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ISSN 0976-5727 (Print) ISSN 2319-8133 (Online) Abbr:J.Comp.&Math.Sci. 2014, Vol.5(2): Pg.199-203
An Oscillatory behavior for Second Order Delay Difference Equation P. Mohankumar1 and A. Ramesh2 1
Professor of Mathematics, Aaarupadaiveedu Institute of Techonology, Vinayaka Missions University, Paiyanoor, Chengalpattu, Kancheepuram District, Tamilnadu, INDIA. 2 Ph.D Scholar, Vinayaka Missions University, Salem INDIA. (Received on: March 28, 2014) ABSTRACT An oscillatory behavior for second order delay difference equation of the form
∆ 2un + f n F (uσ ( n ) , ∆uσ ( n ) ) + G ( n, uσ ( n ) , ∆uσ ( n ) ) = 0, n = 0,1, 2..... The method uses techniques based on normed linear spaces. Example is inserted to illustrate the result. 2010MSC: 39A10. Keywords: Non-linear Difference equation, Neutral delay.
INTRODUCTION
difference
Consider the Second Order Nonlinear delay difference equation of the form
solution
operator
{un }
∆un = un +1 − un
A
of equation (1) is called
oscillatory or converges to zero Where the following conditions are ∆ 2un + f n F (uσ ( n ) , ∆uσ ( n ) ) + G ( n, uσ ( n ) , ∆uσ ( n ) ) assumed = 0, n =to 0,1, 2..... .........(1) hold: C1: f , F : R → R 2 , G: R → R 2 are , ∆uσ ( n ) ) + G ( n, uσ ( n ) , ∆uσ ( n ) ) = 0, n = 0,1, 2..... .........(1) (1) continuous functions. Where { f n } is a sequence of real numbers and
∆ denotes the forward
C 2 : f (n) ≥ 0 for every n ∈ [n0 , +∞]
C 3 : ( F (ϕ ), ⋅) > 0, G(⋅, ϕ , ⋅) ≥ 0 for every ϕ ∈ R 2 with ϕ (n) > 0, n ∈ [−σ , 0] and F ( −ϕ , −ψ ) = − F (ϕ ,ψ ), G (n, −ϕ , −ψ ) = −G (n, ϕ ,ψ ), for every n ∈ [n0 , +∞], ϕ ∈ R 2 ,ψ ∈ R 2 Journal of Computer and Mathematical Sciences Vol. 5, Issue 2, 30 April, 2014 Pages (123 - 257)
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P. Mohankumar, et al., J. Comp. & Math. Sci. Vol.5 (2), 199-203 (2014)
C 4 : F (λϕ , λψ ) ≥ λ n F (ϕ ,ψ ) for every (ϕ ,ψ ) ∈ R 2 , λ ∈ R and p ≥ 0 +∞
C5 :
∑ f ( s) = +∞ s=n
Throughout the sequel we shall denote by R the real line by R n the n one dimensional Euclidean space and by
Proof Let {un } be a non-oscillatory solution of (1). We can assume without loss of generality, the
{un } > 0 for every n ∈ [n0 − σ , +∞). We shall prove that ∆un > 0 for every t of a certain neigborhood of + ∞.
of the norms usually considered in R n . More over given σ ≥ 0 by R will denote the space of all continuous functions mapping the interval [ −σ , 0] into R n endowed with
Since
the sup norm
Which, in the case where
:
un = un −1 + un* , n* ∈ [n − 1, n]....................(1.1) (1.1)
∆un ≤ 0 for every t of a certain neigborhood of + ∞.
implies that
ϕ = sup ϕ (n)
lim ∆un = 0 for every n of a certain neighborhood of + ∞
n∈[ −σ ,1]
n →+∞
Which makes it into Banach space. For any R n -valued function {un } defined on an
I of the real line and any n ∈ I with :
interval
which by (ii),(iii) and the fact that
{un } > 0 for every n ∈ [n0 − σ , +∞). which is impossible Hence, without loss of generality, we can assume that
∆un > 0 for every n ∈ [n0 + ∞). Jτ = {ξ ∈ R : ξ − n ∈ [−σ , 0] ⊆ I ....................(2) (2) we can also prove that ∆u lim n = 0.............................. t →+∞ We shall denote the {un } the function un defined by Using the monotoncity of uσ ( n ) , ∆uσ ( n ) We obtain
ul (n) = u (l + n), n ∈ [−σ , 0] .........................(3) (3)
0<
uσ ( n )
MAIN RESULT Theorem 1 Assume that the equation (1) satisfies the conditions (C1) –(C5) Then every solution of (1) is Oscillatory
∆uσ ( n )
=
∆un −σ ∆un −σ ≤ un un −σ
Which by (1.2) gives
lim n →=∞
∆uσ ( n )
=0
uσ ( n )
Journal of Computer and Mathematical Sciences Vol. 5, Issue 2, 30 April, 2014 Pages (123 - 257)
P. Mohankumar, et al., J. Comp. & Math. Sci. Vol.5 (2), 199-203 (2014)
are equi -continuous and uniformly bounded.
we shall prove now that there exist n0 * ≥ n1 such that the functions
uσ ( n )
∗ σ (n)
,
u ∈ [u
uσ ( n )
ul1 (σ ( n )) ul1 (σ ( n )) =
−
∆u n
ul2 (σ ( n ))
=
(l1 − l2 ) =≤
ul1 (σ ( n ))
To prove the equi-continuity we remark that, if l1 ≥ l2 then
, +∞]
ul2 (σ ( n ))
201
u(l1 + n) u(l2 + n) u (l1 + n) − u (l2 + n) − ≤ ul1 (σ ( n )) ul2 (σ ( n )) ul1 (σ ( n )) ∆u n ul2 (σ ( n ))
(l1 − l2 )
where l2 ≤ n* ≤ l1 from this since
u (l + n) u (l + n) = un un
is increasing and
lim
∆uσ ( n )
n →+∞
uσ ( n )
= 0 we conclude that there exists a
n* ≥ n0 such that u(l1 + n) ul1
−
u (l2 + n ) u l2
≤ l1 − l2 for every l1 , l2 ≥ n0* and every n ∈ [ −σ , 0]
Without loss of generality we assume in the following that n2* = n2 from the above we conclude that there exists an increasing sequence {un }, n = 1, 2,3,... in [n1 + ∞]
u ∆u lim inf F σ , σ u σ uσ
uσ ( n ) ∆uσ ( n ) , = lim F uσ ( n ) uσ ( n )
Obviously ψ = I and we can easily derive by that fact the function {un } as well as the sequence
{un }, n = 1, 2,...
are
such that
lim
n →∞
uσ ( n ) uσ ( n )
=ψ ∈ R
and more over that
= F (ψ , 0)........(1.3) increasing, ψ ( n) > 0 for every n ∈ [ −σ , 0] which implies that F (ψ , 0) > 0 from (1) using (iv) we obtain
both
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P. Mohankumar, et al., J. Comp. & Math. Sci. Vol.5 (2), 199-203 (2014)
n n n n u ∆u 1 ∆un =η − ∑ fs F(us , ∆us ) −∑G(s, us , ∆us ) ≤η − ∑ fs F(us , ∆us ) ≤η − ∑ us fs F s , s ......(1.4) u u s=n0 s=n0 s=n1 s=n0 s s (1.4)
where η is a positive constant.
REFERENCES
Since {un } is increasing, we have
un = un ≥ un0 for every n ∈ [n0 , +∞] and (4) gives
∆un ≤η −u
t p n
us ∆us , ......(1.5) s us
∑ f F u s
s=t0
(1.5)
Which by (1.3) the fact that F (ψ , 0) > 0 and (C5) implies that
lim ∆u n ≤ −∞
n →+∞
This contradiction proves the theorem Example :1. Consider difference equation n
−1 2 1 ∆ 2un + ∆un −1 + un −1 = 0.............(E1) n n
All the assumptions of theorem 1 are satisfied, and hence every solutions of (E1) is oscillatory (using wolfram see below graph)
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