example shows this fact.

Example 2. Let X = R and d(ϖ, ς) = |ς − ϖ| for all ϖ, ς ∈ X. Let θ(t) = t for any
t ≥ 0. Take A = {0}, B = {2n} and C = {n}, with n ≥ 1. We write

Hθ(Π,Ξ)

Hθ(Π, C) +Hθ(C,Ξ)
=
θ(sinh(2n))

2θ(sinh(n))

=
1

2
(en + e−n) → ++∞ asn→ +∞,

which shows that Hθ does not satisfy the triangle inequality, and so Hθ is not a Hausdorff
metric on CB(X).

Definition 4. Let (X, d) a MS. A function f : X → [0,+∞) is termed as lower semi-
continuous if for {ςn} ⊂ X and ς ∈ X, we have

lim
n→+∞

d(ςn, ς) = 0 ⇒ f(ς) ≤ lim inf
n→+∞

f(ςn).

For T : X → CB(X), define fT : X → [0,+∞) by

fT (ς) = d(ς, T ς) for all ς ∈ X.

3. FP results

In this part, we present FP results for some multivalued contractions via θ-hyperbolic
sine functions.

3.1. Multivalued θ-hyperbolic contractions

Jleli and Samet [1] introduced the following class of single-valued mappings.

Definition 5. Let (X, d) be a MS and θ ∈ Θτ for some τ > 0. A mapping T : X → X is
called a θ-hyperbolic contraction on X, if there is k ∈ (0, 1) so that

dθ(Tς, Tϖ) ≤ kdθ(ϖ, ς) (2)

for all ϖ, ς ∈ X.

Also, they established the following FP theorem.

Theorem 2. [1] Let (X, d) be a complete MS and θ ∈ Θτ for some τ > 0. Given
T : X → X so that:

(I) T is a θ-hyperbolic contraction on X;



H. Aydi et al. / Eur. J. Pure Appl. Math, 18 (2) (2025), 6015 6 of 14

(II) For all ϖ, ς ∈ X, if lim
n→+∞

d(Tnϖ, ς) = 0, then there exists a subsequence {Tnkς} of

{Tnς} such that lim
k→+∞

d(T (Tnkς), Tϖ) = 0.

Then, T possesses one and only one FP. Moreover, for all w0 ∈ X, {Tnw0} is convergent
to this unique FP.

We need the next lemma for the rest.

Lemma 1. Let Π,Ξ ∈ CB(X), a ∈ Π and θ ∈ Θτ for some τ > 0. Thus, for each ε > 0,
there is b ∈ Ξ so that

dθ(a, b) ≤ Hθ(Π,Ξ) + ε.

Proof. Let Π,Ξ ∈ CB(X) and a ∈ Π. We have dθ(a,Ξ) ≤ ∆θ(Π,Ξ) ≤ Hθ(Π,Ξ). Then,
for every ε > 0, there is b ∈ B so that

dθ(a, b) ≤ dθ(a,Ξ) + ε.

Consequently,
dθ(a, b) ≤ Hθ(Π,Ξ) + ε.

The next result corresponds to the extension of Theorem 2 to multivalued mappings.
It is stated as follows:

Theorem 3. Let (X, d) be a complete MS and θ ∈ Θτ for τ > 0. Let T : X → CB(X) be
a mapping such that

Hθ(Tς, Tϖ) ≤ kdθ(ϖ, ς), (3)

for all ϖ, ς ∈ X, where k ∈ [0, 1). Assume that, fT is lower semi-continuous, then T has
a FP in X.

Proof.
Let ς0 ∈ X and ς1 ∈ Tς0. When dθ(ς0, ς1) = 0, so by Proposition 1 (i), one gets ς0 = ς1

and so ς0 is a FP of T. Suppose that dθ(ς0, ς1) > 0. Since Tς0, T ς1 ∈ CB(X) and ς1 ∈ Tς0,
using Lemma 1, there is ς2 ∈ Tς1, so that

dθ(ς1, ς2) ≤ Hθ(Tς0, T ς1) +
1− k

2
dθ(ς0, ς1). (4)

If dθ(ς1, ς2) = 0, then by Proposition 1 (i), we have ς1 = ς2 and so ς1 is a FP of T. When
dθ(ς1, ς2) > 0, then by Lemma 1, there is ς3 ∈ Tς2, so that

dθ(ς2, ς3) ≤ Hθ(Tς1, T ς2) +
1− k

2
dθ(ς1, ς2). (5)

Continuing as above, we construct {ςn} ⊂ X so that ςn+1 ∈ T (ςn) with dθ(ςn, ςn+1) > 0
and

dθ(ςn+1, ςn) ≤ Hθ(Tςn, T ςn−1) +
1− k

2
dθ(ςn, ςn−1).



H. Aydi et al. / Eur. J. Pure Appl. Math, 18 (2) (2025), 6015 7 of 14

Then, using (3), we get

dθ(ςn+1, ςn) ≤ kdθ(ςn, ςn−1) +
1− k

2
dθ(ςn, ςn−1) =

1 + k

2
dθ(ςn, ςn−1).

By induction,

dθ(ςn, ςn+1) ≤ (
1 + k

2
)ndθ(ς0, ς1), ∀n ≥ 1.

On the other hand, by (1), and since sinh t ≥ t ∀t ≥ 0, we get

θ(sinh(d(ςn, ςn+1))) ≥ c(sinh(d(ςn, ςn+1)))
τ ≥ c(d(ςn, ςn+1))

τ , ∀n ≥ 1.

This yields to

d(ςn, ςn+1) ≤ ((
1 + k

2
)
1
τ )n(

dθ(ς0, ς1)

c
)
1
τ , ∀n ≥ 1.

Since k ∈ [0, 1) and τ > 0, one has

+∑
n=0

∞((
1 + k

2
)
1
τ )n < +∞.

So, for all p ≥ 0, we have

d(ςn, ςn+p) ≤ d(ςn, ςn+1) + d(ςn+1, ςn+2) + · · ·+ d(ςn+p−1, ςn+p).

That is,

d(ςn, ςn+p) ≤ (
dθ(ς0, ς1)

c
)
1
τ

n+p−1∑
i=n

((
1 + k

2
)
1
τ )n.

By summing the geometric series, we find

+∞∑
i=n

((
1 + k

2
)
1
τ )i → 0 as n→ +∞. (3.4)

The symmetry of d leads to
lim

n,m→+∞
d(ςn, ςm) = 0. (3.5)

This implies that {ςn} is Cauchy in the complete MS (X, d). so {ςn} converges to some
ς∗ ∈ X. Next, the lower semi-continuity of fT yields that

d(x∗, T ς∗) = fT (ς
∗) ≤ lim inf

n→+∞
d(ςn, T ςn) ≤ lim inf

n→+∞
d(ςn, ςn+1) = 0.

Finally, we get d(ς∗, T ς∗) = 0, that is, ς∗ ∈ Tς∗ = Tς∗. Then, ς∗ is a FP of T.



H. Aydi et al. / Eur. J. Pure Appl. Math, 18 (2) (2025), 6015 8 of 14

3.2. θ−hyperbolic contractions via manageable functions

In 2014, a new class of mappings called manageable functions was explored by Du and
Khojasteh [14]. They used this class to obtain some FP theorems. In 2017, Hussain et
al. [15] established some FP theorems in the setting of MSs for contraction mappings via
manageable functions.

Definition 6. [14] A manageable function η : R× R → R is a function so that:

(η1) η(ℏ, ℓ) < ℓ− ℏ for all ℏ, ℓ > 0;

(η2) For each bounded {ℏn} in (0,+∞) and each non-increasing {ℓn} in (0,+∞),

lim sup
n→+∞

ℏn + η(ℏn, ℓn)
ℓn

< 1.

Let M̂an(R) be the set of manageable functions. We give the next two examples.

Example 3. [14] Let ℘ ∈ [0, 1). Then ηk : R2 → R defined by

ηk(ℏ, ℓ) = ℘ℓ− ℏ

is a manageable function.

Example 4. [16] Let η : R× R → R be the function defined by

η(ℏ, ℓ) =

{
ψ(ℓ)− ϕ(ℏ) if (ℏ, ℓ) ∈ [0,+∞)× [0,+∞),

f(ℓ, ℏ) otherwise,

where f : R2 → R is a given function and ψ, ϕ : [0,+∞) → R are two functions sp that

• ψ(ℏ) < ℏ ≤ ϕ(t) for all ℏ > 0, and

• lim supr→ℏ+
ψ(r)
r < 1 for any t ≥ 0.

Then, η ∈ M̂an(R). Indeed, for any s, t > 0,

η(t, s) = ψ(s)− ϕ(t) < s− t,

so, (η1) holds. Let {tn} be a bounded and {sn} be a non-increasing in (0,+∞). Then
limn→+∞ sn exists in [0,+∞). Hence,

lim sup
n→+∞

ψ(sn)

sn
= lim sup

r→t+

ψ(r)

r
< 1.

Thus, we get

lim sup
n→+∞

tn + η(tn, sn)

sn
= lim sup

n→+∞

ψ(sn) + tn − ϕ(tn)

sn
≤ lim sup

n→+∞

ψ(sn)

sn
< 1.

It follows that (η2) holds.



H. Aydi et al. / Eur. J. Pure Appl. Math, 18 (2) (2025), 6015 9 of 14

The next lemma is needful.

Lemma 2. Let (X, d) be a MS, Ξ ∈ CB(X) and k > 0. Let θ ∈ Θτ for some τ > 0. When
a ∈ X and dθ(a,Ξ) < k, then there is b ∈ Ξ so that

dθ(a, b) < k.

Proof. Let a ∈ X and suppose that dθ(a,B) < k. Recall that

dθ(a,B) := inf
b∈B

dθ(a, b).

Suppose in the contrary, for all b ∈ B we have dθ(a, b) ≥ k. Thus, infb∈B dθ(a, b) =
dθ(a,B) ≥ k, which is impossible. So there exists a point b ∈ B such that dθ(a, b) < k.

Our second result for multivalued mappings involves manageable functions.

Theorem 4. Let (X, d) be a complete MS and T : X → CB(X). Let θ ∈ Θτ for some

τ > 0. Suppose there is η ∈ M̂an(R) so that

η(Hθ(Tx, Ty), dθ(ϖ, ς)) ≥ 0 ∀ϖ, ς ∈ X. (6)

If fT is lower semi-continuous, then T admits a FP.

Proof. Let ς0 ∈ X and ς1 ∈ Tς0. When ς1 = ς0 or ς1 ∈ Tς1, one has ς1 is a FP of T .
Otherwise, suppose ς1 ̸= ς0 and ς1 /∈ Tς1. So, dθ(ς0, ς1) > 0 and dθ(ς1, T ς1) > 0. By (6),
we have

η(Hθ(Tς0, T ς1), dθ(ς0, ς1)) ≥ 0. (7)

Define the function λ : R× R → R by

λ(t, s) =

{
t+η(t,s)

s if t, s > 0,

0 otherwise.

By (η1), we have
0 < λ(t, s) < 1 for every t, s > 0. (8)

Also, if η(t, s) ≥ 0, then
0 < t ≤ sλ(t, s) for all t, s > 0. (9)

From (7) and (8), we get

0 < λ(Hθ(Tς0, T ς1), dθ(ς0, ς1)) < 1. (10)

Since dθ(ς1, T ς1) > 0, by using (10), we have

dθ(ς1, T ς1) <
1√

λ(Hθ(Tς0, T ς1), dθ(ς0, ς1))
dθ(ς1, T ς1).



H. Aydi et al. / Eur. J. Pure Appl. Math, 18 (2) (2025), 6015 10 of 14

Using Lemma 2, there is ς2 ∈ Tς1 so that

dθ(ς1, ς2) <
1√

λ(Hθ(Tς0, T ς1), dθ(ς0, ς1))
dθ(ς1, T ς1). (11)

It follows that
dθ(ς1, T ς1) ≤ dθ(ς0, ς1)λ(Hθ(Tς0, T ς1), dθ(ς0, ς1)). (12)

Combining (11) and (12), we get

dθ(ς1, ς2) ≤
√
λ(Hθ(Tς0, T ς1), dθ(ς0, ς1))dθ(ς0, ς1).

Note that ς2 ̸= ς1 because ς1 /∈ Tς1. When ς2 ∈ Tς2, ς2 is a FP of T . Suppose ς2 /∈ Tς2.
Hence, by (6),

η(Hθ(Tς1, T ς2), dθ(ς1, ς2)) ≥ 0.

Since dθ(ς2, T ς2) > 0, by using (10), we have

dθ(ς2, T ς2) <
1√

λ(Hθ(Tς1, T ς2), dθ(ς1, ς2))
dθ(ς1, T ς2).

Lemma 2 implies the existence of a point ς3 ∈ Tς2 such that

dθ(ς2, ς3) <
1√

λ(Hθ(Tς1, T ς2), dθ(ς1, ς2))
dθ(ς2, T ς2).

Similarly, we get

dθ(ς2, ς3) ≤
√
λ(Hθ(Tς1, T ς2), dθ(ς1, ς2))dθ(ς1, ς2).

Continuing the same work, we build {ςn} in X so that for any n ≥ 1,

dθ(ςn, ςn+1) ≤
√
λ(Hθ(Tςn−1, T ςn), dθ(ςn−1, ςn))dθ(ςn−1, ςn). (13)

From (8) and (13), we get 0 < dθ(ςn, ςn+1) < dθ(ςn−1, ςn) for all n, which yields that
{dθ(ςn−1, ςn)} is non-increasing and positive, so it is convergent. Also,

0 < Hθ(Tςn−1, T ςn) < dθ(ςn−1, ςn),

for all n, which yields that {Hθ(Tςn−1, T ςn)} is bounded. From (η2),

lim sup
n→+∞

λ(Hθ(Tςn−1, T ςn), dθ(ςn−1, ςn)) < 1. (14)

Let
λn =

√
λ(Hθ(Tςn−1, T ςn), dθ(ςn−1, ςn)), ∀n ≥ 1. (15)

From (13), we get
dθ(ςn, ςn+1) ≤ λndθ(ςn−1, ςn), ∀n ≥ 1. (16)



H. Aydi et al. / Eur. J. Pure Appl. Math, 18 (2) (2025), 6015 11 of 14

By (14), there are α ∈ (0, 1) and n0 ∈ N so that

λn ≤ α, ∀n ≥ n0.

Hence, by (16), we get

dθ(ςn, ςn+1) ≤ αdθ(ςn−1, ςn), ∀n ≥ n0.

Thus,
dθ(ςn, ςn+1) ≤ αn−n0+1dθ(ςn0−1, ςn0), ∀n ≥ n0.

Moreover, by (1), and sinh(t) ≥ t for all t ≥ 0, one gets

d(ςn, ςn+1) ≤ (α
1
τ )n−n0+1(

dθ(ςn0−1, ςn0)

c
)
1
τ , ∀n ≥ n0.

Now, for m > n ≥ n0, we have

d(ςn, ςm) ≤
m−1∑
i=n

d(ςi, ςi+1) ≤ (
dθ(ςn0−1, ςn0)

c
)
1
τ

+∞∑
i=n

(α
1
τ )i−n0+1 → 0 as n→ +∞.

Thus,
lim

n,m→+∞
d(ςn, ςm) = 0.

So {ςn} is a Cauchy sequence in the complete MS (X, d). Then, there is u ∈ X so that

lim
n→+∞

d(ςn, u) = 0.

By the lower semi-continuity of T , we obtain that u ∈ Tu.

Remark 1. From Theorem 4, several corollaries could be derived following particular
cases of manageable functions.

The next example inspired from Example 3.3 in [1] makes effective Theorem 4. Here,
the theorem of Nadler [10] is not applicable..

Example 5. Let X = {1, 2, 3}. Take d the metric on X given as

d(ϖ, ς) = d(ς,ϖ), d(ς, ς) = 0 ∀ϖ, ς ∈ X

and
d(1, 2) = 1, d(1, 3) = 4, d(2, 3) = 5.

Notice that (X, d) is a complete MS. Choose T : X → CB(X) as

T1 = T3 = {1} and T2 = {1, 3}.



H. Aydi et al. / Eur. J. Pure Appl. Math, 18 (2) (2025), 6015 12 of 14

We point out that T is not a contraction in the sense of Nadler [10]. Indeed,

H(T1, T2) = max{d(1, 1), d(1, 3)} = 4 > 1 = d(1, 2).

We now introduce the mapping θ : [0,+∞) → [0,+∞) defined by

θ(t) =



7t

sinh 1
if 0 ≤ t ≤ sinh 1,

2t

sinh 4
if sinh 1 < t < sinh 4,

5t

4 sinh 5
if t ≥ sinh 4.

Clearly,

θ(t) ≥ 2

sinh 5
t, ∀t ≥ 0,

which shows that θ ∈ Θ1 and θ(0) = 0. Furthermore, take η(t, s) = ks− t for all s, t ∈ R
with k ∈ [25 , 1).
We have

Hθ(T1, T2) = max{dθ(1, 3), dθ(1, 1)} = dθ(1, 3) = θ(sinh 4) = 2

=
2

7
× 7 =

2

7
× θ(sinh 1) =

2

7
dθ(1, 2) ≤

2

5
dθ(1, 2).

Also,

Hθ(T2, T3) = 2 =
2

5
× 5 =

2

5
θ(sinh 5) =

2

5
dθ(2, 3).

Furthermore,

Hθ(T1, T3) = 0 ≤ 2

5
dθ(1, 3),

which implies

Hθ(Tς, Tϖ) ≤ 2

5
dθ(ϖ, ς)∀ϖ, ς ∈ X.

We also have

η(Hθ(Tς, Tϖ), dθ(ϖ, ς)) = kHθ(Tς, Tϖ)− dθ(ϖ, ς) ≥ (k − 2

5
)dθ(ϖ, ς) ≥ 0∀ϖ, ς ∈ X.

Let ς ∈ X and {ςn} ⊂ X so that limn→+∞ d(ςn, ς) = 0. Then, there is n0 ≥ 0 so that
ςn = ς for all n0 ≥ 0. Then Tςn = Tς for all n0 ≥ 0. Henc,e d(ς, T ς) = d(ςn, T ςn) for all
n0 ≥ 0. Finally, we get

d(ς, T ς) = lim inf
n→+∞

d(ςn, T ςn).

Consequently, all required hypothesises of Theorem 4 hold. Here, 1 is a FP of T .



H. Aydi et al. / Eur. J. Pure Appl. Math, 18 (2) (2025), 6015 13 of 14

4. Conclusion

In this work, we initiated the concept of a Hausdorff θ-hyperbolic sine distance function.
We proved two FP results for multivalued contraction mappings, one of Nadler type , and
the second using manageable functions. As open problems, we suggest to prove further
FP results for multivalued mappings, using either different types of control function, like:
(i) implict functions;
(ii) α-admissibility,
or, via generalized metrics.

Acknowledgements

The authors I. Ayoob and N. Mlaiki would like to thank Prince Sultan University for
paying the APC through TAS LAB. The authors declare no conflict of interest

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