Vol 9, No 2 (2018)

(1)

Available online throug

ISSN 2229 – 5046

International Journal of Mathematical Archive- 9(2), Feb. – 2018 284

CLASSIFICATIONS OF SOLUTIONS OF SECOND ORDER

NONLINEAR NEUTRAL DELAY DYNAMIC EQUATIONS OF MIXED TYPE

N. SIKENDER*

1

_{, P. RAMI REDDY}

2

_{, M. VENKATA KRISHNA}

3

AND M. CHENNA KRISHNA REDDY

4

1,3,4

_{Department of Mathematics,}

Osmania University, Hyderabad, Telangana, India.

2

_{Department of Mathematics,}

Khairathabad Govt. Degree College, Hyderabad, Telangana, India.

(Received On: 22-12-17; Revised & Accepted On: 06-02-18)

ABSTRACT

In this paper, the authors classified all solutions of the second-order nonlinear neutral delay dynamic equations of

mixed type into four classes and obtained conditions for the existence / non-existence of solutions in these classes. Examples are included to illustrate the validiation of the main results.

Keywords: second-order, neutral dynamic equations, oscillatory solution, mixed type, weakly oscillatory solution,

asymptotic behavior, time scales.

INTRODUCTION

In this paper, we are concerned to obtain the conditions for the existence / non-existence solutions of a class

M

+,

M

−,

OS

,

WOS

and asymptotic behavior of solutions of a class

M

+ and

M

− of second order nonlinear neutral delay dynamic equations of mixed type

(a(t)(x(t) b(t)x( (t)) c(t)x( (t))) )

∆ ∆

+ α + β +p(t)f (x( (t))) q(t)g(x( (t))) = 0,γ + δ (1) for t∈[t , )₀ ∞_T, where T is a time scale with supT =∞. In what follows, it is always assume that

1. 1

rd 0 T

a∈C ([t , ) , (0, ))∞ ∞ , 2 rd 0 T

b, c∈C ([t , ) , R)∞ and p, q∈C ([t , ) , R)rd 0 ∞T ; 2. f , g∈C(R, R) such that

uf u

( ) > 0

an

ug u

( ) > 0

for

u

≠

0

;

3. α β γ δ ∈, , , C (R, R)_rd are strictly increasin functions such that

α

( )

t

≤

t

,

β

( )

t

≥

t

, γ(t)≤t,

δ

( )

t

≥

t

and limt→∞α(t) =limt→∞β(t) =limt→∞γ(t) =limt→∞δ(t) =∞

Let t_x∈[t , )₀ ∞_T such that

α

(

t

)

≥

t

₀ and

γ

(

t

)

≥

t

₀ for all t∈[t , )_x ∞_T. By a solution of equation (1), we shall mean a function x∈C ([t , ) , R)_rd _x ∞_T which has the properties x(t) b(t)x( (t)) c(t)x( (t))+ α + β ∈C ([t , ) , R)1_rd _x ∞ _T

and 1

rd x T

a(t)(x(t) b(t)x( (t)) c(t)x( (t)))+ α + β ∈C ([t , ) , R)∞ and satisfies equation (1) on [t , )_x ∞ _T. As is customary, a solution of equation (1) is oscillatory solution (OS) if it is neither eventually positive nor eventually negative, otherwise it is called non-oscillatory. A non-oscillatory solution

x

(

t

)

of equation (1) is said to be weakly

oscillatory solution (WOS) if

x

(

t

)

is non-oscillatory and

x

∆

(

t

)

is oscillatory for large value of t∈[t , )₀ ∞_T.

In recent years, there has been much research activity concerning the oscillation, nonoscillation and asymptotic behavior of solutions of various differential equations, difference equations and dynamic equations. For

instance in [7, 8, 9] etc., authors have been studied by classifying all solutions into four classes such as

M

+,

M

−,

OS

,

WOS

and obtained criteria for the existence/non-existence of solutions. In order to extend and generalize the

Corresponding Author: N. Sikender*

1

_,

(2)

papers [7, 8, 9], Rami Reddy et al. [10] were concerned the solutions of existence/non-existence of a class

M

+,

M

−,

OS

,

WOS

of second order nonlinear neutral delay dynamic equation of the form

( ( )( ( )

r t x t

+

p t x

( ) ( ( ))) )

α

t

∆ ∆

+

q t f x

( ) ( ( ( ))) = 0 ,

β

t

₍₂₎ for t∈[t , )₀ ∞_T, subject to the conditions:

4. 1

rd 0 T

r∈C ([t , ) , (0, ))∞ ∞ , 2 rd 0 T p∈C ([t , ) , R∞ ;

5. q∈C ([t , ) , R)_rd ₀ ∞_T and

q

does not vanish eventually;

6. 1

f∈C (R, R) such that

f

satisfies

uf u

( ) > 0

for

u

≠

0

and

f u

'

( )

≥

0

for

u

∈

R

; 7. α β∈, C ([t , ) , R)_rd ₀ ∞_T are strictly increasing functions such that

α

( )

t

≤

t

,

β

( )

t

≤

t

and

_lim

t→∞

α

( ) =

t

∞

=

_lim

t→∞

β

( )

t

.

The study of the oscillation and other asymptotic properties of solutions of neutral delay difference / differential / dynamic equations with positive and negative coefficients attracted a good bit of attention in the last several years. For instance in [16, 17, 18, 20] etc., and the references cited therein, authors have been studied certain criteria of second order nonlinear dynamic equations with positive and negative coefficients. In order to extend and generalize the papers [16, 17, 18, 19], sikender et al. [11] were concerned the solutions of

existence/non-existence of a class

M

+,

M

−,

OS

,

WOS

of second order nonlinear neutral delay dynamic equation of the form

(r(t)(x(t)+p(t)x( (t))) )α ∆ ∆ +q(t)f (x( (t))) h(t)g(x( (t))) = 0β − γ ₍₃₎

for t∈[t , )₀ ∞_T, subject to the conditions:

8. 1

rd 0 T

r∈C ([t , ) , (0, ))∞ ∞ , 2

rd 0 T

p∈C ([t , ) , R)∞ and q, h∈C ([t , ) , R)_rd ₀ ∞ _T ;

9.

f g

,

∈

C R R

( , )

such that

uf u

( ) > 0

and

ug u

( ) > 0

for

u

≠

0

; 10. α β γ ∈, , C ([t , ) , T)_rd ₀ ∞_T are strictly increasing functions such that

( )

t

α

≤

,

β

( )

t

≤

t

,

γ

( )

t

≤

t

and

_lim

t→∞

α

( ) =

t

∞

=

_lim

t→∞

β

( )

t

.

Interest in Functional differential and difference equations is growing due to the development in science and technology and the varied applications in many areas. For examples, equations involving delay and those involving advance and a combination of both arise in nerve conduction (Life Science), organizational behaviour (Social sciences), Signal processing pantograph equations (Mechanical Engineering), to mention a few (see for e.g [12, 13, 14, 15]). Study of such equations has been an active area of research for many researchers and recently an importance is given to the unification of continuous and discrete aspects of analysis on time scales. In [10], the authors considered equation (1) with

c t

( ) = 0

,

q t

( ) = 0

for all t∈[t , )₀ ∞_T and classified all solutions of (1) into four classes and obtained criteria for the existence / non-existence of solutions in these classes. In [11], the authors considered equation (1) with

c t

( ) = 0

,

p t

( )

≥

0

,

q t

( )

≤

0

and

δ

( )

t

≤

t

for all t∈[t , )₀ ∞_T and classified all solutions of (1) into four classes and obtained criteria for the existence / non-existence of solutions in these classes. But, now the authors are so much interested on mixed delays. In [6], the authors considered the neutral differential equations of mixed type of the form

' ' 1 2

(a(t)(x(t) bx(t+ − τ +) cx(t+ τ )) ) +p(t)x (tα − σ +₁) q(t)x (tβ + σ₂) = 0,

0

t≥t > 0

(4) subject to the following conditions:

11. 1

0

a∈C ([t , ), R)∞ and is positive for all

t

≥

t

₀

;

12.

b

and

c

are constants,

τ

₁,

τ

₂,

σ

₁ and

σ

₂ are nonnegative constants; 13.

α

and

β

are the ratio of odd positive integers;

14. p, q∈C ([t , ) , R)._rd ₀ ∞_T

Motivated and inspired by the papers mentioned above, in this paper the authors are interested to study the

solutions of existence / non-existence of a class

M

+,

M

−,

OS

,

WOS

and asymptotic behavior of

M

+ and

M

− of equation (1). Here we consider two cases, namely

p

≤

0

,

q

≥

0

and

p

≥

0

,

q

≤

0

for large t∈[t , )₀ ∞_T, to give sufficient conditions in order that every solution of equation (1) oscillates and to study the asymptotic nature of non-oscillatory solutions of equation (1) with respect to their asymptotic behavior all the solutions of equation (1) may be

priori divided in to the following classes:

1.

M

+

=

{

x

∈

S

:

there exists some _x ₀ _T

1

t ∈[t ,∞) such that

x

(

t

)

x

∆

(

t

)

≥

0

for x T 1 t∈[t , ) }∞ ;

2.

M

−

=

{

x

∈

S

:

there exists some _x _T 1

(3)

3.

OS

=

{

x

∈

S

:

there exists a sequence t_n∈[t , )₀ ∞ _T,

t

_n

→

∞

such that x t x t( ) (_n _n₊₁)≤0};

4.

WOS

= {

x

∈

S

:

x

(

t

)

is non-oscillatory for large t∈[t , )₀ ∞_T, but

x

∆

(

t

)

oscillates}.

With a very simple argument we can prove that M+, M−, OS, WOS are mutually disjoint. By the above

definitions, it turns out that solutions in the class

M

+ are eventually either positive nondecreasing or negative

nonincreasing, solutions in the class

M

− are eventually either positive nonincreasing or negative nondecreasing, solutions in the class

OS

are oscillatory, and finally solutions in the class

WOS

are weakly oscillatory.

In next Section, we mentioned important lemma's which are existing in the literature. In Section 2, we obtain sufficient conditions for the existence/non-existence in the above said classes. In Section 3, we discuss the asymptotic

behavior of solutions in the class of

M

+ and

M

−.

1. IMPORTANT LEMMA’S

Theorem 1.1 (Chain rule) ([1, Theorem 1.90]): Let f : R→R is continuously differentiable and suppose g : T→R

is delta differentiable. Then f g : T →R is delta differentiable and the formula

)

(

}

))

(

)

(

)

(

{

=

)

(

)

(

1

0

f

g

t

h

t

g

t

dh

g

t

g

f



∆

∫

'

+

µ

∆ ∆

Theorem 1.2 ([1, Theorem 1.117]): Let a∈Tk, b∈T and assume f : T×Tk →R is continuous at

(

t

,

t

)

, where

k

t∈T with

t

>

a

. Also assume that

f

∆

(

t

,.)

is rd-continuous on

[

a

,

σ

(

t

)].

Suppose that for each

ε

>

0

there exists a neighbourhood

U

of

t

, independent of

τ

∈

[

a

,

σ

(

t

)]

, such that

| ( ( ), )

f

σ

t

τ

−

f s

( , )

τ

−

f

∆

( , )( ( )

t

τ σ

t

−

s

) |

≤

ε σ

| ( )

t

−

s

|

for all

s

∈

U

where

f

∆ denotes the derivative of

f

with respect to the first variable. Then

1.

( ) :=

t

( , )

a

g t

∫

f t

τ τ

∆

implies ( ) = t ( , ) ( ( ), )

a

g∆ t

∫

f∆ t τ τ∆ + f σ t t

2.

( ) :=

b

( , )

t

h t

∫

f t

τ τ

∆

implies ( ) = b ( , ) ( ( ), )

t

h t∆

∫

f∆ t τ τ∆ − f σ t t .

For more basic concepts in the time scale theory the readers are referred to the books (see [1, 2]).

2. EXISTENCE AND NON-EXISTENCE OF SOLUTIONS IN

M

+,

M

−,

OS

AND

WOS

First, we examine the existence of solutions of equation (1) in the class

M

+.

Theorem 2.1: Assume that

( )

H

1

b

(

t

)

,

c

(

t

)

are non negative and non-decreasing;

( )

H2

p

(

t

)

is non positive and

q

(

t

)

is non negative for t∈[t , ) ;0 ∞T

( )

H3 there exists

M

1

>

0

such that 1

( )

f u M g u

≤ for

u

≠

0;

( )

H4

g

is non-decreasing and

( )

H5 1

0

( ( ) ( )) = limsup_t t

t M p s q s s

→∞

∫

+ ∆ ∞ hold. Then for equation (1), we have

M

= .

ϕ

+

Proof: Suppose that the equation (1) has a solution

x

∈

M

+. Without loss of generality, we may assume that

x t

( ) > 0

and

x

∆

(

t

)

≥

0

for large t∈[t , )₀ ∞ _T (the proof is similar if

x

(

t

)

<

0

and

x

∆

(

t

)

≤

0

for large t∈[t , )₀ ∞ _T). Then

there exists t₁∈[t , )₀ ∞_T such that

x

(

t

)

,

x

(

α

(

t

))

,

x

(

γ

(

t

))

all are positive and

x

∆

(

t

)

,

x

∆

(

α

(

t

))

,

x

∆

(

γ

(

t

))

all are non-negative for all t∈[t , )₁ ∞ _T.

Define,

))

(

)

(

))

(

)

(

)

(

=

)

(

t

x

t

b

t

x

t

c

t

x

t

z

+

α

+

β

(2.1)

(4)

Using (2.1), equation (1) becomes

0 =

)))

(

)

(

)))

(

)

(

))

(

)

(

a

t

z

∆

t

∆

+

p

t

f

x

γ

t

+

q

t

g

x

δ

t

or

)

(

)))

(

)))

(

)

(

=

)))

(

))

(

)

(

t

q

t

x

g

t

x

f

t

p

t

x

g

t

z

t

a

∆ ∆

₋

δ

γ

δ

for t∈[t , )1 ∞T. (2.2)

Now for t∈[t , )₁ ∞ _T,

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

g(x( (t))) g(x( (t)))

∆

∆ ∆ ∆

 

 _δ  _δ

 

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

g(x( (t))) ∆

∆  

+ σ σ _ _

δ  

( ( )

( ))

( ( ( ( ))))

=

( ( ( ))

( ( )))

,

( ( ( )))

( ( ( ))) ( ( ( )))

a t z t

g x

t

a

t z

t

g x

t

g

σ

x

t

g x

t

δ

σ

δ

∆ ∆ ∆

∆





−

_

_





implies that,

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

g(x( (t))) g(x( (t)))

∆

∆ ∆ ∆

 

−

 _δ  _δ

  (a( (t))z ( (t)))

∆

σ σ

1 ' 0

{ g (x( (t)) h (t)(x( (t))) )dh}(x( (t)))

g (x( (t)))g(x( (t)))

∆ ∆

σ

 _δ _{+ µ} _δ _δ 

 

 _δ _δ 

   

∫

or

( )

( ( )

( ))

=

( ( ( )))

a t z t

g x

δ

t

g x

δ

t

∆

∆ ∆ ∆





−









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

∆

σ σ

1 ' 0

{ g (x( (t)) h (t)(x( (t))) )dh}x ( (t)) (t)

g (x( (t)))g(x( (t)))

∆ ∆ ∆

σ

 _δ _{+ µ} _δ _δ _δ 

 

 _δ _δ 

   

∫

(2.3) Therefore,

( )

( ( )

( ))

( ( ( )))

a t z t

g x

δ

t

g x

δ

t

∆ ∆ ∆ ∆





≤









(2.4)

for all t∈[t , )₁ ∞ _T, due to

( )

H

₄ ,

z

∆

(

t

)

≥

0

and

x

∆

(

δ

(

t

))

≥

0

for all t∈[t , )₁ ∞ _T. From (2.2) and (2.4), we have

( )

( ( ( )))

( )

( ( ( )))

a t z t

f x

t

p t

q t

g x

t

g x

t

γ

δ

∆ ∆





≤ −

−









)

(

)))

(

)))

(

)

(

q

t

x

g

t

x

f

t

p

−

≤

γ

≤

−

(

M

1

p

(

t

)

+

q

(

t

))

for t∈[t , )₁ ∞_T, due to

( )

H

₂ -

( )

H

₄ . Integrating the last inequality from

t

₁ to

t

, we obtain

.

))

(

)

(

)))

(

)

(

)

(

)))

(

)

(

)

(

1 1 1 1

1

_M

_p

_s

_q

_s

t

x

g

t

z

t

a

t

x

g

t

z

t

a

t

+

∆

−

≤

−

∆

∫

∆

δ

From

( )

H

₅ , we obtain

,

=

)))

(

)

(

)

(

liminf

−∞

∆ ∞

→

g

x

t

z

t

a

t

δ

which contradicts the assumption

z t

∆

( )

≥

0

for large

t

. Thus, the theorem is proved.

Example 2.2: In Theorem 2.1, the assumption

( )

H

₅ cannot be dropped. For this, suppose T = R, and consider the

following differential equation

' ' 2

1 1

(x(t) (3t 1)x(t 1)) 2(2t 5)x(t )) 2 t  ₊ ₊ ₋ ₊ ₋ ₊      3 3 3 ( 18)

x (t 2)

t (t 2)

− + − −

0,

=

3)

5)

(

5)(

(

28)

10 5)(

(

14

2 2

2

+

x

t

x

t

(2.5)

3 ≥

t

. For this differential equation, all assumptions of Theorem 2.1 hold, but

( )

H

₅ is violated. The equation (2.5) has

(5)

Theorem 2.3: Assume that condition

( )

H

₁ and

( )

H

6

p

(

t

)

is non negative and

q

(

t

)

is non positive for t∈[t , )0 ∞T;

( )

H

7 there exists

M

2

>

0

such that 2

( )

|

( )

g u

M

f u

≤

for

u

≠

0;

( )

H

8

f

is non-decreasing;

( )

H

9 2

0

( ( )

( ))

=

limsup

_t t

t

p s

M q s

s

→∞

∫

+

∆

∞

, hold. Then for equation (1), we have

M

= .

ϕ

+

Proof: Suppose that the equation (1) has a solution

x

∈

M

+. Without loss of generality, we may assume that

( ) > 0

x t

and

x t

∆

( )

≥

0 x t

( ) < 0

and

x

∆

(

t

)

≤

0

for large t∈[t , )₀ ∞_T). Then there exists t₁∈[t , )₀ ∞_T such that

x

(

t

)

,

x

(

α

(

t

))

,

x

(

γ

(

t

))

x

∆

(

t

)

,

x

∆

(

α

(

t

))

,

x

∆

(

γ

(

t

))

all are non-negative for all t∈[t , )₁ ∞ _T.

Define

z

(

t

)

as in Theorem 2.1, and from

( )

H

₁ , we get

z

(

t

)

>

0

and

z

∆

(

t

)

≥

0

for all t∈[t , ) .₁ ∞_T Using (2.1), equation (1) becomes

0 =

)))

(

)

(

)))

(

)

(

))

(

)

(

a

t

z

∆

t

∆

+

p

t

f

x

γ

t

+

q

t

g

x

δ

t

or

)

(

)))

(

)))

(

)

(

=

)))

(

))

(

)

(

t

q

t

x

f

t

x

g

t

p

t

x

f

t

z

t

a

γ

δ

γ

−

∆ ∆

for t∈[t , )₁ ∞ _T (2.6)

Now for t∈[t , )₁ ∞ _T,

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

f (x( (t))) f (x( (t)))

∆

∆ ∆ ∆

 

 _γ  _γ

 

1

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

f (x( (t)))

∆ ∆

+ σ σ

γ

(a(t)z (t))

= (a( (t))z ( (t))) f (x( (t)))

∆ ∆

∆

− σ σ

γ

(f (x( (t)))) , f (x( (t)))f (x( (t)))

∆

σ

 γ 

 _γ _γ 

 

implies that,

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

f (x( (t))) f (x( (t)))

∆

∆ ∆ ∆

 

−

 _γ  _γ

  (a( (t))z ( (t)))

∆

σ σ

1 ' 0

{ f (x( (t)) h (t)(x( (t))) )dh}(x( (t)))

f (x( (t)))f (x( (t)))

∆ ∆

σ

 _γ _{+ µ} _γ _γ 

 

 _γ _γ 

 

 

∫

or

( )

( ( )

( ))

=

( ( ( )))

a t z t

f x

γ

t

f x

γ

t

∆

∆ ∆ ∆





−









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

∆

σ σ

1 _' 0

{ f (x( (t)) h (t)(x( (t))) )dh}x ( (t)) (t)

f (x( (t)))f (x( (t)))

∆ ∆ ∆

σ

 _γ _{+ µ} _γ _γ _γ 

 

 _γ _γ 

 

 

∫

(2.7)

Therefore,

( )

( ( )

( ))

( ( ( )))

a t z t

f x

γ

t

f x

γ

t

∆

∆ ∆ ∆





≤









(2.8)

for all t∈[t , )₁ ∞ _T, due to (H₈),

z

∆

(

t

)

≥

0

and

x

∆

(

γ

(

t

))

≥

0

for all t∈[t , )₁ ∞ _T.

From (2.6) and (2.8), we have

( )

( ( ( )))

( )

( ( ( )))

a t z t

g x

t

p t

q t

f x

t

f x

t

δ

β

γ

∆ ∆

≤ −

−













( ( ( )))

( )

( ( ( )))

g x

t

p t

q t

f x

t

δ

≤ −

−

≤

−

(

p

(

t

)

+

M

2

q

(

t

))

(6)

1 1

2 1

1

( )

( ( )

( ))

.

( ( ( )))

t

a t z t

p s

M q s

s

f x

γ

t

f x

γ

t

∆ ∆

−

≤ −

∫

+

∆

From

( )

H

₉ , we obtain

( )

=

,

liminf

( ( ( )))

t

a t z t

f x

γ

t

∆

→∞

−∞

which contradicts the assumption z t∆( )≥0 for large

t

. Thus, the theorem is proved.

Example 2.4: Let T = R, and consider the following differential equation

2 2

1 ( ( )

(

2 ) (

2)

(2

1) (

9))

3

8

' '

x t

t

t x t

t

x t

t

−

t

+

− +

−

+













3 2 2 3 2

12

x (t 1 / 2)(x (t 1 / 2) 1) t(3t 8) (t 1 / 2) (t t 5 / 4)

+ − − +

− − − + −

5 2 2 5

8 (5

6) = 0,

(3

t

−

8 ) (5

t

+

6)

x

t

+

(2.9)

1 ≥

t

. For this differential equation, all assumptions of Theorem 2.3 hold, but

( )

H

₉ is violated. The equation (2.9) has

a solution

x

(

t

)

=

t

∈

M

+

.

Theorem 2.5: Assume that conditions

( )

H

₂ ,

( )

H

₃ ,

( )

H

₄ ,

( )

H

₅ and

( )

H10

−

1 <

b

(

t

)

+

c

(

t

)

≤

0

for t∈[t , )0 ∞T and

c

(

t

)

is non negative;

( )

H11

M

1

p

(

t

)

+

q

(

t

)

≥

0

for t∈[t , )0 ∞ T;

( )

H12

0

1 =

( )

t

_{a t}

t

∞

∆

∞

∫

hold. Then for equation (1), we have

M

+

= .

ϕ

Proof: Suppose that the equation (1) has a solution x∈M+. Without loss of generality, we may assume that

x t

( ) > 0

and

x t

∆

( )

≥

0

for large t∈[t , )₀ ∞_T (the proof is similar if

x

(

t

)

<

0

and

x

∆

(

t

)

≤

0

for large t∈[t , )₀ ∞_T). Then there exists t₁∈[t , )₀ ∞_T such that

x

(

t

)

,

x

(

α

(

t

))

,

x

(

γ

(

t

))

x t

∆

( )

,

x

∆

( ( ))

α

t

,

x

∆

( ( ))

γ

t

all are non-negative for all t∈[t , )₁ ∞_T.

Define

z

(

t

)

as in Theorem 2.1, we have

))

(

)

(

))

(

)

(

)

(

=

)

(

t

x

t

b

t

x

t

c

t

x

t

z

+

α

+

β

≥

(1

+

b

(

t

)

+

c

(

t

))

x

(

α

(

t

))

>

0

for t∈[t , )₁ ∞_T. From (1), we have

)))

(

))

(

)))

(

)))

(

)

(

=

))

(

)

(

q

t

g

x

t

x

g

t

x

f

t

p

t

z

t

a

δ

γ

₊

−

∆ ∆

)))

(

))

(

)))

(

)))

(

)

(

q

t

g

x

t

x

g

t

x

f

t

p

δ

γ

₊

−

≤

−

(

M

1

p

(

t

)

+

q

(

t

))

g

(

x

(

δ

(

t

)))

≤

0,

for t∈[t , )1 ∞T.

Hence,

a t z t

( )

∆

( )

is non-increasing for t∈[t , )₁ ∞_T.

We claim that,

a

(

t

)

z

∆

(

t

)

≥

0

for t∈[t , )₁ ∞ _T.

If

a

(

t

)

z

∆

(

t

)

<

0

for t∈[t , )₁ ∞_T, then

a

(

t

)

z

∆

(

t

)

≤

a

(

t

₁

)

z

∆

(

t

₁

)

or

.

)

(

1 )

(

)

(

)

(

₁ ₁

t

a

t

z

t

a

t

(7)

Integrating the above inequality from

t

₁ to

t

, we obtain

.

)

(

1 )

(

)

(

)

(

)

(

1 1 1

1

s

a

t

z

t

a

t

z

t

z

t

∆

≤

−

∆

∫

This gives that

z

(

t

)

→

−∞

as

t

→

∞

due to

( )

H

₁₂ , which is a contradiction to

z t

( ) > 0

. Thus

a t z t

( )

∆

( )

≥

0

for

1 T

t∈[t , )∞ . Now proceeding as in the proof of Theorem 2.1, and using

( )

H

₁₁ , we get

.

=

)))

(

)

(

)

(

lim

−∞

∆

∞

→

g

x

t

z

t

a

t

δ

This contradicts the fact that

z t

∆

( )

≥

0

for large t. Hence, the theorem is proved.

Example 2.6: Let T = hZ ,

h

is a ratio of odd positive integers, and consider the following h-difference equation

2

1

2

(

( ( )

(

2 )

(

2 )))

2

t h h

h

e

h

x t

e x t

h

e

x t

h

−

∆

−

+

3

2 12

2 2

1

1 (4

12 )

(

1)

(

) = 0,

2

h h

t h h t

e

x t

h

e

e x t

h

− +

+

−

+

(2.11)

h

t

>

12

. For this difference equation, all assumptions of Theorem 2.5 hold, but

( )

H

₁₂ is violated. The equation

(2.11) has a solution

x

(

t

)

=

e

t

∈

M

+

.

Theorem 2.7: Assume that conditions

( )

H

₆ ,

( )

H

₇ ,

( )

H

₈ ,

( )

H

₉ ,

( )

H

₁₀ ,

( )

H

₁₂ and

( )

H

13 .

p

(

t

)

+

M

2

q

(

t

)

≥

0

for t∈[t , )0 ∞ T hold. Then for equation (1), we have

=

φ

.

+

M

Proof: The proof is similar to that of Theorem 2.5.

Next, we examine the problem of the existence of solutions of equation (1) in the class

M

−:

( )

H

₆ ,

( )

H

₇ ,

( )

H

₈ and

( )

H

14

b

(

t

)

and

c

(

t

)

are non-negative and non-increasing;

( )

H

15

f

(

uv

)

=

f

(

u

)

f

(

v

)

;

( )

H

16

γ

(

t

)

≤

α

(

t

)

;

( )

H

17 the function

)

(

1 u

f

is locally integrable on

(0,

c

)

and

(

−

c

,0)

for some

c

>

0

,

i.e.,

∫

<

∞

)

(

0

_f

_u

du

c

,

∫

−∞

−

₍

₎

>

0

u

f

du

c

;

( )

H

18

t t _T

1 limsup

a(s)f (1 b(s) c(s))

→∞

∫

₊ ₊

(

)

s

2

T(p(v)+M q(v)) v∆ ∆s =∞

∫

for

t

≥

T

.

Then for equation (1), we have

M

−

=

φ

.

Proof: Suppose that equation (1) has a solution

x

∈

M

−. Without loss of generality, we may assume that

x

(

t

)

>

0

and

x

∆

(

t

)

≤

0 x

(

t

)

<

0

and

x

∆

(

t

)

≥

0

for large t∈[t , )₀ ∞ _T). Then

there exists t₁∈[t , )₀ ∞_T such that

x

(

t

)

,

x

(

α

(

t

))

,

x

(

γ

(

t

))

x

∆

(

t

)

,

x

∆

(

α

(

t

))

,

x

∆

(

γ

(

t

))

all

are non-positive for all t∈[t , )₁ ∞ _T. Define

z

(

t

)

as in Theorem 2.1 and using

( )

H

₁₄ , we see that

z

(

t

)

>

0

and

0 )

(

≤

∆

t

(8)

As in the proof of Theorem 2.3, we obtain

s

q

M

s

p

t

x

f

t

z

t

a

t

x

f

t

z

t

a

t

+

∆

−

∆

∫

∆

))

(

)

(

=

)))

(

)

(

)

(

)))

(

)

(

)

(

2 1 1 1 1

γ

or

s

q

M

s

p

t

x

f

t

z

t

a

t

+

∆

−

≤

∫

∆

))

(

)

(

)))

(

)

(

)

(

2 1

γ

or

s

q

M

s

p

t

a

t

x

f

t

z

t

+

∆

−

≤

_∫

∆

))

(

)

(

)

(

1 )))

(

)

(

2 1

γ

for t∈[t , )1 ∞T. (2.12)

We see that

z

(

t

)

≤

(1

+

b

(

t

)

+

c

(

t

))

x

(

γ

(

t

))

. Since

x

is non-increasing and from

( )

H

₁₅ , we see that

)))

(

))

(

)

(

(1

))

(

z

t

f

b

t

c

t

f

x

t

f

≤

+

γ

(2.13)

Combining (2.12) and (2.13), we obtain

)))

(

))

(

)

(

(1

)

(

))

(

)

(

t

x

f

t

c

t

b

f

t

z

t

z

f

t

z

γ

+

≤

∆ ∆

.

))

(

)

(

))

(

)

(

(1

)

(

1

2 1

s

q

M

s

p

t

c

t

b

f

t

a

t

+

∆

+

−

≤

∫

Integrating from

t

₁ to

t

, we obtain

z( t ) z( t )₁

s f (s)

∆

−

_∫

t s

2 t₁ t₁

1

( (p(v) M q(v)) v) s

a(s)f (1 b(s) c(s))

≤ − + ∆ ∆

+ +

∫

or

z( t ) 1 z( t )

s f (s)

∆ ≥

∫

t s

2 t₁ t₁

1

( (p(v) M q(v)) v) s

a(s)f (1 b(s) c(s))+ + + ∆ ∆

∫

Using

( )

H

₁₈ , we obtain

,

=

)

(

limsup

(1)

)

(

∞

∆

∫

∞

→

f

s

t z t z t

which is a contradiction to

( )

H

₁₇ . Hence, the theorem is proved.

( )

H

₂ ,

( )

H

₃ ,

( )

H

₄ ,

( )

H

₅ ,

( )

H

₁₁ ,

( )

H

₁₂ and

( )

H

19

b

,

c

are non negative constants,

( )

H

20

f

is nonincreasing hold.

Then every solution of equation (1) is either oscillatory or weakly oscillatory.

Proof: From Theorem 2.1, it follows that for equation (1), we have

M

+

=

ϕ

. To complete the proof, it suffices to

show that for equation (1),

M

−

=

φ

. Let

x

∈

M

−, then

x

(

t

)

>

0

and

x

∆

(

t

)

≤

0

for t∈[t , )₁ ∞_T. Define

z

(

t

)

as

in Theorem 2.1, we have

z

(

t

)

>

0

and

z

∆

(

t

)

≤

0

for t∈[t , )₁ ∞_T due to (H₁₉). Using (2.1), equation (1) becomes

( ( )

a t z t

∆

( )) =

∆

−

p t f x

( ) ( ( ( )))

γ

t

−

q t g x

( ) ( ( ( )))

δ

t

( ( ( )))

=

( )

( )) ( ( ( ))

( ( ( )))

f x

t

p t

q t g x

t

g x

t

γ

_δ

δ

−













( ( ( )))

(

( )

( )) ( ( ( )))

( ( ( )))

f x

t

p t

q t g x

t

g x

t

δ

_δ

γ

≤ −

−

(9)

Hence,

( ( )

a t z t

∆

( ))

∆

≤

0

for t∈[t , )₁ ∞ _T, due to (H₁₁).

This implies that,

a

(

t

)

z

∆

(

t

)

is nonincreasing for t∈[t , )₁ ∞ _T. Then there exists k(< 0)∈R and t₂∈[t , )₁ ∞_T such that

k

t

z

t

a

(

)

∆

(

)

≤

or

)

(

1 )

(

t

a

k

t

z

∆

≤

t

₂ to

t

, we obtain

.

)

(

1 )

(

)

(

2

s

a

k

t

z

t

z

t

∆

≤

−

_∫

This implies that

z

(

t

)

→

−∞

as

t

→

∞

. Which gives a contradiction to

z

(

t

)

>

0

.

Hence, the theorem is proved.

( )

H

₆ ,

( )

H

₇ ,

( )

H

₈ ,

( )

H

₁₂ ,

( )

H

₁₃ ,

( )

H

₁₉ and

( )

H

21

g

is nonincreasing hold.

Then every solution of equation (1) is either oscillatory or weakly oscillatory.

Proof: The proof is similar to that of Theorem 2.9 and hence complete the proof.

Remark 2.11: In Theorem 2.9 and Theorem 2.10, we used

f

is nonincreasing and

g

is nondecreasing respectively.

In this cases

f

and

g

are defined on R {0}− only.

3. BEHAVIOR OF SOLUTIONS IN

M

+ AND

M

−

( )

H

₆ ,

( )

H

₇ ,

( )

H

₈ ,

( )

H

₁₄ ,

( )

H

₁₅ ,

( )

H

₁₆ and

( )

H

₁₈ hold. Then every solution −

∈

M

t

x

(

)

, we have

lim

t→∞

x

(

t

)

=

0.

Proof: The argument used in the proof of Theorem 2.8, again leads that

.

=

)

(

1 limsup

(1)

)

(

∆

∞

∫

∞ →

s

f

t z

t z t

This implies that,

lim

t→∞

z

(

t

)

=

0

. But

z

(

t

)

>

x

(

t

)

for all t∈[t , )1 ∞ T.

This shows that

_lim

t→∞

x

(

t

)

=

0.

Hence complete the proof of the theorem.

( )

H

₁ ,

( )

H

₂ ,

( )

H

₃ ,

( )

H

₄ ,

( )

H

₁₁ and

( )

H

22

1 +

b

(

t

)

+

c

(

t

)

is bounded,

( )

H

23 →∞

∫

+

∫

∆

)

∆

=

∞

)

(

1 (

))(

(

)

(

limsup

1

v

s

v

a

s

q

s

p

M

s

T t

T

t for all

T

≥

t

0 hold.

Then every solution of (1) in the class

M

+ is unbounded.

Proof: Let

x

be a solution of (1) such that

x

∈

M

+. Proceeding as in the proof of Theorem 2.1, by defining

z

(

t

)

as

in (2.1), we have

z

(

t

)

>

0

,

z

∆

(

t

)

≥

0

for all

t

∈

[

t

1

,

∞

)

due to

( )

H

1 . Then from (1), we obtain (2.2). Consider the

function

1

( )

1 ( ) =

,

( ( ( )))

( )

t

a t z t

w t

s

g x

δ

t

a s

∆

(10)

we have,

t t₁

1 1

w (t) = (a(t)z (t)) s

g(x( (t))) a(s)

∆ ₋ ∆ ∆ _{∆ −}

 _δ 



∫



t t₁

1 1

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

g(x( (t))) a(s) ∆

∆  

σ σ _ ∆ _

δ



∫



f (x( (t))) = (p(t)

g(x( (t)))

γ δ

t t

t₁ t₁

1 1 1

q(t)) s (a( (t))z ( (t)) s

a(s) g(x( (t))) a(s)

∆

∆ _   _

+ ∆ − σ σ _ _ ∆ _ _

δ _ _

 

∫

( t ) t₁

1 1

(a( (t))z ( (t)) s ,

a(s) g(x( (t)))

∆ σ

∆ _   _

− σ σ _ _{∆ } _ _

δ   

∫

 implies that ₁ t t₁ 1

w (t) (M p(t) q(t)) s

a(s) ∆ _≥ ₊ _{∆ −}

∫

1 (a(t)z (t)) a(t)g(x( (t))) ∆  ₋

 _δ 

 

( t ) t₁

1 1

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

a(s) g(x( (t)))

∆ σ

∆ _   _

σ σ _{∆ } _

 _ δ _ 



∫



+

−

∆

+

∫

∆

)))

(

)

(

)

(

1 ))

(

)

(

=

1 1

t

x

g

t

z

s

a

t

q

t

p

M

t t

δ

( t ) t₁

1 1

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

a(s) g(x( (t)))

∆ σ

∆

   

σ σ ∆

   _δ 

  



∫



₁

1

1 ( )

(

( )

( ))

.

( )

( ( ( )))

t

z t

M p t

q t

s

a s

g x

δ

t

∆

≥

+

∫

∆ −

t

₁ to

t

, we get

1 1 1

1 ( )

( )

(

( )

( ))

.

( )

( ( ( )))

t s t

t t t

z

s

w t

Mp s

q s

s

a

τ

g x

δ

s

∆

≥

∫

+

∫

∆ ∆ −

∫

∆

(3.14)

As the function

( )

( ( ( )))

z t

g x

δ

t

∆

is positive for t∈[t , )₁ ∞_T, then

1

( )

lim

( ( ( )))

t t t

z

s

g x

δ

s

∆ →∞

∫

∆

exists. Assume that 1

( )

=

<

.

lim

( ( ( )))

t t t

z

s

k

g x

δ

s

∆

→∞

∫

∆

∞

Taking into account (H23), and from (3.14) we get

,

=

)

(

limsup

∞

∞ →

t

w

t

which gives a contradiction, because

w

is negative for all t∈[t , )₁ ∞ _T. Thus,

.

=

)))

(

)

(

lim

1

∞

∆

∆ ∞

→

∫

g

x

s

z

t t t

δ

(3.15) Consequently, 1 1 1

( )

1 ( )

lim

( ( ( )))

t t t t t

z

s

z

s

g x

δ

s

g x

δ

t

∆ ∆ →∞

∆ ≤

∆

∫

)).

(

)

(

)))

(

1 =

₁ 1

t

z

t

z

t

x

g

δ

−

From (3.15), we get

.

=

)

(

lim

∞

∞ →

z

t

(3.16)

Since

z

(

t

)

=

x

(

t

)

+

b

(

t

)

x

(

α

(

t

))

+

c

(

t

)

x

(

β

(

t

))

and

x

is non-negative, we have

).

(

))

(

(1

)

(

t

p

t

x

t

z

≤

+

Thus from (3.16), we have

.

=

)

(

lim

∞

∞ →

x

t

This completes the proof of the theorem.

(11)

Remark 3.3: In this paper we obtained conditions for the non-existence of solutions in the classes

M

+,

M

− and the

existence of solutions in the classes OS and WOS. It would be interesting to extend the results of this paper to the following equation

(a(t)(x(t) b(t)x( (t)) c(t)x( (t))) )+ α + β ∆ ∆+p(t)f (x( (t)))γ +q(t)g(x( (t))) = r(t)δ for t∈[t , )₀ ∞ _T

where

r

(

t

)

is a rd-continuous function.

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