J. Nig. Soc. Phys. Sci. 4 (2022) 1021

Journal of the
Nigerian Society

of Physical
Sciences

System of Non-Linear Volterra Integral Equations in a
Direct-Sum of Hilbert Spaces

Jabar S. Hassana,∗, Haider A. Majeedb, Ghassan Ezzulddin Arifb

a Department of Mathematics, College of Science, Salahaddin University - Erbil / Iraq
bDepartment of Mathematics, Tikrit University, College of Education for Pure Sciences, Tikrit/ Iraq

Abstract

We use the contraction mapping theorem to present the existence and uniqueness of solutions in a short time to a system of non-linear Volterra
integral equations in a certain type of direct-sum H[a, b] of a Hilbert space V [a, b]. We extend the local existence and uniqueness of solutions to
the global existence and uniqueness of solutions to the proposed problem. Because the kernel function is a transcendental function in H[a, b] on
the interval [a, b], the results are novel and very important in numerical approximation.

DOI:10.46481/jnsps.2022.1021

Keywords: system of non-linear integral equations, reproducing kernel Hilbert spaces, fixed point theorem

Article History :
Received: 20 August 2022
Received in revised form: 12 September 2022
Accepted for publication: 13 September 2022
Published: 01 October 2022

c© 2022 The Author(s). Published by the Nigerian Society of Physical Sciences under the terms of the Creative Commons Attribution 4.0 International license

(https://creativecommons.org/licenses/by/4.0). Further distribution of this work must maintain attribution to the author(s) and the published article’s title, journal citation, and DOI.

Communicated by: B. J. Falaye

1. Introduction

Non-linear Volterra integral equations have many applica-
tions in several fields, such as physics, chemistry, biology, and
engineering. For example, particle transport problems in as-
trophysics theory, electrostatics, potential theory, mathemati-
cal problems of radiative steady state, heat transfer problems,
and many other mathematical modeling are described by Volt-
tera integral equations [1-9]. In this paper, we introduce the
following system of non-linear Volterra integral equations for
t ∈ [a, b]:

f (t) = α(t) +
∫ t

a
F(t, s, f (s), g(s))d s, (1)

∗Corresponding author tel. no:
Email address: jabar.hassan@su.edu.krd (Jabar S. Hassan )

g(t) = β(t) +
∫ t

a
G(t, s, f (s), g(s))d s, (2)

where α,β ∈ H[a, b] is a direct sum of reproducing kernel
Hilbert space V [a, b] consisting of those absolutely continuous
functions whose derivative is square-integrable on [a, b]. F and
G are given functions that satisfy fixed regularity conditions. f
and g are unknown functions that need to be determined.

Recently, reproducing kernel Hilbert space methods have
been widely studied by many researchers to solve linear and
non-linear problems such as partial and ordinary differential
equations, as well as integral, fractional, and integral differen-
tial equations [4,7,9]. Obviously, considering the existence and
uniqueness of solutions to such kinds of problems is very im-
portant in pure and applied mathematics. In view of the fact
that most real phenomena and non-linear problems in the world
can not be solved analytically, researchers use numerical meth-
ods to obtain their approximate and numerical solutions in an

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Hassan et al. / J. Nig. Soc. Phys. Sci. 4 (2022) 1021 2

appropriate space.
In this work, we examine the local and global existence and

uniqueness of solutions to a system of non-linear Volterra inte-
gral equations in the reproducing kernel Hilbert space H[a, b].
This space is a very favorable space in numerical approxima-
tion since its reproducing kernel function is a transcendental
function in [a, b].

2. Preliminary Notation

This section is assigned to present basic notation, defini-
tions, and theorems which will be used later.

Definition 1. Let S , ∅. A Hilbert space H of continuous
real-valued functions f : S → R is called reproducing kernel
Hilbert space if there exists a function K : S × S → R in H
such that
〈 f (·), K(·, s)〉H = f (s), and K(·, s) ∈ H for all f ∈ H and all
s ∈ S . Such a function K = K(·, ·) is said to be a reproducing
kernel function of H [4,6].

Definition 2. Let V [a, b] be the space of all absolutely contin-
uous functions f : [a, b] → R such that f ′ ∈ L2[a, b] [4,6].

Theorem 1. The function space V [a, b] equipped with the in-
ner product [4]

〈 f1, f2〉V [a,b] = f1(a) f2(a) +
∫ b

a
f ′1 (t) f

′
2 (t)dt,

and associated with the norm

|| · || =
√
〈·, ·〉V [a,b],

is a reproducing kernel Hilbert space and the reproducing ker-
nel function K = K(·, ·) is defined by:

K(t,τ) =
1

2 sinh(b − a)
(

cosh(τ + t − b + a)

+ cosh(| τ− t | −b − a)
)
.

Definition 3. The function space

H[a, b] = V [a, b] ⊕ V [a, b],

consists of those functions ~h : [a, b] → R2 where ~h = (h1, h2)
such that h1 and h2 belong to V [a, b].

Definition 4. The inner product of the space H[a, b] is defined
by :

〈~f ,~g〉H[a,b] = 〈 f1, g1〉V [a,b] + 〈 f2, g2〉V [a,b],

where f = ( f1, f2) and g = (g1, g2). Such a space is called
direct sum of the reproducing kernel Hilbert space V [a, b].

3. Existence and Uniqueness

In this section, we discuss the Banach fixed point theorem to
show the local and global existence and uniqueness of solutions
to (1)-(2). To do this first, we need to introduce some basic
tools.

Let ~h ∈ H[a, b] and let A = {(t, s) : a ≤ s ≤ t ≤ b}. Define
maps T~h : [a, b] → R and L~h : [a, b] → R by:

T~h(t) =
∫ t

a
F
(
t, s,~h(s)

)
d s,

L~h(t) =
∫ t

a
G
(
t, s,~h(s)

)
d s,

such that the following conditions are hold for (k = 0, 1):

C1) ∂
k

∂tk F and
∂k

∂tk G are uniformly bounded functions on A×
R2.

C2) For some positive constants M and N such that

i)
∣∣∣∣ ∂k∂tk F(x, s, ~f1(s)) − ∂k∂tk F(y, s, ~f2(s))∣∣∣∣ 6 M (|x − y| +
‖~f1 − ~f2‖2

)
;

ii)
∣∣∣∣ ∂k∂tk G(x, s, ~g1(s)) − ∂k∂tk G(y, s, ~g2(s))∣∣∣∣ 6 N (|x − y| +
‖~g1 − ~g2‖2

)
.

Theorem 2. Let ~h ∈ H[a, b]. Then T~h ∈ H[a, b].

We first assert that T~h is absolutely continuous in [a, b]. By
condition (C1) for (k=0); F is uniformly bounded on A× R2
and condition (C2) part (i) there are positive constants M and
M1.

Let I j = {[a j, b j]}nj=1 be a finite collection of non-over lap-
ping intervals in [a, b], and let ε > 0 such that:

n∑
j=1

∣∣∣∣b j − a j∣∣∣∣ < ε(
M1(b − a) + M

).
Since,

n∑
j=1

∣∣∣∣T~h(b j) − T~h(a j)∣∣∣∣ = n∑
j=1

∣∣∣∣ ∫ b j
a

F
(
b j, s,~h(s)

)
d s

−

∫ a j
a

F
(
a j, s,~h(s)

)
d s

∣∣∣∣
=

n∑
j=1

∣∣∣∣ ∫ a j
a

F
(
b j, s,~h(s)

)
d s

+

∫ b j
a j

F
(
b j, s,~h(s)

)
d s

−

∫ a j
a

F
(
a j, s,~h(s)

)
d s

∣∣∣∣
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Hassan et al. / J. Nig. Soc. Phys. Sci. 4 (2022) 1021 3

6
n∑

j=1

∫ a j
a j

∣∣∣∣F(b j, s,~h(s)) − F(a j, s,~h(s))∣∣∣∣d s
+

∫ b j
a j

∣∣∣∣F(b j, s,~h(s))d s∣∣∣∣
6

n∑
j=1

∫ a j
a

M1
∣∣∣∣b j − a j∣∣∣∣d s + ∫ b j

a j
Md s

=

n∑
j=1

(
M1(a j − a)|b j − a j| + M|b j − a j|

)
6

(
M1(b − a) + M

) n∑
j=1

∣∣∣∣b j − a j∣∣∣∣
< ε.

Hence, T~h is absolutely continuous on in [a, b]. Next, we want
to show ∂

∂t T
~h(·) ∈ L2[a, b]. Leibniz rule implies for almost

every t ∈ [a, b] that

∂

∂t
T~h(t) =F

(
t, t,~h(t)

)
+

∫ t
a

∂

∂t
F
(
t, s,~h(s)

)
d s.

Then,

∫ b
a

∣∣∣∣ ∂
∂t

T~h(t)
∣∣∣∣2dt = ∫ b

a

∣∣∣∣∣F(t, t,~h(t))
+

∫ t
a

∂

∂t
F
(
t, s,~h(s)

)
d s

∣∣∣∣∣2dt
≤ 2

∫ b
a

∣∣∣∣F(t, t,~h(t))∣∣∣∣2dt
+ 2

∫ b
a

∣∣∣∣∣
∫ t

a

∂

∂t
F
(
t, s,~h(s)

)
d s

∣∣∣∣∣2dt.
It follows from condition (C1) for (k=0,1 ) there are positive
constants N, D and the Cauchy-Schwartz inequality that∫ b

a

∣∣∣∣ ∂
∂t

T~h(t)
∣∣∣∣2dt ≤ 2 ∫ b

a
N2dt

+ 2
∫ b

a

( ∫ t
a

( ∂
∂t

F
(
t, s,~h(s)

))2
d s

∫ t
a

12d s
)
dt,

implies∫ b
a

∣∣∣∣ ∂
∂t

T~h(t)
∣∣∣∣2dt ≤ 2 ∫ b

a
N2dt

+ 2
∫ b

a

( ∫ t
a

( ∂
∂t

F
(
t, s,~h(s)

))2
d s

∫ t
a

12d s
)
dt

6 2N2(b − a) + 2(b − a)
∫ b

a

∫ t
a

( ∂
∂t

F(t, s,~h(t)
)2

d sdt

6 2N2(b − a) + 2(b − a)
∫ b

a

∫ b
a

D2d sdt

= 2N2(b − a) + 2D2(b − a)3

< ∞.

Therefore, T~h belongs to H[a, b] by definitions (2) and (3).
Similar arguments one can use to show that L~h belongs to
H[a, b].

Theorem 3. Let ~f ∈ H[a, b]. Then L ~f ∈ H[a, b].

The proof is analogous to the proof of Theorem 2. Set

α,β ∈ H[a, b]. Define operators Γ : H[a, b] → H[a, b] and
Λ : H[a, b] → H[a, b] such that:

Γ~h(t) = α(t) + T~h(t);

Λ~h(t) = β(t) + L~h(t);

for all ~h ∈ H[a, b].

We divide the interval [a, b]into N equally sub-intervals a ≤
t0 < t1 < ... < tn ≤ b; where 4t = t j − t j−1 j = 1, 2, ..., N and
4t = b−aN . The inner product in H[t j, t j + 4t] is defined by:

〈~f ,~g〉H[t j,t j +4t] = 〈 f1, g1〉V [t j,t j +4t] + 〈 f2, g2〉V [t j,t j +4t],

for all ~f ,~g ∈ H[t j, t j + 4t]. As a result, we see that the
operators Γ : H[t j, t j + 4t] → H[t j, t j + 4t] and Λ : H[t j, t j +
4t] → H[t j, t j + 4t] become

Γ~h(µ) = α(µ) +
∫ µ

t j
F
(
µ, s,~h(s)

)
d s;

Λ~h(µ) = β(µ) +
∫ µ

t j
G
(
µ, s,~h(s)

)
d s;

for all µ ∈ H[t j, t j + 4t].

Lemma 1. Let ~h ∈ H[t j, t j + 4t] and 4t < 1 [7]. Then∥∥∥~h∥∥∥
2
≤
√

2
∥∥∥~h∥∥∥

H[t j,t j +4t]
.

remark 1. Assume that α(t j) = β(t j) for all j = 0, 1, ..., N − 1.

Theorem 4. Let ~h1,~h2 ∈ H[t j, t j + 4t]. Then∥∥∥∥Γ~h1 − Γ~h2∥∥∥∥
H[t j,t j +4t]

6 δ(4t)
∥∥∥∥~h1 −~h2∥∥∥∥

H[t j,t j +4t]
,

where δ(4t) ≤ C
√
4t, for some positive constant C.

3



Hassan et al. / J. Nig. Soc. Phys. Sci. 4 (2022) 1021 4

Since∥∥∥∥Γ~h1 − Γ~h2∥∥∥∥2H[t j,t j +4t] = (Γ~h1(t j) − Γ~h2(t j))2
+

∫ t j +4t
t j

( ∂
∂t

Γ~h1(t) −
∂

∂t
Γ~h2(t)

)2
dt

=
(
α(t j) −β(t j)

)2
+

∫ t j +4t
t j

(
F
(
t, t, ~h1(t)

)
− F

(
t, t, ~h2(t)

)
+

∫ t
ti

[ ∂
∂t

F
(
t, s, ~h1(s)

)
−
∂

∂t
F
(
t, s, ~h2(s)

)]
d s

)2
dt

Implies,∥∥∥∥Γ~h1 − Γ~h2∥∥∥∥2H[t j,t j +4t] 6 2
∫ t j +4t

t j

(
F
(
t, t, ~h1(t)

)
− F

(
t, t, ~h2(t)

))2
dt

+ 2
∫ t j +4t

t j

( ∫ t
ti

[ ∂
∂t

F
(
t, s, ~h1(s)

)
−
∂

∂t
F
(
t, s, ~h2(s)

)]
d s

)2
dt

By (C2) we get constants M1, M2 such that∥∥∥∥Γ~h1 − Γ~h2∥∥∥∥2
H[t j,t j +4t]

≤ 2
∫ t j +4t

t j
M21

∥∥∥∥~h1(t) −~h2(t)∥∥∥∥2
2
dt

+ 2
∫ t j +4t

ti

( ∫ t
ti

M2
∥∥∥∥~h1(s) −~h2(s)∥∥∥∥

2
d s

)2
dt.

Then,

∣∣∣∣∣∣∣∣Γ~h1 − Γ~h2∣∣∣∣∣∣∣∣2
H[t j,t j +4t]

6 2
∫ t j +4t

ti
M21 M

2
3

∥∥∥~h1 −~h2∥∥∥22dt
+ 2

∫ t j +4t
ti

( ∫ t
ti

M2 M3
∥∥∥~h1 −~h2∥∥∥2d s)2dt

6 2M21 M
2
3

∥∥∥~h1 −~h2∥∥∥22 (4t)
+

2
3

M22 M
2
3

∥∥∥~h1 −~h2∥∥∥22 (4t)3
= 4t

(
2M21 M

2
3 +

2
3

M22 M
2
3 (4t)

2
)∥∥∥∥~h1 −~h2∥∥∥∥2

2
.

By using Lemma 1 that

∣∣∣∣∣∣∣∣Γ~h1 − Γ~h2∣∣∣∣∣∣∣∣2 H[t j,t j +4t] 6 δ2(4t) ∥∥∥∥~h1 −~h2∥∥∥∥2H[t j,t j +4t],
Therefore,

∥∥∥∥Γ~h1 − Γ~h2∥∥∥∥
H[t j,t j +4t]

6 δ(4t)
∥∥∥∥~h1 −~h2∥∥∥∥

H[t j,t j +4t]
,

Where δ(4t) < C
√
4t, and

C =
√

2M23
(
M21 +

1
3 M

2
2 4

2 t
)
<

√
2M23

(
M21 +

1
3 M

2
2

)
if 4t < 1.

Theorem 5. Let f, g ∈ H[t j, t j + 4t]. Then∥∥∥∥Λ f − Λg∥∥∥∥
H[t j,t j +4t]

6 σ(4t)
∥∥∥∥ f − g∥∥∥∥

H[t j,t j +4t]
,

where σ(4t) ≤ C
√
4t, for some positive constant C.

The proof is similar to the proof of Theorem 4.

Theorem 6. Let F and G satisfy conditions (C1) and (C2).
Then there exists a unique solution ~h = ( f, g) ∈ H[a, b] to (1)
and (2).

For all ~h = ( f, g) in the space H[a, b]. It is clear that ~h 7→ Γ~h
and ~h 7→ Λ~h are maps from H[t j, t j +4t] into H[t j, t j +4t]. From
Theorems 4 and 5; since 4t is an arbitrary positive constant and
if we pick 4t small enough such that 4t < 1C2 then we conclude
that δ(4t) < 1 and σ(4t) < 1. Therefore, by Theorems 4 and 5
the operators Γ and Λ are contraction mapping on H[t j, t j +4t],
respectively. It is clear

(
H[t j, t j +4t],‖·‖H[t j,t j +4t]

)
is a complete

matrix space. Hence, the Banach contraction mapping theorem
guarantees that the operators Γ and Λ have a unique fixed point
~h = ( f, g) in H[t j, t j + 4t].

Let ε(4t) = min{δ(4t),σ(4t)}. The existence and unique-
ness of solutions in the entire interval [a, b] for (1) and (2) can
be achieved by iterating the local existence result. This is ac-
complished by taking [a,ε(4t)], [ε(4t), 2ε(4t)], ...[nε(4t), b].

4. conclusion

We studied the local and global existence and uniqueness
of solutions to a system of non-linear Volterra integral equa-
tions (1)-(2) in the reproducing kernel Hilbert spaces V [a, b]
and H[a, b]. The results are very significant in numerical meth-
ods since the reproducing kernel function of the space V [a, b]
is a smooth function on the compact interval [a, b] and it can be
used to solve a wide variety of linear and nonlinear problems.

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