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International Journal of Engineering & Technology, 7 (4.10) (2018) 1050-1053
International Journal of Engineering & Technology
Website: www.sciencepubco.com/index.php/IJET
Research paper
Oscillation of Second-Order Quasilinear Generalized Difference
Equations
V.Srimanju
1
*, Sk.Khadar Babu
2
, V.Chandrasekar
3
1,2
Vellore Institute of Technology, Vellore – 632 014, Tamil Nadu, India
3
Thiruvalluvar University College of Arts and Science, Thennangur - 604 408, Tamil Nadu, India
*Corresponding author E-mail:srimanjushc@gmail.com
Abstract
Authors present sufficient conditions for the oscillation solutions of the generalized perturbed quasilinear difference equation
( )
1
(( 1) ) (( 1) ) (( 1) ) ( , ( )) = ( , ( ), ( ))
i i
i i i
i i i i i i i i i i i i i i i ii i a k j v k j v k j F k j vk j Gk j vk j v i i k i ii i i i i j
−
− + − + − + + + + + + +
where 0< <1 i i i , [ 0, ) i i i k . Examples are illustrates the importance of our results are also included.
Keywords: Generalized difference equation; Oscillation; Quasilinear.
1. Introduction
Difference equations represent a captivating mathematical field,
has rich field of the applications in such diverse disciplines as
population dynamics, operations research, ecology, economics,
biology etc. For the background of difference equations and its
application in diverse fields with examples, see [1,13,20,27],
based on assumption ( )= ( 1) ( ), [ 0, ). i i i i i i uk uk uk k ii i i + −
Though some authors [1],[19] have recommended the definition of
as
( )= ( ) ( ), ( 0, ),
i
uk i i i i i i i i i i uk uk + − (iE)
no notable progress have been taken on this line. But in [14] the
authors took up the definition of i as given in (E), and given
many important results and applications. They labelled the opera-
tor i defined by (E) as i and its inverse by
1
i
−
, many inter-
esting results in number theory were obtained. Qualitative proper-
ties like rotator, expanding, shrinking, spiral and web like were
established by extending theory of i to complex function, for
the solutions of difference equations involving
i
i in [2-12,14-
18,21-26].
In the sequel, in this paper we consider the generalized perturbed
quasilinear difference equation
( )
1
(( 1) ) (( 1) ) (( 1) )
i i
i i i
i i i i i i i i a k j v k j v ki i i j i
−
− + − + − +
( , ( )) = ( , ( ), ( ))
[ 0, , )
i
F k j vk j Gk j v i i ii i i i i ii i i i k j vk j i i
i i i k
+ + + + + +
(1)
where 0< <1 i i i , ( ) ak i i i j + is an eventually positive real
valued function, and
i
is the generalized forward difference
operator defined as
( )= (( 1) ) ( ) i i i i i i i i i i vk j vk j vk j + + + − + .
By a solution of (1), we mean a nontrivial real valued function
( ) vk i i i j + satisfying (1) for [ 0, ) i k i i . A solution ( ) vk i i i j +
is said to be oscillatory if it is neither eventually positive nor nega-
tive, and nonoscillatory otherwise.
2. Main Results
Throughout this paper we assume that there exist real valued func-
tions ( ) qk i i i j + , ( ) pk i i i j + and a function : if iR i R → such
that
(i). ( )>0 xf x i i i for all 0 ix i ;
(ii). ( ) ( )= ( , )( ) fx fy g i i i i i i ii i xy x y − − for , 0 ixyi i , where g is
a nonnegative function; and
(iii)
( , )
( )
( )
F k j x j
qk j
f x j
i i ii i
i i i
i i i
+ +
+
+
,
( , , )
( )
( )
i i ii ii i
i i i
i i
Gk j x j y j
pk j
f ji x
+ + +
+
+
for , 0 ixyi i .
The conditions used in the main results are listed as follows:
1/
1
= ,
(( 1) )
i i
a k
i
i
i i i i i j
− +
(2)
1
( ( ) ( )) = ,
i
k i i
i i i qk j pk j i i i i
=
+ − +
(3)
1
( ( ) ( )) < ,
i
k i i
i i i qk j pk j i i i i
=
+ − +
(4)
1/
= = 1
0
1
(( ) ( )) = ,
lim
( )
i i k
i i i
k
r k s r i
i
qs j ps
i
i i i
i
j
r j i a
→
+
+ − +
+
(5)