ON ASYMPTOTICAL BEHAVIORS OF SOLUTIONS OF EMDEN-FOWLER
ADVANCED DIFFERENTIAL EQUATION
Roman Koplatadze
Department of Mathematics of Tbilisi State University, Tbilisi, Georgia
Abstract. In the paper the following differential equation

u (n) (t )  p(t ) u ( (t )) sign u ( (t ))  0
(0.1)
is considered, where p  Lloc ( R ; R) ,   C( R ; R ) ,  (t )  t for t  0 and
0    1 . New sufficient conditions for the equation (0.1) to have Property A or
Property B are established.
1. INTRODUCTION
Consider the differential equation

u (n) (t )  p(t ) u ( (t )) sign u ( (t ))  0
(1.1)
where 0    1 , p  Lloc ( R ; R) ,   C( R ; R ) and  (t )  t for t  R .
Oscillatory properties of the equation (1.1) in the case where n  2 and  (t )  t
were first studied by Atkinson [1]. Kiguradze [2] studied the analogous problem for
the higher order ordinary differential equation (  (t )  t ) in the case where n is even
and   1 . For the ordinary differential equation Ličko and Švec [7] proved
necessary and sufficient conditions for both even and odd n and 0    1 and   1 .
For the delayed equation (  (t )  t ) the analogous topics were quite thoroughly
investigated in [4] while for the advanced equation (1.1) (  (t )  t ) attempts were
made [3] for obtaining optimal conditions for oscillation of solutions. In the works [5]
and [6] results essentially different from those presented in [3] were proved.
Moreover, some results therefrom are substantial generalizations of some results of
[3].
It will always be assumed that either the condition
(1.2)
p(t )  0 for t  R
or
(1.3)
p(t )  0 for t  R
is fulfilled.
Let t 0  R . A function u : [t 0 ,)  R is said to be a proper solution of the
equation (1.1) if it is locally absolutely continuous along with its derivatives up to the
order n  1 inclusive, sup| u(s) |: s  t  0 for t  t 0 and it satisfies (1.1) almost
everywhere on [t 0 ,) .
A proper solution u : [t 0 ,)  R of the equation (1.1) is said to be oscillatory if it
has a sequence of zeros tending to   . Otherwise the solution u is said to be
nonoscillatory.
Definition 1.1. We say that the equation (1.1) has Property A if any of its proper
solutions is oscillatory when n is even and either is oscillatory or satisfies
u (i ) (t )  0 as t   (i  0,1,, n  1)
7-31
(1.4)
when n is odd.
Definition 1.2. We say that the equation (1.1) has Property B if any of its proper
solutions either is oscillatory or satisfies (1.4) or
u (i ) (t )   as t   (i  0,1,, n  1)
(1.5)
when n is even, and either is oscillatory or satisfies (1.5) when n is odd.
In the present paper sufficient conditions of new type will be given for the equation
(1.1) to have Property A or B.
Let t 0  R and l  {1,, n  1} . By U l ,t 0 we denote the set of all proper
solutions u : [t 0 ,)  R of the equation (1.1) satisfying the condition
u (i ) (t )  0 for t  t 0 (i  0,1,, l  1),
(1) i  l u (i ) (t )  0 for t  t 0 (i  l ,, n  1),
(1.6l)
2. SUFFICIENT CONDITIONS OF NONEXISTENCE OF MONOTONE
SOLUTIONS
Theorem 2.1. Let the condition (1.2) ((1.3)) be fulfilled, l  {1,, n  1} with
l  n odd ( l  n even) and

n l
 (l 1)
| p (t ) | dt  .
 t ( (t ))
(2.1l)
0
If, moreover, for some   [0,  ] and k  N

n  l 1  
( (t ))  (l 1) l , k ( (t ))  | p(t ) | dt  ,
 t
(2.2l)
0
then for any t 0  R we have Ul ,t 0  Ø, where
1
 1 
t 
 1 
 n  l 1 ( ( ))  (l 1) | p( ) | d ds 
 l!(n  l )!  

0 s


 l ,1 (t )  
,
t 
1
 l , i (t ) 
 n  l 1  l , i 1 ( ( ))  ( ( ))  (l 1) | p( ) | d ds


l!(n  l )! 0 s


(i  2,, k ).
(2.3l)
Remark 2.1. (2.1l) is a necessary condition for the equation (1.1) to have a solution
of the type (1.6l).
Corollary 2.1. Let the condition (1.2) ((1.3)) be fulfilled, l  {1,, n  1} with l  n
odd ( l  n even) and (2.1l) hold. If, moreover, for some k  N

n  l 1
( (t ))  (l 1)  l , k ( (t ))  | p(t ) | dt  ,
 t
0
then for any t 0  R we have Ul ,t 0  Ø, where  l, k is defined by (2.3l).
Corollary 2.2. Let the condition (1.2) ((1.3)) be fulfilled, l  {1,, n  1} with
l  n odd ( l  n even) and
7-32

n  l 1 
( (t ))  (l 1) | p(t ) | dt  ,
 t
0
then for any t 0  R we have Ul ,t 0  Ø.
Remark 2.2. Corollary 2.1 is a generalization of Theorem 1.1 [3].
Theorem 2.2. Let the condition (1.2) ((1.3)) be fulfilled, l  {1,, n  1} with l  n odd
( l  n even) and for some   [0,  ] and k  N
 1   l , k ( (t )) 


  0,
t


where  l, k is defined by (2.3l). Then the condition (2.1l) is necessary and sufficient
for Ul ,t 0  Ø for any t 0  R .
lim inf t
t  
Corollary 2.3. Let the condition (1.2) ((1.3)) be fulfilled, l  {1,, n  1} with
l  n odd ( l  n even) and for some k  N
 l , k ( (t )) 
 0.
t
Then the condition (2.1l) is necessary and sufficient for Ul ,t 0  Ø for any t 0  R .
lim inf
t  
3. DIFFERENTIAL EQUATIONS WITH PROPERTIES A AND B
Theorem 3.1. Let the condition (1.2) be fulfilled and for any l  {1,, n  1} with
l  n odd the condition (2.1l) hold as well as the condition (2.2l) for some   [0,  ]
and k  N , where  l, k is defined by (2.3l). If, moreover, for add n

n 1
| p (t ) | dt  ,
 t
(3.1)
0
then the equation (1.1) has Property A.
Theorem 3.2. Let the condition (1.3) be fulfilled and for any l  {1,, n  1} with
l  n even the condition (2.1l) hold as well as the condition (2.2l) for some   [0,  ]
and k  N , where  l, k is defined by (2.3l). If moreover, for even n (3.1) holds, then
the equation (1.1) has Property B.
Theorem 3.3. Let (1.2) ((1.3)) be fulfilled, n be even ( n be odd) and
  (t )
 0.
t
If moreover, for some   [0,  ] and k  N
lim inf
t  
(3.2)

n  2   
1, k ( (t ))  | p(t ) | dt  ,
 t
0
where 1, k is defined by (2.31), then the equation (1.1) has Property A (B).
Corollary 3.1. Let (1.2) and (3.2) ((1.3) and (3.2)) be fulfilled, n be even ( n be odd)
and for some   (0,1)
7-33


n2
| p ( s ) | ds  0.
lim inf t  s
t  
(3.3)
t
Then the equation (1.1) has Property A ( B).
Corollary 3.2. Let (1.2) be fulfilled and for some  ,   (0,1)
lim inf
  (t )
t
t 
 0,
(3.4)


 ( n  2)
| p ( s ) | ds  0.
lim inf t  s
t  
If moreover, for some   (0,  )
   ( n  2) 
 t
1 
(   )
1 
t
p (t ) dt  ,
0
then the equation (1.1) has Property A.
Corollary 3.3. Let (1.3) be fulfilled and for some  ,   (0,1) (3.4) holds and


1  ( n  3)
| p ( ) | d  0.
lim inf t  
t  
t
If moreover, for some   (0,  )
  1  ( n  3) 
 t
1 
(   )
1 
| p (t ) | dt  ,
0
then the equation (1.1) has Property B.
Corollary 3.4. Let (1.2) ((1.3)) be fulfilled, n be even ( n be odd) and for some
 0
lim inf
  (t )

t   exp( t )
 0,
(3.5)
If, moreover,

n2
| p( s ) | ds  0,
lim inf t  s
t  
t
then the equation (1.1) has Property A (B).
Theorem 3.4. Let (1.2), (3.1) and (3.2) ((1.3), (3.1) and (3.2)) be fulfilled, n be odd
( n be even) and for some   [0,  ] and k  N

n  3   
( (t ))   2, k ( (t ))  | p (t ) | dt  ,
 t
0
where  2, k is defined by (2.32). Then the equation (1.1) has Property A (B).
Corollary 3.5. Let (1.2) and (3.2) ((1.3) and (3.2)) be fulfilled, n be odd ( n be even)
and for some   (0,1)


n 3

lim inf t  s ( ( s)) | p( s) | ds  0.
t  
t
7-34
Then the equation (1.1) has Property A ( B).
Corollary 3.6. Let (1.2) and (3.1) ((1.3) and (3.1)) be fulfilled as well as (3.5) for
some,   0 , n be odd ( n be even) and

n 3

lim inf t  s ( ( s)) | p( s) | ds  0.
t  
t
Then the equation (1.1) has Property A ( B)..
Theorem 3.5. Let (1.2) and (3.1) be fulfilled and
lim sup
t  
  (t )
t
 .
(3.6)
If, moreover, for some   [0,  ] and k  N

 
( (t ))  ( n  2)  n 1, k ( (t ))  p (t ) dt  ,
 t
0
where  n 1, k is defined by (2.3n-1), then the equation (1.1) has Property A.
Theorem 3.6. Let (1.3), (3.1) and (3.6)
kN
be fulfilled and for some   [0,  ] and

1   
( (t ))  ( n  3)  n  2, k ( (t ))  | p(t ) | dt  .
 t
0
Then the equation (1.1) has Property B.
REFERENCES
1. Atkinson F. V.: 'On second order nonlinear oscillations'. Pacific J. Math 1955 5 (1)
643-47.
2. Kiguradze I. T.: 'On oscillatory solutions of some ordinary differential equations'.
Soviet Math. Dokl. 1962 144 33-36.
3. Kiguradze I. T. and Stavroulakis I. P.: 'On the oscillation of solutions of higher
order Emden-Fowler advanced differential equations'. Appl. Anal. 1998 70 (1-2)
97-112.
4. Koplatadze R.: 'On oscillatory properties of solutions of functional differential
equations'. Mem. Differential Equations Math. Phys. 1994 3 1-179.
5. Koplatadze R.: 'On Oscillatory properties of solutions of generalized EmdenFowler type differential equations'. Proc. A. Razmadze Math. Inst. 2007 145 117121.
6. Koplatadze R.: 'On asymptotic behaviors of solutions of almost linear and essential
nonlinear functional differential equations'. Nonlinear analysis Theory, Methods
and Applications (submit).
7. Ličko I. and Švec M.: 'Le caractère oscillatorie des solutions de 'equation
y n  f ( x) , n  1 ' Crech. Math. J. 1963 13 481-489.
7-35