On the-approximate-solution-of-a-nonlinear-singular-integral-equation

World Academy of Science, Engineering and Technology
International Journal of Mathematical, Computational Science and Engineering Vol:3 No:11, 2009

On the Approximate Solution of a
Nonlinear Singular Integral Equation
Nizami Mustafa, Cemal Ardil

AbstractIn this study, the existence and uniqueness of the
solution of a nonlinear singular integral equation that is defined on a
region in the complex plane is proven and a method is given for
finding the solution.
KeywordsApproximate solution, Fixed-point principle,
Nonlinear singular integral equations, Vekua integral operator

International Science Index 35, 2009 waset.org/publications/358

I. INTRODUCTION

A

S it is known, the application area of nonlinear singular
integral equation is so extensive such as the theories of
elasticity, viscoelasticity, thermo elasticity, hydrodynamics,
fluid mechanics and mathematical physics and many other
fields [1]-[6].
Furthermore, the solution of the seismic wave equation
that has a great importance in elastodynamics is investigated
by reducing it to the solution of nonlinear singular integral
equation by using Hilbert transformation [7].
All of these studies contribute to update of investigations
about the solution of the nonlinear singular integral equation
by using approximate and constructive methods.
In this study, the approximate solution of nonlinear singular
integral equations that is defined on a bounded region in
complex plane is discussed.
It is known that the investigation of the problems of the
Dirichlet boundary-value problems for many nonlinear
differential equation systems which have partial derivatives
and defined on a planar region G ⊂ ℂ can be reduced to the
investigation of the problem of a nonlinear singular integral
equation which is in the following form [8], [9], [10]-[14]

o

∂G is the boundary of the region G, G is the set of interior
points of G and

D = {( z , ϕ , t , s ) : z ∈ G , ϕ , t , s ∈ ℂ} = G × ℂ 3
are as
given, f , g : D0 → ℂ and F : D → ℂ are the known
functions,
is
any
scalar
parameter,
λ ∈ℝ
h ∈ H α (G )(0 < α < 1) and for z = x + iy, ζ = ξ + iη
followings are the known Vekua integral operators [15].

TG h(.)( z ) = (−1 / π ).∫∫ h(ζ ).(ζ − z ) −1 dξdη ,
G

Π G h(.)( z ) = (−1 / π ) ∫∫ h(ζ ).(ζ − z ) − 2 dξdη
G

In this study, by taking base, a more useful modified variant
of Schauder and Banach fixed point principles, the existence
and uniqueness of the solution of the Equation (1) is proven
with more weak conditions on the functions f , g and F.
Furthermore, iteration is given for the approximate solution
of the Equation (1) and it is shown that the iteration is
converging to the real solution.
II. BASIC ASSUMPTIONS AND AUXILIARY RESULTS
Further, throughout the study, if not the opposite said we
take the set G ⊂ ℂ as bounded and simple connected region.
o

For

G is to be the set of interior points of the region G and
o

ϕ ( z) =
(1)
λ.F ( z , ϕ ( z ), TG f (., ϕ (.))( z ), Π G g (., ϕ (.))( z )).

∂G is to be the boundary of G let G = ∂G U G .
If, for every z1 , z 2 ∈ G there exist H > 0 and α ∈ (0,1]
numbers such that

ϕ ( z1 ) − ϕ ( z 2 ) ≤ H . z1 − z 2

Here, let
o


D0 = ( z , ϕ ) : z ∈ G = ∂G U G, ϕ ∈ ℂ


= G ×ℂ

then it is said that the function

α

ϕ :G → ℂ

satisfies the

Holder condition on the set G with exponent α . We will
show the set of all functions that satisfies Holder condition on
the set G with exponent α with H α (G ) .

Nizami Mustafa is with the Mathematics Department, University of
Kafkas, Kars, 36100 Turkey (corresponding author to provide phone: +90474-2128898; fax +90-474-2122706; e-mail nizamimustafa@mynet.com)
Cemal Ardil is with the National Academy Aviation, AZ 1056 Baku,
Azerbaijan (e-mail: cemalardil@gmail.com).

For

(H

20

α

ϕ ∈ H α (G )

)

and 0 < α < 1 , the vector space

(G ); . α is a Banach space with the norm


ϕ
Here,

ϕ

= ϕ

α

∞

Hα (G )

≡ ϕ

∞

+ H (ϕ , α ; G ).

ϕ

= max{ϕ ( z ) : z ∈ G },

H (ϕ , α ; G ) =

sup
z1 , z 2 ∈G , z1 ≠ z 2

where

{ϕ ( z ) − ϕ ( z ) }
1

2




∫∫ ϕ (ζ )

p

G

1 < p, the vector space

p

with the following norm,

ϕ
Further,

p

= ϕ

L p (G )



p
≡  ∫∫ ϕ (ζ ) dξdη 


G


throughout

the

study


+ m 2 ϕ1 − ϕ 2

l1 z1 − z 2

(5)

2
2 +αp

α

and

.ϕ

p > 1 the

αp
2 +αp
p

.

(6)

p ) = max{M 1 (α , p ), M 2 (α , p)} ,
−1
p

M 1 (α , p ) = 2.( n(α , p )) + (π .( n(α , p )) ) ,
take

(2)

2

M 2 (α , p ) = (2. 4 ).( 4 − 1) −1 .(n(α , p ))α ,
n(α , p ) = (α . p. π )

p

−p
2 +α . p

Lemma 2.3.
For D0 R

= {( z , ϕ ) ∈ D0 : z ∈ G , ϕ ≤ R}, R > 0,

Bα (0; R) = { ∈ H α (G ) : ϕ ≤ R}, α ∈ (0,1)
ϕ

and ϕ ∈ Bα (0; R ) ,

f , g : D0 R → ℂ let

f1 ( z ) = f ( z , ϕ ( z )), g 1 ( z ) = g ( z , ϕ ( z )), z ∈ G . Then
for the functions f 1 , g1 : G → ℂ the following inequalities
are true

f 1 ( z ) ≤ m 0 + m 2 . R, z ∈ G ,
α

f 1 ( z1 ) − f 1 ( z 2 ) ≤ (m1 + m2 .R). z1 − z 2 , (7)

(3)

+ n 2 ϕ1 − ϕ 2

z 1 , z 2 ∈ G,

g1 ( z ) ≤ n0 + n2 .R, z ∈ G,

F ( z1 , ϕ1 , t1 , s1 ) − F ( z 2 , ϕ 2 , t 2 , s 2 ) ≤
α

∞

p

g ( z1 , ϕ1 ) − g ( z 2 , ϕ 2 ) ≤
α

≤ M (α , p ). ϕ

p

f ( z1 , ϕ 1 ) − f ( z 2 , ϕ 2 ) ≤

n1 z1 − z 2

Here, M (α ,

.

following inequalities be satisfied

m1 z1 − z 2

ϕ

α ∈ (0,1)

α

D = {( z , ϕ , t , s ) : z ∈ G , ϕ , t , s ∈ ℂ} = G × ℂ 3 . Besides,
for
every
and
z1 , z 2 ∈ G
( z k , ϕ k ) ∈ D0 , ( z k , ϕ k , t k , s k ) ∈ D , for k = 1,2 we
suppose that the number
α ∈ (0,1) and the positive
constants m1 , m 2 , n1 , n 2 , l1 , l 2 , l 3 , l 4 are exist such that

α

+ (πε 2 ) −1 / p . ϕ p ,

1/ p

we

D0 = {( z , ϕ ) : z ∈ G , ϕ ∈ ℂ} = G × ℂ and

α

d = sup{ z1 − z 2 : z1 , z 2 ∈ G}.


dξdη < +∞ ,


(L (G ); . ) is a Banach space
p

≤ 2.ε α . ϕ

Lemma 2.2. For ϕ ∈ H α (G ) ,
following inequality is held

are as given.
For L p (G ) = ϕ : G → ℂ :

∞

+ l 2 ϕ1 − ϕ 2 +

α

g1 ( z1 ) − g 2 ( z 2 ) ≤ (n1 + n2 .R ). z1 − z 2 , (8)

(4)

z 1 , z 2 ∈ G.

l 3 t1 − t 2 + l 4 s1 − s 2
Here,

m0 = max{ f ( z ,0) : z ∈ G },

We will denote the set of functions that satisfy the
conditions
(2),
(3)
and
(4)
with
H α ,1 (m1 , m2 ; D0 ), H α ,1 (n1 , n2 ; D0 ) and

H α ,1,1,1 (l1 , l 2 , l 3 , l 4 ; D) respectively.
Now, let us give some supplementary lemmas for the
theorems about the existence and uniqueness of the solution of
the Equation (1).
Lemma 2.1. If

ϕ ∈ H α (G ) , α ∈ (0,1)

then for every

n0 = max{ g ( z ,0) : z ∈ G } .
The proof of the Lemma 2.3 is obvious from the Inequalities
(2), (3) and the assumptions of the lemma.
Corollary 2.1. If the assumptions of the Lemma 2.3 are
satisfied then for, f 1 , g 1 ∈ H α (G ),

p > 1 and ε ∈ (0, d ) the following inequality is held

21

α ∈ (0,1) we have


f1

α

≤ m0 + m1 + 2 Rm 2 ,

g1

α

≤ n0 + n1 + 2 Rn2 .

F ( z ) ≤ l 0 + l 2 .R + l3 .L1 + l 4 .L2 , z ∈ G ,

(9)

F ( z1 ) − F ( z 2 ) ≤ (l1 + l 2 .R + l 3 .L1 + l 4 .L2 ) ×
α

Now, for

f ∈ H α ,1 (m1 , m2 ; D0 ), g ∈ H α ,1 (n1 , n2 ; D0 )

and ϕ ∈ H α (G ) ,

α ∈ (0,1) let us define the functions

~ ~
f 1 , g 1 : G → ℂ as given below
~
f 1 ( z ) = TG f (., ϕ (.))( z ),
~
g ( z ) = Π g (., ϕ (.))( z ).
1

z1 − z 2 , z1 , z 2 ∈ G .
Here, l 0

= F (.,0,0,0) ∞ .

The proof of the Lemma 2.5 is obvious from the definition of
(10)

the function F : G → ℂ , the Inequality (13) and from the
assumption of lemma.

G

For

the

bounded

operators

TG , Π G : H α (G ) → H α (G ) , α ∈ (0,1)
the following norms:

TG


ΠG

α

α

{

≡ sup TG ϕ

α

: ϕ ∈ H α (G ), ϕ

α

{

α

: ϕ ∈ H α (G ), ϕ

α

F

}

≤1

(11)

≡ sup Π Gϕ

satisfied then for

F ∈ H α (G ) we have

let us define

}

≤1

α

≤ L = 2{max(l 0 , l1 ) + l 2 .R + l3 .L1 + l 4 .L2 } .

satisfied then the operator A that is defined with

A(ϕ )( z ) =

(12)
[11], [14].

λ.F ( z , ϕ ( z ), (TG o f1 )( z ), (Π G o g1 )( z ), z ∈ G

Lemma 2.4. If

(14)

f ∈ H α ,1 (m1 , m2 ; D0 ), g ∈ H α ,1 (n1 , n2 ; D0 )

ϕ ∈ Bα (0; R ) , α ∈ (0,1)
~ ~
f 1 , g 1 ∈ H α (G ) and we have
~
f1

α

≤ L1 ,

then

in

and

this

transforms the sphere Bα (0; R ) to the sphere Bα (0; λ L ) .

case
satisfied and if

~
g1

α

≤ L2 .

(13)

Here,

L1 = (m0 + m1 + 2m2 .R ). TG

,
α

L2 = (n0 + n1 + 2n2 .R ). Π G

λ L ≤ R then the operator A that is defined

with 14) transforms the sphere Bα (0; R ) to itself.
Lemma 2.6. The sphere Bα (0; R ) ,

α

set of the space ( H α (G );

.

Proof.

. ∞ ).

From the definition of the sphere Bα (0; R ) ,

α ∈ (0,1)

The proof of the Lemma 2.4 is obvious from the definitions of

α ∈ (0,1) is a compact

for

ϕ ∈ Bα (0; R )

we

have

ϕ

∞

≤ R,

~ ~
the functions f 1 , g 1 : G → ℂ and corollary 2.1.

therefore, it is clear that the sphere Bα (0; R ) is uniform

Lemma 2.5. Let

bounded in the space ( H α (G );

ε > 0 if we take δ = (ε / R)1 / α then for every z1 , z 2 ∈ G
and ϕ ∈ Bα (0; R ) when
z1 − z 2 < δ we have

f ∈ H α ,1 ( m1 , m 2 ; D0 ), g ∈ H α ,1 ( n1 , n 2 ; D0 ) ,

F ∈ H α ,1,1,1 (l1 , l 2 , l3 , l 4 ; DR ) and DR =

{( z, ϕ , t , s) ∈ D

0

: z ∈ G , ϕ ≤ R, t ≤ R, s ≤ R}
.

F : G → ℂ that is defined as
F ( z ) = F ( z , ϕ ( z ), (TG o f1 )( z ), (Π G o f1 )( z ) ),

for the function

z ∈ G and for ϕ ∈ Bα (0; R) , α ∈ (0,1)
we have

. ∞ ) . Furthermore, for every

ϕ ( z1 ) − ϕ ( z 2 ) ≤ R. z1 − z 2

Then

α

< ε . From here we can see

that the elements of the sphere Bα (0; R ) are continuous at
same order. Thus, as required by the Arzela-Ascoli theorem
about compactness the sphere Bα (0; R ) is a compact set of
the space ( H α (G );

. ∞ ).

Corollary 2.5. The sphere Bα (0; R ) ,
complete subspace of the space ( H α (G );

22

α ∈ (0,1) is a
. ∞).


~
l 2 ϕ ( z ) − ϕ ( z ) +



~
≤ λ .l3 TG ( f (., ϕ (.)) − f (., ϕ (.)))( z ) + .


~
l 4 Π G ( g (., ϕ (.)) − g (., ϕ (.)))( z ) 

Now, let us define the following two transformations, which
in
the
space ( H α (G ); . ∞ ) .
~ ∈ H (G ) , α ∈ (0,1) ,
let
For ϕ , ϕ
p >1
α
~
~
~
~
d ∞ (ϕ , ϕ ) = ϕ − ϕ ∞ , d p (ϕ , ϕ ) = ϕ − ϕ p . Then it is

are

easy

to

show

that

the

transformations

d ∞ : H α (G ) → [0,+∞), d p : H α (G ) → [0,+∞)

As for this, from the assumptions on the functions
f , g and F and in accordance with Minkowski inequality
we have,

are



~
 ∫∫ A(ϕ )( z ) − A(ϕ )( z ) p dxdy 


G


both defines a metric on the space H α (G ) , α ∈ (0,1) ,

( H α (G ); . ∞ ) and

consequently, it can be easily seen that

Lemma 2.7.. The convergence is equivalent for the metrics
d ∞ and d p in the subspace Bα (0; R ) , α ∈ (0,1) , p > 1 .

[


inequalities, for all

α ∈ (0,1) .

]

.d ∞ (ϕ 0 , ϕ n )

ϕ 0 , ϕ n ∈ Bα (0; R), n = 1,2,...

Now, for bounded operators

 l 2 + m2 l3 . TG L (G )
p
≤ λ .
n l . Π
 2 4 G L (G )
p


TG , Π G : L p (G ) → L p (G ), p > 1 let us define the
following norms

{

TG

Lp (G )

≡ sup TG ϕ

ΠG

L p (G )

≡ sup{ Π G ϕ

p

:ϕ
p

p

:ϕ

}

≤1,
p

1/ p

l ϕ − ϕ +
~

 2

p


~
≤ λ . l3 . TG L ( G ) . f (., ϕ (.)) − f (., ϕ (.)) p + 
p


~
 l 4 . Π G L ( G ) . g (., ϕ (.)) − g (., ϕ (.)) p 
p



Proof of Lemma 2.7 is direct result of (6) and

d p (ϕ 0 , ϕ n ) ≤ π .(d / 2)

≤λ×

p
~
 l ϕ ( z ) − ϕ ( z ) +


2


~
 l T ( f (., ϕ (.)) − f (., ϕ (.)))( z ) +  dxdy 

 ∫∫  3 G

G 
~ (.)))( z ) 
 l 4 Π G ( g (., ϕ (.)) − g (., ϕ


 


( H α (G ); . p ) are metric spaces.

2 1/ p

1/ p

+
. ϕ − ϕ
~




p

and therefore we obtain

≤ 1}

1/ p



~
 ∫∫ A(ϕ )( z ) − A(ϕ )( z ) p dxdy  ≤


G

~
λ . l 2 + m 2 l 3 . TG L ( G ) + n 2 l 4 . Π G L ( G ) . ϕ − ϕ p .

[11], [14].

(

Lemma 2.8. Let

f ∈ H α ,1 ( m1 , m 2 ; D0 R ), g ∈ H α ,1 ( n1 , n 2 ; D0 R )

p

)

p

From this inequality, it is seen that the Inequality (15) is true.
With this, the lemma is proved.

F ∈ H α ,1,1,1 (l1 , l 2 , l3 , l 4 ; DR ) , α ∈ (0,1) , p > 1 . In
~
this case, for ϕ , ϕ ∈ B (0; R ) the operator A that is defined

Lemma 2.9. If the assumptions of the Lemma 2.8 are satisfied

with the Equality (14) satisfies the following inequality

and

and

α

~
~
d p ( A(ϕ ), A(ϕ )) ≤ λ M 3 ( p ).d ∞ (ϕ , ϕ ) . (15)

(

Here,

M 3 ( p ) = l 2 + m2 l 3 . TG

(m(G) )

1/ p

L p (G )

+ n2 l 4 . Π G

L p (G )

)×

Proof.
For z ∈ G
assumption of the lemma and definition of the operator A we
have

G

G

the

operator

n→∞

lim d ∞ ( A(ϕ n ), A(ϕ 0 )) = 0 . From the Inequality (6)

n →∞

can write

A(ϕ n ) − A(ϕ 0 )

∞

≤
2

~
A(ϕ )( z ) − A(ϕ )( z ) = λ ×

F ( z , ϕ ( z ), TG f (., ϕ (.))( z ), Π G g (., ϕ (.))( z ) ) −
~
~
~
F ( z , ϕ ( z ), T f (., ϕ (.))( z ), Π g (., ϕ (.))( z ) )

Then

A : Bα (0; R) → Bα (0; R) , α ∈ (0,1) is a continuous
operator for the metric d ∞ .
Proof. Let ϕ 0 , ϕ n ∈ Bα (0; R ), n = 1,2,... α ∈ (0,1)
and lim d ∞ (ϕ n , ϕ 0 ) = 0 . We want to show that

.
~
and ϕ , ϕ ∈ Bα (0; R ) from the

λ .L ≤ R.

2
M (α , p ) A(ϕ n ) − A(ϕ 0 ) α +αp × .

A(ϕ n ) − A(ϕ 0 )

αp
2 +αp
p

Thus, due to the Inequality (15) we will have

23

we


2

A(ϕ n ) − A(ϕ 0 )
( λ .M 3 ( p ))

αp
2 +αp

∞

≤ M (α , p )(2 R) 2+αp ×

.(d ∞ (ϕ n , ϕ 0 ))

αp
2 +αp

.

From this inequality, it can easily be seen that if
lim d ∞ (ϕ n , ϕ 0 ) = 0 then
n→∞

lim d ∞ ( A(ϕ n ), A(ϕ 0 )) = 0 .

q ∈ [0,1) such that
ρ 2 ( Ax, Ay ) ≤ q.ρ 2 ( x, y ) .
In this case, the operator equation x = Ax has only
unique solution x* ∈ X and for x 0 ∈ X to be any initial
approximation, the velocity of the convergence of the
sequence ( x n ) to the solution x* is determined by the
following inequality

ρ 2 ( x n , x* ) ≤ q n .(1 − q ) −1 .ρ 2 ( x1 , x0 )

n →∞

III. MAIN RESULTS
Now, we can give the theorem about the existence of the
solution of the Equation (1).
Theorem 3.1. If

f ∈ H α ,1 ( m1 , m 2 ; D0 R ), g ∈ H α ,1 ( n1 , n 2 ; D0 R ) and

Here, the terms of the sequence are defined by

xn =

Ax n−1 , n = 1,2,... .
On the basis of this theorem and preceding information,
we can give the following theorem about the uniqueness of
the solution of the Equation (1) and also about finding the
solution.

F ∈ H α ,1,1,1 (l1 , l 2 , l 3 , l 4 ; DR ) , α ∈ (0,1) and λ L ≤ R

.

Theorem 3.3. If

then the nonlinear singular integral equation (1) has at least
one solution in the sphere Bα (0; R ) .

F ∈ H α ,1,1,1 (l1 , l 2 , l 3 , l 4 ; DR ) , α ∈ (0,1) ,

Proof. If

f ∈ H α ,1 ( m1 , m 2 ; D0 R ), g ∈ H α ,1 ( n1 , n 2 ; D0 R ) ,

λ L ≤ R then the operator A which is defined

λ L ≤ R , l = λ .(l 2 + m2 l3 . TG

with the Formula (14) transforms the convex, closed and
compact set Bα (0; R ) to itself in the Banach space

n2 l 4 . Π G

( H α (G ); . α ) , α ∈ (0,1) . Thus, the operator A is a

L p (G )

+

integral

compact operator in the space H α (G ) . In addition to this,
since the operator A is also continuous on the
set Bα (0; R ) , it is completely continuous operator. In that
case, due to the Schauder fixed-point principle the operator
A has a fixed point in the set Bα (0; R ) . Consequently, the
Equation
(1)
has
a
solution
in
the
space H α (G ) , α ∈ (0,1) .
With this, the theorem is proved.
Now, we will investigate the uniqueness of the solution
of the Equation (1) and the problem that how can we find
the approximate solution. For this, we will use a more
useful modified version of the Banach fixed-point principle
for the uniqueness of the solution of operator equations
[16].
Theorem 3.2. Suppose the following assumptions are
satisfied:
i. ( X , ρ 1 ) is a compact metric space;
ii. In the space X, every sequence that is
convergent for the metric ρ1 is also
convergent for a second metric ρ 2 that is
defined on X;
iii. The operator A : X → X is a contraction
mapping for the metric

ρ 2 , that is, for

L p (G

) < 1, p > 1 then the nonlinear singular

equation (1) has only unique solution
ϕ * ∈ Bα (0; R) and for ϕ 0 ∈ Bα (0; R) to be any initial
approximation, this solution can be found as a limit of the
sequence (ϕ n ), n = 1,2,... whose terms are defined as
below

 z , ϕ n −1 ( z ), TG f (., ϕ n −1 (.))( z ), 
,

Π G g (., ϕ n −1 (.))( z )



ϕ n ( z ) = λ .F 


z ∈ G , n = 1,2,... .
(16)
Furthermore, for the velocity of the convergence the following
evaluation is right,

d p (ϕ n , ϕ * ) ≤ l n .(1 − l ) −1 .d p (ϕ1 , ϕ 0 ), n = 1,2,... .
(17)
Proof. In order to prove the first part of the theorem it is
sufficient to show that the assumptions i-iii of the Theorem 3.2
is satisfied. In accordance with Lemma 2.6 and Corollary 2.5
~
~
for ϕ , ϕ ∈ Bα (0; R ) and d p (ϕ , ϕ ) = ∞

~
ϕ −ϕ

∞

,

(Bα (0; R); d ∞ )

is a compact metric space.

Therefore, for X = Bα (0; R ) and ρ1 = d ∞ the assumption
i. of the Theorem 3.2 is satisfied.
If we take ρ 2 = d p , p > 1 , from the Lemma 2.7 it is
obvious that the assumption ii. of the Theorem 3.2 is satisfied.

every x, y ∈ X there exist a number

24


Now, let us show that when l < 1 then the operator A is a
contraction transformation with respect to the

d p metric and

so we have shown that the assumption iii. of the Theorem 3.2
is satisfied.
For any ϕ1 , ϕ 2 ∈ Bα (0; R ) , same as the proof of the
Lemma 2.8 the following inequality can be proven

A(ϕ1 ) − A(ϕ 2 )
Here,

l = λ .(l 2 + m2 l3 . TG

Therefore, for

p

L p (G )

ϕ1 , ϕ 2 ∈ Bα (0; R)

≤ l . ϕ1 − ϕ 2
+ n2 l 4 . Π G

From the Inequality (19), it can be seen that the sequence
(ϕ n ), n = 1,2,... is a Cauchy sequence with respect to the
metric d p . Therefore, since

(B

(0; R); d p ) is a complete

α

metric space there exist the limit

ϕ * ∈ Bα (0; R)

such that

lim ϕ n = ϕ * or lim d p (ϕ n , ϕ * ) = 0.
n→∞

n→ ∞

From (18), for every natural number n since
p

d p (ϕ n +1 , A(ϕ * )) = d p ( A(ϕ n ), A(ϕ * )) ≤

.

L p (G

l.d p (ϕ n , ϕ * ) ,

).
we have

we can write

lim d p (ϕ n +1 , A(ϕ * )) = 0 ,

n→∞

d p ( A(ϕ1 ), A(ϕ 2 )) ≤ l.d p (ϕ1 , ϕ 2 ) .

(18)


Thus, when l < 1 it can be seen from the Inequality (18) that
the operator A is a contraction mapping with respect to the
d p metric that is defined on the Bα (0; R ) .

Bα (0; R ) is
a compact metric space; ii. In the space Bα (0; R ) , every
sequence that is convergent for the metric d ∞ is also
convergent for the metric d p that is defined on Bα (0; R )

and consequently,

ϕ * = A(ϕ* ) . With this, it can be seen that the operator
equation ϕ ( z ) = A(ϕ ( z )), z ∈ G , consequently, the
solution of the Equation (1) is the limit of the sequence whose
terms
are
defined
by

Thus, from the assumptions of the theorem; i.

A : Bα (0; R ) → Bα (0; R) is a contraction
mapping with respect to the norm d p .
and iii.

Therefore, because of the Theorem 3.2 the Equation (1) has
only unique solution ϕ * ∈ Bα (0; R ) .
Now, let us show that for ϕ 0 ∈ Bα (0; R ) to be an initial
approximation point, this solution is the limit of the sequence
(ϕ n ), n = 1,2,... whose terms are defined by (16) or by

ϕ n ( z ) = A(ϕ n−1 ( z )), z ∈ G , n = 1,2,... .
For every

n = 1,2,... from the Inequality (18) we have
d p (ϕ n +1 , ϕ n ) = d p ( A(ϕ n ), A(ϕ n −1 )) ≤
l.d p (ϕ n , ϕ n −1 )

so from here we can write

ϕ n ( z ) = A(ϕ n−1 ( z )), z ∈ G , n = 1,2,...
because

z ∈ G , n = 1,2,... to the solution ϕ * ( z ) is given

Corollary 3.1.

In the space

sequence (ϕ n (z ) ), z ∈ G ,
defined by (16) (or defined by

Bα (0; R ), α ∈ (0,1) the
n = 1,2,... whose terms are

ϕ n ( z ) = A(ϕ n −1 ( z )), z ∈ G , n = 1,2,... ) also converges
to the unique solution of the Equation (1) with respect to the
metric d ∞ .
ACKNOWLEDGMENT
The authors would like to express sincerely thanks to the
referees for their useful comments and discussions.
REFERENCES
[1]

[2]

[3]

d p (ϕ m , ϕ 0 ) ≤ (l m −1 + l m− 2 + ... + l + 1)d p (ϕ1 , ϕ 0 )
we will have,

m →∞

by the Inequality (17).
From Theorem 3.3 and Lemma 2.7 we will have the
following result.

So, for any natural numbers m and n we can write
Furthermore, because

lim ϕ n + m ( z ) = ϕ * ( z ) and lim l m = 0 from the

m →∞

(ϕ n ( z ) ),

and from this inequality , we will have

d p (ϕ n + m , ϕ n ) ≤ l n .d p (ϕ m , ϕ 0 ).

. Furthermore, as

Inequality (19) the convergence velocity of the sequence

d p (ϕ n +1 , ϕ n ) ≤ l.d p (ϕ n , ϕ n −1 )

d p (ϕ n+1 , ϕ n ) ≤ l n .d p (ϕ1 , ϕ 0 ) .

lim ϕ n +1 = A(ϕ * ) in other words

n→∞

[4]

d p (ϕ n + m , ϕ n ) ≤ (1 − l m ).(1 − l ) −1 .l n .d p (ϕ1 , ϕ 0 ). (19)

25

E.G. Ladopoulas, “On the numerical evaluation of the general type of
finite-part singular integrals and integral equations used in fracture
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E.G. Ladopoulas, “On a new integration rule with the Gegenbauer
polynomials for singular integral equations used in the theory of
elasticity”, Ing. Archiv 58, 1988, pp.35-46.
E.G. Ladopoulas, “On the numerical solution of the finite-part singular
integral equations of the first and the second kind used in fracture
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E.G. Ladopoulas, “On the solution of the finite-part singular integrodifferential equations used in two-dimensional aerodynamics”, Arch.
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[5]

[6]

[7]

[8]

[9]

[10]
[11]
[12]

[13]


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[16]

E.G. Ladopoulas, “The general type of finite-part singular integrals and
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equations arising in two-dimensional fluid mechanics”, J. Nonlinear
Analysis, 42, 2000, pp.277-290.
E.G. Ladopoulas, “Nonlinear singular integral equations
elastodynamics by using Hilbert Transformations”, J. Nonlinear
Analysis: Real World Applications 6 2005, pp.531-536.
L. Bers and L. Nirenberg , “On a representation for linear elliptic
systems with discontinuous coefficients and its applications”,
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pp.111-140.
B.V. Bojarskii, “Quasiconformal mappings and general structural
properties of system of nonlinear elliptic in the sense of Lavrentev”,
Symp. Math.18, 1976, pp.485-499.
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applications. Pergamon Press, London, 1985.
V.N. Monahov, Boundary value problems with free boundaries for
elliptic systems. Nauka, Novosibirsk , 1977.
A.S. Mshim Ba and W. Tutschke, Functional-analytic methods in
complex analysis and applications to partial differential equations.
World Scientific, Singapore, New Jersey, London, 1990.
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einer komplexen Normalform“, Mat. Nachr. 75, 1976, pp.283-298.
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integrals”, Acta Mathematica 88, 1952, pp.85- 139.
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nonlinear singular integral equations. Nauka, Moscow, 1980.

26

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