Integrated Intelligent Research (IIR)
International Journal of Computing Algorithm Volume: 06 Issue: 01 June 2017 Page No.5-15 ISSN: 2278-2397
On the Oscillation of Higher Order Impulsive Neutral Partial Differential Equations with Distributed Deviating Arguments V. Sadhasivam1, T. Raja2, T. Kalaimani3 PG and Research Department of Mathematics, Thiruvalluvar Government Arts College, Rasipuram, Namakkal (Dt), Tamil Nadu, India. (Affiliated to Periyar University, Salem) E-mail: ovsadha@gmail.com, trmaths19@gmail.com, kalaimaths4@gmail.com 1,2,3
Abstract — In this work, we consider a class of boundary value problems associated with even order nonlinear impulsive neutral partial functional differential equations with continuous distributed deviating arguments and damping term. Necessary and Sufficient conditions are obtained for the oscillation of solutions using impulsive differential inequalities and integral averaging scheme with Robin boundary condition. Examples are specified to point up our important results. 2010 Mathematics Subject Classification: 35B05, 35L70, 35R10, 35R12.
and the references they are cited. Consequently, it is required to study with impulse effect on higher order partial differential equations. In the monographs, J. H. Wu [22], N. Yoshida [23] provided several fundamental theories and applications of partial functional differential equations to population ecology, generic repression, climate models, viscoelastic materials, control problems, coupled oscillators, beam equations and structured population models. Strong interest on these mathematical models, we formulated this higher order problem. In this effort, we begin oscillation criteria for even order impulsive neutral partial differential equation with damping which was not formerly studied. In the consequence, the main results of this manuscript is the generalization of earlier results studied in [2,8,9,10,11,21] with additional force components along the system such as impulse, damping and distributed delay. Distributed delay is broad case of constant delay and the authors provided in the monographs [6, 7].
Keywords - Neutral partial differential equations; Oscillatio; Impulse; Distributed delay; Damping. I. INTRODUCTION The oscillation of impulsive and non-impulsive parabolic and hyperbolic equations has been extensively studied in the literature, we refer the readers to the papers [1, 4, 8, 12, 14-19]
In this article, we consider the following impulsive system with damping of the form a m 1 b b p (t ) g ( t , ) u ( x , ( t , )) d ( ) q ( t , ) u ( x , ( t , )) d ( ) u ( x,t) m 1 a a t n = a (t ) u ( x , t ) b j ( t ) u ( x , j ( t )), t t k , ( x , t ) [0, ) G j =1 (i) (i) u ( x, tk ) u ( x, tk ) (i) = Ik x, tk , , k = 1,2, , i = 0,1,2, , m 1 (i) (i) t t m 1 u ( x,t) r (t ) m 1 t t
b
g ( t , ) u ( x , ( t , )) d ( )
(1)
where is a bounded domain in R N with a piecewise smooth boundary and Equation (1) is supplemented by the following Robin boundary conditions, (x)
u ( x, t)
( x ) u ( x , t ) = 0,
where is the outer surface normal vector to
and
is the Laplacian in the Euclidean space
( x , t ) [0, )
, C , [0, ) ,
2
(x)
R
N
.
(2) 2
(x) 0
.
In the sequel, we assume that the following hypotheses ( H ) hold: (H1)
r ( t ) C [0, ), (0, ) , r ( t ) 0
,
p ( t ) C ([0, ), R )
,
t
(H2)
a ( t ), b j ( t ) PC ([0, ), [0, ))
continuous in
t
(H3) g ( t , ) C lim t
,
j = 1,2, , n
, where
with discontinuities of first kind only at m
([0, ) [ a , b ], [0, ))
j ( t ) = , j = 1,2, , n
,
ds =
R ( t ) = exp
t
t
r ( s ) p ( s )
0
r(s)
represents the class of functions which are
t = t k , k = 1,2,
, and left continuous at
q ( t , ) C ([0, ) [ a , b ], [0, ))
, a, b are non-positive constants with
5
, where
R (s)
0
PC
1
a b.
,
.
piecewise
t = t k , k = 1,2,
j ( t ) C ([0, ), R )
ds
,
. j (t ) t
,