Mathematical Theory and Modeling ISSN 2224-5804 (Paper) ISSN 2225-0522 (Online) Vol.2, No.7, 2012 unbounded. The same property holds for
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)( ) ((
(
if
)( ) ( ) )
)(
(
)
We now state a more precise theorem about the behaviors at infinity of the solutions 376 377
Behavior of the solutions If we denote and define Definition of ( )( )(
(a)
)
( )(
)
)
( )(
( )(
)
)
( )(
( )(
)
)
( )( ) :
four constants satisfying
( )(
)
(
)(
)
(
)(
)
(
)( ) (
( )(
)
(
)(
)
(
)(
)
(
)( ) (
)
)(
( )
Definition of ( )( (b) By ( )(
)
)
equations ( ( )( )
By ( ̅ )( ( )( ) (
) ( )
)
)(
(c) If we define ( )(
)
(
)(
( )(
)
( )(
and ( )( )(
(
)
( )(
)
( )
(
)
( ̅ )(
( ̅ )( ) ( )( )
Definition of (
(
)
( )
)
Definition of ( ̅ )(
( )(
)
(
)(
)
) )
(
( ̅ )(
)
)( ) (
( ( )
(
)
( )(
)(
)
)
)
378
)(
)
)(
(
)
and (
)
)
the roots of
( )
(
( )
)
)
( )(
( )(
)
)
380
by
( )(
)
( )(
)
( )(
)
( )(
)
)
the roots of the equations ( ) ( )( ) ( ) ( )
( )( ) :-
( )(
)
( ̅ )(
)
)
( ̅ )(
)
)
( )(
(
)(
)
)
)(
( )(
and ( (
(
)
( )
379
)
)(
(
)
( )(
)
( ̅ )(
)
( )(
)
and analogously ( )(
the
( ) ( )
( ̅ )( ) :
( )(
)
( )(
)
)
:
( )
(
( )(
)
and respectively ( ̅ )( ) ( ̅ )( ( )( ) and ( )( ) ( ( ) )
)(
)(
)
( )
( )
(
(
( ) ( )
)
)(
)
)( ) (
(
and respectively (
)
(
)(
)
381
)( )(
)
)
( )(
( )(
)
)
(
)(
( ̅ )(
)
)
)(
(
)
(
)(
)
)(
)
(
)(
)
( ̅ )(
( ̅ )(
)
(
)(
)
where (
(
)
)
( )(
)
( )(
)
(
)(
)
)(
)
( ̅ )(
)
382
are defined respectively Then the solution satisfies the inequalities ((
)( ) (
)( ) )
( )
(
383 )( )
149