New oscillation criteria for second order advanced neutral differential equations

New oscillation criteria for second order advanced neutral differential equations

Xhevair Beqiri

State University of Tetova

Alit Ibraimi

State University of Tetova

Abstract

In this paper we present new criteria for oscillation of advanced neutral differential

A3) (t) C1 ([t

0

0

lim (t) .

t

, ),) , (t) t ,

equations second order of the form

[r(t)[((a(t)x(t) b(t)x( (t)) x(

t t0 >0 (1)

where the coefficient r(t) is nonnegative continuous function , a(t), b(t) and c(t) are continuous function which filled certain conditions .

The conclusion is based also on building functions where are involved coefficients of

equation , positive functions (t) and the

In following we set

0z(t) a(t)x(t) b(t)x( (t)) . By a solution of equation (1) we consider a function

x(t), t [t x ,) [t0 ,} which is twice

continuously differentiable and satisfies equation

(1) on the given interval. We consider only non- trivial solutions . A solution x(t) of (1) is said to be oscillatory if there exists a sequence

n n1 0

n n1 0

{ } of points in the interval [t , }, such

positive function of Philo H (t, s) .

that lim and

n

n

n

x(n ) 0 ,

n N ,

Here , by using the generalized Riccati technique we get a new oscillation criteria for (1).

Key words: oscillation , differential equation, second order, interval, criteria etc.

Introduction

otherwise it is said to be non-oscillatory. An equation is said to be oscillatory if all its solutions are oscillatory, otherwise it is considered that is non-oscillatory solution .

Lemma 1. If x(t) is a positive solution of (1)

then exists t1 [t0 , ) such the corresponding function

Let consider and create new oscillation of

z(t) a(t)x(t) b(t)x( (t))

satisfies

(2)

advanced neutral differential equations second order of the form

t t1

z(t) 0 ,

z'(t) 0 ,

z''(t) 0 for

[r(t)((a(t)x(t) b(t)x( (t)) )']'c(t) x( (t) 0

eventually.

Proof: Assume that the function x(t) is a

t t0

where is a quotient of odd positive integers and is a even number.

positive solution of (1) . Then from (1) follow that exists t1 [t0 , ) such that

We assume that

A1) a(t) 0 , c(t) 0 , 0 b(t) 1,

((r(t)(z'(t)) )' c(t) x( (t) 0

for

1

1

A2) r(t) 0 , 1

t t1

from where we get that the function

t0 r (s)

r(t)(z ' (t)) is decreasing for t t

1

and we

claim that r(t)(z ' (t)) 0 or

r(t)(z ' (t))

0 . If we let

r(t)(z ' (t)) 0

Since ''(w)

1

1

Aw

0 , we have

on t t1 then exists t2 t1 , such that 2

r(t)(z'(t)) r(t )(z'(t

2 2

))

0 , for all

that the function (w) attains to max value on

t t2 from where

1

at wmax , i. e. (wmax ) is a max value of function (w) and

1

1

z'(t) (r(t2 )) z'(t2 )

(r(t))

(wmax )

( 1)

1

B 1

A

and we can write the inequality

Integrating this from t2 to t we have

1

Bw Aw

B 1

.

1 t 1

( 1) 1 A

1

1

z(t) z(t2 ) (r(t2 ) z'(t2 ) ds

t2 (r(s))

Consider (2) we have

we can see that z(t) , where t .

x(t) 1 [z(t) b(t)x( (t))]

This contradicts because z(t) 0 we have

a(t)

r(t)(z ' (t)) 0 , from where z'(t) 0.

from where

From (1) we get

((r(t)(z'(t)) )' 0

r'(t)(z'(t)) r(t)(z't)) 1 z''(t) 0

from where

z''(t) 0.

This complete the proof.

1

x( (t)) 1 [z( (t)) b( (t))x( ( (t)))]

a( (t))

for x(t) 0 , (t) t and x'(t) 0 , also from (2) we get

x( (t)) x(t) and

x( (t)) z( (t))

finally

Lemma 2. Let () Bw Aw , A>0,

x( (t)) 1 [z( (t)) b( (t))z( (t))]

and B are constants, is a quotient of odd positive integers. Then function attains its

maximum value on at

x( (t))

a( (t))

1

[z( (t))(1 b( (t))] .

wmax

B

( 1) A

and

Now define

a( (t))

1

r(t)(z'(t))

max( w) B .

w(t) v(t)

z ( (t))

, for

Proof.: From

( 1) 1

A

1 1

t t0 0

(3)

'(w) B

Aw

and

differenting (3) and using (1) we see that

w'(t) v'(t)

'(w) 0 , we get

w

( 1) 1

B 1

A .

r(t)(z'(t))

z ( (t))

• v(t)

  (r(t)z' (t))

  z ( (t))

  v(t)

  r(t)z' (t)z 1 (t)z'( (t)) '(t)

  z2 ( (t))

  w'(t)

  v'(t) w(t) v(t)

  H (t, s)

  t

  p (t, s)

  H (t, s)

  and

  v(t) c(t)(1 p( (t)) z' ( (t))

  z'( (t)) '(t)

  H (t, s) h (t, s)

  H (t, s)

  a ( (t))z ( (t))

  w(t)

  z( (t)) s 2

  w'(t)

  v'(t)

  w(t)

  Teorem 1. Assumed that A1) A3) hold

  .Assume that exists a positive differentiable

  v(t)

  function v(t) and a function H (t, s) X and if

  c(t)(1 p( (t))

  1 '(t)

  there exist (a, b) [t

  , ),c (a, b) , such that

  v(t) w(t)w (t) 0

  a ( (t)) 1 1 c

  c(t)(1 p( (t))

  r ( (t))v (t)

  1 [H (t, s)v(s)L(s)

  H (c, a) a

  for L(t)

  a ( (t)) 0

  H (t, s)v'(s) p (t, s) H (t, s) ]ds

  we obtain

  v'(t)

  ( 1) 1 v (s)H (t, s) ' (s)

  c

  c

  1 b

  w'(t)

  w(t)

  v(t)

  H (b, c) [H (b, s)v(s)L(s)

  1

  '(t)

  H (b, s)v'(s) h (b, s) H (b, s)

  v(t)L(t) w

  (t) 1 1

  2 ]ds 0 (6)

  ( 1) 1 v (s)H (b, s) ' (s)

  (4)

  for

  B v'(t) ,

  r ( (t))v (t)

  A '(t)

  then every solution of eq. (1) is osillatory.

  Proof: Suppose to the contrary, that x(t) be a non-oscillatory solution of (1) , say x(t) 0 on

  v(t)

  1 1

  [t , ) from where z(t) 0 on [t , ) .

  v (t)r ( (t)) 0 0

  we have

  1(t )

  If inequation (5) multiplying with H (t, s) and integrate from c to t where t (c, b), s (c, t)

  we have

  w'(t) v(t)L(t) B(t)w(t) A(t)w

  now to consider lemma 2, we have t t

  B 1

  H (t, s)v(s)L(s)ds H (t, s)w'(s)ds

  w'(t) v(t)L(t) c

  ( 1) 1 A

  c

  1

  from where

  t w'(s)

  t W

  (s) ' (s)H (t, s)

  ds

  ds

  c

  c

  H (t, s) w(s) w(s)ds 1 1

  (v'(t)) 1 r( (t))

  w'(t) v(t)L(t)

  ( 1) 1 v (t)( '(t)) t

  c r ( (s))v (s)

  t

  t t0 0

  (5)

  H (t, s)v(s)L(s)ds w(s)H (t, s)

  We say that a function H (t, s) belons to the class X if

  i) H C(D,[0, )) ;

  1. H (t,t) 0 and H (t, s) 0 , for

     s t ;

     c

     t

     p (t, s)

     c

     t w'(s)

     c

     H (t, s)w(s)ds

     1

     t W (s) '(s)H (t, s)

  2. H has continuous partial

  H (t, s) w(s) w(s)ds 1

  1 ds

  c

  derivatives on first and second variable

  c r ( (s))v (s)

  t t v'(s) c c v'(s)

  H (t, s)v(s)L(s)ds w(c)H (t, c) [(H (t, s) v(s) H (s, t)v(s)L(s)ds w(c)H (s, t) [(H (s, t) v(s)

  c c t t

  • h (t, s)

H (t, s))w(s) W

1

(s) '(s)H (t, s)]ds

1

W

W

(s) '(s)H (s, t)

2 1 1

p (s, t)

H (s, t))w(s) 1

1 ]ds

From Lemma2 for

r ( (s))v (s)

A '(s)H (t, s) ,

From Lemma2 for

r ( (s))v (s)

A '(s)H (s, t) ,

1 1 1 1

r ( (s))v (s)

r ( (s))v (s)

2

2

B H (t, s) v'(s) h (t, s)

H (t, s)

B H (s, t) v'(s) h (s,t)

H (s,t)

we have

t

v(s)

we have

c

v(s) 1

H (t, s)v(s)L(s)ds w(c)H (t, c)

c

H (s, t)v(s)L(s)ds w(c)H (s, t)

t

t H (t, s)v'(s) h (t, s)

H (t, s)

c H (s, t)v'(s) h (s, t)

H (s, t)

2

1

ds

(7)

1

1

ds

(9)

c ( 1) v (s)H

(t, s) ' (s)

t ( 1) v (s)H

(s, t) ' (s)

Let t a

in (9) ) and dividing it by H(c,a)

Let t b in (7 and dividing it by H(b,c) we get

we obtain

a

a

1 c

1 b

H (b, c) H (b, s)v(s)L(s)ds w(c)

H (c, a) H (t, s)v(s)L(s)ds w(c)

c 1 c H (t, s)v'(s) h (t, s)

H (t, s)

(10)

b H (b, s)v'(s) h (b, s) H (b, s)

1

1

ds

2 ds

c

c

( 1) 1 v (s)H (b, s) ' (s)

(8)

H (c, a) a ( 1)

v (s)H

(t, s) '

(s)

If (5)multiplying with

H (s, t)

and integrate

Adding (8) and (10) we have the following inequality

a

a

1 c

over (t, c) where t (a, c), s (t, c) we get

c c

H (c, a) [H (c, s)v(s)L(s)

H (s, t)v(s)L(s)ds H (s, t)w'(s)ds

H (c, s)v'(s) p (c, s) H (c, s) ]ds +

t t

1

( 1) 1 v (s)H (c, s) ' (s)

c w'(s)

c W

(s) ' (s)H (s, t) 1 b

H (s, t) w(s) w(s)ds 1

1 ds [H (b, s)v(s)L(s)

t t r ( (s))v (s)

H (b, c) c

c c c

H (b, s)v'(s) h (b, s)

H (b, s)

H (s, t)v(s)L(s)ds w(s)H (s, t)

t t

p (t, s)

t

H (s, t)w(s)ds

2

( 1) 1 v (s)H (b, s) ' (s)

]ds 0

c w'(s)

1

c W

(s) '(s)H (s, t)

Which contradict to the condition (6) , therefore

H (s, t) w(s) w(s)ds 1 1 ds

t t r ( (s))v (s)

, every solution of equation (1) be oscillatory . The proof is complete.

Corollary 1: Let assume that A1, A2, A3 hold. If

1 t

1 t

lim sup [(t s)v(s)L(s)

t

lim sup [H (t, s)v(s)L(s)

H (t, a)

t k

t k

(t s)v'(s) 1

H (t, s)v'(s) h (t, s)

H (t, s)

1

]ds

(15)

1 ]ds 0

(11)

( 1) v (s)(t s) '

(s)

0

0

( 1) 1 v (s)H (t, s) ' (s)

and

for any H X ,

v C1 ([t , ),(0, )) and

t

t

for all

1

k t0 , then every solution of (1) is

lim sup [H (t, s)v(s)L(s)

t H (t, c) k

oscillatory.

Proof: From (14) and (15) for

H (t, s)v'(s) p (t, s) H (t, s) ]ds 0

(12)

H (t, s)

H (t, s)

( 1) 1 v (s)H (t, s) ' (s)

t 1 ,

s 1

we have

0

0

for any H X , v C1 ([t , ),(0, )) and for all k t0 , then every solution of (1) is oscillatory.

Proof: For k t0 , from (11) if we take

k a , and c a , we get

1 c

lim sup [H (t, s)v(s)L(s)

t H (c, a) a

(14) respectively (15). The proof is complete.

Reference

1. J. Dzurina, Oscillation theorems for second order advanced neutral differential equations, Mathematical institute, Slovac Academy of sciences, 2011, p. 61- 71

2. Xh.Beqiri, E. Koci, Oscillation criteria for second order differential equations, British

Journal of Science, 2012, 73 -80.

H (t, s)v'(s) p (t, s) H (t, s)

[3]A. A. Soliman, R. A. Sallam, A. M. Hassan,

( 1) 1 v (s)H (t, s) ' (s)]ds

(13)

Oscillation criteria of second order nonlinear neutral differential equations, International

journal of applied mathematical research, 2012,

From (12) for

1

k c

b

and for any

b c

p. 314 322.

[4] M. M. A. El-Sheikh, R. A. Sallam, D. I.

lim sup [H (b, s)v(s)L(s)

t H (t, c) c

H (t, s)v'(s) p (t, s) H (t, s) ]ds 0

( 1) 1 v (s)H (t, s) ' (s)

If adding (12) to (13) ,we obtain the inequality of the theorem 1. Now, the proof is complete.

If for H (t, s) (t s) , t s t0 , we have the following corollary.

Corollary 1. Let assume that A1, A2, A3 hold. If

1 t

lim sup [(t s)v(s)L(s)

t

Elimy,Oscillation criteria for nonlinear second order damped differential equations , Int. Jour. of nonl. sc. 2010, 297-307.

[5]J. Dzurina, E. Thandapani, S. Tamilvanan, Oscillation of solutions to third- order half-linear neutral differential equations, electronic journal of differential equations, 2012, p. 1-9.

[6]T. Li, E. Thandapani, J. Graef, Oscillation of third-order neutral retarded differential equations, international journal of pure and applied mathematics, 2012, p. 511 520.

[7]Xh. Beqiri, New Oscillation Criteria For Second Order Nonlinear Differential Equations, reasearch inventy, International journal of

enginering and science, 2013,

t k

(t s)v'(s) 1

( 1) 1 v (s)(t s) ' (s)

and

]ds

(14)

p. 36 41.