International Journal of Analysis and Applications

Volume 19, Number 5 (2021), 760-772

URL: https://doi.org/10.28924/2291-8639

DOI: 10.28924/2291-8639-19-2021-760

FIXED POINT THEOREMS FOR CONTRACTIVE MAPPINGS IN PROBABILISTIC

MODULAR SPACES

SHAHNAZ JAFARI1,∗, MARYAM SHAMS1 AND ASIER IBEAS2

1Department of Pure Mathematics, Shahrekord University, Iran

2Department of Telecommunications and System Engineering, Faculty of Engineering, Universitat

Autonoma de Barcelona, 08193 Bellaterra, Cerdanyola del Valles, Barcelona, Spain

∗Corresponding author: jafari.shahnaz@yahoo. com

Abstract. In this paper we introduce the concept of contractive maps and prove some related fixed point

theorems in probabilistic modular spaces. In addition, we investigate the existence of common fixed points

for a finite linear combination of contractive mappings. Finally, some results concerned with the convergence

properties of sequences defined by contractive maps in probabilistic modular spaces are also given.

1. Introduction

In recent times, fixed point theory has become an important tool in pure and applied sciences, such as

biology [1], chemistry [2], engineering and physics , to cite just a few. The Banach’s fixed point theory, widely

known as the contraction principle, is an important tool in the theory of metric spaces [3], [4]. Moreover,

since the location of the fixed point can be obtained by means of an iterative process it can be implemented

on a computer to find the fixed point of contraction mappings easily. The fixed point theory has been widely

developed and extended to very general classes of spaces such as [5], [7], [16]. The concept of modular space

was firstly introduced by Nakano [8] and it was later generalized by Musielak and Orlicz [9]. Many authors

have worked ever since on the fixed point theory in modular spaces, see [10], [11], [12]. In 2009, Nourouzi

introduced in [13] the notion of probabilistic modular space according to Menger’s probabilistic approach [6].

Received June 20th, 2021; accepted July 13th, 2021; published August 12th, 2021.

2010 Mathematics Subject Classification. 47H09, 47H10.

Key words and phrases. probabilistic modular space; contrative map; fixed point; linear combination; converging sequences.

©2021 Authors retain the copyrights

of their papers, and all open access articles are distributed under the terms of the Creative Commons Attribution License.

760

https://doi.org/10.28924/2291-8639
https://doi.org/10.28924/2291-8639-19-2021-760


Int. J. Anal. Appl. 19 (5) (2021) 761

In this paper we introduce the concept of ϕ-contractive maps in probabilistic modular spaces and prove some

related fixed theorems. There are no such results in probabilistic modular spaces and this paper contributes

to fill in this gap. Moreover, the existence of fixed points for a finite linear combination of ϕ-contractive

mappings in a probabilistic modular space is also investigated. Finally, we will provide some results concerned

with the convergence properties of iterative sequences defined by ϕ-contractive maps in probabilistic modular

spaces.

2. Preliminaries

We denote the function min by ∧, Z+ = {z ∈ Z : z > 0}, Z0+ = Z+ ∪{0}, R+ = {z ∈ R : z > 0},

R0+ = R+ ∪ {0}.We also will denote the set of all non-decreasing functions f : R −→ R+0 satisfying

inft∈R f(t) = 0, and supt∈R f(t) = 1 by ∆. The latter properties imply that limt→∞f(t) = 1. The set of

those distribution functions such that f(0) = 0 is denoted by ∆+. The space ∆+ is partially ordered by the

usual pointwise ordering of functions, and has a maximal element �0, defined by

�0(t) =


 0 t 6 0,1 t > 0.

Definition 2.1. Let X be an arbitrary vector space. A functional ρ : X → [0,∞] is called modular if for

any arbitrary x,y ∈ X:

(ii) ρ(x) = 0, iff x = 0,

(ii) ρ(αx) = ρ(x), for every scalar α with |α| = 1,

(iii) ρ(αx + βy) 6 ρ(x) + ρ(y) for all x,y ∈ X, and α,β ∈ R+0 , α + β = 1.

Definition 2.2. [13] A probabilistic modular space (briefly, PM-space) is a pair (X,µ) in which X is a

real vector space and µ is a mapping from X into ∆ (for x ∈ X, the function µ(x) is denoted by µx, and

µx(t) is the value of µx at t ∈ R) satisfying the following conditions:

(i) µx(0) = 0,

(ii) µx(t) = 1, for all t > 0 iff x = 0,

(iii) µ−x(t) = µx(t), for all x ∈ X,

(iv) µαx+βy(s + t) ≥ µx(s) ∧µy(t) for all x,y ∈ X, and α,β,s,t ∈ R+0 , α + β = 1.

Definition 2.3. [13] We say that (X,µ) is β-homogeneous, where β ∈ (0, 1] if,

µαx(t) = µx(
t
|α|β ), for every x ∈ X,t > 0 and α ∈ R\{0}.



Int. J. Anal. Appl. 19 (5) (2021) 762

Example 2.1. Let ρ : X → X be defined by ρ(x) = 1
α
|x|, for every α ∈ R\{0}. It is easy to see that ρ is a

modular on X. Define

µx(t) =


 0 t 6 0,t

t+ρ(x)
t > 0

for all x ∈ X. Then (X,µ) is a β–homogeneous PM-space, for β = 1.

Example 2.2. Consider the real vector space X with µx defined as:

µx(t) =


 0 t 6 ρ(x),1 t > ρ(x),

where ρ is a modular on X. Then (X,µ) is a PM-space.

Definition 2.4. [13] Let (X,µ) be a PM-space.

1) A sequence {xn} in X is said to be µ-convergent to a point x ∈ X and denoted by xn −→ x, if for every

t > 0 and r ∈ (0, 1), there exists a positive integer k such that µxn−x(t) > 1 − r for all n ≥ k. In this case,

the point x ∈ X is said to be the µ-limit of the µ-converging sequence {xn}.

2) A sequence {xn} in X is said to be µ-Cauchy sequence, if for every t > 0 and r ∈ (0, 1), there exists a

positive integer k such that µxn−xm(t) > 1 −r for all n,m ≥ k.

3) The modular space (X,µ) is said to be µ-complete if each µ-Cauchy sequence in X is µ-convergent to a

point of X.

Lemma 2.1. [13] Let (X,µ) be a PM-space. Then the µ-limit of any µ-convergent sequence is unique.

Lemma 2.2. [13] The operations +, . in the β-homogeneous P-modular space (X,µ) are continuous.

Definition 2.5. [13] Let (X,µ) and (Y,ν) be two PM-spaces. A mapping T from (X,µ) to (Y,ν) is said

to be sequentially PM-continuous (probabilistic modular continuous) at x ∈ X if T(xn)
ν−→ T(x) for every

sequence {xn} of points in X that converges to x ∈ X, xn
µ−→ x.

The definition below is introced by Sherstnev in 1963, [14].

Definition 2.6. A random normed space (briefly, RN-space) is a triple (X,ν,T) where X is a vector space,

T is a continuous t-norm, and ν is a mapping from X into ∆+ (for x ∈ X, if νx denotes the value of x ∈ X,

the following conditions hold:

(i) νx(t) = ε0(t), for all t > 0 iff x = 0,

(ii) ναx(t) = νx(
t
|α|), for every x ∈ X,t > 0 and α ∈ R\{0}.

(iii) νx+y(s + t) ≥ T(νx(s),νy(t)) for all x,y ∈ X and s,t ≥ 0.

Theorem 2.1. Let (X,ν,T) be a RN-space with t-norm T(a,b) = min(a,b) for all a,b ∈ X. Then (X,ν)

is a PM-space.



Int. J. Anal. Appl. 19 (5) (2021) 763

Proof.

(1) ν−x(t) = ν(−1)x(t) = νx(
t
|−1|) = νx(t), for all x ∈ X.

(2) Let x,y ∈ X, α,β,s,t ∈ R+0 and α + β = 1. Hence

ναx+βy(t) = ναx+βy((α + β)t)

≥ T(ναx(αt),νβy(βt))

= T(νx(
αt

α
),νy(

βt

β
))

= T(νx(t),νy(t))

≥ T(νx(
t

2
),νy(

t

2
))

= νx(
t

2
) ∧νy(

t

2
).(2.1)

�

3. Fixed point theorems for ϕ-contractive mappings

In this section we define the notion of ϕ-contractive mapping in probabilistic modular spaces and prove

some fixed point theorems related to this concept. Let us introduce the following definition:

Definition 3.1. A function ϕ : [0,∞) → [0,∞) is said to be a Φ-function if it satisfies the following

conditions:

(i) ϕ(t) is continuous,

(ii) ϕ(t) is strictly monotone increasing and ϕ(t) →∞ as t →∞,

(iii) ϕ(αt) 6 αφ(t), for all α ∈ (0, 1) and t ≥ 0.

It is easy to see that the condition (iii) of Definition 3.1 is equivalent to the following one:

ϕ(0) = 0.

Example 3.1. ϕ(t) = k tr, is a simple example of Φ-function for k > 0 and r > 1.

Lemma 3.1. A direct consequence of condition (iv) of Definition 2.2 is:

µΣn
i=1

αixi(t) ≥ µx1 (
t

n
) ∧µx2 (

t

n
) · · ·∧µxn(

t

n
)︸ ︷︷ ︸

n

,(3.1)

for all x1,x2, ...,xn ∈ X and αi, t ∈ R+0 with Σ
n
i=1αi = 1.



Int. J. Anal. Appl. 19 (5) (2021) 764

Proof. It is obtained by induction as follows:

µΣn
i=1

αixi(t) = µΣn−1
i=1

αixi+αnxn
(
(n− 1)t

n
+
t

n
)

= µ
(Σ
n−1
i=1

αi
Σ
n−1
i=1

αixi

Σ
n−1
i=1

αi

+αnxn)
(
(n− 1)t

n
+
t

n
)

≥ µ
Σ
n−1
i=1

αixi

Σ
n−1
i=1

αi

(
(n− 1)t

n
) ∧µxn(

t

n
)

≥ µx1 (
t

n
) ∧µx2 (

t

n
) · · ·∧µxn(

t

n
)︸ ︷︷ ︸

n

.(3.2)

�

Definition 3.2. Let (X,µ) be a probabilistic modular space (PM-space). A mapping T : X −→ X is said

to be ϕ-contractive if

µTx−Ty(ϕ(t)) ≥ µl(x−y)(ϕ(
t

c
)),(3.3)

where l,c ∈ (0, 1) and ϕ ∈ Φ.

It is easy to see that every ϕ-contractive map is sequentially PM-continuous. In fact, if xn → x, hence,

for every t > 0 and r ∈ (0, 1), there exists N such that µxn−x(t) > 1 − r for all n ≥ N. Therefore we get

µTxn−Tx(ϕ(t)) ≥ µxn−x(ϕ(
t
c
)) > 1 −r.

Remark 3.1. We can see that the definition 3.2 generalizes the previous ones introduced in [15].

Theorem 3.1. Let (X,µ) be a β-homogeneous µ-complete PM-space and T : X −→ X be a ϕ-contractive

map. Then T has a unique fixed point x∗ ∈ X and the iterative sequence {Tn(x0)}, generated by the initial

element x0 ∈ X, converges to the fixed point x∗ ∈ X as n →∞ .

Proof. Choose x ∈ X arbitrarily. We first prove that {Tn(x)} is a µ-Cauchy sequence. Let s > 0 be given.

Since ϕ is continuous at 0 and ϕ(0) = 0, we can find t > 0 such that ϕ(t) < s. Hence, we have:

µTnx−Tn+px(s) ≥ µTnx−Tn+px(ϕ(t))

≥ µl(Tn−1x−Tn+p−1x)(ϕ(
t

c
))

= µTn−1x−Tn+p−1x(l
−βϕ(

t

c
))

≥ µTn−1x−Tn+p−1x(ϕ(
t

c
))

...

≥ µx−Tpx(l−βϕ(
t

cn
))

≥ µx−Tpx(ϕ(
t

cn
)).(3.4)



Int. J. Anal. Appl. 19 (5) (2021) 765

On the other hand, we have:

µx−Tpx(ϕ(
t

cn
)) = µ(x−Tx)+(Tx−Tpx)(ϕ(

t

cn
))

≥ µ2(x−Tx)(
1

2
ϕ(

t

cn
)) ∧µ2(Tx−Tpx)(

1

2
ϕ(

t

cn
))

≥ µx−Tx(
1

2β+1
ϕ(

t

cn
)) ∧µTx−Tpx(

1

2β+1
ϕ(

t

cn
))

≥ µx−Tx(ϕ(
t

2β+1cn
)) ∧µTx−Tpx(ϕ(

t

2β+1cn
))

≥ µx−Tx(ϕ(
t

2β+1cn
)) ∧µl(x−Tp−1x)(ϕ(

t

2β+1cn+1
))

= µx−Tx(ϕ(
t

2β+1cn
)) ∧µx−Tp−1x(l−βϕ(

t

2β+1cn+1
))

≥ µx−Tx(ϕ(
t

2β+1cn
)) ∧µx−Tp−1x(ϕ(

t

2β+1cn+1
)).(3.5)

By induction we get:

µx−Tpx(ϕ(
t

cn
)) ≥ µx−Tx(ϕ(

t

2β+1cn
)) ∧µx−Tx(ϕ(

t

22(β+1)cn+1
)) ∧·· ·

∧µx−Tx(ϕ(
t

2p(β+1)cn+p−1
)).(3.6)

According to property (ii) of Φ-function and since µ(∞) = 1, from (3.4) and (3.6) we get limn→∞µTnx−Tn+px(s) =

1. Since X is µ-complete, there exists x∗ ∈ X with limn→∞Tnx∗ = x∗.

We will prove now that x∗ is a fixed point of T . The ϕ-contractivity of T yields sequentialy PM-continuity.

Therefore, x∗ = limn→∞T
n+1x∗ = limn→∞T(T

nx∗) = Tx∗; i.e x∗ is a fixed point of T . In order to prove

that the fixed point if unique, assume that there exists another fixed point y∗ ∈ X such that y∗ = Ty∗.

Hence, Tnx∗ = x∗ and Tny∗ = y∗, and there exists t1 > 0 such that µy∗−x∗ (t1) = a < 1. Then,

a = µy∗−x∗ (t1) ≥ µy∗−x∗ (ϕ(t))

= µTny∗−Tnx∗ (ϕ(t))

≥ µy∗−x∗ (l−nβϕ(
t

cn
)).(3.7)

Letting n →∞ in (3.7), according the property (ii) of Φ-function and since µ(∞) = 1, we get a ≥ 1, that is

contradiction. Therefore y∗ = x∗. �

The following Theorem shows that a linear combination of a family of ϕ-contrative mappings possesing a

common fixed point has a fixed point and it can be calculated by using an interative process.

Theorem 3.2. Let (X,µ) be a β-homogeneous µ-complete PM-space and fi : X −→ X (i = 1, 2, · · · ,m)

be a finite family of ϕ-contractive maps for ϕ ∈ Φ and c ∈ (0, 1
m

). Define f =
∑m
i=1 λifi, where λi ∈ [0, 1],

Σmi=1λi = 1. Then f has fixed point x
∗ ∈ X, which is common to each linear operator’s one and the iterative

sequence {fn(x)} defined by the initial element x0 ∈ X, converges to x∗ ∈ X.



Int. J. Anal. Appl. 19 (5) (2021) 766

Proof. Since fi have a common fixed point x
∗ ∈ X, then:

f(x∗) = λ1f1(x
∗) + λ2f2(x

∗) + ... + λmfm(x
∗) = (λ1 + λ2 + ... + λm)x

∗ = x∗,(3.8)

This means that x∗ is a fixed of f (and common to ech operator’s fixed point). Now we prove that f is a

ϕ-contractive map. We have:

µfx−fy(ϕ(t)) = µ
∑
m
i=1

λifixi−
∑
m
i=1

λifiy(ϕ(t))

≥ µf1x−f1y(
1

m
ϕ(t)) ∧µf2x−f2y(

1

m
ϕ(t)) ∧·· ·∧µfnx−fny(

1

m
ϕ(t))︸ ︷︷ ︸

m

≥ µf1x−f1y(ϕ(
t

m
)) ∧µf2x−f2y(ϕ(

t

m
)) ∧·· ·∧µfnx−fny(ϕ(

t

m
))︸ ︷︷ ︸

m

≥ µl(x−y)(ϕ(
t

mc
)) ∧µl(x−y)(ϕ(

t

mc
)) ∧·· ·∧µl(x−y)(ϕ(

t

mc
))

≥ µl(x−y)(ϕ(
t

mc
))

= µl(x−y)(ϕ(
t

k
)),(3.9)

where k ∈ (0, 1). Hence, f is ϕ-contractive and according to Theorem 3.1, the sequence {fn(x0)} converges

to the fixed point x∗ ∈ X for any arbitrary initial element x0. �

The subsequent results are concerned with the convergence properties of ϕ-contractive maps. PM-space.

Lemma 3.2. The following property hold:

If Tn : X → X,∀n ∈ Z+ are continuous and {Tn} uniformly converges to {T}, then {Tmn } uniformly converge

to {Tm}, ∀m ∈ Z+.

Proof. We prove these properties with induction. Assume that {Tjn} converge to {Tj}, as n → ∞, for all

1 ≤ j ≤ m and for any given m ∈ Z+. We have:

µ
T
j
n(Tnx)−Tj(Tx)

(t) ≥ µ
2(T

j
n(Tnx)−Tj(Tnx))

(
t

2
) ∧µ2(Tj(Tnx)−Tj(Tx))(

t

2
)(3.10)

since Tn : X → X is continuous and {Tjn} converge to {Tj}, there exists a big enough n such that

µTj(Tnx)−Tj(Tx)(
t
2
) > 1−λ and µ

T
j
n(Tnx)−Tj(Tnx)

( t
2
) > 1−λ, for any given λ ∈ (0, 1). Thus, from (3.10), we

have

µ
T
j+1
n x−Tj+1x

(t) > (1 −λ) ∧ (1 −λ) = 1 −λ.

Thus, Tj+1n converge to T
j+1 as n →∞, for all 1 6 j 6 m.

�

Theorem 3.3. Let (X,µ) be a β-homogeneous µ-complete PM-space and {Tn} be a sequence of sequentially

PM-continuous operators with Fix(Tn) = {x∗n}, such that:



Int. J. Anal. Appl. 19 (5) (2021) 767

(i) {Tn} uniformly converge to T for some T : X −→ X.

(ii) T is ϕ-contractive, with T(x∗) = x∗.

Then {x∗n}→ x∗.

Proof. According to the definition of convergence in PM-space, we show that limn→∞µx∗n−x∗ (t) = 1, for

every t > 0. In this way, we have:

µx∗n−x∗ (ϕ(t)) = µTmn x∗n−Tmx∗ (ϕ(t))

≥ µ2(Tmn x∗n−Tmx∗n)(ϕ(
t

2
)) ∧µ2(Tmx∗n−Tmx∗)(ϕ(

t

2
))

≥ µ2(Tmn x∗n−Tmx∗n)(ϕ(
t

2
)) ∧µx∗n−x∗ (ϕ(

t

2β+1cm
)), ∀n,m ∈ Z0+,(3.11)

If we take the limit m →∞ in (3.11) we get limn→∞µx∗n−x∗ (ϕ(t)) = 1. Thus, {x
∗
n}→{x∗}.

�

Theorem 3.4. Let (X,µ) be a β-homogeneous µ-complete PM-space and {Tn} be a sequence of ϕ-contractive

operators Tn : X → X for some l,c ∈ (0, 1), ϕ ∈ Φ with Fix(Tn) = {x∗n}. Moreover, let T : X −→ X be a

ϕ-contractive mapping with Fix(T) = {x∗}. Assume the following properties hold:

(a) {Tn} converge to T ,

(b) There exists a subsequence {x∗nm} of {x
∗
n}, converging to a point z ∈ X.

Then z = x∗ and the iterated sequence generated by xn+1 = Tnxn converges to the fixed point x
∗, for any

given x0 ∈ X and n ∈ Z+

Proof. We first prove that {x∗nm} converge to x
∗. Proceed by assuming, since {x∗nm}→{z} and {Tn}→ T,

for any given δ ∈ (0, 1) and t > 0 there exists N1(∈ Z0+) = N1(δ,t) such that for n,m ≥ N1, µx∗nm−z(ϕ(t)) >

1 − δ and µTnmz−Tz(ϕ(t)) > 1 − δ , where ϕ ∈ Φ. Therefore,

µx∗nm−Tz(ϕ(t)) = µTnmx
∗
nm
−Tz(ϕ(t))

≥ µ2(Tnmx∗nm−Tnmz)(ϕ(t)) ∧µ2(Tnmz−Tz)(ϕ(t))

≥ µx∗nm−z(ϕ(
t

2βc
)) ∧µTnmz−Tz(2

−βϕ(t))

≥ (1 − δ) ∧ (1 − δ)

= (1 − δ).(3.12)



Int. J. Anal. Appl. 19 (5) (2021) 768

This result means that {x∗nm} converges to Tz. Hence, by the uniqueness of the limit, we get Tz = z implying

that z = x∗. Also for α ∈ (0, 2c) we have:

µxn+1−x∗ (ϕ(t)) = µTnxn−Tx∗ (ϕ(t))

≥ µ2(Tnxn−Tnx∗)(αϕ(t)) ∧µ2(Tnx∗−Tx∗)((1 −α)ϕ(t))

≥ µxn−x∗ (ϕ(
αt

2βc
)) ∧µ2(Tnx∗−Tx∗)((1 −α)ϕ(t)).(3.13)

On the other hand:

µxn−x∗ (ϕ(
αt

2βc
)) ≥ µxn−1−x∗ (ϕ(

α2t

22(β+1)c2
)) ∧µ2(Tnx∗−Tx∗)((1 −α)ϕ(

αt

2βc
)).(3.14)

By induction from (3.13) and (3.14), we have

µxn+1−x∗ (ϕ(t)) ≥ µx0−x∗ (ϕ(
αnt

2n(β+1)cn
)) ∧µ2(Tnx∗−Tx∗)((1 −α)ϕ(t))

∧µ2(Tnx∗−Tx∗)((1 −α)ϕ(
αt

2βc
)) ∧·· ·

∧µ2(Tnx∗−Tx∗)((1 −α)ϕ(
αnt

2n(β+1)cn
)).(3.15)

Letting n →∞ in (3.15) we get lim µxn+1−x∗ (ϕ(t)) ≥ 1, i.e {xn}→ x∗. �

Theorem 3.5. Let (X,µ) be a β-homogeneous µ-complete PM-space, and {Tn} be a sequence of operators

such that {Tn} are ϕ-contractive for some l,c ∈ (0, 1) and ϕ ∈ Φ. Assume that {Tn} converge to T for some

T : X −→ X. Then the following properties hold:

(a) T is ϕ-contractive for some c ∈ (0, 1
4
),

(b) {x∗n}→ x∗, where Fix(Tn) = {x∗n}, ∀n ∈ Z+, and Fix(T) = {x∗},

(c) The iterative sequence generated by xn+1 = Tnxn converges to x
∗, for any given x0 ∈ X arbitrary and

n ∈ Z+.

Proof. First we prove that T is ϕ-contractive. For any x,y ∈ X we have:

µTnx−Tny(ϕ(t)) ≥ µl(x−y)(ϕ(
t

c
)).(3.16)

Additionally,

µTx−Ty(ϕ(t)) = µTx−Tnx+Tnx−Ty(ϕ(t))

≥ µ2(Tx−Tnx)((1 −α)ϕ(t)) ∧µ2(Tnx−Ty)(αϕ(t))

≥ µTx−Tnx(
ϕ((1 −α)t)

2β
) ∧µTnx−Ty(

ϕ(αt)

2β
).(3.17)



Int. J. Anal. Appl. 19 (5) (2021) 769

On the other hand

µTnx−Ty(
ϕ(αt)

2β
) ≥ µ2(Tnx−Tny)(

ϕ(αt)

2β+1
) ∧µ2(Tny−Ty)(

ϕ(αt)

2β+1
)

≥ µTnx−Tny(ϕ(
αt

22(β+1)
)) ∧µTny−Ty(

ϕ(αt)

22(β+1)
)

≥ µl(x−y)(ϕ(
c−1αt

22(β+1)
)) ∧µTny−Ty(

ϕ(αt)

22(β+1)
).(3.18)

By using equation (3.17) and (3.18) we have

µTx−Ty(ϕ(t)) ≥ µTx−Tnx(
ϕ((1 −α)t)

2β
) ∧µl(x−y)(ϕ(

c−1αt

22(β+1)
)) ∧µTny−Ty(

ϕ(αt)

22(β+1)
).(3.19)

Letting n →∞ in (3.19), we get

µTx−Ty(ϕ(t)) ≥ 1 ∧µl(x−y)(ϕ(
c−1αt

22(β+1)
)) ∧ 1

= µl(x−y)(ϕ(
c−1t

22(β+1)
))

≥ µl(x−y)(ϕ(
t

k
)),(3.20)

where k ∈ (0, 1
4
). Hence T is ϕ-contractive. Finally, according to Theorem 3.3, {x∗n} converges to x∗ and by

the same method of proof of Theorem 3.4, {xn} converges to x∗.

�

Remark 3.2. Every probabilistic modular space (X,µ) induces a probabilistic metric space (X,F,∧) with

F : X ×X → ∆ via Fx,y = µx−y for all x,y ∈ X.

4. Numerical examples

In this section we present some numerical examples in order to illustrate the main results discussed in the

previous sections.

Example 4.1. Let X = R and µx(t) = tt+ρ(x) , x,y ∈ X, t > 0, where ρ(x) = |x| is a modular functional on

X. Define a mapping f : R → R by f(x) = x
8

for all x ∈ R. Let ϕ(t) = 2 t2. Then f is ϕ-contractive with

the constants l = 1
2

and c ≥ 1
2

. Indeed, for x,y ∈ R, we have

µfx−fy(ϕ(t)) =
2t2

2t2 + 1
8
|x−y|

, µl(x−y)(ϕ(
t

c
)) =

2t2

c2

2t2

c2
+ 1

2
|x−y|

.

It is easy to see that

2t2

2t2 + 1
8
|x−y|

≥
2t2

c2

2t2

c2
+ 1

2
|x−y|

,

for all c ∈ [ 1
2
, 1). Accordingly, f is ϕ-contractive and it has a unique fixed point, as predicted by Theorem

3.1. In addition, it is easy to check that x = 0 is the fixed point of f.



Int. J. Anal. Appl. 19 (5) (2021) 770

Example 4.2. Let X = R and µx(t) = tt+ρ(x) , x,y ∈ X, t > 0, where ρ(x) = |x| is a modular functional

on X. Define the mappings fi : R → R by f1(x) = x4 and f2(x) =
x
2

for all x ∈ R. Let ϕ(t) = t, l = 2
3

and c = 3
4

. Define f = 1
2
f1 +

1
2
f2. We can see that f1 and f2 are ϕ-contractive maps. We prove that

f(x) = 1
2
f1x +

1
2
f2x =

3x
8

is ϕ-contractive.

µfx−fy(ϕ(t)) =
t

t + 3
8
|x−y|

, µl(x−y)(ϕ(
t

c
)) =

4
3
t

4
3
t + 2

3
|x−y|

,

Thus,

t

t + 3
8
|x−y|

≥
4
3
t

4
3
t + 2

3
|x−y|

,

Consequently, f is a ϕ-contractive map and it has a unique fixed point, as predict by Theorem 3.2. It is easy

to check that x = 0 is the fixed point of f.

Example 4.3. Let X = R and µx(t) = tt+ρ(x) , x,y ∈ X, t > 0, where ρ(x) = |x| is a modular functional on

X. Let ϕ(t) = t and define Tnxn =
(n+1)xn

(2n+3)(1+x2n)
. We show that Tn is ϕ-contractive. We have:

µTnx−Tny(ϕ(t)) =
t

t +
(n+1)|x−y|

(2n+3)(1+x2)(1+y)

=
(2n + 3)(1 + x2)(1 + y)t

(2n + 3)(1 + x2)(1 + y)t + (n + 1)|x−y|
,

and

µl(x−y)(ϕ(
t

c
)) =

t
c

t
c

+ l|x−y|
=

t

t + lc|x−y|
.

So the condition (3.3) becomes:

(2n + 3)(1 + x2)(1 + y)t

(2n + 3)(1 + x2)(1 + y)t + (n + 1)|x−y|
≥

t

t + lc|x−y|
.(4.1)

Eq. (4.1) leads to 2n+3
n+1

(1 + x2)(1 + y) ≥ 1
lc

, that holds for every l,c ∈ (0, 1). Hence Tn is ϕ-contractive.

On the other hand we have:

T = lim
n→∞

Tn = lim
n→∞

(n + 1)x

(2n + 3)(1 + x2)
=

x

2(1 + x2)
.

Similarly to the above method, we can see that T is also ϕ-contractive. Therefore, Theorem 3.5 holds and

the iterative scheme:

xn+1 =
(n + 1)xn

(2n + 3)(1 + x2n)
(4.2)

converges to the unique fixed point of T . It is easy to check out that x∗ = 0 is the fixed point of T . Figure 1

shows the evolution of the iterative scheme (4.2) for different initial conditions. In Figure 1, we can observe

that the sequence {xn} converges to zero as predicted by Theorem 3.5.



Int. J. Anal. Appl. 19 (5) (2021) 771

0 1 2 3 4 5 6 7

Iteration (n)

-3

-2

-1

0

1

2

3

x
n

Figure 1. Evolution of the sequence of iterates for different initial conditions

5. Conclusions

This paper has introduced the concept of ϕ-contractive maps in probabilistic modular spaces. Further-

more, the existence of fixed points for these operators in probabilistic modular spaces is investigated as well.

Afterwards, the results are extended to a finite linear combination of ϕ-contractive mappings. Finally, we

also investigate some convergence properties of sequences constructed by these operators which are either

convergent to either a ϕ-contractive map.

Conflicts of Interest: The author(s) declare that there are no conflicts of interest regarding the publication

of this paper.

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	1. Introduction
	2. Preliminaries
	3. Fixed point theorems for  -contractive mappings
	4. Numerical examples
	5. Conclusions
	References