J. Comp. & Math. Sci. Vol.3 (2), 191-195 (2012)
Oscillatory Behavior of Second Order Neutral Delay Difference Equations B. SELVARAJ1 and G. GOMATHI JAWAHAR2 1
Dean, Department of Science and Humanities, Nehru Institute of Engineering and Technology, Coimbatore-641105, India. 2 Assistant Professor, Department of Mathematics, Karunya University, Coimbatore-641114, India (Received on : March 23, 2012) ABSTRACT In this paper, we obtain some sufficient conditions for the oscillation of second order neutral delay difference equation of the form
∆2 ( y n + p n y n − k ) + q n f ( y n −l ) = 0, n ∈ N ( n0 )
,
where k,l >0. Keywords: Neutral Delay Difference Equation, Oscillatory solution.
1.1 INTRODUCTION Difference equations with discrete and continuous arguments are playing a fundamental role in nonlinear phenomena and in process occurring in various drastically different systems. In the past few decades the study of difference equations has already drawn a great deal of attention, not only among mathematicians themselves, but from various other disciplines as well. Many statements concerning the theory of linear differential equations are also valid for the corresponding difference equations. To a certain extent, the growing interest in
difference equations may be also attributed to their simplicity. With the use of a computer one can easily discover that difference equations posses fascinating properties with a great deal of structure and regularity. Ofcourse all computer observations and predictions must also be proven analytically. Difference equations are often rearranged as a recursive formula so that a system output can be computed from the input signal and past outputs. In this paper, we obtain some sufficient condition for the oscillation of
Journal of Computer and Mathematical Sciences Vol. 3, Issue 2, 30 April, 2012 Pages (131-247)
192
B. Selvaraj, et al., J. Comp. & Math. Sci. Vol.3 (2), 191-195 (2012)
second order neutral equation of the form
delay
difference
decreasing function.
∆2 ( y n + p n y n − k ) + q n f ( y n − l ) = 0 , n ∈ N ( n0 )
∆2 z n = −q n f ( y n −l ) < 0. Hence ∆zn is a
(1.1.1)
Also, ∆z n = z n+1 − z n . Hence ∆z n = y n+1 + p n+1 y n−k +1 − ( y n + p n y n−k ) > 0
Where k,l >0. Here we assume the following conditions. H1: { p n } is an increasing sequence.
{qn } is an positive sequence.
H2: H3:
f is a continuous function such
that f (u) ≥ u and f (uv) ≥ f (u). f (v) 1.2 EXISTENCE OF OSCILLATORY SOLUTIONS In this section, we study the structure of the oscillatory solutions of the equation 1.1.1. Theorem 1. 2.1 Suppose Qn = qn f (1 − pn−l ) and
Since {z n } is eventually positive, ∆zn > 0 Also, y n = z n − p n y n−k
y n = z n − pn ( z n − k − pn − k y n − k )
y n > z n − pn z n − k . Hence
yn > (1 − pn ) z n .
Hence for some l > 0, y n−l > (1 − pn−l ) z n−l Using H3, f ( yn−l ) ≥ f (1 − pn−l ) f ( z n−l ) . From the equation(1.1.1),
∆2 zn = −qn f ( yn−l ) ≤ − qn f (1− pn−l ) f ( zn−l ) 2 Hence ∆ z n ≤ −Qn f ( z n−l ) .
2 Using H3, ∆ z n ≤ −Qn z n−l .
lim
m→∞
m
sup ∑ Qn = ∞, n = m0
Then every solution of the equation (1.1.1)
Therefore ∆ z n + Qn z n−l ≤ 0. 2
Define wn =
ρ n ∆z n z n −l
(1.2.1)
where ρn = max(k, l )
.
Then wn > 0
oscillates.
Equation (1.2.1) becomes,
Proof Suppose { y n } is a nonoscillatory solution, without loss of generality we may assume that { y n } is eventually positive. Let z n = y n + pn y n−k From the equation (1.1.1),
∆(
wn z n −l
ρn
) +
Qn z n−l ≤ 0. ρ ∆(wn zn−l ) − wn zn−l ∆ρn Hence n ≤ −Qn zn−l ρn+1ρn ∆( wn z n −l ) wn z n−l ∆ρ n − ≤ − Qn z n−l . ρ n +1 ρ n ρ n +1
Journal of Computer and Mathematical Sciences Vol. 3, Issue 2, 30 April, 2012 Pages (131-247)
B. Selvaraj, et al., J. Comp. & Math. Sci. Vol.3 (2), 191-195 (2012)
wn +1∆z n−l + z n −l ∆wn
ρ n +1
−
wn z n −l ∆ρ n
wn +1 ∆z n −l (1.2.2) ρ n +1 z n −l ρ n+1 ρ n Taking summation from m0 to m , for
− Qn z n−l .
m
∑
n = m0
m
∑
wn ( ρ n +1 − ρ n )
m
∑
wn ( ρ n +1 − ρ n )
n = m0
m
∑
n = m0
ρ n +1 ρ n
wn ( ρ n +1 − ρ n )
ρ n +1 ρ n
−
ρ n +1 ρ n
n = m0
wm+1 − w m0 ≤
∑
∑
n = m0
wn +1 ∆z n −l ρ n +1 z n −l
n = m0
wn +1 ∆z n −l ) ) ρ n +1 z n −l
≤ wm0 −
m
∑
Qn
n = m0
Proof Let y n be an eventually positive solution of the equation (1.2.3). Hence there exists
n1 ∈ N (n0 ) such that yn >0 and y n−l >0
Corollary 1.2.1
for n ≥ n1
If f ( yn −l ) = yn−l , pn = 0, equation (1.1.1) becomes
∆2 y n + qn y n −l = 0, n ∈ N (n0 )
∑
( Qn +
n = m0
m
+
∑ Qn −
m
-
wn +1 ( z n −l +1 − z n −l ) ρ n +1 z n −l
m
m
n = m0
m → ∞, w m → −∞. which is a when contradiction to the assumption . Hence every solution of the equation (1.1.1) is an oscillatory solution.
If
− Qn −
some m > m0 ,
wn +1 − wn ≤
wm +1 −
wn ∆ρ n
∆wn ≤
≤
ρ n +1 ρ n
193
..
From the equation (1.2.3), .
(1.2.3)
yn is an eventually positive solution of
the equation (1.2.3), then there exists
n ∈ N ( n0 )
∆2 yn = −qn yn−l < 0, n ∈ N (n0 ) ,
(1.2.4)
Hence ∆y n+1
< ∆y n . So ∆yn is eventually decreasing. Since q n is a positive function, the decreasing function ∆y n is either eventually positive or eventually negative.
Such that
Suppose there exists
yn ≥ yn−k > 0, ∆yn ≥ 0, ∆2 y n < 0
n2 ≥ n1
∆y n2 < 0.
Journal of Computer and Mathematical Sciences Vol. 3, Issue 2, 30 April, 2012 Pages (131-247)
such that
194
B. Selvaraj, et al., J. Comp. & Math. Sci. Vol.3 (2), 191-195 (2012)
Taking summation from n2
to
s
the
equation (1.2.4) becomes, s
∑ ∆y
n = n2
n +1
≤
n = n2
Hence we have,
y s +2
n
y s+2 − y s+1
is an eventually positive
∆x n + q n x n −l −1 ≤ 0.
s
∑ ∆y
Clearly y n solution of
Taking summation from n − l − 1
≤ y n2 +1 − y n2
s,
we have
≤ ∆y n2 +ys+1
s
Hence as n → ∞, y n → −∞.
to
0≥
∑ {( x
n −l −1
n +1
− x n ) + q n x n −l −1 }
Which is a contradiction. Hence ∆yn ≥ 0,for all n∈N(n0).
≥
x s +1 − xn−l −1 +
s
∑
n − l −1
q n x n −l −1
Corollary 1.2.2 If lim
s
s
n→ ∞
sup
∑ qn > 1
=
n − l −1
then every solution of the equation (1.2.3)
x s +1
+(
∑
n −l −1
q n − 1) x n −l −1
Hence, lim
n →∞
oscillates.
s
sup ( x s +1 + (
Proof
∑
n − l −1
Suppose to the contrary that yn is a nonoscillatory solution of the solution of the equation (1.2.3). Define xn Hence
= ∆y n
y n−l ≥ xn−l −1
s
Therefore lim sup n →∞
∑
n −l −1
q n ≤ 0.
This is a contradiction to our assumption. Hence every solution of the equation (1.2.4) has oscillatory solution.
y n −l − y n −l −1 = x n −l −1
Therefore
q n − 1) x n −l −1 ) ≤ 0.
1.3 EXAMPLES (1.2.5)
Example 1.3.1 From the equations (1.2.5) and (1.2.4), we get
Consider equation
∆x n = ∆2 y n = −q n y n −l ≤ − qn x n −l −1
∆2 y n + 4 y n−3 = 0, n > 0
the
second
order
Journal of Computer and Mathematical Sciences Vol. 3, Issue 2, 30 April, 2012 Pages (131-247)
difference
B. Selvaraj, et al., J. Comp. & Math. Sci. Vol.3 (2), 191-195 (2012)
Here
l = 3, q n = 3 s
Also, lim sup n→∞
∑q
n − l −1
>1
n
Hence all the conditions of the corollary 1.2.2 are satisfied. Hence all its solutions are oscillatory. One such solution is (-1) n.
Example 1.3.2 Consider
the
second
order
difference
equation
∆2yn +
4(n+1) yn−3 =0,n>3 n−3
Here
l = 3, q n =
4(n + 1) ,n > 3 n −3 s
Also, lim sup n→∞
∑q
n − l −1
n
>1
Hence all the conditions of the corollary 1.2.2 are satisfied. Hence all its solutions are oscillatory. One such solution is n(-1)n.
REFERENCES 1. R. P. Agarwal , ‘Difference Equations and Inequalities’, Marcel Dekker,
195
New York, (1992). 2. R. P. Agarwal, M. M. S. Manuel and E. Thandapani, ‘Oscillation and Nonoscillation of Second order Neutral Delay Difference Equations’, Appl. Maths. Vol 10, 103-109. 3. R. P. Agarwal, Martin Bohner, R. Grace, Donal O’ Regan, ‘Discrete Oscillation Theory’, CMIA Book Series, Vol 1, ISBN: 977-5945, 19-4. 4. R. P. Agarwal and P. J. Y. Wong, Topics in Difference ‘Advanced Equations’, kluwer, Dordrecht, 10 (1997). 5. R.P. Agarwal, P.J.Y. Wong, ‘Oscillation Theorems and Existence of Positive Monotone Solutions for Second Order Nonlinear Difference Equations,’ Mathematical Computer Modeling 21, 63-84 (1995). 6. S. R. Grace, ‘Oscillation theorems for Nonlinear Differential Equations of Second Order’, J. Math. Anal. Appl.171, 220-241 (1992). 7. S. R. Grace, M. P. Chen and B. S. Lalli, ‘Oscillations of Second order Neutral Differential Equations’, Sea. Bull. Math.14, 91-101 (1990). 8. B. Szmanda, ‘Oscillation of Solutions of Second Order Difference Equations’, Publications De L’ Institute Mathematique Nouvelle Series 27(41), 237-239 (1980). 9. Y.G. Sun and S. H. Saker, ‘Oscillation for Second order Nonlinear Neutral Delay Difference Equations’, Applied Mathematics and Computation 163, 909-918 (2005).
Journal of Computer and Mathematical Sciences Vol. 3, Issue 2, 30 April, 2012 Pages (131-247)