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Appl. Appl. Math.
ISSN: 1932-9466
Applications and Applied
Mathematics:
An International Journal
(AAM)
Vol. 7, Issue 1 (June 2012), pp. 201 – 210
Oscillation of Neutral Partial Dynamic Equations
Deniz Uçar
Department of Mathematics
Usak University
Usak, Turkey
deniz.ucar@usak.edu.tr
Yaşar Bolat
Department of Mathematics
Afyon Kocatepe University
Afyonkarahisar, Turkey
yasarbolat@aku.edu.tr
Received: March 13, 2011; Accepted: April 30, 2012
Abstract
This paper is concerned with the oscillation of solutions of a certain more general neutral type
dynamic equation. We establish within the necessary and sufficient conditions for the oscillation
of its solutions.
Keywords: Oscillation; Partial Dynamic Equation; Neutral Type
MSC 2010: 34K40, 34N05, 35B05
1. Introduction and Preliminaries
The theory of "dynamic equations" unifies the theories of differential equations and difference
equations and it also extends these classical cases to cases "in between." The general idea is to
prove a result for a dynamic equation where the domain of the unknown function is a so-called
time scale, which may be an arbitrary closed subset of the reals. This way results not only related
to the set of real numbers or set of integers but those pertaining to more general time scales are
obtained. They give rise to plenty of applications, among them the study of population dynamic
models which are discrete in season (and may follow a difference scheme with variable step-size
or often modeled by continuous dynamic systems), die out, say in winter, while their eggs are
201
202
Deniz Uçar and Yaşar Bolat
incubating or dormant, and then in season again, hatching gives rise to a nonoverlapping
population Bohner and Peterson (2001). In this paper we obtain some necessary and sufficient
conditions for oscillation of neutral delay partial dynamic equation of the form
é u ( x , t )- a u
êë
( x , f (t ))ùúû
D
2
= a (t ) L u ( x , t ) +
s
å
a k (t ) L u
( x , b k (t ))
(1)
k =1
-
n
å
q j (t ) u
( x , g j (t )) ,
j=1
where t0 Î  , sup { } = ¥, ( x, t ) Î W´[t0 , ¥) º G and L is Laplacian in n . We assume
throughout this paper that
(H1) a is constant with 0 < a < 1 ;
(H2) a, ak , q j Î Crd ([t0 , ¥) ,  + ) , k = 1, 2,..., s; j = 1, 2,..., n ;
(H3
f, b , g Î Crd ([t0 , ¥) , )
are
unbounded
increasing
functions
satisfying
f (t ) , b (t ) , g (t ) £ t for all sufficiently large t .
The intervals with a  index below are used to denote the intersection of the usual interval with
 ; i.e., [t0 , ¥) := [t0 , ¥) Ç  for convenience. Consider the following boundary condition:
u  N  x, t   0 ,
( x, t ) Î ¶W´[t0 , ¥) ,
(2)
where N is the exterior normal vektor to  .
Definition 1. The function u Î Crd2 (G ) Ç Crd1 (G ) is said to be a solution of the problem (1) and
(2), if it satisfies (1) in the domain G and boundary condition (2).
Definition 2. The solution u ( x, t ) of the problem (1) and (2) is said to be oscillatory in the
domain G = W´[t0 , ¥) , if for any positive number tm , there exists a point ( x0 , t0 ) Î W´ éêëtm , ¥)

such that the condition u ( x0 , t0 ) = 0 holds.
We will give a short introduction to the time scales calculus which will be used in the next
sections
AAM: Intern. J., Vol. 7, Issue 1 (June 2012)
203
1.1. Basic Definitions
We start with the definitions given by Hilger (1988) to define the exponential function on a time
scale.
Definition 3. A function f :    is called rd-continuous provided it is continuous at rightdense points in  and its left-sided limits exist (finite) at left-dense points in  .
Definition 4. Let a, b Î  and f Î Crd .
i) If  =  , then
b
b
a
a
 f t t   f t dt ,
where the integral on the right is the usual Riemann integral from calculus.
ii) If a ,b consist of only isolated points, then
b

a
 ta ,b    t  f  t  , a  b

f  t  t  0,
ab

  tb ,a    t  f  t  , a  b.
Some useful definitions about partial dynamic equations from Bohner, M. and Guseinov, G. Sh.
(2004) are the followings.
Definition 5. Let n   be fixed. Further, for each i  1,..., n let i denote a time scale, that is,
i is a nonempty closed subset of the real numbers  . Let us set
 n  1 x ... x n  t   t1 ,..., tn  : ti  i for all i  1,..., n .
We call  n an n - dimensional time scale.
Definition 6. Let si and ri denote, respectively, the forward and backward jump operators in i .
Remember that for u  i the forward jump operator  i : i  i is defined by
si (u ) = inf {v Î i : v > u }
and the backward jump operator i : i  i is defined by
ri (u ) = sup {v Î i : v < u } .
204
Deniz Uçar and Yaşar Bolat
If i has a left-scattered maximum M , then we define i  i \ M  , otherwise i  i . If i
has a right-scattered minimum m , then we define  i   i \ m , otherwise  i   i .
Definition 7. Let f :  n   be a function. The partial delta derivative of f with respect to
ti  i is defined as the limit
lim
f  t1 ,..., ti 1 ,  i  ti  , ti 1 ,...tn   f  t1 ,..., ti 1 , si , ti 1 ,...tn 
 i  ti   si
si ti
si  i  ti 
,
provided that this limit exists as a finite number, and is denoted by any of the following symbols:
f  t1 ,..., tn  f  t  f
,
,
 t  , ftii  t  .
 i ti
 i ti
 i ti
2. Main results
To obtain our main results, we need the following lemmas.
Lemma 1. Assume that y (t ) > 0 , y D (t ) < 0 and qi ( s ) ¹ 0 for s Î éêët , T * ùúû and t ³ T where T *
satisfies that gi (T * ) = t . Let b = max {f (t ) , T - min {gi (t )}} , and assume that the integral
inequality
¥
z (t ) ³ a z (f (t )) + ò qi ( s ) z (gi ( s )) Ds, t ³ T
(3)
t
has a continuous positive solution y : [T - b, ¥)  (0, ¥) with T - b Î  . Then the
corresponding integral equation
¥
x (t ) = a x (f (t )) + ò qi ( s ) x (gi ( s )) Ds, t ³ T
t
has a continuous positive solution x : T  b,  T  0,   .
Proof:
Define a set of functions L and a mapping  as follows:
L = {w Î Crd ([T - b, ¥) ,  + ) : 0 £ w (t ) £ 1, t ³ T - b} ,
(4)
AAM: Intern. J., Vol. 7, Issue 1 (June 2012)
205
¥
ì 1 é
ù
ï
ï
ê
ú , t ³T
ï
a
w
f
t
z
f
t
q
s
w
g
s
z
g
s
s
+
D
(
)
(
)
(
)
(
)
(
)
(
)
(
)
(
)
(
)
i
i
i
ò
ï
ú
ï z (t ) êëê
t
ûú
(w)(t ) = í
ï
ï
t -T + b
t -T + b
ï
w)(T ) + 1,
T -b £ t £ T.
(
ï
ï
b
b
ï
î
It is easy to see from (3) that L Ì L and (w)(t ) > 0 on [T - b, T ) for any w Î L . Define a
sequence
{wk (t )}
in L as follows: w0 (t ) = 1 and wk +1 (t ) = (wk )(t ) , k = 0,1, 2,... , for
t ³ T - b . From (3), by induction, we have
0 £ wk +1 (t ) £ wk (t ) £ 1, k = 0,1, 2,..., t ³ T - b.
Then, w (t ) = lim wk (t ) , for t ³ T - b , exists and
k ¥
¥
ù
1 éê
ú , for t ³ T ,
+
D
w (t ) =
a
w
f
t
z
f
t
q
s
w
g
s
z
g
s
s
(
)
(
)
(
)
(
)
(
)
(
)
(
)
(
)
(
)
i
i
ò i
ú
z (t ) êêë
úû
t
w (t ) =
t -T + b
t -T + b
, for T - b £ t £ T .
( w)(T ) +1b
b
Set x (t ) = w (t ) z (t ) . Then x (t ) satisfies (4) and x (t ) > 0 for t Î [T - b, T ) . Clearly, x (t ) is
continuous on [T - b, T ] . Then, in view of (4), we see that x (t ) is continuous on [T - b, ¥) .
Finally, it remains to show that x (t ) > 0 for t ³ T - b . Assume that there exists t * ³ T such
that x (t ) > 0 for T - b £ t £ t * and x (t * ) = 0 . Thus by (4), we have
¥
0 = x (t * ) = a x (f (t * )) + ò qi ( s ) x (gi ( s )) Ds, t ³ T
t*
which implies that a = 0 and qi ( s ) x (gi ( s )) = 0 for all s ³ t * which contradicts a Î (0,1) ,
q j Î Crd ([t0 , ¥) ,  + ) and qi ( s ) ¹ 0 for s Î éëêt , T * ùûú . Therefore, x (t ) > 0 on [T - b, ¥) .
Lemma 2. Every solution of the dynamic equation
D
n
é v (t ) - av (f (t ))ù + å q (t ) v (g (t )) = 0, t ³ t ,
i
i
0
êë
úû
i =1
(5)
where a Î (0,1) , qi Î Crd ([t0 , ¥) ,  + ) and gi Î Crd ([t0 , ¥) , ) is oscillatory, if and only if
the inequality
206
Deniz Uçar and Yaşar Bolat
n
D
é v (t ) - av (f (t ))ù + å q (t ) v (g (t )) £ 0, t ³ t
i
i
0
ëê
ûú
(6)
i =1
has no eventually positive solutions.
Proof:
The sufficiency is obvious. To prove the necessity, we assume that (5) has an eventually positive
solution v (t ) . Set y (t ) = v (t ) - av (f (t )) . Since 0    1 and v (t ) is an eventually positive
solution, it is easy to see that y t   0 eventually and
D
n
y D (t ) = éê v (t ) - av (f (t ))ùú = -å qi (t ) v (gi (t )) £ 0 .
ë
û
i =1
Integrating (6) from t to  , we have
¥
y (t ) ³ ò qi ( s ) v (gi ( s )) Ds .
t
That is,
¥
v (t ) ³ av (f (t )) + ò qi ( s ) v (gi ( s )) Ds .
t
By Lemma 1, the corresponding integral equation
¥
z (t ) = a z (f (t )) + ò qi ( s ) v (gi ( s )) Ds
t
also has a positive solution z t  . Clearly, z t  is an eventually positive solution of (5),
contradicting the assumption.
Theorem 1. Every solution of the problem (1) is oscillatory in G if and only if the inequality (6)
has no eventually positive solutions.
Proof:
i) Sufficiency. Suppose to the contrary that there is a nonoscillatory solution u  x, t  of the
problem (1) and (2) which has no zero in W´[t0 , ¥) for some t0 ³ 0 . Without loss of generality,
we may assume that u  x, t   0, u ( x, f (t )) > 0, u ( x, bk (t )) > 0 and u ( x, g j (t )) > 0 in
AAM: Intern. J., Vol. 7, Issue 1 (June 2012)
207
W´ 1 = [t1 , ¥) , t1 ³ t0 , k = 1, 2,..., s; j = 1, 2,..., n . Integrating (1) with respect to x over the
domain W , we have
é
ù
ê u ( x, t ) Dx - a u ( x, f (t )) Dx ú
ò
êò
ú
W
ëê W
ûú
D2
s
= a (t ) ò Lu ( x, t ) Dx + å ak (t ) ò Lu ( x, bk (t )) Dx
W
k =1
W
n
(7)
- å q j (t ) ò u ( x, g j (t )) Dx,
j =1
W
for t ³ t1 . From the Green's formula on time scales, Bohner et al. (2007) and boundary condition
(2), it follows that
 Lu  x, t  x   u  x, t  S  0 ,
N

t ³ t1
(8)

and
ò Lu ( x, b
W
k
(t )) Dx = ò u D ( x, bk (t )) DS = 0 , t ³ t1 ,
N
(9)
¶W
where DS is the surface element on ¶W . Let
vt    u  x, t x for t ³ t1 .

Combining (7)-(9), we have
D
n
é v (t ) - av (f (t ))ù + å q (t ) v (g (t )) = 0, t ³ t .
1
i
i
ëê
ûú
(10)
i =1
This shows that v (t ) > 0 is a solution of (5). But by lemma 2 and the condition that the inequality
(6) has no eventually positive solutions, we obtain that every solution of (5) is oscillatory. This is
a contradiction.
ii) Necessity. Suppose that inequality (6) has an eventually positive solution. By lemma 2, we
obtain that (5) has a nonoscillatory solution. Without loss of generality, we may assume that vt 
for t ³ t0 is a solution of (5). From (5), we have
D
n
é v (t ) - av (f (t ))ù + å q (t ) v (g (t )) = 0, t ³ t .
0
i
i
êë
úû
i =1
Notice that Lv (t ) = 0 , Lv (bk (t )) = 0 for t ³ t0 , from (10) we get
208
Deniz Uçar and Yaşar Bolat
é v (t ) - av (f (t ))ù
ëê
ûú
D2
s
n
k =1
j =1
= a (t ) Lv (t ) + å ak (t ) Lv (bk (t )) - å q j (t ) v (gi (t )).
This shows that u  x, t   vt   0 satisfies (1). It is easy to see that u DN = 0 , ( x, t ) Î W´[t0 , ¥) .
Hence u  x, t   vt   0 is a nonoscillatory solution of the problem (1) and (2), which is a
contradiction. This completes the proof.
Using lemma 2 and Theorem 1, we can obtain the following corollary.
Corollary 1. Every solution of the problem (1) and (2) is oscillatory in G if and only if every
solution of the dynamic equation (5) oscillates.
By (10) and
w (t ) = v (t ) - v (f (t )) ,
(12)
we have
n
n
j =1
j =1
wD (t ) £ -å q j (t ) v (g j (t )) £ -å q j (t ) w (g j (t )).
for t ³ t1 . Then we obtain the inequality
n
wD (t ) + å q j (t ) w (g j (t )) £ 0, t ³ t1 .
(13)
j =1
Our aim is to establish a sufficient condition for the oscillation of all solutions of the inequality
(13).
Consider the linear delay dynamic equation of the form
n
wD (t ) + å q j (t ) w (g j (t )) = 0 .
(14)
j =1
where for j = 1, 2,..., n ,
q j Î Crd ([t0 , ¥) ,  + ) , and lim q j t   q j .
t 
(15)
Therefore, we have the limiting equation
D
n
w (t ) + å q j w (g j (t )) = 0 .
j =1
(16)
AAM: Intern. J., Vol. 7, Issue 1 (June 2012)
209
Zhang et al. (2002) studied the oscillation of the following delay differential equations on time
scales
xD (t ) + p (t ) x (t (t )) = 0 ,
t ³ t0 ,
where p Î Crd (,  + ) , t Î Crd (, ) and t (t ) < t for t Î  and sup { } = ¥ . They have
proved the following result.
Proposition 1. Assume that 1- p (t ) m (t ) > 0, t Î  . Define a set
E = {l | l > 0, 1- l p (t ) m (t ) > 0} .
If
ìï t
üïïü
ïìï
ïï
ïï
ï
lim sup sup íl exp í ò xm(s) (-l p ( s )) Dsïï
ýý < 1 ,
t0 ¥ t >t l Î E ï
ïït t
ïï
0
ïîï
(
)
îï
þïïï
þï
then all solutions of equation are oscillatory.
By this proposition we have the following theorem.
Theorem 2. Assume that 1- å j=1 q j (t ) m (t ) > 0, t Î  . Define a set
n
{
}
E = l | l > 0, 1- l å j=1 q j (t ) m (t ) > 0 .
n
If
ìï
ìï t
üïüï
ïï
n
ïï
ïï
lim sup sup íl exp í ò xm(s) -l å j=1 q j ( s ) Dsïï
ýý < 1 ,
t0 ¥ t >t l Î E ï
ïïg t
ïï
0
ïîï
ïï
ïþþï
îï j ( )
(
)
then all solutions of equation (1) are oscillatory.
Corollary 2. Assume that (15) holds and that every solution of the limiting equation (16)
oscillates. Then every solution of (14) also oscillates.
Example 1. Let  =  and t Î [0, ¥) . Consider the following neutral delay parabolic
differential equation
210
Deniz Uçar and Yaşar Bolat
ù
æ
1
¶ é
pö
êu ( x, t ) - u ( x, t - p )ú = Lu ( x, t ) + et Lu çç x, t - ÷÷÷
çè
úû
2
2ø
¶t êë
æ
æ
3p ö
pö
+ et Lu çç x, t - ÷÷÷ - 2u çç x, t - ÷÷÷ ,
çè
çè
2ø
2ø
(17)
( x, t ) Î (0, p )´[0, ¥) with boundary condition
¶u (0, t ) ¶u (p, t )
=
=0.
¶x
¶x
p
p
1
3p
, f (t ) = t - p , b1 (t ) = t - , b2 (t ) = t - , g (t ) = t - , a (t ) = 1 ,
2
2
2
2
p
a1 (t ) = a2 (t ) = et , f1 (t ) = and q = 2 . Then we see that all the assumptions of the theorems
2
are satisfied. Thus we obtain that every solution of the problem (17) oscillates in (0, p )´[0, ¥) .
It is easy to see that a =
In fact u ( x, t ) = cos x.sin t is such a solution of the problem (17).
Acknowledgement
The authors are thankful to Professor Haydar Akca for his useful suggestions.
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