@ Appl. Gen. Topol. 22, no. 1 (2021), 91-108doi:10.4995/agt.2021.13902
© AGT, UPV, 2021

Convexity and boundedness relaxation for fixed

point theorems in modular spaces

Fatemeh Lael a and Samira Shabanian b

a
Department of Mathematics, Buein Zahra Technical University, Buein Zahra, Qazvin, Iran.

(f lael@bzte.ac.ir)
b
Microsoft Research (samira.shabanian@microsoft.com)

Communicated by S. Romaguera

Abstract

Although fixed point theorems in modular spaces have remarkably

applied to a wide variety of mathematical problems, these theorems

strongly depend on some assumptions which often do not hold in prac-

tice or can lead to their reformulations as particular problems in normed

vector spaces. A recent trend of research has been dedicated to studying

the fundamentals of fixed point theorems and relaxing their assump-

tions with the ambition of pushing the boundaries of fixed point theory

in modular spaces further. In this paper, we focus on convexity and

boundedness of modulars in fixed point results taken from the literature

for contractive correspondence and single-valued mappings. To relax

these two assumptions, we seek to identify the ties between modular

and b-metric spaces. Afterwards we present an application to a par-

ticular form of integral inclusions to support our generalized version of

Nadler’s theorem in modular spaces.

2010 MSC: 46E30; 47H10; 54C60.

Keywords: modular space; fixed point; correspondences; b-metric space.

1. Introduction

Compared to 1922 when Banach fixed point theorem has been proved [8],
certainly, fixed point theory now plays a significant and meaningful role in
both the field of Mathematics and many real-life applications due to providing

Received 21 June 2020 – Accepted 18 January 2021

http://dx.doi.org/10.4995/agt.2021.13902


F. Lael and S. Shabanian

a general framework which opens the door to the development of many other
approaches. In one approach, fixed point theory in modular spaces has received
a lot of attention after being proposed as a generalization of normed spaces [39,
40, 42, 45, 46]. A growing literature on fixed point theorems in Modular spaces
deals with rigorous formulations and proofs of many interesting problems which
are applicable in a wide variety of settings, including Quantum Mechanics,
Machine Learning and etc. Fixed point theory in modular spaces has its root
in [27] by using some constructive techniques for single-valued mappings. This
work has been widely cited as the inspiration for a variety of fixed point work
along with [25, 26]. This line of work was extended by several works in a variety
of ways. In one successful approach, in 1969, Nadler proposed the Banach
contraction principle for multivalued mappings of in modular spaces [41]. A
wide range of extensions was subsequently proposed by various authors, based
on different relaxations [1, 18, 19, 31, 50, 51, 53]. Furthermore, authors in [34]
focus on a particular case of multivalued mappings in modular spaces with a
key property of modulars, additivity. Then, [3] explores the existence of fixed
points of a specific type of G-contraction and G-nonexpansive mappings in
modular function spaces.

In another approach, in 1993, Czerwik in [15, 16] proposed the first Ba-
nach’s fixed point theorem for both single and multivalued mapping in b-metric
spaces, introduced by Bourbaki and Bakhtin [7, 13]. Then, authors in [30] ex-
tended it for some particular types of contractions in the context of b-metric
spaces. Along this direction, many researchers studied the extension of various
well known fixed point results for various types of contractive mapping in the
framework of b-metric spaces [12, 33, 47, 48, 57].

Although fixed point theory is shown to be successful in challenging problems
and has contributed significantly to many real-world problems, various fixed
point theorems strongly are proved under strong assumptions. In particular,
in modular spaces, some of these assumptions can lead to having some induced
norms. So, some assumptions that often do not hold in practice or can lead to
their reformulations as a particular problem in a normed vector spaces. A recent
trend of research has been dedicated to studying the fundamentals of fixed
point theorems and relaxing their assumptions with the ambition of pushing
the boundaries of fixed point theory in modular spaces further [2, 18, 27].

The aim of the work presented in this paper is to contribute to a deeper
understanding of fixed point results in modular spaces and to improve their
conditions and assumptions by addressing the open questions and challenges
outlined in the literature by identifying the ties between modular spaces and
b-metric spaces. In a bird-eyes view, the paper starts with Section 2 which is
a brief introduction to modular and b-metric spaces along with the required
concepts. Afterwards we describe the relation between these two particular
spaces. Section 3 introduces some techniques and ways of improving some
current fixed point results. Finally, an application to integral inclusions is
provided in Section 5.

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Convexity and boundedness relaxation for fixed point theorems in modular spaces

2. Background

This section will serve as an introduction to some fundamental concepts
of modular and b-metric spaces. A detailed introduction can be found, for
example, in the textbooks [4, 5, 7, 32, 40, 52].

2.1. Modular spaces. A modular space is a pair (X,ρ) where X is a real
linear space and ρ is a real valued functional on X which satisfies the conditions:

(1) ρ(x) = 0 if and only if x = 0,
(2) ρ(−x) = ρ(x),
(3) ρ(αx + βy) ≤ ρ(x) + ρ(y), for any nonnegative real numbers α,β with

α + β = 1.

The functional ρ is called a modular on X. There are many arguably im-
portant special instances of well known spaces in which these properties are
fulfilled [44, 45, 46, 54]. Interestingly, it is shown that a modular induces a
vector space Xρ = {x ∈ X : ρ(αx) → 0 as α → 0} which is called a modular
linear space. Furthermore, Musielak and Orlicz in [39, 45, 46] naturally pro-
vide the first definitions of the following key concepts in a modular space (X,ρ):

D1. A sequence xn in B ⊆ X is said to be ρ-convergent to a point x ∈ B if
ρ(xn − x) → 0 as n → ∞.
D2. A ρ-closed subset B ⊆ X is meant that it contains the limit of all its
ρ-convergent sequences.
D3. A sequence xn in B ⊆ X is said to be ρ-Cauchy if ρ(xm − xn) → 0 as
m,n → ∞.
D4. A subset B of X is said to be ρ-complete if each ρ-Cauchy sequence in B
is ρ-convergent to a point of B.
D5. ρ-bounded subsets: A subset B ⊆ Xρ is called ρ-bounded if sup

x,y∈B

ρ(x−

y) < ∞.
D6. ρ-compact subsets: A ρ-closed subset B ⊆ X is called ρ-compact if any
sequence xn ∈ B has a ρ-convergent subsequence.

For a modular space (X,ρ), the function ωρ which is said growth function
[17] is defined on [0,∞) as follows:

ωρ(t) = inf{ω : ρ(tx) ≤ ωρ(x) : x ∈ X,0 < ρ(x)}.

It is easy to show that when (X,ρ) satisfies ωρ(2) < ∞, then every ρ-
convergent sequence in (X,ρ) is ρ-Cauchy. Also, we note that in such case
every ρ-compact set is ρ-bounded and ρ-complete [35].

2.2. b-metric spaces. Now, we turn our attention to another important and
related space in the sense that it can be induced by a modular, namely b-
metric spaces. It is shown that a modular induces some well known operators
of which we are interested in b-metrics; a b-metric on a nonempty set X is
a real function d : X × X → [0,∞) such that for a given real number s ≥ 1
satisfies the conditions:

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F. Lael and S. Shabanian

(1) d(x,y) = 0 if and only if x = y,
(2) d(x,y) = d(y,x),
(3) d(x,z) ≤ s[d(x,y) + d(y,z)], for all x,y,z ∈ X,

the pair (X,d) is called a b-metric space. As stated in [56], it is true that
a b-metric space is not necessarily a metric space. With s = 1, however, a
b-metric space is a metric space. Furthermore, many concepts like convergent
and Cauchy sequences, complete spaces, closed sets and etc are easily defined
in b-metric spaces [11] which are denoted by b- convergent and b- Cauchy
sequences, b- complete spaces, b- closed sets. Moreover, it has been shown that
a lot of metric fixed point theorems can be extended to b-metric spaces [11, 58],
although b-metric, in the general case, is not continuous, lim

n→∞
xn = x does not

necessarily imply lim
n→∞

d(xn,y) = d(x,y) (see [6, 21] for further details).

The notion of b-metric spaces were introduced to reach the generalization
of some known fixed point theorems for single valued mappings and correspon-
dences [9, 10, 15, 16].

In the following example, we generalize some examples which are mentioned
in [6].

Example 2.1. Suppose that (X,d) is a b-metric space with s ≥ 1. Then
(X,dr) is a b-metric space, for all r ∈ R+. Since from the general form of
Holder’s inequality [55], for every x,y,z ∈ X and r ∈ R+ with 1 + 1

r
≥ 1, we

get

d(x,y) ≤ s(d(x,z) + d(z,y)) ≤ (2s)(dr(x,z) + dr(z,y))
1

r ,

that is,
dr(x,y) ≤ (2s)r(dr(x,z) + dr(z,y)).

This implies that, dr is a b-metric. Since every metric d is a b-metric, then
dr is a b-metric. However, dr is not necessarily to be a metric. For example,
if d(x,y) = |x − y| is the usual Euclidean metric, d2(x,y) = |x − y|2 is not a
metric on R.

We note that a modular ρ induces a b-metric. In fact, for such modular, we
can define

d(x,y) = ρ(x − y).

Then, d is a b-metric with s = ωρ(2) and when (X,ρ) is a ρ-complete space, then
(X,d) is a b-complete space. Actually, ρ-Cauchy sequence and ρ-convergent
sequence are equivalent to b-Cauchy sequence and b-convergent sequence re-
spectively. We recall that for any subset C of (X,ρ) a correspondence f on a
set C, denoted by f : C ։ X assigns to each a ∈ C a (nonempty) subset f(a)
of X and an element x ∈ C is said to be a fixed point if x ∈ f(x). A corre-
spondence f is called continuous if xn → x and yn → y and yn ∈ f(xn) imply
y ∈ f(x). For a correspondence f we define d(a,f(b)) = inf{d(a,y) : y ∈ f(b)}
and distρ(a,f(b)) = inf{ρ(a − y) : y ∈ f(b)}. Also, Hausdorff distance is
defined as

Hρ(A,B) = max{sup
a∈A

distρ(a,B), sup
b∈B

distρ(A,b))}

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Convexity and boundedness relaxation for fixed point theorems in modular spaces

where A and B are subsets of C.

2.3. Relevant Literature. Much work has been done on the problem of the
fixed point existence for single-valued mappings and in general correspondence
in modular spaces [18, 24, 29]. Over the years, multiple authors have analyzed
various conditions which suffice to guarantee the existence of fixed points for a
board class of functions in modular spaces. Arguably, the following (H1)-(H4)
conditions are specified to be some of the most common and popular ones in
modular spaces:

(H1) ∆2-condition: A modular ρ is said to satisfy the ∆2-condition [45, 46]
if ρ(2xn) → 0, whenever sup

n
ρ(xn) → 0 as n → ∞.

(H2) ∆2-type condition: A modular ρ is said to satisfy the ∆2-type condi-
tion [45, 46] if there exists k > 0 such that ρ(2x) ≤ kρ(x) for all x ∈ Xρ.
(H3) s̃-convex modulars: If condition (3) in the modular definition is re-
placed by ρ(αx + βy) ≤ αs̃ρ(x) + βs̃ρ(y) for all α,β ∈ [0,∞) with αs̃ + βs̃ = 1
with an s̃ ∈ (0,1], the modular ρ is called an s̃-convex modular [22]. In partic-
ular, a 1-convex modular is simply called convex.
(H4) Fatou property: A modular ρ has the Fatou property [14] if ρ(x) ≤
lim inf ρ(xn), whenever xn → x.

Some excellent overviews of (H1)-(H4) conditions are provided in [27, 22, 54].
It is shown that a modular ρ implies that

‖x‖ρ = inf{a > 0 : ρ(
x

a
) ≤ 1},

defines an F-norm on Xρ. Specifically, if ρ is convex, ‖ · ‖ρ is a norm and
it is frequently called the Luxemburg norm [23]. Note that a modular space
determined by a function modular ρ will be called a modular function space
and will be denoted by Lρ. Then, it is not difficult to show that ‖ · ‖ρ is an
F-norm induced by ρ. More importantly, (Lρ,‖ · ‖ρ) is a complete space.

Being able to define such norm in a real vector space can lead to a smooth
proof for many fixed point theorems in very specific modular spaces. For in-
stance, an earlier Work on this topic goes back to Theorem 2-2 of [27] which
was proposed in the early 1990s:

Theorem 2.2 ([27]). Let ρ be a function modular satisfying the ∆2-condition
and let B be a ‖ · ‖ρ-closed subset of Lρ. Let T : B → B be a single-valued
mapping such that ρ(T(f) − T(g)) ≤ kρ(f − g) where f,g ∈ B and k ∈ (0,1).
Then T has a fixed point if supn(2T

n(f0)) < 1.

Since then, there has been significant work on extending and improving this
result further in many ways.

Ait Taleb and Hanebaly present some example illustrating that the following
result (Theorem I-1 of [2]) tends to be more applicable than Theorem 2.2:

Theorem 2.3 ([2]). Suppose that Xρ is a ρ-complete modular space where ρ
is an s̃-convex modular satisfying the ∆2-condition and has the Fatou property.

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F. Lael and S. Shabanian

Moreover, assume that B is a ρ-closed subset of Xρ and T : B → B is a
single-valued mapping such that there are c,k ∈ R+ that c > max{1,k} and
ρ(c(T(x) − T(y))) ≤ ks̃ρ(x − y) where x,y ∈ B. Then T has a fixed point.

However, it should be stressed that Theorem 2.2 is not generalized by The-
orem 2.3. As mentioned in [2], we can ask what if it is mentioned that they
are unable to prove whether the conclusion of Theorem 2.3 is true if we have
c = 1 and 0 < k < 1. We note that our main result, Theorem 3.3, replied to
this open question, also it generalized Theorem 2.2, when wρ(2) < ∞.

Various extensions were subsequently proposed by various authors, based
on different relaxations that require: A recent extension of Theorem 2.2 to the
cases of correspondence maps appeared in 2006 in Theorem 3-1 of [18] as:

Theorem 2.4 ([18]). Let ρ be a convex function modular satisfying the ∆2-type
condition, B a nonempty ρ-bounded ρ-closed subset of Lρ, and f : B ։ B a
ρ-closed valued correspondence such that there exists a constant k ∈ [0,1) that

Hρ(f(f1),f(f2)) ≤ kρ(f1 − f2),

where f1,f2 ∈ B. Then f has a fixed point.

Additionally, in 2009, this result is improved to Theorem 2-1 of [34]:

Theorem 2.5 ([34]). Let ρ be a convex modular satisfying ∆2-type condition
and B ⊂ Lρ be a nonempty ρ-closed ρ-bounded subset of the modular space Lρ.
Then any closed valued correspondence f : B ։ B such that for f1,f2 ∈ B
and f3 ∈ f(f1), there is f4 ∈ f(f2) such that ρ(f3 − f4) ≤ kρ(f2 − f1), where
k ∈ (0,1), has a fixed point.

In both Theorems 2.4 and 2.5, it is assumed that the correspondence defined
on a ρ-bounded subset of a modular space (X,ρ) with convex modular. In [35]
Theorem 2-5, the correspondence has ρ-compact set values.

Theorem 2.6 ([35]). Let B be a ρ-bounded subset of ρ-complete space (X,ρ).
Let f : B ։ B be a correspondence with ρ-compact values that for each x,y ∈ C
and z ∈ f(x), there exists w ∈ f(y) such that

ρ(z − w) ≤ kρ(x − y),

where 2kwρ(2)
2 < 1. Then f has a fixed point.

Our main result, Theorem 3.3, is definitely a generalization of Theorems 2.4,
2.5, 2.6.

In the following sections, we provide certain conditions under which we can
guarantee the existence of fixed points for myriad mappings and some strong
assumptions such as the convexity of modulars and the ρ-boundedness of the
domain of a correspondence are relaxed which can lead to making our theorems
much stronger and more applicable.

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Convexity and boundedness relaxation for fixed point theorems in modular spaces

3. Main results

In this section, we focus on the case of the ρ-complete modular space X
and consider f : B ։ B is a correspondence with ρ-closed valued where B
is a ρ-closed subset of X. Further, we assume ωρ(2) < ∞. Also, to ease the
notation let us now denote X = (X,ρ).

To prove our main results, we make use of the following lemma suggested
by the reviewer.

Lemma 3.1 ([36]). Let (X,d) be a b-metric space with s ≥ 1 and {xn} be a
b-convergent sequence in X with lim

n→∞
xn = x. Then for all y ∈ X,

s−1d(x,y) ≤ lim
n→∞

inf d(xn,y) ≤ lim
n→∞

supd(xn,y) ≤ sd(x,y).

We also use the following lemma taken from the literature to obtain our
main results.

Lemma 3.2 ([38]). A sequence {xn} in a b-metric space (X,d) is a b-Cauchy
sequence if there exists k ∈ [0,1) such that

d(xn,xn+1) ≤ kd(xn−1,xn),

for every n ∈ N.

Now we can state one of our main results which is an equivalent of Nadler’s
theorem in [41] in a modular space. We would like to highlight that while the
convexity of ρ is required for both theorems 2.4 and 2.5, we show that it can
be removed.

Theorem 3.3. Consider k ∈ [0,1) and for every y ∈ B, there exists w ∈ f(y)
such that ρ(z − w) ≤ kρ(x − y) for every x ∈ B and z ∈ f(x). Then f has a
fixed point.

Proof. Take x0 ∈ B and x1 ∈ f(x0). We know from our assumption that for
every n ≥ 1 there exists xn+1 ∈ B such that xn+1 ∈ f(xn) and

d(xn+1,xn) ≤ kd(xn,xn−1),

where d(x,y) = ρ(x − y) is the b-metric induced by the modular ρ. Note that
by Lemma 3.2, {xn} is a b-Cauchy sequence in the ρ-complete space B. Which
means that there exists x ∈ B such that xn → x as n → ∞.

On the other hand, from our assumption, it is true that for every xn ∈
f(xn−1), there exists yn ∈ f(x) such that

ρ(yn − xn) ≤ kρ(x − xn−1).

It implies that limρ(yn −xn) = 0, and as a result we have limd(xn,yn) = 0. By
Lemma 3.1, s−1 limd(yn,x) ≤ limd(yn,xn). Therefore, limd(yn,x) = 0 which
means limρ(yn − x) = 0. Since yn ∈ f(x) and f(x) is ρ-closed, it concludes
our proof. �

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F. Lael and S. Shabanian

Example 3.4. Define the modular ρ : ℓ∞(R) → R as follows:

ρ((xn)) = sup
n∈N

|xn|,

for every (xn) ∈ ℓ∞(R) and the correspondence f : ℓ∞(R) ։ ℓ∞(R) by

f((xn)) = {(
1

i
,
x1

2
,
x2

2
, . . .) : i ∈ N}.

So (ℓ∞(R),ρ) is a ρ-complete space and the correspondence f satisfies the
contraction of Theorem 3.3. Indeed, for (xn),(yn) ∈ ℓ∞(R), and

(zn) = (
1

i
,
x1

2
,
x2

2
, . . .) ∈ f((xn)),

there is (wn) = (
1
i
,
y1
2
,
y2
2
, . . .) ∈ f((yn)) such that

ρ((zn) − (wn)) = sup
n

|
xn − yn

2
| ≤

1

2
sup
n

|xn − yn| =
1

2
ρ((xn) − (yn))

Thus all assumptions of Theorem 3.3 are fulfilled and f has many fixed points
such as {(1

i
, 1
2i
, 1
4i
, . . .) : i ≥ 2}.

The following result shows that Theorem 3.3 can be even further generalized:

Theorem 3.5. Consider for every x, y ∈ B and z ∈ f(x), there exists w ∈
f(y) such that

ρ(z − w) ≤ k max{ρ(x − y),αρ(x − z),αρ(y − w),
β

2
(ρ(x − w) + ρ(y − z))},

where α,β ∈ [0,1], and k ∈ [0,1). Then f has a fixed point if one of the
following assumptions satisfies:

i: f is continuous.
ii: ρ is continuous i.e. limρ(xn) = ρ(x) as xn → x.
iii: kβωρ(2) < 1.

Proof. Let us define a sequence {xn} with x0 ∈ B, x1 ∈ f(x0) and xn+1 ∈ f(xn)
such that

ρ(xn+1 − xn) ≤ k max{ρ(xn − xn−1),αρ(xn − xn−1),

αρ(xn+1 − xn),
β

2
ρ(xn−1 − xn+1)},(3.1)

for every n ≥ 1. As becomes clear by equation (3.1), the right hand side of this
equation is not αρ(xn − xn+1) or αρ(xn − xn−1). Now it is easy to see that
(3.2)

ρ(xn+1 − xn) ≤ k max{ρ(xn − xn−1),
βωρ(2)

2
(ρ(xn−1 − xn) + ρ(xn − xn+1))},

Now we distinguish the two cases whether the right hand side of equation
(3.2) is ρ(xn − xn−1) or

βωρ(2)

2
(ρ(xn−1 − xn) + ρ(xn − xn+1)).

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Convexity and boundedness relaxation for fixed point theorems in modular spaces

If the former is the case, then using equation (3.2), we have ρ(xn+1 − xn) ≤
kρ(xn − xn−1). The latter leads to

ρ(xn+1 − xn) ≤
kβωρ(2)

2 − kβωρ(2)
ρ(xn − xn−1).

Now, we are ready to derive a new upper bound for equation (3.2) as:

ρ(xn+1 − xn) ≤ max{k,
kβωρ(2)

2 − kβωρ(2)
}ρ(xn − xn−1).

Thus it follows from Lemma 3.2 and ρ-completeness of B that there exists
x ∈ B such that xn → x as n → ∞.

The proof is obviously complete under assumption (i). Now assume (ii)
holds. We know that for xn, there exists yn ∈ f(x) such that

ρ(xn − yn) ≤ k max{ρ(xn−1 − x),αρ(xn−1 − xn),

αρ(yn − x),
β

2
(ρ(xn−1 − yn) + ρ(x − xn))}.(3.3)

By looking at the definition of the ρ-convergent sequences, it becomes clear
that lim

xn→x
ρ(xn − x) = 0 and lim

xn→x
ρ(xn − xn−1) = 0. Now (ii) leads to

lim
xn→x

ρ(xn − yn) = ρ(x − yn). Using this and equation (3.3), it is easy to

derive that

lim
xn→x

ρ(xn − yn) ≤ k max{ lim
xn→x

αρ(yn − xn), lim
xn→x

βρ(yn − xn)

2
}

≤ k lim
xn→x

ρ(yn − xn),

which yields lim
xn→x

ρ(xn − yn) = 0. Now from the facts that f(x) is ρ-closed

and

ρ(yn − x) ≤ ωρ(2)(ρ(xn − x) + ρ(yn − xn)),

we have limn→∞ yn = x ∈ f(x).
Finally, if assumption (iii) is satisfied, it is possible to repeat the proof

presented for assumption (ii) to get equation (3.3). Therefore, it follows that

ρ(xn − yn) ≤ k max{ρ(xn−1 − x),αρ(xn − xn−1),αρ(yn − x),

β

2
(ρ(xn−1 − yn) + ρ(x − xn))},

≤ k max{αωρ(2)ρ(yn − xn),
β

2
(ρ(xn−1 − yn) + ρ(x − xn))},

≤ k max{αωρ(2)ρ(yn − xn),

β

2
[ωρ(2)(ρ(xn − xn−1) + ρ(xn − yn)) + ρ(x − xn)]},

≤ max{kαωρ(2),
kβωρ(2)

2
}ρ(xn − yn).

This implies that limn→∞ ρ(xn − yn) = 0, by Lemma 3.1, limρ(yn − x) = 0
and then x ∈ f(x). �

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F. Lael and S. Shabanian

Example 3.6. For a function p : N → [1,∞), define the vector space

ℓp(·) = {(xn) ∈ R
N : Σ∞n=1|λxn|

p(n) < ∞, for some λ > 0},

and the modular ρ : ℓp(·) → R by

ρ((xn)) = Σ
∞
n=1|xn|

p(n),

for every (xn) ∈ ℓp(·). Now define the corespondence f : ℓp(·) ։ ℓp(·) by

f((xn)) = {(
sinx1

2
,
x2

3
, · · · ,

xn

n + 1
, . . .),(

cosx1

2
,
x2

3
, · · · ,

xn

n + 1
, . . .)},

for every (xn) ∈ ℓp(·). For (xn),(yn) ∈ ℓp(·) and (zn) = (
sinx1

2
, x2

3
, · · · , xn

n+1
, . . .),

there is (wn) =
siny1

2
,
y2
3
, · · · , yn

n+1
, . . .) such that

ρ((zn) − (wn)) = |
sinx1 − siny1

2
|p(1) + Σ∞n=2|

xn − yn
n + 1

|p(n),

≤
1

2
max{ρ((xn) − (yn)),ρ((xn) − (zn)),ρ((yn) − (wn)),

1

2
(ρ((xn) − (wn)) + ρ((yn) − (zn)))}.

Otherwise (zn) = (
cosx1

2
, x2

3
, · · · , xn

n+1
, . . .), there is (wn) =

cosy1
2
,
y2
3
, · · · , yn

n+1
, . . .)

such that

ρ((zn) − (wn)) = |
cosx1 − cosy1

2
|p(1) + Σ∞n=2|

xn − yn
n + 1

|p(n),

≤
1

2
max{ρ((xn) − (yn)),ρ((xn) − (zn)),ρ((yn) − (wn)),

1

2
(ρ((xn) − (wn)) + ρ((yn) − (zn)))}.

Thus all assumptions of Theorem 3.5 are fulfilled and f has a fixed point (0).

From now on, for the sake of clearness of notation, we consider ψ to be a
continuous and nondecreasing self-map on [0,∞) such that ψ(t) = 0 if and only
if t = 0. This notation has been taken from the literature [28] and it was shown
that, under mild assumptions, fixed point exists for many families.

Theorem 3.7. Let k ∈ [0,1), α ≥ 0 and for every x,y ∈ B we have

Hx,y =
distρ(x,f(y)) + distρ(y,f(x))

2wρ(2)
.

Furthermore, suppose that for every y ∈ B there is w ∈ f(y) such that

ψ(
1

k
ρ(z − w)) ≤ ψ(S(x,y)) + αψ(I(x,y)),

for every x ∈ B and z ∈ f(x) where

S(x,y) = max{ρ(x − y),distρ(x,f(x)),distρ(y,f(y))
1 + distρ(x,f(x))

1 + ρ(x − y)
,Hx,y},

© AGT, UPV, 2021 Appl. Gen. Topol. 22, no. 1 100



Convexity and boundedness relaxation for fixed point theorems in modular spaces

and

I(x,y) = min{distρ(x,f(x)) + distρ(y,f(y)),distρ(x,f(y)),distρ(y,f(x))}.

Then f has a fixed point, provided that it satisfies one of the following condition:

i: f is continuous.
ii: ρ is continuous.
iii: kwρ(2) < 1.

Proof. Let x0 ∈ X and x1 ∈ f(x0). By assumption, there exists x2 ∈ f(x1)
such that

ψ(
1

k
ρ(x2 − x1)) ≤ ψ(S(x1,x0)) + αψ(I(x1,x0)).

Thus, one can define a sequence of {xn} in B such that

(3.4) ψ(
1

k
ρ(xn − xn+1)) ≤ ψ(S(xn−1,xn)) + αψ(I(xn−1,xn)),

where xn ∈ f(xn−1). On the other hand, taking into account that
limn→∞ I(xn−1,xn) = 0, we have

S(xn−1,xn) = max{ρ(xn−1 − xn),distρ(xn−1,f(xn−1)),

distρ(xn,f(xn))
1 + distρ(xn−1,f(xn−1))

1 + ρ(xn−1 − xn)
,Hxn−1,xn},

≤ max{ρ(xn−1 − xn),ρ(xn − xn+1),

ρ(xn−1 − xn+1) + ρ(xn − xn)

2wρ(2)
},

= max{ρ(xn−1 − xn),ρ(xn − xn+1)}.

The right side of this inequality can be either ρ(xn−1 − xn) or ρ(xn − xn+1).
However, the latter follows that

ψ(
1

k
ρ(xn − xn+1)) ≤ ψ(ρ(xn − xn+1)) + αψ(0),

which gives a contradiction

ρ(xn − xn+1) ≤ kρ(xn − xn+1),

based on the fact that ψ is nondecreasing. Therefore, we have

ψ(
1

k
ρ(xn − xn+1)) ≤ ψ(ρ(xn−1 − xn)) + αψ(0) = ψ(ρ(xn−1 − xn)).

It leads to ρ(xn − xn+1) ≤ kρ(xn−1 − xn) for every n ∈ N. Lemma 3.2 implies
that there exists x ∈ B such that xn → x. Now, our goal is to prove x ∈ f(x).

Obviously, x is a fixed point of f if (i) holds. By considering assumption
(ii), it becomes obvious that x ∈ f(x). For xn ∈ f(xn−1), there is qn ∈ f(x)
such that

ϕ(
1

k
ρ(qn − xn)) ≤ ϕ(S(xn−1,x) + lϕ(I(xn−1,x)).

© AGT, UPV, 2021 Appl. Gen. Topol. 22, no. 1 101



F. Lael and S. Shabanian

We have

S(xn−1,x) = max
{

ρ(xn−1 − x),distρ(xn−1,f(xn−1)),

distρ(x,f(x))
1 + distρ(xn−1,f(xn−1))

1 + ρ(xn−1 − x)
,

distρ(xn−1,f(x)) + distρ(x,f(xn−1))

2wρ(2)
},

≤ max{ρ(xn−1 − x),ρ(xn−1 − xn),

distρ(x,f(x))
1 + ρ(xn−1 − xn)

1 + ρ(xn−1 − x)
,

wρ(2)(distρ(x,f(x)) + ρ(x − xn−1)) + ρ(x − xn)

2wρ(2)
}.

This implies that limS(xn−1,x) ≤ distρ(x,f(x)). Therefore

limϕ(
1

k
ρ(qn − xn)) ≤ ϕ(distρ(x,f(x))) ≤ ϕ(lim ρ(x − qn)).

So lim ρ(qn−xn) ≤ k limρ(x−qn). Since ρ is continuous and xn → x, limρ(qn−
x) ≤ kρ(x−qn). Thus limρ(qn −x) = 0. This implies that, since f(x) is closed
and qn ∈ f(x), we have x ∈ f(x). If condition (iii) is provided, since we have

limϕ(
1

k
ρ(qn − xn)) ≤ ϕ(distρ(x,f(x))),

≤ ϕ(ρ(x − qn)),

≤ ϕ(wρ(2)(ρ(x − xn) + ρ(xn − qn))).

Therefore lim ρ(xn − qn) ≤ kwρ(2) limρ(xn − qn). Now, kwρ(2) < 1 implies
limρ(xn − qn) = 0. Therefore, by Lemma 3.1, qn → x. Thus x ∈ f(x). �

4. Addressing some open questions and challenges

A series of research articles addressing challenges and hitherto open ques-
tions in the context of fixed point theory in modular spaces has been presented
[2, 49]. Our aim is to contribute to a deeper understanding of fixed point theo-
rems in modular spaces and to extend them to further general cases. A way of
doing this is to address open problems taken from the literature. For instance,
Radenović et. al. in [49] considered the following open problem

If T : B → B is a single valued mapping such that

ρ(T(x)−T(y)) ≤ k max{ρ(x−y),ρ(x−T(x)),ρ(y−T(y)),ρ(x−T(y)),ρ(y−T(x))},

for every x,y ∈ B where B ⊆ X and k ∈ R, then under what constraints does T
have a fixed point? and can answer this question under the constraints that T :
B → B is a single valued mapping and k ∈ (0, 1

wρ(2)(1+wρ(2))
). However, there

is no answer to this question in the case of multi-valued T or k ≥ 1
wρ(2)(1+wρ(2))

.

The open questions that have arisen address open questions outlined in [49]
existence of a fixed point for some certain maps:

© AGT, UPV, 2021 Appl. Gen. Topol. 22, no. 1 102



Convexity and boundedness relaxation for fixed point theorems in modular spaces

Existing of a fixed point for T has been successfully shown if
k ∈ (0, 1

wρ(2)(1+wρ(2))
) (given by Radenovic et. al. in [49]). Interestingly,

this question can be reformulated for correspondence, and we prove it with
k ∈ [0, 1

2wρ(2)
] in the next theorem.

Theorem 4.1. Let f be a correspondence that for each x, y ∈ B and z ∈ f(x),
there exists w ∈ f(y) such that

ρ(z − w) ≤ k max{ρ(x − y),ρ(x − z),ρ(y − w),ρ(x − w),ρ(y − z)},

where 2kwρ(2) < 1. Then f has a fixed point.

Proof. We first find x1 ∈ f(x0) for an arbitrary x0 ∈ B. By assumption, for
every n ≥ 1 there exists xn+1 ∈ f(xn) such that
(4.1)
ρ(xn+1 −xn) ≤ k max{ρ(xn −xn−1),ρ(xn+1 −xn),ρ(xn−1 −xn+1),ρ(xn −xn)},

It follow that

ρ(xn+1 − xn) ≤ k max{ρ(xn − xn−1),ρ(xn+1 − xn),

wρ(2)(ρ(xn−1 − xn) + ρ(xn − xn+1))},

≤ kwρ(2)(ρ(xn−1 − xn) + ρ(xn − xn+1)

which implies that

ρ(xn+1 − xn) ≤ k
′
ρ(xn−1 − xn).

where k′ =
kwρ(2)

1−kwρ(2)
. Note that k′ is never larger than one Since k < 1

2wρ(2)
.

In addition, from the fact that B is a ρ-complete set and xn is a ρ-Cauchy
sequence by Lemma 3.2, there exists x ∈ B such that xn → x.

On the other hand, for every xn, there exists yn ∈ f(x) such that

ρ(xn − yn) ≤ k max{ρ(xn−1 − x),ρ(xn−1 − xn),ρ(yn − x),ρ(yn − xn−1),

ρ(xn − x)},

≤ k max{ρ(xn−1 − x),wρ(2)(ρ(yn − xn) + ρ(xn − x)),

ρ(xn−1 − xn),wρ(2)(ρ(yn − xn) + ρ(xn − xn−1)),ρ(xn − x)}.

Hence, it becomes obvious that limρ(xn −yn) = 0 by Lemma 3.1, x ∈ f(x). �

5. Application to integral inclusions

As outlined in the introduction, a modular fixed point theorem can be used
for providing sufficient (but not necessary) conditions for finding a real contin-
uous function u defined on [a,b] such that

(5.1) u(t) ∈ v(t) + γ

∫ b

a

G(t,s)g(s,u(s))ds, t ∈ [a,b],

where γ is a constant, g : [a,b] × R ։ [a,b] is lower semicontinuous, G :
[a,b] × [a,b] → [0,∞) and v : [a,b] → R are given continuous functions.

© AGT, UPV, 2021 Appl. Gen. Topol. 22, no. 1 103



F. Lael and S. Shabanian

For simplicity we introduce the following shorthand notations. We use X =
C[a,b] to denote all real continuous functions defined on [a,b], gu : [a,b] ։ [a,b]
where gu(s) = g(s,u(s)) and a modular ρ defined on X as

ρ(u) = max
a≤t≤b

| u(t) |2 .

It is not difficult to prove that (X,ρ) is a ρ-complete modular space. Now the
aforementioned integral inclusion problem (5.1) can be reformulated as u is a
solution of problem (5.1) if and only if it is a fixed point of f : X ։ X defined
as

f(u) = {x ∈ X : x(t) ∈ v(t) + γ

∫ b

a

G(t,s)g(s,u(s))ds, t ∈ [a,b]}.

Now we show under the following mild assumptions:

i: for all x,y ∈ X and wx(t) ∈ gx(t), there exists hy(t) ∈ gy(t) such that

|wx(t) − hy(t)|
2 ≤

1

2s
| x(t) − y(t) |2, t ∈ [a,b],

ii: max
a≤t≤b

∫ b

a
G2(t,z)dz ≤ 1

b−a
,

iii: | γ |≤ 1,

the correspondence f has a unique fixed point. So, we assume x,y ∈ X and
w ∈ f(x) by definition, we have

w(t) ∈ v(t) + γ

∫ b

a

G(t,s)g(s,x(s))ds = v(t) + γ

∫ b

a

G(t,s)gx(s)ds.

By Michael’s selection theorem (see Theorem 1 in [38]), it follows that there
exists a continuous single valued mapping wx(s) ∈ gx(s) that w(t) = v(t) +

γ
∫ b

a
G(t,s)wx(s)ds. According to assumption (i), for wx(s) ∈ gx(s), there is an

hy(s) ∈ gy(s) such that

|wx(s) − hy(s)|
2 ≤

1

2s
| x(s) − y(s) |2,

for all s ∈ [a,b]. We define

h(t) = v(t) + γ

∫ b

a

G(t,s)hy(s)ds

which means that

h(t) ∈ v(t) + γ

∫ b

a

G(t,s)gy(s)ds.

© AGT, UPV, 2021 Appl. Gen. Topol. 22, no. 1 104



Convexity and boundedness relaxation for fixed point theorems in modular spaces

Therefore h ∈ f(y). Using the Cauchy-Schwarz inequality and conditions (i-iii),
we have

ρ(w − h) = max
a≤t≤b

|w(t) − h(t)|
2
,

= max
a≤t≤b

| v(t) + γ

∫ b

a

G(t,s)wx(s)ds − (v(t) + γ

∫ b

a

G(t,s)hy(s)ds) |
2
,

= | γ |2 max
a≤t≤b

|

∫ b

a

G(t,s)(wx(s) − hy(s))ds |
2,

≤ | γ |2 max
a≤t≤b

{

∫ b

a

G2(t,s)ds

∫ b

a

| wx(s) − hy(s) |
2 ds

}

,

= | γ |2
{

max
a≤t≤b

∫ b

a

G
2(t,s)ds

}

.
{

∫ b

a

| wx(s) − hx(s) |
2
ds
}

,

≤
| γ |2

b − a

{ 1

2s

∫ b

a

| x(s) − y(s) |2 ds
}

,

≤
| γ |2

2s(b − a)

∫ b

a

max
a≤s≤b

| x(s) − y(s) |2 ds,

=
| γ |2

2s
max
a≤s≤b

| x(s) − y(s) |2,

=
1

2s
ρ(x − y).

Theorem 3.3 implies that f has a unique fixed point u ∈ X, that is, the integral
inclusion (5.1) has a solution which belongs to C[a,b].

6. Conclusion

Our main results show that strong assumptions such as convexity and bound-
edness of modulars in fixed point results for contractive correspondence and
single-valued mappings can be relaxed by making use of some ties between
modular and b-metric spaces. Our approach in this work includes a unifying
view on fixed point results to yield some assumptions which are more likely to
hold in practice and reformulations as particular normed vector space problems
are no longer required. In particular, a generalized version of Nadler’s theorem
along with an application in modular spaces is presented.

Acknowledgements. The authors gratefully acknowledge the reviewer and

the editor for their useful observations and recommendations.

© AGT, UPV, 2021 Appl. Gen. Topol. 22, no. 1 105



F. Lael and S. Shabanian

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