CONTENTS

J. Duhok Univ., Vol.14, No.1 (Pure and Eng. Sciences), Pp 25-29, 2011
EXISTENCE AND UNIQUENESS SOLUTION FOR NONLINEAR
VOLTERRA INTEGRAL EQUATION
RAAD. N. BUTRIS * and AVA SH. RAFEEQ **
* Dept. of Mathematics, Faculty of Education, University of Zakho ,Kurdistan Region-Iraq
** Dept. of Mathematics, Faculty of Science, University of Duhok ,Kurdistan Region-Iraq
(Received: February 14, 2010; Accepted for publication: November 28, 2010)
ABSTRACT
In this paper, we study the existence and uniqueness solution for nonlinear Volterra integral equation, by using
both methods ( Picard Approximation ) and (Banach Fixed Point Theorem). Also these methods could be developed
and extended throughout the study.
KEYWORDS: Existence and Uniqueness Solution; Volterra Integral Equation; Non-linear; Picard Approximation; Banach
Fixed Point Theorem.
I
INTRODUCTION
ntegral equations are encountered in
various fields of science and numerous
applications (oscillation theory, fluid dynamics,
electrical engineering, etc.).
Exact (closed-form) solutions of integral
equations play an important role in the proper
understanding of qualitative features of many
phenomena and processes in various areas of
natural science. Lots of equations of physics,
chemistry and biology contain functions or
parameters which are obtained from experiments
and hence are not strictly fixed. Therefore, it is
expedient to choose the structure of these
functions so that it would be easier to analyze
and solve the equation. As a possible selection
criterion, one may adopt the requirement that the
model integral equation admit a solution in a
closed form. Exact solutions can be used to
verify the consistency and estimate errors of
various numerical, asymptotic, and approximate
methods. Recently, [2,3,6].
Pachpztte [5] studied the global existence of
solutions of some volterra integral and integrodifferential
equations of the form
t
x ( t ) �h( t ) ��k(
t , s ) g ( s, x ( s )) ds ,
and
0
t
'
x t f t x t k t s g s x s ds
( ) � ( , ( ), � ( , ) ( , ( )) ),
0
with initial condition x(0) � x . o
Tidke [7] investigated the existence of global
solutions to first-order initial-value problems,
with non-local condition for nonlinear mixed
Volterra-Fredholm integro differential equations
in Banach spaces of the form.
t b
'
x () t f ( t , x ( t ), k ( t , s, x ( s)) ds, h( t , s, x ( s)) ds )
0 0
� � �
with non-local condition
x (0) �g( x ) � x . o
Consider the following non linear system of
Volterra integral equations which has the form :
t s
( , ) ( ) ( , ( , ), ( , �) ( �, ( �, )) �,
o o
a
o
��
o
x t x F t f s x s x G s g x x d
� �� �
�
b( s)
as ( )
g ( �, x ( �, x )) d� ) ds,
o
(1)
where x D
n
R
n
domain subset of Euclidean space R .
Let the vectors functions
f ( t , x , y , z ) � ( f ( t , x , y , z ), f ( t , x , y , z ),..., f ( t , x , y , z ))
� � D is a closed and bounded
1 2
g ( t , x ) � ( g1( t , x ), g2( t , x ),..., g n ( t , x ))
and
Fo ( t ) � ( Fo1( t ), Fo2( t ),..., Fon ( t ))
are defined and continuous in the domain
( t , x , y , z ) �[ a, b] �D �D1 �D 2 � ( ��, � ) �D �D1 �D
(2)
2
where 1 D and D 2 are closed and bounded
m
domains subsets of Euclidean space R .
Suppose that the functions f ( t , x , y , z ) and
g ( t , x ) satisfy the following inequalities :
f ( t , x , y , z ) �M, g ( t , x ) � N
(3)
f ( t , x , y , z ) �f( t , x , y , z ) �Kx�x�Ly� y
1 1 1 2 2 2 1 2 1 2
�Qz1� z 2
(4)
g ( t , x ) �g( t , x ) �Hx� x
(5)
1 2 1 2
for all t �[ a, b] , x , x 1, x 2 �D , y , y 1, y 2 � D1
,
z , z 1, z 2 � D2
.
where M and N are positive constant vectors
and K , L , Q and H are positive constant
matrices. Let G(t , s) is an (n � n) positive
matrix which is defined and continuous in the
n
25