The Existence of Maximal and Minimal Solution of Quadratic Integral Equation

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The Existence of Maximal and Minimal Solution of Quadratic Integral
Equation
Amany M. Moter
Lecture, Department of Computer Science, Education College, Kufa University, Najaf, Iraq
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Abstract - In this paper, we study existence of solution of
quadratic integral equations
( ) ( ) ( ) ∫ ( ) ( )
, - ( )
by using Tychonoff fixed point theorem. Also existence
maximal and minimal solution for equation ( ).
Key words: quadratic integral equation, maximal and
minimal solution, Tychonoff Fixed Point Theorem.
1. Introduction
In fields, physics and chemistry, they can be use quadratic
integral equations (QIEs) in their applications, for
examples: the theory of radiative transfer, traffic theory,
kinetic theory of gases and neutron transport and in many
other phenomena.
The paper ([1-10]) studied quadratic integral equations.
Thus, we study solvability of the following quadratic
integral equation:
( ) ( )
( ) ∫ ( ) ( ) , - ( )
2. Preliminaries
We need in our work the following fixed point theorems
and definitions
Definition 1[11]: A set is said to be convex if,
, - and ( )
If and , - then ( ) is said to be a
convex combination of and .
Simply says that is a convex set if any combination of
every two elements of is also in .
Theorem 2 (Tychonoff Fixed Point Theorem) [12]:
suppose B is a complete, locally convex linear space and S
is a closed convex subset of B. Let a mapping be
continuous and ( ) . If the closure of ( ) is compact,
then T has a fixed point.
Theorem 3 (Arzel ̀ -Ascoli Theorem) [13]: Let be a
compact metric space and ( ) the Banach space of real or
complex valued continuous functions normed by
‖ ‖ | ( )|
If * + is a sequence in ( ) such that is uniformly
bounded and equicontinuous, then ̅ is compact.
To prove the existence of continuous solution for quadratic
integral equation( ), we let , -, , - be the
space of Lebesgue integrable function and be the set of
real numbers.
3. Existence of solution
We study the existence of at least one solution of the
integral equation ( ) under the following assumptions:
( ) - is continuous and there a exist
function
Such that | ( )| (| |)
( )- , - , - is continuous for two variables t
and s such that:
| ( )| for all , -
Where is constant ( ).
( ) is bounded function and satisfies
Carath ́odory condition, Also there exist continuous
function satisfying
| ( ( ))| ( ) ( ) for all , - and
( ) There exists a constant , - such that

( ) ∫ ( )
And *| ( ) ( )| , -+
Now we can formulate the main theorem
Theorem 4: if the assumptions ( ) and ( ) are satisfied,
then the quadratic integral equation of Volterra type has at
least one solution , -.
Proof: Let be set of all continuous function on interval
, - denoted by , -, it is a complete locally convex
linear space that has been proved in [12], and define the
set by
* | ( )| + , -
Where ( )
Clearly is nonempty, bounded and closed, but we will
prove that the set convex.
Let and , -, then we have
‖ ( ) ‖ ‖ ‖ ( )‖ ‖
( )
Then ( ) which means that is convex
set.
To show that let , then
| ( )| | ( ) ( ) ∫ ( ) ( ) |
| ( )| | ( )| ∫ | ( )|| ( )|
| ( )| | ( )| ∫ | ( )| ( ) ( )
( ) ( ) ∫ ( )
( )
This means that is closed and by similar steps we can
prove
Consider the operator :
( ) ( )
( ) ∫ ( ) ( ) , -
| ( )|
| ( ) ( ) ∫ ( ) ( ) |
( )
Then implies to
Now, Let and | | then
| ( ) ( )|
| ( ) ( ) ∫ ( ) ( ) ( )
( ) ∫ ( ) ( ) |
| ( ) ( ) ∫ ( ) ( )
( ) ( ) ∫ ( ) ( )
( ) ( ) ∫ ( ) ( )
( ) ( ) ∫ ( ) ( )
( ) ( ) ∫ ( ) ( )
( ) ( ) ∫ ( ) ( )

( ) ( ) ∫ ( ) ( )
( ) ( ) ∫ ( ) ( )
( ) ( ) ∫ ( ) ( )
( ) ( ) ∫ ( ) ( ) |
| ( ) ( )|
| ( ) ( )| ∫ | ( )|| ( )|
| ( )| ∫ | ( )|| ( )|
| ( )| ∫ | ( ) ( )|| ( )|
| ( )| ∫ | ( )|| ( ) ( )|
We have
| ( ) ( )| ( ) ∫ ( ) ∫ | ( )|
( ) ∫ ( )
as .
This means that the function is equi-continuous
on, -. By using Arzela-Ascoli theorem, we can say that
is compact.
Tychonoff fixed point theorem is satisfied all its conditions,
then the operator has at least one fixed point. This
completes the proof.
4 Maximal and minimal solution
Definition 5: [14] let ( ) be a solution of equation ( ) then
( ) is said to be a maximal solution of equation ( ) if
every solution of ( ) on , - satisfies the inequality
( ) ( )
A minimal solution ( ) can be defined in a similar way by
reversing the above inequality i.e ( ) ( )
The following lemma important to prove the existence of
maximal and minimal solution of equation ( ).
Lemma 6: suppose that ( ) satisfies the assumption ( )
of theorem 1 and let ( ) ( ) be continuous function on
, - satisfying
( ) ( ) ( ) ∫ ( ) ( )
( ) ( ) ( ) ∫ ( ) ( )
And one of them is strict.
Let ( ) is nondecreasing function in then
( ) ( ) , - ( )
Proof: Let conclusion ( ) be false, then there exists such
that
( ) ( ) ( )
And
( ) ( ) , -
From the monotonicity of in , we get
( ) ( ) ( ) ∫ ( ) ( )
( ) ( ) ∫ ( ) ( )
That implies to
( ) ( )
This is contradiction with ( ), then
( ) ( ).
Next, we prove the existence maximal and minimal
solution of quadratic integral equation ( ). So, we have the
next theorem.
Theorem: let all conditions of theorem 1 be satisfied and if
( ) is nondecreasing functions in , then there exist
maximal and minimal solutions of equation ( ).
Proof: for the existence of the maximal solution
let be given and
( ) ( )
( ) ( )

From equation (1) we obtain that:
( ) ( ( ) )
( ) ∫ ( )( ( ) )
( ) ( ) ∫ ( ) ( ) ( )
Clearly the functions ( ) and ( ) satisfy
assumptions ( ), ( ) then equation ( ) has a continuous
solution on ( ).
Let and be such that then
( ) ( ) ( ) ∫ ( ) ( )
( ( ) ) ( ) ∫ ( )( ( )
) ( )
Also
( ) ( ) ( ) ∫ ( ) ( )
( ( ) )
( ) ∫ ( )( ( ) ) ( )
( ) ( ( ) )
( ) ∫ ( )( ( ) ) ( )
Applying lemma 6 to ( ) and ( ) we have
( ) ( ) , -
According to the previous of the theorem 1, we conclude
that equation ( ) is equi-continuous and uniformly
bounded, through it we use the Arzela-Ascoli theorem so,
there exists a decreasing sequence such that as
, and ( ) exists uniformly in and we
denote this limit by ( ). From the continuity of the
functions and in the second argument, we get
( ) ( ( )) as
( ) ( ( )) as
and
( ) ( )
( ( )) ( ) ∫ ( ) ( ( ))
which implies that ( ) is a solution of equation ( ).
Now, we can prove that ( ) is the maximal solution of
quadratic integral equation ( )
Let ( ) be any solution of equation ( ), then
( )
( ) ( ) ∫ ( ) ( ) ( )
and
( ) ( ( ) )
( ) ∫ ( )( ( ) )
( )
( ) ( ) ∫ ( ) ( ) ( )
by Lemma 6 and equations ( ) ( ) we get
( ) ( ) , -
From the uniqueness of the maximal solution (see [14] and
[15]), it is clear that ( ) tends to ( ) uniformly in , -
as .
In the same manner we can prove the existence of the
minimal solution.
4. Conclusion:
Equation ( ) has a maximal and minimal solution after we
proved the existence of at least one solution by using
Tychonoff Fixed Point Theorem under 4 assumptions.
References
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Theorems of some quadratic Integral Equation”. J. Math.
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[2] J.Bana ́ and A. Martianon, “Monotonic Solution of a
quadratic Integral equation of

Volterra Type”. Comput. Math. Apple., 47 (2004), 271-
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