International Journal of Analysis and Applications

Volume 17, Number 5 (2019), 771-792

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

DOI: 10.28924/2291-8639-17-2019-771

SOME FIXED POINT THEOREMS IN MENGER PROBABILISTIC PARTIAL

METRIC SPACES WITH APPLICATION TO VOLTERRA TYPE INTEGRAL

EQUATION

AMIR GHANENIA, MAHNAZ KHANEHGIR, REZA ALLAHYARI∗ AND MOHAMMAD

MEHRABINEZHAD

Department of Mathematics, Mashhad Branch, Islamic Azad University, Mashhad, Iran

∗Corresponding author: rezaallahyari@mshdiau.ac.ir

Abstract. In this paper, we introduce the notion of Menger probabilistic partial metric space and prove

some fixed point theorems in the framework of such spaces. Some examples and an application to Volterra

type integral equation are given to support the obtained results. Finally, we apply successive approximations

method to find a solution for a Volterra type integral equation with high accuracy.

1. Introduction

The concept of a Menger probabilistic metric space (briefly, Menger PM-space) was initiated by Menger

[15]. The idea of Menger was to use a distribution function instead of a nonnegative number for the value

of a metric.

The study of this space was expanded rapidly with the pioneering works of Schweizer and Sklar [20],

Stevens [25] and some of their coworkers. In 1972, Sehgal and Bharucha-Reid [23] obtained a generalization of

the Banach contraction principle on a complete Menger space. Since then, a number of mathematicians have

made a substantial contribution to the theoretical development of Menger PM-spaces (see [4,6–10,12,17,26]).

Received 2019-04-19; accepted 2019-08-14; published 2019-09-02.

2010 Mathematics Subject Classification. 47H10, 54H25.

Key words and phrases. Fixed point; Menger probabilistic metric space; Partial metric space; Volterra type integral equation;

Successive approximations method.

c©2019 Authors retain the copyrights

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

771

https://doi.org/10.28924/2291-8639
https://doi.org/10.28924/2291-8639-17-2019-771


Int. J. Anal. Appl. 17 (5) (2019) 772

On the other hand, Matthews [14] introduced the notion of partial metric space as a part of the study of

denotational semantics of dataflow networks. In recent years, some scholars have investigated the topological

properties of partial metric spaces and have established some fixed point results in these spaces (see [1, 3, 5,

16, 19, 22]).

In this work, using the concepts of Menger probabilistic metric space and partial metric space, we establish

a new concept of Menger probabilistic partial metric space. We present some fixed point theorems in these

spaces. Some examples and an application to Volterra type integral equation are given to illustrate the

usability of our results. Finally, we apply successive approximations method to find an approximate solution

for a Volterra type integral equation with high accuracy.

First, we recall some basic definitions and facts which will be used further on. We denote by R the set of

real numbers and R+ the set of nonnegative real numbers.

Definition 1.1. [14] A partial metric on a nonempty set X is a mapping p : X ×X → R+ such that for

all x,y,z ∈ X:

(p1) x = y if and only if p(x,x) = p(x,y) = p(y,y),

(p2) p(x,x) ≤ p(x,y),

(p3) p(x,y) = p(y,x),

(p4) p(x,y) ≤ p(x,z) + p(z,y) −p(z,z).

A pair (X,p) is called a partial metric space, if X is a nonempty set and p is a partial metric on X.

It is clear that if p(x,y) = 0, then from (p1) and (p2), x = y. But if x = y, p(x,y) may not be 0. A basic

example of a partial metric space is the pair (R+,p), where p(x,y) = max{x,y} for all x,y ∈ R+ (see [16]).

Each partial metric p on a nonempty set X generates a T0 topology τp on X whose base is the family of

open p-balls {Bp(x,ε) : x ∈ X,ε > 0}, where Bp(x,ε) = {y ∈ X : p(x,y) < ε + p(x,x)} for all x ∈ X and

ε > 0.

Definition 1.2. [14] Let (X,p) be a partial metric space, {xn} be any sequence in X and x ∈ X. Then:

(i) The sequence {xn} is said to be convergent to x with respect to τp if

lim
n→∞

p(xn,x) = p(x,x).

(ii) The sequence {xn} is said to be a Cauchy sequence in (X,p), if lim
n,m→∞

p(xn,xm) exists and is finite.

(iii) (X,p) is said to be a complete partial metric space if for every Cauchy sequence {xn} in X, x ∈ X

exists such that

lim
n,m→∞

p(xn,xm) = lim
n→∞

p(xn,x) = p(x,x).

Notice that in a partial metric space the limit of a convergent sequence may not be unique.



Int. J. Anal. Appl. 17 (5) (2019) 773

In the sequel, we recall the definition of n-th order t-norm. Shi et al. [24] gave the following definition of

n-th order t-norm.

Definition 1.3. A mapping T : Πni=1[0, 1] → [0, 1] is called an n-th order t-norm if the following conditions

are satisfied:

(i) T(0, 0, . . . , 0) = 0, T(a, 1, 1, . . . , 1) = a for all a ∈ [0, 1],

(ii) T(a1,a2,a3, . . . ,an) = T(a2,a1,a3, . . . ,an) = T(a2,a3,a1, . . . ,an)

= . . . = T(a2,a3,a4, . . . ,an,a1),

(iii) ai ≥ bi, i = 1, 2, 3, . . . ,n imply T(a1,a2,a3, . . . ,an) ≥ T(b1,b2,b3, . . . ,bn),

(iv) T(T(a1,a2,a3, . . . ,an),b2,b3, . . . ,bn)

= T(a1,T(a2,a3, . . . ,an,b2),b3, . . . ,bn)

= T(a1,a2,T(a3,a4, . . . ,an,b2,b3),b4, . . . ,bn)

= ...

= T(a1,a2,a3, . . . ,an−1,T(an,b2,b3, . . . ,bn)).

When n = 2, we have a binary t-norm, which is commonly known as t-norm.

The following are three basic continuous 3-th order t-norms:

(1) The minimum 3-th order t-norm, say TM , defined by TM (a,b,c) = min{a,b,c}.

(2) The product 3-th order t-norm, say TP , defined by TP (a,b,c) = abc.

(3) The Lukasiewicz 3-th order t-norm, say TL, defined by TL(a,b,c) = max{a + b + c− 2, 0}.

These t-norms are related in the following way: TL ≤ TP ≤ TM .

Definition 1.4. [21] A function F : R → R+ is called a distribution if it is increasing left-continuous with

inf
t∈R

F(t) = 0 and sup
t∈R

F(t) = 1.

The set of all distribution functions is denoted by D+. A special distribution function is given by

H(t) =




0, t ≤ 0,

1, t > 0.

Definition 1.5. [21] A Menger probabilistic metric space (briefly, Menger PM-space) is a triple (X,F,T)

where X is a nonempty set, T is a continuous t-norm, and F is a mapping from X ×X into D+ such that,

if Fx,y denotes the value of F at the pair (x,y), the following conditions hold:

(PM1) Fx,y(t) = H(t) if and only if x = y,

(PM2) Fx,y(t) = Fy,x(t),

(PM3) Fx,y(t + s) ≥ T(Fx,z(t),Fz,y(s)) for all x,y,z ∈ X and s,t ≥ 0.



Int. J. Anal. Appl. 17 (5) (2019) 774

According to [21], the (ε,λ)-topology in a Menger PM-space (X,F,T) is a family of neighborhoods Nx of

a point x ∈ X given by Nx = {Nx(ε,λ) : ε > 0,λ ∈ (0, 1)}, where Nx(ε,λ) = {y ∈ X : Fx,y(ε) > 1 − λ}.

The (ε,λ)-topology is a Hausdorff topology.

Definition 1.6. [21] Let (X,F,T) be a Menger PM-space. Then:

(i) A sequence {xn} in X is said to be convergent to x in X if, for every ε > 0 and λ > 0, a positive

integer N exists such that for each n ≥ N, Fxn,x(ε) > 1 −λ.

(ii) A sequence {xn} in X is called a Cauchy sequence if, for every ε > 0 and λ > 0, a positive integer

N exists such that for each n,m ≥ N, Fxn,xm (ε) > 1 −λ.

(iii) A Menger PM-space is said to be complete if every Cauchy sequence in X is convergent to a point

in X.

Example 1.1. Let X = R+, T(a,b) = ab
a+b−ab if a,b ∈ (0, 1] and T(a,b) = 0, if a = 0 or b = 0. Define

F : X ×X →D+ by

Fx,y(t) =




t
t+|x−y|, if t > 0,

0, if t ≤ 0

for all x,y ∈ X. Then (X,F,T) is a complete Menger PM-space.

Definition 1.7. [10] A function φ : R+ → R+ is said to be a Φ-function if it satisfies the following

conditions:

(i) φ(t) = 0 if and only if t = 0,

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

(iii) φ is left-continuous in (0,∞),

(iv) φ is continuous at 0.

From now on, we mean by Φ the class of all φ-functions and by Ψ the class of continuous functions

ψ : R+ → R+ such that ψ(0) = 0 and ψn(an) → 0, whenever an → 0 as n →∞.

2. Main result

In this section, first we describe the new concept of Menger probabilistic partial metric space. Then we

improve some fixed point results of Gopal et al. [10], in the setup of Menger probabilistic partial metric

spaces.

Definition 2.1. A probabilistic partial metric space is an ordered pair (X,F) where X is a nonempty set,

F : X×X →D+ is given by (x,y) 7→ Fx,y, such that the following conditions are satisfied for all x,y,z ∈ X

and t ∈ R+:



Int. J. Anal. Appl. 17 (5) (2019) 775

(PPM1) Fx,y(t) = Fx,x(t) = Fy,y(t) if and only if x = y,

(PPM2) Fx,y(t) = Fy,x(t),

(PPM3) Fx,x(t) ≥ Fx,y(t),

(PPM4) If Fx,z(t1) = 1, Fz,y(t2) = 1 and Fz,z(t3) = 1 for t1, t2, t3 ∈ R+, then

Fx,y(t1 + t2 + t3) = 1.

It is clear that every probabilistic metric space is a probabilistic partial metric space. However, the

converse of this fact needs not hold. For example, x = y does not imply Fx,y(t) = H(t). See the following

example.

Example 2.1. Let (X,p) be a partial metric space. If F : X ×X →D+ is a mapping defined by

Fx,y(t) = H(t−p(x,y)), ∀x,y ∈ X,t ∈ R,

then (X,F) is a probabilistic partial metric space. Obviously, x = y does not imply Fx,y(t) = H(t).

Definition 2.2. A Menger probabilistic partial metric space is a triple (X,F,T), where (X,F) is a proba-

bilistic partial metric space, T is a continuous 3-th order t-norm and the following inequality holds:

Fx,y(t1 + t2 + t3) ≥ T(Fx,z(t1),Fz,y(t2),Fz,z(t3)) (2.1)

for all x,y,z ∈ X and all t1, t2, t3 ∈ R+.

Remark 2.1. Let (X,F) be as Example 2.1. Then (X,F,TM ) is a Menger probabilistic partial metric space

induced by (X,p).

Example 2.2. Let (X,p) be a partial metric space. Define a mapping F : X ×X →D+ by

Fx,y(t) =




t
t+p(x,y)

, if t > 0,

0, if t ≤ 0

for all x,y ∈ X. Then (X,F,TM ) is a Menger probabilistic partial metric space.

Definition 2.3. Let (X,F,T) be a Menger probabilistic partial metric space. Then:

(i) A sequence {xn} in X is said to be convergent to x in X if, for each t > 0, lim
n→∞

Fxn,x(t) = Fx,x(t).

(ii) A sequence {xn} in X is called a Cauchy sequence if, for each t > 0,

lim
m,n→∞

Fxm,xn (t) exists.

(iii) A Menger probabilistic partial metric space is said to be complete if for every Cauchy sequence {xn}

in X, a point x ∈ X exists such that lim
m,n→∞

Fxm,xn (t) = lim
n→∞

Fxn,x(t) = Fx,x(t).

(iv) A sequence {xn} is called G-Cauchy if for each p ∈ N and t > 0, lim
n→∞

Fxn,xn+p (t) exists.



Int. J. Anal. Appl. 17 (5) (2019) 776

(v) The space (X,F,T) is called G-complete if for every G-Cauchy sequence {xn} in X, a point x ∈ X

exists such that lim
n→∞

Fxn,xn+p (t) = lim
n→∞

Fxn,x(t) = Fx,x(t).

Definition 2.4. (see also [10]) Let X be a nonempty set, f : X → X be a mapping and β,γ : X × X ×

(0,∞) → (0,∞) be two functions. Then f is said to be (β,γ)-admissible if for all x,y ∈ X and all t > 0 we

have

β(x,y,t) ≤ 1 implies β(fx,fy,t) ≤ 1

and

γ(x,y,t) ≥ 1 implies γ(fx,fy,t) ≥ 1.

Definition 2.5. Let (X,F,T) be a Menger probabilistic partial metric space, f : X → X be a given mapping

and β,γ : X×X×(0,∞) → (0,∞) be two functions. We say that f is a (β,γ)-admissible ψ-type contractive

mapping if f is a (β,γ)-admissible mapping, satisfying in the following inequality

γ(fx,fy,t)
( 1
Ffx,fy(φ(ct))

− 1
)
≤ β(x,y,t)ψ

( 1
Fx,y(φ(t))

− 1
)

(2.2)

for all x,y ∈ X and all t > 0 such that Fx,y(φ(t)) > 0, where c ∈ (0, 1), φ ∈ Φ and ψ ∈ Ψ.

According to Gopal et al. [10, Theorem 2.1], we present a new fixed point theorem in the Menger proba-

bilistic partial metric spaces.

Theorem 2.1. Let (X,F,T) be a G-complete Menger probabilistic partial metric space and f : X → X be

a (β,γ)-admissible ψ-type contractive mapping satisfying the following conditions:

(i) x0 ∈ X exists such that β(x0,fx0, t) ≤ 1 and γ(x0,fx0, t) ≥ 1 for all t > 0,

(ii) if {xn} is a sequence in X such that β(xn−1,xn, t) ≤ 1 and γ(xn,xn+1, t) ≥ 1 for all n ∈ N, and for all

t > 0 and xn → x as n →∞, then β(xn,x,t) ≤ 1 and γ(xn,fx,t) ≥ 1 for all n ∈ N and for all t > 0.

Then f has a fixed point.

Proof. Let x0 ∈ X be such that condition (i) holds. We define inductively the sequence {xn} in X by

xn+1 = fxn, for n = 0, 1, 2, . . . . We may suppose that xn+1 6= xn for each n, otherwise f has obviously

a fixed point. We conclude from (β,γ)-admissibility of the mapping f, the condition (i), and by induction

that β(xn−1,xn, t) ≤ 1 and γ(xn,xn+1, t) ≥ 1 for all n ∈ N and all t > 0. From properties of the function

φ, it is possible to find some t > 0 such that Fx0,x1 (φ(t)) > 0. It implies that Fx0,x1 (φ(
t
c
)) > 0, too. From

(2.2), we have that

1

Fx1,x2 (φ(ct))
− 1 ≤ γ(fx0,fx1, t)

( 1
Ffx0,fx1 (φ(ct))

− 1
)

≤ β(x0,x1, t)ψ
( 1
Fx0,x1 (φ(t))

− 1
)

≤ ψ
( 1
Fx0,x1 (φ(t))

− 1
)
.



Int. J. Anal. Appl. 17 (5) (2019) 777

Repeating the above procedure successively r times (r < n), we obtain

1

Fxn,xn+1 (φ(c
rt))
− 1 ≤ ψn−r

( 1
Fxr,xr+1 (φ(

crt
cn−r

))
− 1
)
. (2.3)

Since ψn(an) → 0, whenever an → 0, then from (2.3), for any positive real number r we have

Fxn,xn+1 (φ(c
rt)) → 1, as n →∞. (2.4)

Now, let ε > 0 be given and choose r > 0 so that φ(crt) < ε. Regarding (2.4) we deduce

Fxn,xn+1 (ε) → 1, as n →∞, for every ε > 0. (2.5)

On the other hand, we can write

Fxn,xn+p (ε) ≥ T
(
Fxn,xn+1 (

ε

3
),Fxn+1,xn+1 (

ε

3
),Fxn+1,xn+p (

ε

3
)
)

≥ . . .

≥ T
(
Fxn,xn+1 (

ε

3
),Fxn,xn+1 (

ε

3
),T(Fxn+1,xn+2 (

ε

9
),Fxn+1,xn+2 (

ε

9
),

T(. . . ,T(Fxn+p−2,xn+p−1 (
ε

3p−1
),Fxn+p−2,xn+p−1 (

ε

3p−1
),Fxn+p−1,xn+p (

ε

3p−1
)))
)
.

On making n →∞ and in view of (2.5), for any positive integer p, we have

Fxn,xn+p (ε) → 1, as n →∞, for every ε > 0.

It follows that {xn} is a G-Cauchy sequence. Since (X,F,T) is G-complete, {xn} is convergent and

lim
n→∞

Fxn,xn+p (t) = lim
n→∞

Fxn,u(t) = Fu,u(t) (2.6)

for some u ∈ X. Furthermore, we get

Ffu,u(ε) ≥ T
(
Ffu,xn+1 (

ε

3
),Fu,xn+1 (

ε

3
),Fxn+1,xn+1 (

ε

3
)
)
. (2.7)

Taking into account the continuity of φ at zero, t1 > 0 exists such that φ(t1) <
ε
3
. From (2.6), n0 ∈ N exists

such that Fxn,u(φ(t1)) > 0 for all n ≥ n0. Hence, applying (ii) we derive that

1

Fxn+1,fu(
ε
3
)
− 1 ≤ γ(fxn,fu,t1)

( 1
Ffxn,fu(φ(t1))

− 1
)
≤ β(xn,u,t1)ψ

( 1
Fxn,u(φ(

t1
c

))
− 1
)

for all n ≥ n0. Now, letting n →∞, since ψ(0) = 0 and by the continuity of function ψ, we obtain

Fxn+1,fu(
ε

3
) → 1, as n →∞.

From (2.1) and (2.6) Fxn+1,xn+1 (
ε
3
) → 1 as n →∞, too. Passing n to infinity in the relation (2.7) it follows

that Ffu,u(ε) = 1, for each ε > 0. Thus fu = u. This completes the proof. �



Int. J. Anal. Appl. 17 (5) (2019) 778

Example 2.3. Let X = R+. Define F : X ×X →D+ by

Fx,y(t) =
t

t + max{x,y}

for all x,y ∈ X and for all t > 0. Define the mapping f : X → X by

fx =




x
2
, if x ∈ [0, 1),

2x, otherwise

and the functions β and γ from X ×X × (0,∞) into (0,∞) by

β(x,y,t) =
t

t + |x−y|
,

γ(x,y,t) =




t
t+2(x+y)

, if x,y ∈ [0, 1),

1
4
( t
t+x+y

), otherwise

for all t > 0. Now, consider self-mappings φ and ψ on R+ defined by φ(t) = ψ(t) = t and let c = 1
2

.

Obviously, f is (β,γ)-admissible. To show that f is a (β,γ)-admissible ψ-type contractive mapping, we have

to check the condition (2.2).

To do this, we distinguish three cases:

case I: If 0 ≤ x ≤ y < 1, then

γ(fx,fy,t)
( 1
Ffx,fy(φ(ct))

− 1
)

=
y

t + |x + y|

≤ β(x,y,t)ψ
( 1
Fx,y(φ(t))

− 1
)

=
y

t + |x−y|
.

case II: If x ∈ [0, 1) and y /∈ [0, 1], then

γ(fx,fy,t)
( 1
Ffx,fy(φ(ct))

− 1
)

=
2y

2t + |x + 4y|
≤ β(x,y,t)ψ

( 1
Fx,y(φ(t))

− 1
)

=
y

t + |x−y|
.

case III: If x,y /∈ [0, 1), then

γ(fx,fy,t)
( 1
Ffx,fy(φ(ct))

− 1
)

=
y

t + |2x + 2y|
≤ β(x,y,t)ψ

( 1
Fx,y(φ(t))

− 1
)

=
y

t + |x−y|
.

It can be easily verified that all conditions of Theorem 2.1 hold, and therefore f has a fixed point.

We denote by Fix(f) the set of fixed points of f. In what follows, we give a sufficient condition for the

uniqueness of the fixed point in Theorem 2.1.

(H): For all u,v ∈ Fix(f) and for all t > 0 there exists z ∈ X such that

β(z,fz,t) ≤ 1 with β(u,z,t) ≤ 1 and β(v,z,t) ≤ 1,



Int. J. Anal. Appl. 17 (5) (2019) 779

and

γ(z,fz,t) ≥ 1 with γ(u,z,t) ≥ 1 and γ(v,z,t) ≥ 1.

Similar to this condition was already considered in the paper [10].

Theorem 2.2. Adding condition (H) to the hypotheses of Theorem 2.1, we obtain that f has a unique fixed

point.

Proof. Let u,v ∈ Fix(f). From condition (H), z ∈ X exists such that

β(z,fz,t) ≤ 1 with β(u,z,t) ≤ 1 and β(v,z,t) ≤ 1,

and

γ(z,fz,t) ≥ 1 with γ(u,z,t) ≥ 1 and γ(v,z,t) ≥ 1.

Taking into account (β,γ)-admissibility of f, we obtain

β(fz,f2z,t) ≤ 1, β(u,fz,t) ≤ 1, β(v,fz,t) ≤ 1

and

γ(fz,f2z,t) ≥ 1, γ(u,fz,t) ≥ 1, γ(v,fz,t) ≥ 1.

Consequently, by induction, we get

β(zn,zn+1, t) ≤ 1, β(u,zn, t) ≤ 1, β(v,zn, t) ≤ 1

and

γ(zn+1,zn+2, t) ≥ 1, γ(u,zn+1, t) ≥ 1, γ(v,zn+1, t) ≥ 1

for all t > 0, where zn = f
nz. Then, using (2.2) we derive

1

Fu,fnz(φ(ct))
− 1 ≤ γ(u,zn, t)

( 1
Ffu,f(fn−1z)(φ(ct))

− 1
)

≤ β(u,zn−1, t)ψ
( 1
Fu,fn−1z(φ(t))

− 1
)

≤ ψ
( 1
Fu,fn−1z(φ(t))

− 1
)
.

It follows that

1

Fu,fnz(φ(ct))
− 1 ≤ ψn

( 1
Fu,z(φ(

t
cn−1

))
− 1
)

for all n ∈ N.

Letting n → ∞, we obtain lim
n→∞

Fu,zn+1 = 1. A similar argument shows that lim
n→∞

Fv,zn+1 = 1. Using

these facts, it is easily can be shown that Fu,v(t) = 1 for any t > 0. It implies that u = v, and the proof is

complete. �



Int. J. Anal. Appl. 17 (5) (2019) 780

Example 2.4. Let X = R+ and Fx,y(t) = H(t − max{x,y}) for all x,y ∈ X and for all t > 0. Clearly,

(X,F,Tp) is a G-complete Menger probabilistic partial metric space. Define the mapping f : X → X by

f(x) =
x

2

and the functions γ and β from X ×X × (0,∞) into (0,∞) by

β(x,y,t) =




2, if x,y ∈ (2,∞),

t
t+|x−y|, otherwise

γ(x,y,t) =




1
3
, if x,y ∈ (5,∞),

5, otherwise

for all t > 0. Also suppose that φ,ψ : R+ → R+ defined by φ(t) = ψ(t) = t and let c = 1
2

. Now, it can be

easily shown that all the hypotheses of Theorem 2.2 are satisfied and so f has a unique fixed point.

In the sequel, we first introduce the concept of (β,γ)-contractive mapping of type (I) and then we describe

a fixed point theorem concerned with these kinds of contractions in the framework of Menger probabilistic

partial metric spaces.

Definition 2.6. Let (X,F,T) be a Menger probabilistic partial metric space and f : X → X be a given
mapping. We say that f is a (β,γ)-contractive mapping of type (I), if functions β : X×X×(0,∞) → (0,∞)
and γ : X ×X × (0,∞) → (0,∞) exist such that

β(x,y,t)Ffx,fy(φ(t)) ≥ γ(fx,fy,
t

c
) min

{
Fx,y(φ(

t

c
)),Fy,fy(φ(

t

c
)),Fy,fx(φ(

t

c
))
}

(2.8)

for all x,y ∈ X and all t > 0, where φ ∈ Φ and c ∈ (0, 1).

Example 2.5. Let X = {0, 2, 4} and F be as in Example 2.3. Clearly, (X,F,TM ) is a complete Menger

probabilistic partial metric space. Define the mapping f : X → X by f(0) = 0, f(2) = 4 and f(4) = 2. Also,

define two functions β and γ from X ×X × (0,∞) into (0,∞) by

β(x,y,t) =




2, if x = y,

1
2
, otherwise

γ(x,y,t) =




1, if x = y = 2, or x = y = 4,

2, if x = y = 0,

1
x+y+1

, otherwise.

Now, consider φ : R+ → R+ defined by φ(t) = t and let c = 1
2

. Then f is a (β,γ)-contractive mapping of

type (I).



Int. J. Anal. Appl. 17 (5) (2019) 781

Two useful following lemmas help us to prove Theorem 2.3.

Lemma 2.1. Let (X,F,T) be a Menger probabilistic partial metric space and φ be a Φ-function. Then the

following statement holds.

If for x,y ∈ X and c ∈ (0, 1) we have Fx,y(φ(t)) ≥ Fx,y(φ( tc )) for all t > 0, then x = y.

Proof. The proof is similar to [4, Lemma 2.9]. �

Lemma 2.2. Let (X,F,T) be a Menger probabilistic partial metric space. Then the function F is a lower

semi-continuous function of points, i.e., for every fixed t > 0 and every two sequences {xn}, {yn} in X such

that lim
n→∞

Fxn,x(t) = Fx,x(t) = 1 and lim
n→∞

Fyn,y(t) = Fy,y(t) = 1 it follows that lim inf
n→∞

Fxn,yn (t) = Fx,y(t).

Proof. Let t > 0 and ε > 0 be given. Since Fx,y is left-continuous at t, so h exists such that 0 < 2h < t and

Fx,y(t) − Fx,y(t − 2h) < ε. Put Fx,y(t − 2h) = a. Taking into account continuity of T and T(a, 1, 1) = a,

there is a real number l in (0, 1), fulfills

T(a,l, l) > a−
ε

3
and T(a−

ε

3
, l, l) > a−

2ε

3
.

On the other hand, by our assumptions, an integer Mh,l exists such that

Fxn,x(
h

2
) > l and Fyn,y(

h

2
) > l,

whenever n > Mh,l. Now, by (2.1) and (PPM3)

Fxn,yn (t) ≥ T(Fxn,y(t−h),Fy,yn (
h

2
),Fy,yn (

h

2
)) (2.9)

and

Fxn,y(t−h) ≥ T(Fxn,x(
h

2
),Fx,y(t− 2h),Fx,x(

h

2
)) > T(a,l, l) > a−

ε

3
. (2.10)

Thus, on combining (2.9) and (2.10), we have

Fxn,yn (t) ≥ T(a−
ε

3
, l, l) > a−

2ε

3
> Fx,y(t) −ε.

This completes the proof. �

Now, we present a new version of [10, Theorem 3.2] due to Gopal et al. in the Menger probabilistic partial

metric spaces.

Theorem 2.3. Let (X,F,T) be a complete Menger probabilistic partial metric space, which satisfies T(a,a,a) ≥

a with a ∈ [0, 1]. Let f : X → X be a (β,γ)-contractive mapping of type (I) satisfying the following condi-

tions:

(i) f is (β,γ)-admissible,

(ii) there exists x0 ∈ X such that β(x0,fx0, t) ≤ 1 and γ(x0,fx0, t) ≥ 1 for all t > 0,



Int. J. Anal. Appl. 17 (5) (2019) 782

(iii) if {xn} is a sequence in X such that β(xn−1,xn, t) ≤ 1 and γ(xn,xn+1, t) ≥ 1 for all n ∈ N, and all

t > 0 and xn → x as n →∞, then β(xn−1,x,t) ≤ 1 and γ(xn,fx,t) ≥ 1 for all n ∈ N and all t > 0.

Then f has a fixed point.

Proof. Since T(a,a,a) ≥ a for all a ∈ [0, 1], then T ≥ TM . Let x0 ∈ X be such that (ii) holds and define

a sequence {xn} in X so that xn+1 = fxn, for n = 0, 1, . . . . We suppose xn+1 6= xn for all n = 0, 1, . . .,

otherwise f has trivially a fixed point. From (i), (ii) and by induction, we get β(xn−1,xn, t) ≤ 1 and

γ(xn,xn+1, t) ≥ 1 for all n ∈ N and all t > 0. By continuity of φ at zero, for each t > 0 we can find r > 0

such that t > φ(r) and therefore using (2.8) and (PPM3) we have

Fxn,xn+1 (t) ≥ β(xn−1,xn,r)Ffxn−1,fxn (φ(r))

≥ γ(xn,xn+1,
r

c
) min

{
Fxn−1,xn (φ(

r

c
)),Fxn,fxn (φ(

r

c
)),Fxn,fxn−1 (φ(

r

c
))
}

≥ min
{
Fxn−1,xn (φ(

r

c
)),Fxn,xn+1 (φ(

r

c
))
}
.

We will show that

Fxn,xn+1 (φ(r)) ≥ Fxn−1,xn (φ(
r

c
)). (2.11)

If we assume that Fxn,xn+1 (φ(
r
c
)) is the minimum, then from Lemma 2.1, we deduce that xn = xn+1. This

is in contradiction with the assumption xn 6= xn+1 and so Fxn−1,xn (φ(
r
c
)) is the minimum i.e., inequality

(2.11) holds. Now, from (2.11), it follows that

Fxn,xn+1 (t) ≥ Fxn,xn+1 (φ(r)) ≥ Fxn−1,xn (φ(
r

c
)) ≥ . . . ≥ Fx0,x1 (φ(

r

cn
)),

that is,

Fxn,xn+1 (t) ≥ Fx0,x1 (φ(
r

cn
)), (2.12)

for arbitrary n ∈ N. Next, let m,n ∈ N with m > n, then by (2.1) and (PPM3) we get

Fxn,xm ((2(m−n) − 1)t) ≥ T(Fxn,xn+1 (t),Fxn,xn+1 (t),Fxn+1,xm ((2(m−n) − 3)t)).

By strictly increasing of φ and also making use of (2.12) we have

Fxn,xm ((2(m−n) − 1)t) ≥ min
{
Fxn,xn+1 (t), . . . ,Fxm−1,xm−2 (t),Fxm−1,xm (t)

}
≥ min

{
Fx0,x1 (φ(

r

cn
)), . . . ,Fx0,x1 (φ(

r

cm−2
)),Fx0,x1 (φ(

r

cm−1
))
}

= Fx0,x1 (φ(
r

cn
)).

Since φ( r
cn

) →∞ as n →∞, for fixed ε ∈ (0, 1), n0 ∈ N exists such that for each n ≥ n0, Fx0,x1 (φ(
r
cn

)) >

1 −ε. This implies that, for every m > n ≥ n0,

Fxn,xm ((2(m−n) − 1)t) > 1 −ε.



Int. J. Anal. Appl. 17 (5) (2019) 783

By the arbitrariness of t > 0 and ε ∈ (0, 1), we deduce that {xn} is a Cauchy sequence in the complete

Menger probabilistic partial metric space (X,F,T). Thus, u ∈ X exists such that

lim
m,n→∞

Fxm,xn (t) = lim
n→∞

Fxn,u(t) = Fu,u(t). (2.13)

We are going to show that u ∈ Fix(f). Indeed, we have

Ffu,u(2t) ≥ T
(
Ffu,xn (φ(r)),Fxn,u(2t− 2φ(r)),Fxn,xn (φ(r))

)
≥ min

{
Ffu,xn (φ(r)),Fxn,u(2t− 2φ(r))

}
.

We may assume that xn 6= fu for all n ∈ N, since otherwise if xn = fu for infinitely many values of n, then

u = fu and hence the proof is finished. Now, from (2.13), for any arbitrary ε ∈ (0, 1) and n large enough,

we get Fxn,u(2t− 2φ(r)) > 1 − ε. Hence, Ffu,u(2t) ≥ min{Ffu,xn (φ(r)), 1 − ε}. Since ε > 0 is arbitrary, it

yields that Ffu,u(2t) ≥ Ffu,xn (φ(r)). Next, using (2.8) we deduce

Fu,fu(2t) ≥ Fxn,fu(φ(r))

≥ β(xn−1,u,r)Ffxn−1,fu(φ(r))

≥ γ(fxn−1,fu,
r

c
) min

{
Fxn−1,u(φ(

r

c
)),Fu,fu(φ(

r

c
)),Fu,fxn−1 (φ(

r

c
))
}

≥ min
{
Fxn−1,u(φ(

r

c
)),Fu,fu(φ(

r

c
)),Fu,xn (φ(

r

c
))
}
.

By taking the limit infimum on both sides of the above inequality and applying Lemma 2.2, we have

Fu,fu(2t) ≥ lim
n→∞

inf Fxn,fu(φ(r))

≥ lim
n→∞

inf min
{
Fxn−1,u(φ(

r

c
)),Fu,fu(φ(

r

c
)),Fu,xn (φ(

r

c
))
}

≥ min
{

1 −ε,Fu,fu(φ(
r

c
)), 1 −ε

}
.

Finally, since ε ∈ (0, 1) is arbitrary, then Ffu,u(φ(r)) ≥ Fu,fu(φ( rc )). From Lemma 2.1, we conclude that

u = fu and so we achieve our desired goal. �

Example 2.6. Let X = [0, 1], p(x,y) = max{x,y} and (X,F,T) be as in Example 2.1. Then (X,F,T) is a

complete Menger probabilistic partial metric space. Define the mapping f : X → X by

fx =




x
4
, if x ∈ [0, 1

2
) ∪ ( 1

2
, 1],

0, if x = 1
2

,

and the functions β and γ from X ×X × (0,∞) into (0,∞) by

β(x,y,t) =
t + max{x,y}
t + min{x,y}

,



Int. J. Anal. Appl. 17 (5) (2019) 784

γ(x,y,t) =
t

x + y + t
.

We consider φ : R+ → R+ defined by φ(t) = t and let c = 1
2

. It is routine to see that all the hypotheses

of Theorem 2.3 are satisfied, and therefore f has a fixed point.

Theorem 2.4. Adding condition (H) to the hypotheses of Theorem 2.3, we obtain that f has a unique fixed

point.

Proof. Let u,v ∈ X be such that u = fu and v = fv. From condition (H), z ∈ X exists such that

β(z,fz,t) ≤ 1 with β(u,z,t) ≤ 1 and β(v,z,t) ≤ 1, and γ(z,fz,t) ≥ 1 with γ(u,z,t) ≥ 1 and γ(v,z,t) ≥ 1.

By virtue of the fact that f is (β,γ)-admissible and using induction, we derive

β(zn,zn+1, t) ≤ 1, β(u,zn, t) ≤ 1, β(v,zn, t) ≤ 1,

and

γ(zn+1,zn+2, t) ≥ 1, γ(u,zn+1, t) ≥ 1, γ(v,zn+1, t) ≥ 1

for all t > 0, where zn = f
nz (n ∈ N). By continuity of φ, r > 0 exists such that t > φ(r) and therefore we

have

Fu,zn+1 (t) ≥ Fu,zn+1 (φ(r))

≥ β(u,zn,r)Ffu,fzn (φ(r))

≥ γ(fu,fzn,
r

c
) min

{
Fu,zn (φ(

r

c
)),Fzn,zn+1 (φ(

r

c
)),Fzn,fu(φ(

r

c
))
}

≥ min
{
Fu,zn (φ(

r

c
)),Fzn,zn+1 (φ(

r

c
))
}
.

Now, we consider following cases:

case I. If Fzn,zn+1 (φ(
r
c
)) is the minimum, then by (2.8) and (PPM3), it follows that

Fu,zn+1 (φ(r)) ≥ Fzn,zn+1 (φ(
r

c
))

≥ β(zn−1,zn,
r

c
)Ffzn−1,fzn (φ(

r

c
))

≥ γ(zn,zn+1,
r

c2
) min

{
Fzn−1,zn (φ(

r

c2
)),Fzn,fzn (φ(

r

c2
)),Fzn,zn (φ(

r

c2
))
}

≥ min
{
Fzn−1,zn (φ(

r

c2
)),Fzn,zn+1 (φ(

r

c2
))
}
.

Now, if Fzn,zn+1 (φ(
r
c2

)) is the minimum for some n ∈ N, then by Lemma 2.1, we deduce that zn = zn+1.

Applying (PPM3), we get Fu,zn+1 (φ(r)) ≥ Fzn,zn+1 (φ(
r
c2

)) ≥ Fu,zn+1 (φ(
r
c2

)), then u = zn+1. Consequently



Int. J. Anal. Appl. 17 (5) (2019) 785

β(v,u,t) ≤ 1 and γ(fv,fu,t) ≥ 1 for all t > 0 and thus we have

Fv,u(φ(t)) ≥ β(v,u,t)Ffv,fu(φ(t))

≥ γ(fv,fu,
t

c
) min

{
Fv,u(φ(

t

c
)),Fu,fu(φ(

t

c
)),Fu,fv(φ(

t

c
))
}

≥ Fv,u(φ(
t

c
)).

Again, by Lemma 2.1, we have u = v.

On the other hand, if Fzn−1,zn (φ(
r
c2

)) is the minimum, then

Fzn,zn+1 (φ(
r

c
)) ≥ Fzn−1,zn (φ(

r

c2
)) ≥ . . . ≥ Fz0,z1 (φ(

r

cn+1
)),

and, letting n → ∞, we get Fzn,zn+1 (φ(
r
c
)) → 1. Therefore lim

n→∞
Fu,zn+1 (t) = 1. A similar method shows

that lim
n→∞

Fv,zn+1 (t) = 1. By virtue of these facts, we get Fu,v(t) = 1 for each t > 0. Hence, u = v.

case II. Suppose that Fu,zn (φ(
r
c
)) is the minimum, then we get

Fu,zn+1 (φ(r)) ≥ Fu,zn (φ(
r

c
)) ≥ Fu,zn−1 (φ(

r

c2
)) ≥ . . . ≥ Fu,z0 (φ(

r

cn+1
)).

A similar argument as above shows that u = v, and the proof is complete. �

In the sequel, we first introduce the concept of (β,γ)-contractive mapping of type (II) and then we describe

a fixed point theorem concerned with these kinds of contractions in the setup Menger probabilistic partial

metric spaces.

Definition 2.7. Let (X,F,T) be a Menger probabilistic partial metric space and f : X → X be a given

mapping. We say that f is a (β,γ)-contractive mapping of type (II), if functions β : X×X×(0,∞) → (0,∞)

and γ : X ×X × (0,∞) → (0,∞) exist such that

β(x,y,t)Ffx,fy(φ(t)) ≥ γ(fx,fy,
t

c
) min

{
Fx,fx(φ(

t

c
)),Fy,fy(φ(

t

c
))
}

(2.14)

for all x,y ∈ X, for all t > 0, where c ∈ (0, 1) and φ ∈ Φ.

Now, we present a new version of [10, Theorem 3.4] due to Gopal et al. in the Menger probabilistic partial

metric spaces.

Theorem 2.5. Let (X,F,T) be a complete Menger probabilistic partial metric space and f : X → X be a

(β,γ)-contractive mapping of type (II). Suppose that the following conditions hold:

(i) f is (β,γ)-admissible,

(ii) x0 ∈ X exists such that β(x0,fx0, t) ≤ 1 and γ(x0,fx0, t) ≥ 1 for all t > 0,

(iii) for each sequence {xn} in X such that β(xn−1,xn, t) ≤ 1 and γ(xn,xn+1, t) ≥ 1, for all n ∈ N and all

t > 0, k0 ∈ N exists such that β(xm−1,xn−1, t) ≤ 1 and γ(xm,xn, t) ≥ 1, for all m,n ∈ N with m > n ≥ k0

and for all t > 0,



Int. J. Anal. Appl. 17 (5) (2019) 786

(iv) if {xn} is a sequence in X such that β(xn−1,xn, t) ≤ 1 and γ(xn,xn+1, t) ≥ 1 for all n ∈ N and all

t > 0 and xn → x as n →∞, then β(xn−1,x,t) ≤ 1 and γ(xn,fx,t) ≥ 1 for all n ∈ N and all t > 0.

Then f has a fixed point. In addition, if condition (H) holds, then f has a unique fixed point.

Proof. Let x0 ∈ X be such that (ii) holds. Define a sequence {xn} in X such that xn+1 = fxn for all

n = 0, 1, . . . . We may suppose that xn+1 6= xn for all n = 0, 1, . . ., otherwise f has trivially a fixed point.

By (i) and (ii), and applying induction, we get β(xn−1,xn, t) ≤ 1 and γ(xn,xn+1, t) ≥ 1 for all n and all

t > 0. By continuity of φ at zero, for each t > 0, r > 0 exists such that t > φ(r), thus β(xn−1,xn,r) ≤ 1

and γ(xn,xn+1,
r
c
) ≥ 1. It follows from condition (2.14) and (PPM1) that

Fxn,xn+1 (t)) ≥ Ffxn−1,fxn (φ(r))

≥ β(xn−1,xn,r)Ffxn−1,fxn (φ(r))

≥ γ(xn,xn+1,
r

c
) min

{
Fxn−1,xn (φ(

r

c
)),Fxn,xn+1 (φ(

r

c
)),
}

≥ min
{
Fxn−1,xn (φ(

r

c
)),Fxn,xn+1 ((

r

c
))
}
.

Now, if Fxn,xn+1 (φ(
r
c
)) is the minimum, then Fxn,xn+1 (φ(r)) ≥ Fxn,xn+1 (φ(

r
c
)) and so by Lemma 2.1, xn =

xn+1, which contradicts the assumption xn 6= xn+1. Thus Fxn−1,xn (φ(
r
c
)) is the minimum, and so

Fxn,xn+1 (t) ≥ Fxn−1,xn (φ(
r

c
)) ≥ . . . ≥ Fx0,x1 (φ(

r

cn
)).

Letting n →∞, then

Fxn,xn+1 (t) → 1. (2.15)

We claim that {xn} is a Cauchy sequence. Suppose the contrary. Then there exist ε > 0, λ ∈ (0, 1) for which

we can find subsequences {xm(s)} and {xn(s)} of {xn} such that n(s) is the smallest index for which

s < m(s) < n(s), Fxm(s),xn(s) (ε) ≤ 1 −λ, Fxm(s),xn(s)−1 (ε) > 1 −λ.

By the properties of φ, ε1 > 0 exists such that

φ(ε1) < ε.

We deduce that Fxm(s),xn(s) (φ(ε1)) ≤ 1 − λ, so {xn} is not Cauchy sequence with respect to φ(ε1) and λ.

Thus, increasing sequences of integers m(s) and n(s) exist such that n(s) is the smallest index for which

s < m(s) < n(s), Fxm(s),xn(s) (φ(ε1)) ≤ 1 −λ, Fxm(s),xn(s)−1 (φ(ε1)) > 1 −λ. (2.16)

Take a real number η such that 0 < η < φ( ε1
c

) −φ(ε1). From (2.16) it follows that

Fxm(s),xn(s)−1 (φ(
ε1
c

) −η) > 1 −λ.



Int. J. Anal. Appl. 17 (5) (2019) 787

Then, for any 0 < λ1 < λ < 1, by (2.15) it is possible to find a positive integer N1 such that for all s > N1,

we have

Fxm(s)−1,xm(s) (η) > 1 −λ1, Fxn(s)−1,xn(s) (η) > 1 −λ1. (2.17)

From (2.17) it follows that

Fxm(s)−1,xm(s) (φ(
ε1
c

)) ≥ Fxm(s)−1,xm(s) (η) > 1 −λ1 > 1 −λ. (2.18)

A similar relation holds when one substitutes xm(s)−1 and xm(s) with xn(s)−1 and xn(s), respectively. On

the other hand, from (2.18) we observe that

1 −λ ≥ Fxm(s),xn(s) (φ(ε1)) = Ffxm(s)−1,fxn(s)−1 (φ(ε1))

≥ β(xm(s)−1,xn(s)−1,ε1)Ffxm(s)−1,fxn(s)−1 (φ(ε1))

≥ γ(fxm(s)−1,fxn(s)−1,
ε1
c

) min
{
Fxm(s)−1,xm(s) (φ(

ε1
c

))Fxn(s)−1,xn(s) (φ(
ε1
c

))
}

> γ(fxm(s)−1,fxn(s)−1,
ε1
c

) min{1 −λ, 1 −λ}

≥ 1 −λ.

This is a contradiction, therefore {xn} is a Cauchy sequence in the complete Menger probabilistic partial

metric space. Hence, u ∈ X exists such that lim
m,n→∞

Fxm,xn (t) = lim
n→∞

Fxn,u(t) = Fu,u(t). Now, we show that

u is a fixed point of f. We have

Ffu,u(2t) ≥ T(Ffu,xn (φ(r)),Fxn,u(2t− 2φ(r)),Fxn,xn (φ(r)). (2.19)

By continuity of φ, r > 0 exists such that t > φ(r). Furthermore, for arbitrary δ ∈ (0, 1), n0 ∈ N exists such

that for all n ≥ n0, we get

Fxn,u(2t− 2φ(r)) > 1 − δ. (2.20)

Hence, from (2.15), (2.19) and (2.20), we obtain that

Ffu,u(2t) ≥ T(Ffu,xn (φ(r)), 1 − δ, 1 − δ).

Since δ > 0 is arbitrary and T is continuous, we can write Ffu,u(2t) ≥ Ffu,xn (φ(r)). Without loss of

generality we may assume that xn 6= fu for all n ∈ N, otherwise if for infinitely many values of n, xn = fu,

then u = fu, and hence the proof is finished. Applying (2.14) and (iv), we derive

Fu,fu(2t) ≥ Fxn,fu(φ(r))

≥ β(xn−1,u,r)Ffxn−1,fu(φ(r))

≥ γ(fxn−1,fu,
r

c
) min

{
Fxn−1,fxn−1 (φ(

r

c
)),Fu,fu(φ(

r

c
))
}
.



Int. J. Anal. Appl. 17 (5) (2019) 788

Letting n → ∞ in the above inequality, we get Ffu,u(φ(r)) ≥ Fu,fu(φ( rc )). Thus u = fu by Lemma 2.1.

Hence f has a fixed point. Moreover, if (H) holds, then by using a similar technique as in the proof of

Theorem 2.4 one can see that u is a unique fixed point of f. �

3. Application to integral equation

Here, in this section, we wish to study the existence of a solution to a Volterra type integral equation, as

an application of our results.

Let k > 0 be an arbitrary fixed number. Consider the following Volterra type integral equation:

x(t) = g(x(t)) +

∫ t
0

Ω(t,s,x(s))ds, t ∈ [0,k] (3.1)

where g ∈ C(R) and Ω ∈ C([0,k] × [0,k] ×R,R). Equip C([0,k],R) with the partial metric

p(x,y) = max
t∈[0,k]

{|x(t)|, |y(t)|}, x,y ∈ C([0,k],R).

Now, we define the mapping F : C([0,k],R) ×C([0,k],R) →D+ by

Fx,y(t) = H(t−p(x,y)), t > 0, x,y ∈ C([0,k],R). (3.2)

We know that (C([0,k],R),F,TM ) is a complete Menger probabilistic partial metric space.

Theorem 3.1. Let k > 0, g ∈ C(R), and Ω ∈ C([0,k] × [0,k] ×R,R), satisfying the following conditions:

(i) ‖Ω‖∞ = sup
t,s∈[0,k],x∈C([0,k],R)

| Ω(t,s,x(s)) |< ∞,

(ii) c1,c2 ∈ (0, 1) exist such that for all x ∈ C([0,k],R), all r ∈ R, and all t,s ∈ [0,k] we have

|g(r)| ≤ c1|r|, |Ω(t,s,x(s))| ≤
c2

k
|x(s)|,

and c = c1 + c2 < 1. Then, the Volterra type integral equation (3.1) has a solution x
∗ ∈ C([0,k],R).

Proof. Consider the complete Menger probabilistic partial metric space (C([0,k],R),F,TM ) defined as (3.2).
Now, we define the mapping f : C([0,k],R) → C([0,k],R) by

f(x)(t) = g(x(t)) +

∫ t
0

Ω(t,s,x(s))ds.

For each x,y ∈ C([0,k],R) we have

p(f(x),f(y)) = max
t∈[0,k]

{|f(x)(t), |f(y)(t)|}

≤ max
t∈[0,k]

{|g(x(t))| +
∫ t
0
|Ω(t,s,x(s))|ds, |g(y(t))| +

∫ t
0
|Ω(t,s,y(s))|ds}

≤ c( max
t∈[0,k]

{|x(t)|, |y(t)|})

= cp(x,y)

≤ c max{p(x,y),p(y,f(y)),p(y,f(x))}.



Int. J. Anal. Appl. 17 (5) (2019) 789

Applying (3.2), for any r > 0, we derive

Ff(x),f(y)(r) = H
(
r −p(f(x),f(y))

)
≥ H

(r
c
− max{p(x,y),p(y,f(y)),p(y,f(x))}

)
= min

{
Fx,y(

r

c
),Fy,f(y)(

r

c
),Fy,f(x)(

r

c
)
}

for all x,y ∈ C([0,k],R). Thus all conditions of Theorem 2.3 are satisfied, when φ(r) = r for all r > 0 and

β(x,y,t) = γ(x,y,t) = 1 for all x,y ∈ C([0,k],R) and t > 0. Therefore, f has a fixed point x∗ ∈ C([0,k],R),

which is the solution of the integral equation (3.1). �

Example 3.1. Consider the following Volterra type integral equation

x(t) =
1

3
cos(x(t)) +

∫ t
0

ln(t + s + 1
2

) sin(e−3tx(s))

3 cosh 5
√

2s + 1
ds. (3.3)

Observe that Eq. (3.3) is a special case of the Eq. (3.1) when g(r) = 1
3

cos(r) (r ∈ R), and Ω(t,s,x(s)) =
ln(t+s+ 1

2
) sin(e−3tx(s))

3 cosh 5
√
2s+1

(t,s ∈ [0, 1]). Obviously, all the conditions of Theorem 3.1 are satisfied. Then the

Volterra type integral equation (3.3) has at least one solution which belongs to the space C([0, 1],R).

4. numerical results

Numerical methods can help us to investigate the solutions of differential and integral equations (see

for instance [2, 11, 18]). In this section, we use a numerical method to find an approximate solution for a

Volterra type integral equation. For this purpose, we use successive approximations method (SAM) [2] to

find a solution for the Example 3.1.

The SAM, also called the Picard iteration method provides a scheme that can be used for solving initial

value problems or integral equations. This method solves any problem by finding successive approximations

to the solution by starting with an initial guess as x0(t), called the zeroth approximation. As will be seen,

the zeroth approximation is any selective real-valued function that will be used in a recurrence relation to

determine the other approximations. The most commonly used values for the zeroth approximations are 0,

1, or t. Of course, other real values can be selected as well.

Given the nonlinear Volterra type integral equation

x(t) = f(x(t)) + λ

∫ t
a

K(t,s,x(s))ds,

where x is the unknown function to be determined, K(t,s,x(s)) is the kernel, and λ is a parameter. The

successive approximation method introduces the recurrence relation

x̃0(t) = any selective real valued function, (4.1)

x̃n+1(t) = f(x̃n(t)) + λ

∫ t
a

K(t,s, x̃n(s))ds, n ≥ 0.



Int. J. Anal. Appl. 17 (5) (2019) 790

The solution is determined by using the limit

x̃(t) = lim
n→∞

x̃n+1(t).

Now, we discuss on the solutions of Example 3.1. To this end, consider the following nonlinear equation

x(t) = F(x(t)) + G(x(t)), (4.2)

where

F(x(t)) =
1

3
cos(x(t)), (4.3)

and

G(x(t)) =

∫ t
0

ln(t + s + 1
2

) sin(e−3tx(s))

3 cosh 5
√

2s + 1
ds. (4.4)

Applying SAM, we solve the nonlinear Volterra type integral equation (3.3). To this aim, put t = 0 in

equation (3.3). We get x(0) = 1
3

cos(x(0)) or equivalently α = 1
3

cos(α). It gives us α = 0.3176508287. Now,

we choose x̃0(t) = 0.3176508287 in (4.1), by doing two steps of successive approximations method we find

x̃2(t) and consider it as an approximation for x(t) (see Fig. 1). Since equation (3.3) is nonlinear, it is difficult

to proceed this method further. In order to see how good is this approximation, we put x̃2(t) in the left and

right hand sides of equation (3.3) instead of x(t), and consider the difference of these values as error. we

define

errSAM (t) := |x̃2(t) − (F(x̃2(t)) + G(x̃2(t)))|.

In table (1), we have calculated x̃2(t) and errSAM (t) at different values of t. The error graph of errSAM (t)

is also plotted in Figure (2) in the interval [0, 1].

Table 1. The values of x̃2(t) and error related to SAM for different values of t

t ũ2(t) errSAM (t)

0 3.17e-1 e-10

0.1 3.14e-1 2.93e-5

0.2 3.15e-1 2.86e-5

0.3 3.16e-1 1.23e-5

0.4 3.17e-1 1.1e-5

0.5 3.19e-1 3.66e-5

0.6 3.21e-1 6.22e-5

0.7 3.23e-1 8.66e-5

0.8 3.24e-1 1.09e-4

0.9 3.25e-1 1.31e-4

1 3.27e-1 1.51e-4



Int. J. Anal. Appl. 17 (5) (2019) 791

Figure 1. graph of approximate solution x̃ in [0,1]

Figure 2. absolute error graph of approximate solution x̃ in [0,1]

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	1. Introduction
	2. Main result
	3. Application to integral equation
	4. numerical results
	References