Oscillation criteria of second-order non-linear dynamic equations with integro forcing term on time scales

Introduction and Preliminaries

The oscillation theory of differential equation and difference equation has been receiving a lot of attention in the past few decades. It is a very important concept in the qualitative behaviour of the solutions of both types of equations. Many researchers have taken an interest in studying the oscillation and non-oscillation criteria of the solution of both differential and difference equations [1-3]. The problem of obtaining the sufficient conditions for oscillation of both types of equations takes much time. Therefore, it is necessary to find a single way for both equations.

The time scale theory which removes this ambiguity has been first introduced by a German mathematician Stefan Hilger in his Ph.D. dissertation (1988) [4]. The significance of this theory is that it does not only avoid the dual analysis but also harmonize both continuous and discrete calculus.

Time scale is a non-empty closed subset of the real numbers (i.e., R), e.g., set of natural numbers (N) , integers (Z) , Cantor set, set of harmonic numbers etc. In this way, the results do not only relate to the set of real numbers or set of integers but also pertain to more general times scales. The three most popular examples of calculus on time scales are differential calculus (T = R), difference calculus (T = Z), and quantum calculus (T = q ^Z U { 0 }, q> 1), for more details see [5,6] and references therein. It has many applications in various fields, e.g., population dynamics, economics, neural networks, quantum physics and social science etc. [7,8].

We briefly recall some basic definitions, useful Theorems, Lemmas, assumptions and basic facts of time scales etc.

Definition 1. [5,6] For t E T , we define a forward jump operator a : T ^ T by a ( t ) := inf {s E T : s > t}. if t < a ( t ) aiul t <  sup T as well as t = a ( t ) , then t is right-scattered and right dense respectively. A backward jump operator p : T ^ T by p ( t ) := sup {s E T : s < t}, if t > p ( t ) asul t >  inf T as w<'ll as t = p ( t ) , then t is left-scattered and left dense respectively. The graininess operator p : T ^ [0 , to ) is defined by p ( t ) = a ( t ) — t.

Remark 1. We put inf ф = sup T (i.e., a ( t ) = t if T has a maximum t ), sup ф = inf T (i.e.. p ( t ) = t if T lias a, m illinium t).

Definition 2. [5,6] A function f : T ^ R is called rd-continuous provided it is continuous at all right-dense points in T and its left-sided limit exists (finite) at left-dense points in T, denoted by C_rd = C_rd (T) = C_rd (T , R) . We define some notations as follows:

+C^A(T) = { q : q ( t ) is positive rd-continuous function and q ^A( t ) E C _rd (T)} . +C^A+(T) = { q : q ( t ) , q ^A( t ) are positive rd-соиtinuous functions} .

Definition 3. [5,6] A function G : T ^ R is called an anti-derivative of g : T ^ R , provided G ^A( t ) = g ( t ) Vt E T Thicn Va,b E T, the Cauchy integral is defined by

g ( s )A( s ) = G ( b ) — G ( a ) .

Define

T ^κ

T — {l}, if T lias a left-scattered maximum l, T ^K = T , otherwise.

Definition 4. [5,6] For a function f : T ^ R a nd t E T ^K, we define f ^A ( t ), to be a number (provided it exists) with the property that given any e >  0, there exists a neighborhood A = ( t — 5,t + 5 ) Q T for some 5 >  0 such that

|^{[ f ( a (} t )) ^{— f ( r )] — f A(} t ^{)[ a (} t ) ^{— r} ] | < ^el^{a (} t ) ^{— r}| ^{Vr E} A.

Thus, we call f ^л( t ) the delta or HU ger derivative of f at t. f is also called differentiable at t.

• Theorem 1. [5,6] For the functions g, f : T ^ R a nd t E T ^K, we have the following:

• 1.    If f is differentiable at t. then f is continuous at t:

• 2.    If f is continuous at t and t is right-scattered, then f has a delta, derivative at t and

  f ^A( t ) =

  
  

• 3.    If t is right-dense, then f is differentiable at t if the limit

f ( a ( t )) — f ( t ) ;

p ⁽ t )        ’

f A( t ) = lim fyH—f M s→t    t - s

• exists and has a finite value;

• j. If f is differentiable at t. then

• 5.    If f and g both arc. differentiable at t. then the product fg : T ^ R is differentiable at t and

• 6.    If g ( t ) g ( a ( t )) = 0 гиг th g ( t ) = 0 , th en ft) is differe ntiable at t and

^f ° = ^{f ( a (} t )) = ^{f (} t ) + p ⁽ t ) ^{f A(} t );

⁽ fg ^{) л} ( t ) = f ^{л (} t ) g ⁽ t ) + f ^{( a (} t )) g ^{л (} t ) = f ⁽ t ) g ^{л (} t ) + f ^{л (} t ) g ^{( a (} t ));

⁽ f ) ^л _{( t} ) = f ^л ( t ) g ( t ) - f ( t ) g ^л ( t ) g)                g ⁽ t ) g ^{( a (} t ))

Definition 5. [5,6] A function q : T ^ R is called regressive if 1 + p ( t ) q ( t ) = 0 , Vt G T . We denote the collections of all functions h : T ^ R which are rd-continuous and regressive by R and. R ⁺ = {q G R : 1 + p ( t ) p ( t ) >  0 for all t G T }.

Definition 6. [5,6] (Time scale version of exponential function). If q G R, then we define the exponential function by eq (t, s) = exp ( ^ n^(т) (q (T)) A/) ,

∀ t ∈ T , s ∈ T ^κ ,

where nh(z) is the cylinder transformation, which is defined by nh (z) =

log (1+hz) h z,

if h = 0 , if h = 0 .

Definition 7. [5,6] If q G R, then the first order linear dynamic equation y л( t) = q (t) y (t)

(i)

is called regressive.

Theorem 2. [5,6] Suppose that (1) is regressive and fix tо G T. Then eq(.,to) is a solution of the initial value problem y л( t )= q (t) y (t). y (to) = 1

on T .

Theorem 3. [5,6] If (1) is regressive, then eq ( ..t ₀) is the only solution of (2).

Theorem 4. [5,6] If p,q G R, then

• 1.    e ₀ ( t, s ) = 1 aiul e_p ( t, t ) = 1;

• 2.    ep ⁽ a ⁽ t ) . s ) = ⁽¹ + p ⁽ t ) p ⁽ t )) ep ⁽ t. s );

• -I eep ⁽ t, s ) = ep (t,s )^;

A ep ⁽ t, ^s ) eq ⁽ t, s )     ^ep®q ⁽ t, ^s );

; ( eph Г

p ( t ) . ^ep ⁽ -.s ).

G. If q E R + , then eq ( t, s ) >  0 for all t E T •

In this paper, we study the oscillation criteria of the second order non-linear dynamic equation with integro forcing term on time scale T:

У ^ЛЛ( t ) + ^вУ ^Л( t ) = B ⁽ t ) У ⁽ t ) + H ^Л ( t, У ⁽ t ) , У   J ⁽ t ^- s ) H ⁽ s, У ⁽ s )) △ s) ,

where B, J : T ^ R are the ftmction of t, and the forcing terms H : T x R2 ^ R and H : T x R ^ R. For the alternate form of equation (3), we multiply (3) by a function r" (t). We have r" (t) y ЛЛ( t) + er" (t) У Л( t )= r" (t) B (t) y (t)+r" (t) HЛ tt,y (t), I- J (t - s) H ( s,y(s))△ s^) •

From above relation, we obtain r"(t)УЛЛ(t) + rЛ(t)УЛ(t) -rЛ(t)УЛ(t) + er"(t)УЛ(t) =

|

{Z

}

= r" ⁽ t ) В ⁽ t ) У ⁽ t ) + r" ⁽ t ) H ^Л ^ ,У ⁽ t ) , j J ⁽ t — s ) H ⁽ s^,y ⁽ s )) △ s) ,

which is equivalent to

( r ( t ) У ^Л( t ))^Л + ( er" ( t ) - r ^Л( t )) У ^Л( t ) ==

= B ( t ) r" ( t ) y ( t ) + r" ( t ) H ^Л tt,y ( t ) , I J ( t - s ) H ( s,y ( s )) △ s) .

We study oscillation criteria of (4). A non-trivial у(t) such that у(t) E CЛ([ty, to)t), r(t)yЛ(t) E CЛ([ty, to)t) for certain ty > 11 and satisfying (4) for ty < t is called 11011-oscillatory if it is eventually positive or eventually negative, otherwise it is called oscillatory. In other words, it is said to be oscillatory if it has an arbitrarily large number of zeros, i.e., there exists a sequence {s„} such that lim sn = to as well as y(sn) = 0, Vn. A dynamic n→∞ equation is called oscillatory if every solution is oscillatory, and non-oscillatory otherwise. Our attention is restricted to those solutions of (4) which exist on some [ty, to)t and satisfy sup{ly(t)| : t > t*} > 0 for any ty < t*.

In the past few years, there have been many research activities concerning the oscillation of solutions of various forced second-order dynamic equations on time scales. The oscillation criteria of the second order linear and non-linear dynamic equations on time scales have been studied by many researchers, and we have plenty of essential papers, articles etc. [9-14]. In particular: In (2003), Erbe, et. al. [13] considered linear damped dynamic equation x ЛЛ( t) + p (t) x Л( t) + q (t) x (t) = 0, and non-linear equation x ЛЛ( t) + p (t) x Л "(t) + q (t)(fox")(t) = 0,

to establish the oscillation criteria. In (2007) , Saker, et. al. [14] gerneralized (5) as follows:

(r(t) x л( t ))Л + p(t) x л °(t) + q(t)(fox°)(t) = 0,(6)

and studied an oscillation criteria. Finally, in (2008), Erbe, et. al. [15] have considered a non-linear damped delay dynamic equation by introducing a constant term в, where в is a quotient of odd positive integers and studied the sufficient conditions for oscillation:

(r(t)(x Л( t))в)Л + p(t)(xл °(t))e + q(t) f(x(T(t))) = 0,(7)

which extend and improve the results [13,14].

In (2004) , Bohner, et. al. [10] considered a second-order perturbed dynamic equation:

(r(t)(xл(t))Y)л + F(t, x"(t)) = G(t, x(t), xл(t)),(8)

where 7 is a positive 0dd integer. In (2006) , Agarwal, et. al. [9] modified equation (8)

(r(t)(xл(t))Y)Л + F(t, x(t)) = G(t, x(t), xл(t)), |---------{z} and both have established the sufficient conditions for oscillation. In (2010), Chen, et. al. [11] considered a dynamic equation with damping on time scale

((x л( t))Y )л + p (t)(x л( t))Y + q (t) f (x (a (t)) = 0, and studied the oscillation criteria as well as established the Kamenev type and the Philos-type oscillation criteria of it. Throughout this paper we denote [a, то) ПT = [a, to)t, where sup(T) = to.

This paper is organized as follows: In this section, we establish some necessary Lemmas. In the next section, we use Riccati transformation technique to establish the sufficient conditions for oscillation of (4) and also establish the Kamenev type oscillation criteria. Finally, we consider another second order dynamic equation with deviating argument and establish the same for oscillation.

It will be convenient to make the following notations:

Q ( t ) =

er° ( t ) -

r ( t )

r ^л( t )

-------, D ( t, s ) = e    Q ( t )   ( t, s ) , M ( t ) = ( p ( t ) - B ( t )) r ( t ) ,

1 - m ( t ) Q ( t )

B i( t ) =

( 5 ^л( t ) - 5° ( t ) Q ( t )Э( t ))

5 ft t )

¹    ( 5° ( t )

-

5 ( t )

D° ( t,s * ) 5 ( t )

D ( t,s * ) 5° ( t ) Q ⁽ t ) ■

5° ( t Ж t )                   D° ( t,s * ) 5 ( t )    \

, ²⁽ )      5 ²( t ) r ( t ) ,     2⁽ )        r ( t )( 5° ( t ))² D ( t,s* ) ’

To establish our results we use the following assumptions:

(U1) B : [ 1 1 , to ) t ^ R is a, negative rd-continuous function and p : [ 1 1 , to ) _t ^ R is a positive rd-continuous function;

(U₂) B : [ 1 1 , to ) t ^ R is an rd-continu<)us function and p : [ 1 1 , to ) _t ^ R is a positive rd-continuous function such that p ( t ) — B ( t ) >  0;

(U₃) Q ^: [ 1 1 , to ) t ^ Ris a positive rd-continuous function such that 1 — ц ( t ) Q ( t ) >  0 , and r ( t ) G + C ^A rd (T);

(U4) H : T x R2 is a Д-derivative finction (w.r.t first variable) such that n ( t ) H ^A( t,n ( t ) ,( ( t )) <  0 ,   Vn ( t ) G R v { 0 }, V( ( t ) G R , Vt G T;

(U5) |H ^A( t,n ( t ) ,( ( t )) |> P ( t ) In ( t ) I,   Vn ( t ) G R v { 0 },  V( ( t ) G R , t G T;

∞

(Ue) / t1

_r ( t ) e-Q ⁽ t )⁽ t, ^{s )д} t

∞.

Now we establish two Lemmas, which will be used in the next section.

Lemma 1. Let y(t) be an eventually positive solution of (4). Assume that (U3) — (U6) and either (U1) or (U2) holds. Then there, exists s* > 11 stick that y (t) > 0, y A( t) > 0 ат id (D (t, s*) r (t) y A( t ))A < 0 Vs* < t.              (9)

Proof. Since y ( t ) is an eventually posith^-e solution of (4). Take 12 G [ 1 1 , to ) _t such that y ( t ) >  0 on [ 12, to ) t . In view of eqiration (4) and ( U 1) — ( U 5) , we have

( r ( t ) y ^A( t )) ^a + Q ( t ) r ( t ) y ^A( t ) < — ( p ( t ) — B ( t )) A ( t ) y ( t ) = —M ( t ) y ( t ) <  0       (10)

011 [12, to)t. Now. by dividing equation (10) by a, 11011-negative function 1 — ц(t)Q(t), we get

( r W y ^ t )^

^{1 —} Ц ⁽ t ) ^{Q (} t )

+

Q ( t )

^{1 —} ц ⁽ t ) ^{Q (} t )

r ( t ) y ^A ( t ) <  0

or

( D ^{( t,s}* ) r ⁽ t ) y ^A( t ))^A <  ⁰ .

Then D ( t, s* ) r ( t ) y ^A( t ) is an eventually decreasing function and thus it is eventually of one sign, i.e. it is either eventually positive or eventually negative. We assert that D ( t, s* ) r ( t ) y ^A( t ) is eventually non-negative. Suppose it is eventually negative, then there exist 1 3 > 12 and a, coiistant K 1 <  0 such that

D ( t, s* ) r ( t ) y ^A( t ) < K 1 <  0 .

Integrating the above relation from 1 ₃ 10 t , we obtain

y(t) < y(t 3) + K1 [ z W1--- Д x = t 13 r(x)D(x,s*)

= y ( 1 3) + K1    -—-e-Q ( x )( x, s* )Д x ^ —to

1 3 ^{r (} x )

as t ^ to,

which is a contradiction since y ( t ) >  0 for all 12 < t. Hence, D ( t,s* ) r ( t ) y ^A( t ) is eventually 11011-negative, i.e.. y ^A( t ) is eventually positive function or there exists s* G T such that y ^A( t ) >  0 for all t G [ s*, to )_t.

Thus, we have

y ( t ) >  0 , y ^A( t ) >  0 and  ( D ( t, s* ) r ( t ) y ^A( t ))^A <  0 ,    Vs* < t.

Lemma 2. Let Lemma 1 hold. Father more, assume r ^Л( t ) >  0 • Then ^{a (} t ’ <% < ¹ ,

(П)

tuhcrc Э ⁽ t ) = - s ^- ^ ( t ) •

Proof. From (9), we obtain

y ( t ) >  0 , y ^Л( t ) >  0 , y ^ЛЛ( t ) <  0 •

Also from (12), we have

y ⁽ t ) = y ⁽ t ) ^— y ⁽ s* ) =     y ^Л( П ^)A П > y ^Л( t ⁾⁽ t ^— s* ) •

From (13) and y^CT ( t ) = y ( t ) + ц ( t ) y ^Л( t ) , we get

9( t ) < -yt- <  1 • y^CT ( t ) "

□

• 1.    Main Results

In this section, we give some new oscillation criteria for equation (4).

Theorem 5. Assume that (U3) — (U6) and either (U1) or (U2) hold. Furthermore, there exists 6(t) G +CЛ(T)fno^ necessary rd-continuity of 5Л) such that for all sufficiently large s∗, limsup [ [M(()6(() — A^. A( = TO tH^   s*                4 A 2( 4)

or

limsup     M (() 6 (() — t→∞   s∗

r ( 4 ) ( 6 ^Л( 4 ) - 6 ( 4 ) Q ( 4 ))² 1

4 6 ( 4 )           ]

-

A 4 = to.

Then, Proof.

every solution of equation (4) is oscillatory.

Suppose to the contrary that y(t) is a non-oscillatory solution of equation (4). We may assume without loss of generality that y(t) is eventually positive. By Lemma 1, we have

y ( t ) >  0 , y ^Л( t ) >  0 and ( D ( t, s* ) r ( t ) y ^Л( t ))^Л <  0    Vs* < t.

Now define a function w ( t ) by the Riccati substitution

W ( t )= 6 ( t )           , t > s* > t_b

y ( t )                       ¹

Then, w ( t ) >  0 and

W Л( t ) = ( 6 ( t ) rti^ )^Л = ( r ( t ) y Л( t )) ' ( ⁶-l )^Л + ( r ( t ) y Л( t ))Л ( tfl ) • y ( t )                                  y ( t )                              y ( t )

From (17). w, ha^ w„ ₍ t ) ,        , ₍ t ) y ^Л ₍ t ))                 ⁽ 6 ( t ))

w ⁽ t ) = ha I ^{6 (} t ^{1 —} ""WF) ^{+ ( r (} t ) y ⁽ t ⁾⁾ l y ( t )) •

From (10) and (17), we obtain wЛ(t) < w"() FЛ(t) — ^SyM) — (M(t)y(t) + Q(t)(r(t)yЛ(t))) №) (t)                   y (t)                                                       y (t)

6 ^Л( t ) w" ( t )     w" ( t ) w ( t )

= —M ( t ) 6 ( t ) — Q ( t ) w ( t ) + 6^" ( t )

-

r ( t ) 6^" ( t ) ^.

Since D ( t, s* ) r ( t ) y ^Л( t ) is eventually decreasing function, then

D" ( t, s* )( r ( t ) y ^Л( t )) " < D ( t, s* ) r ( t ) y ^Л( t ) ,   as t < a ( t ) .

From (17), we obtain

D" ⁽ t, s* ) y" ⁽ t ) ^{6 (} t ) w" ⁽ t ) ⁽⁾ -     D ( t,s* ) y ( t ) 6" ( t )

.

Also, since y ( t ) is eventually increasing function, then we get

y ( t ) < y" ⁽ t ) •

From

(19) and (20), we have

D" ( t,s* ) 6 ( t ) w" ( t ) w ⁽ t ’ >    D ( t,s- ) 6" ( t )

or

/ D ^{( t,s}* ) ^{6 (} t ) w" ⁽ t )

^-Q ( t ) w ( t ) <       d _{( г} ^* ) 6_{a (} t )    Q ( t ) •

From

(19) and (21), we obtain

w ^Л( t ) < —M ( t ) 6 ( t ) —

( w" ( t ))² D" ( t, s* ) 6 ( t ) - r ( t ) D ( t,s* )( 6" ( t ))²

^D" ⁽ t,s* ) ^{6 (} t ) w" ⁽ t )          ^{6 Л(} t ) w" ⁽ t ) _

D ( t, s* ) 6" ( t )    Q ⁽ t ⁾⁺     6" ( t )

■ = —M ( t ) 6 ( t ) + A i( t ) w" ( t ) - ( w" ( t ))² A 2( t ) =

—M ( t ) 6 ( t ) —

w" ( t ) VM)

-

A 1 ( t )

2    A 2 ( t )

A ²1( t )

⁺ 4 A 2( t ) •

Hence wЛ (t) < —

M ( t ) 6 ( t ) —

A 1( t ) "

4 A ₂( t )_ '

Integrating (23) from s* tо t , we obtain

t

J, M ( t ) 6 ( t )

-

AA ^ t ) ^{д t}< w ⁽ s* ) - w ⁽ t ) < w ⁽ s* ) < ^’

for all large t . Which is a contradiction because of relation (15). Hence, the proof is complete due to asumption (15).

Now we prove the result using (16). From equations (19), (20) and ID ( t, s* ) > D ( t, s* ) , we obtain

, .      5 (t) w7 (t).

w(t> >•

Similarly, from (23) and (25), we obtain

[M(5)5(5) _ r(5)(5чo-5(5)Q(5A] .-w4(t)•

Integrating equation (16) from s* tо t , we get

Г* [            r ( 5 ) ( 5 ^Д( 5 ) - 5 ( 5 ) Q ( 5 >f1л<< И

J* ^M ( ⁵ ) ⁵ ( ⁵ )--'---4^-------— ^{A 5}< ^w ( ^s ) ^{- w} ( t ) < ^w ( ^s ) < TO (27)

for sufficiently large t, which is a contradiction as (16) holds. Hence, the proof is complete due to assumption (16). Thus, the whole proof is complete.

□

As an immediate consequence of Theorem (5), we have the following corollaries for different values of 5 ( t ).

For 5 ( t ) = constant (say, C >  0), t > 1 1, we have the following result.

Corollary 1. Assume that (U3) — (U5) and either (U1) or (U2) hold. Furthermore, assume r ft \w.r(5)D'(<’S*)Q2(5)1д< hmsup j^ M (5)--^^   Д 5 = TO or r     ft \wtr(5)Q2(5)U limsup     M(5)------- A 5 = to.

t^^   s*              ⁴

Then, every solution of equation (4) is oscillatory.

For 5 ( t ) = t ² , we have following result.

Corollary 2. Assume that ( U 3) — ( U ₅) and either ( U 1) or ( U ₂) hold. Furthermore, assume

t

lim sup t→∞

s ∗

M ( ( ) ( ² —

( ^ ± £ O ( ^ ( € ))²

-

D^{7 (} <,s* ) < ²

D ( e,s* )( a ( ( )) ² Q ⁽ ^ J

∗

4 D⁷ ( (,s* ) ( ²

r ( ( )( a ( ( )) ⁴ D ( (,s* )

^A ( = TO

or llmsup ft M(^)(2 — r<5X5 + a<:) — 52Q(S»21 A^ = to. t^^   s*                        45

Then, every solution of equation (4) is oscillatory.

We add one more condition, i.e. r ^л( t ) >  0 0r r ( t ) G ±C⁴±(T) in Theorem (5) to obtain some new oscillation criteria of equation (4).

A f = to.

Then, every solution of equation (4) is oscillatory.

Proof. In view of Theorem (5) and equation (17), we obtain

( r ⁽ t ) y ^A( t ) )         ( ^{r (} t ’ У ^Л( t ’ V

w ⁽ t ’= ^{5 (} t^{5 (} t ’I •

By using equation (10), Lemma 2 and after manipulation, we obtain wA(t) < — M(t)5'(t)Э(t) + Bi(t)w(t) — B2(t)w2(t), which is equivalent to wA(t) < — M(t)5'(t)3(t) — w(t)VBW — ffif + B^.

2 v B 2( t )      ^{4 B} 2⁽ t ’

Integrating the above equation from s* tо t , we obtain

[t к ( f ) 5' ( f )3( f ) — B^ s*                        ^{4 B} 2 ^{( f} ’

A f < w ( s* ) — w ( t ) < w ( s* ) <  to,

for all large t , which is a contradiction due to (28). Hence, the proof is complete.

□

Now as a special case if 5 ( t ) is positive constant, we obtain a result:

Corollary 3. Assume that ( U 3) — ( U 6) and either ( U 1) or ( U2 ) hold. Moreover, r ^A( t ) >  0 or r ( t ) G ⁺C d+ (T) and.

limsup Г3( t ) M ( f ) — r ^{( f} ’ Q ^{2 ( f} ’ A f = to.                    (30)

t^^ s*                    ⁴

Then, every solution of equation (4) is oscillatory.

Now we establish Kamenev-type oscillation criteria for (4). We need the following result of [16] (( t — s ) ^m )^{A s}< —m ( t — a ( s )) ^{m- 1} <  0 fc>r m >  1 aiid a ( s ) < t to estalolish our results.

Theorem 7. Assume that (U3) — (U6) and either (U1) or (U2) holds. Furthermore, there exists 5(t) G +Cda(T’f/wt necessary 1 cl-continuity of 5^) such that. for N > 1 and for all. sufficiently large s*, lim31P t^ j*l t— f ’" [M (f ’5 (f ’— ® ]A f = to 1311

or

М‘„                   r ( £ ) ( 5 A £ ) - ⁵ ( £ ) Q ( £ ))21 А г

^«Up с L ( t^{- £} > M ( ^£ ) ^{5 ( £ )}--4 5£ ------- ^{Д £} =

Then, every solution of (4) is oscillatory.

Proof. In view of Theorem 5, from (23), we have wA(t) < - [M(t)5(t) - AM].

Thus

Г ( t - £ ) ^S w ^A ( £ )Д £ <- Л t - £ ) ^S M ( £ ) 5 ( £ ) - AP) Д £.         (33)

J 8*                                  J 8*                                      2 ^{( 5} ) J

We know

I*t ( t - £ ) ^S w ^A ( £ )Д £ = - ( t - s’ ) ^S w ( s’ ) - Г (( t - £ ) ^S ) ^{A 5} w ( a ( £ ))Д £,

and using the remark (3 . 3) in [16], we have

(( t - £ ) * ) ^{A 5}< -N ( t - a ( £)) ^- 1 ¹ <  0 , a ( £ ) < t aiid N >  1 .                (35)

From equations (33), (34) and (35), we obtain

- / * ( t - £ ) * M ( £ ) 5 ( £ ) - Ж1 a £ >  / * ( t - £ ) *w ^A ( £ )Д £ =

- ( t - s ^- ) ^K w ( s’ ) -

- j* ((t- £)* )a 5 w(a(£))a £ >- (t- s’ ) Kw ( s’ ),

j₌\ ( t - £ ) ^S [ M ( £ ) 5 ( £ ) - A M ] Д £ <  ( t - s’ ) ^S w ( s’ ) . Thus

^Hm . ™p ^ / * ⁽ t -^£ ) ^S M ^{( £} ) ^{5 ( £} ) ^- 4 A B))

Д £ <  lim sup

→∞

(1 - it

w ( s’ ) < TO.

Which contradicts assumption (31). To prove the result using (32), we apply the above process for equation (26), then we get a contradiction since we have (32). Hence, the proof is complete.

□ Theorem 8. Assume that (U3) - (U6) and either (U1) or (U2) holds. Moreover, either rA(t) > 0 оr r(t) G +CA+(T) and. the.re. exists 5(t) G +CA(T) fno^ necessary rd-continuity of 5A) such that. for N > 1 and for all syfficiently large s’, limsup 1 /'(t - £)* M(£)6”(£)Э(£) - Bh т>«з t Js*                                4B2(£)

Д £ = to.

Then, every solution of equation (4) is oscillatory.

Proof of the above Theorem is same as proof of Theorem 8 .

Remark 2. The result of Theorem 8 and Theorem 9 holds for any 5 ( t ) G + C^A(T) (not necessary rd-continuity of 5 ^A). Thus, by these Theorems, we can immediately obtain corollaries with different choice of 5 ( t ).

Let us consider a second order non-linear dynamic equation with deviating argument on an arbitrary time scale T :

y ^AA( t ) + ву ^A( t ) = B ( t ) y ( t ) + w ^A( t,y ( t ) ,y ( w 1 ( t,y ( t)))),

where в >  0 , w 1( t, у ( t )) = b 1( t, у ( b ₂( t, ••• ,y ( bm ₀ ( t, у ( t ))) ••• ))) , B is a, function of t, and the forcing terms W ar id bi,i = 1 , 2 , • • • , m 0 .

Remark 3. If we replace the integro forcing term H

(t,y ⁽ t ) , j J ⁽ t - s ) H ⁽ s,y ^{( s} )) △ ^s^)

(in 3) by the deviating argument W ( t, у ( t ) ,y ( w 1( t,у ( t )))) (in 37) , then in the similar

manner we can obtain oscillation criteria of (37). Thus, all the above Theorems and corollaries can be achieved for the second order dynamic equation (37), i.e., all the results will remain the same for it.