Skip to main content

Oscillatory behavior of solutions of third-order delay and advanced dynamic equations

Abstract

In this paper, we consider oscillation criteria for certain third-order delay and advanced dynamic equations on unbounded time scales. A time scale is a nonempty closed subset of the real numbers. Examples will be given to illustrate some of the results.

MSC:34N05, 39A10, 39A21.

1 Introduction

In this paper, we are concerned with oscillation criteria for solutions of the third-order delay and advanced dynamic equations

( 1 a ( t ) ( x Δ ( t ) ) α ) Δ Δ +q(t)f ( x [ g ( t ) ] ) =0
(1.1)

and

( 1 a ( t ) ( x Δ ( t ) ) α ) Δ Δ =q(t)f ( x [ g ( t ) ] ) +p(t)h ( x [ k ( t ) ] )
(1.2)

on [ t 0 , ) T such that t 0 T and t 0 0, where α is the ratio of two positive odd integers, a,p,q C rd ( [ t 0 , ) T ,(0,)) with

a 1 α (s)Δs=,
(1.3)

and g,k C rd (T,T) are nondecreasing functions such that g(t)<t<k(t) and lim t g(t)=. We also assume that f,hC(R,R) such that xf(x)>0, xh(x)>0, f(x) and h(x) are nondecreasing for x0 satisfying

f(xy)f(xy)f(x)f(y)if xy>0
(1.4)

and

h(xy)h(xy)h(x)h(y)if xy>0.
(1.5)

By a solution of (1.1) (or (1.2)) we mean a function x C rd 1 ( [ T x , ) T ,R), T x t 0 , which has the property that (1/α) ( x Δ ) α C rd 2 ( [ T x , ) T ,R) and satisfies (1.1) (or (1.2)) for all large t T x . A nontrivial solution is said to be nonoscillatory if it is eventually positive or eventually negative and it is oscillatory otherwise. A dynamic equation is said to be oscillatory if all its solutions are oscillatory.

Since we are interested in the oscillatory behavior of solutions of (1.1) and (1.2) near infinity, we assume throughout this paper that our time scale is unbounded above. An excellent introduction of time scales calculus can be found in the books by Bohner and Peterson [1] and [2].

The purpose of this paper is to extend the oscillation results given in [3] and [4]. Oscillation criteria for third-order dynamic equations have recently been studied in [58]. Other papers related with oscillation of higher-order dynamic equations can be found in [9, 10]. Well-known books concerning the oscillation theory are [11, 12].

For simplification, we define the following operators:

L 0 x(t)=x(t), L 1 x(t)= 1 a ( t ) ( L 0 Δ x ( t ) ) α , L 2 x(t)= L 1 Δ x(t), L 3 x(t)= L 2 Δ x(t).

Thus (1.1) and (1.2) can be written as

L 3 x(t)+q(t)f ( x [ g ( t ) ] ) =0

and

L 3 x(t)=q(t)f ( x [ g ( t ) ] ) +p(t)h ( x [ k ( t ) ] ) ,

respectively. In what follows we use the following notation. For (t,s,T) [ s , ) T × [ T , ) T × [ t 0 , ) T

A(t,s)= s t a 1 α (u) ( u s ) 1 α Δu
(1.6)

and

B(t,s)= s t a 1 α (u) ( t u ) 1 α Δu.
(1.7)

2 Oscillation criteria for (1.1)

In this section, we investigate some oscillation criteria for solutions of the third-order delay equation (1.1).

Theorem 2.1 Let (1.3) and (1.4) hold and assume that

f ( u 1 α ) u c>0
(2.1)

for u0 and a constant c. If for t [ t 0 , ) T

lim sup t g ( t ) t q(s)f ( A ( g ( s ) , t 0 ) ) Δs> 1 c
(2.2)

and

lim sup t g ( t ) t q(s)f ( B ( g ( t ) , g ( s ) ) ) Δs> 1 c ,
(2.3)

where A and B are defined as in (1.6) and (1.7), respectively, then (1.1) is oscillatory.

Proof Let x be a nonoscillatory solution of (1.1) and assume that without loss of generality x(t)>0 for t [ t 0 , ) T and so L 3 x(t)0 eventually for t [ t 0 , ) T . Therefore, there exists a t 1 [ t 0 , ) T such that L 1 x(t) and L 2 x(t) are of one sign for all t [ t 1 , ) T . We now distinguish the following two cases:

  1. (I)

    L 1 x(t)>0 and L 2 x(t)>0 eventually;

  2. (II)

    L 1 x(t)<0 and L 2 x(t)>0 eventually.

We now start with the first case.

  1. (I)

    Assume that L 1 x(t)>0 and L 2 x(t)>0 for t [ t 1 , ) T . Then we have

    t 1 t L 2 x(s)Δs= L 1 x(t) L 1 x( t 1 ) L 1 x(t),t [ t 1 , ) T .

Since L 2 x is nonincreasing, we have

(t t 1 ) L 2 x(t) L 1 x(t),t [ t 1 , ) T .

This implies that

x Δ (t) a 1 α (t) ( t t 1 ) 1 α L 2 1 α x(t),t [ t 1 , ) T .

Integrating the above inequality from t 1 to t, we have

x(t) L 2 1 α x(t) t 1 t a 1 α (s) ( s t 1 ) 1 α Δs=A(t, t 1 ) L 2 1 α x(t),t [ t 1 , ) T ,

where A is defined as in (1.6). Hence there exists t 2 [ t 1 , ) T such that

x [ g ( t ) ] A ( g ( t ) , t 1 ) L 2 1 α x [ g ( t ) ] ,t [ t 2 , ) T .
(2.4)

From (2.4) and (1.4) in (1.1) we obtain

L 3 x(t)q(t)f ( A ( g ( t ) , t 1 ) ) f ( L 2 1 α x [ g ( t ) ] ) ,t [ t 2 , ) T .

Integrating the above inequality from g(t) to t, we obtain

L 2 x [ g ( t ) ] g ( t ) t q(s)f ( A ( g ( s ) , t 1 ) ) f ( L 2 1 α x [ g ( s ) ] ) Δs,t [ t 2 , ) T

or

L 2 x [ g ( t ) ] f ( L 2 1 α x [ g ( t ) ] ) g ( t ) t q(s)f ( A ( g ( s ) , t 1 ) ) Δs,t [ t 2 , ) T .

Dividing both sides of the above inequality by f( L 2 1 α x[g(t)]), taking the lim sup of both sides as t and using (2.2), we obtain a contradiction to (2.1).

  1. (II)

    Assume that L 1 x(t)<0, L 2 x(t)>0 for t [ t 1 , ) T . Then we have

    s t L 2 x(u)Δu= L 1 x(t) L 1 x(s) L 1 x(s),(t,s) [ s , ) T × [ t 1 , ) T .

Since L 2 x is nonincreasing, we have

L 1 x(s)(ts) L 2 x(t),(t,s) [ s , ) T × [ t 1 , ) T

or

x Δ (s) a 1 α (s) ( t s ) 1 α L 2 1 α x(t),(t,s) [ s , ) T × [ t 1 , ) T .

Integrating the above inequality from s to t we obtain

x(s)B(t,s) L 2 1 α x(t),(t,s) [ s , ) T × [ t 1 , ) T ,
(2.5)

where B is defined as in (1.7). Then there exists t 2 [ t 1 , ) T such that

x [ g ( s ) ] B ( g ( t ) , g ( s ) ) L 2 1 α x [ g ( t ) ] , ( g ( t ) , g ( s ) ) [ g ( s ) , ) T × [ t 2 , ) T .
(2.6)

Integrating (1.1) from g(t) to t and using (1.4) along with the above inequality we have

L 2 x [ g ( t ) ] g ( t ) t q(s)f ( x [ g ( s ) ] ) Δsf ( L 2 1 α x [ g ( t ) ] ) g ( t ) t q(s)f ( B ( g ( t ) , g ( s ) ) ) Δs.

Dividing the above inequality by f( L 2 1 α x[g(t)]), taking the lim sup of both sides of the above inequality as t and using (2.3), we obtain a contradiction to (2.1). □

For the bounded solutions of (1.1) we have the following, which is immediate from Theorem 2.1.

Corollary 2.2 In addition to (1.3) and (1.4), assume that (2.1) and (2.3) hold. Then all bounded solutions of (1.1) are oscillatory.

Now we prove the following result.

Theorem 2.3 Let (1.3) and (1.4) hold and assume that

lim u 0 u f ( u 1 α ) =0.

If

lim sup t g ( t ) t q(s)f ( B ( g ( t ) , g ( s ) ) ) Δs>0,

where B is defined as in (1.7), then all bounded solutions of (1.1) are oscillatory.

Proof Let x be a nonoscillatory bounded solution of (1.1). Without loss of generality assume that x is positive. Since x satisfies Case (II) in the proof of Theorem 2.1, we have (2.6). Integrating (1.1) from g(t) to t and using (1.4) along with (2.6) yields

L 2 x [ g ( t ) ] g ( t ) t q(s)f ( x [ g ( s ) ] ) Δsf ( L 2 1 α x [ g ( t ) ] ) g ( t ) t q(s)f ( B ( g ( t ) , g ( s ) ) ) Δs.

By dividing the above inequality by f( L 2 1 α x[g(t)]) and taking the lim sup of both sides of the resulting inequality as t, we obtain a contradiction. The proof is now complete. □

We now consider a special case of (1.1) of the form

( 1 a ( t ) ( x Δ ( t ) ) α ) Δ Δ +q(t) x β [ g ( t ) ] =0,
(2.7)

where β is a ratio of two odd positive integers, and we obtain some oscillatory criteria.

Theorem 2.4 Let αβ and assume that

q(s) A β ( g ( s ) , t 0 ) Δs=
(2.8)

and

q(s) B β ( g ( s ) , t 0 ) Δs=,
(2.9)

where A and B are defined as in (1.6) and (1.7), respectively. Then (2.7) is oscillatory.

Proof Let x be a nonoscillatory solution of (2.7). Without loss of generality, assume that x is positive. We consider two cases as we did in the proof of Theorem 2.1.

  1. (I)

    Assume that L 1 x(t)>0 and L 2 x(t)>0 for t t 1 . Using (2.4) in (2.7), we have

    L 2 Δ x(t)=q(t) x β [ g ( t ) ] q(t) A β ( g ( t ) , t 1 ) L 2 β α x [ g ( t ) ] ,g(t) [ t 2 , ) T .

Integrating the above inequality from g(t) to u and letting u, we obtain

L 2 1 β α x [ g ( t ) ] g ( t ) q(s) A β ( g ( s ) , t 1 ) Δs.

Since αβ and the right hand side of the above inequality is infinity by (2.8), we obtain a contradiction to the facts that L 2 x is positive and nondecreasing.

  1. (II)

    Assume that L 1 x(t)<0, L 2 x(t)>0 for t t 1 . Then we have (2.6). Using (2.6) in (2.7), for (g(t),g(s)) [ g ( s ) , ) T × [ t 1 , ) T , we have

    L 2 Δ x [ g ( t ) ] =q(t) x β [ g ( t ) ] q(t) B β ( g ( t ) , g ( s ) ) L 2 1 α x [ g ( t ) ] .

Integrating the above inequality from g(t) to u and letting u, we obtain

L 2 1 β α x [ g ( t ) ] g ( t ) q(s) B β ( g ( s ) , t 1 ) Δs.

Since αβ and the right hand side of the above inequality is infinity by (2.9), we obtain a contradiction to the facts that L 2 x is positive and nondecreasing. □

3 Oscillation criteria for (1.2)

In this section, we investigate some oscillation criteria for solutions of the third-order delay equation (1.2).

Theorem 3.1 Let (1.3)-(1.5) and (2.1) hold. Also assume that

h ( u 1 α ) u b>0
(3.1)

for u0 and a constant b. If for t [ t 0 , ) T

lim sup t t k ( t ) q(s)h ( A ( k ( s ) , k ( t ) ) ) Δs> 1 b ,
(3.2)

and

lim sup t g ( t ) t q(s)f ( B ( g ( t ) , t 0 ) ) Δs> 1 c ,
(3.3)

where A and B are defined as in (1.6) and (1.7), respectively, then (1.2) is oscillatory.

Proof Let x be an eventually positive solution of (1.2). Then L 3 x(t)0 eventually and so L 1 x(t) and L 2 x(t) are eventually of one sign. We now distinguish the following two cases:

  1. (I)

    L 1 x(t)>0 and L 2 x(t)>0 eventually;

  2. (II)

    L 1 x(t)>0 and L 2 x(t)<0 eventually.

  1. (I)

    Assume that L 1 x(t)>0 and L 2 x(t)>0 for t [ t 1 , ) T . Then we have

    s t L 2 x(τ)Δτ= L 1 x(t) L 1 x(s) L 1 x(t),(t,s) [ s , ) T × [ t 1 , ) T .

Since L 2 x is nondecreasing, we have

L 1 x(t)(ts) L 2 x(s),(t,s) [ s , ) T × [ t 1 , ) T .

This implies that

x Δ (t) a 1 α (t) ( t s ) 1 α L 2 1 α x(s),(t,s) [ s , ) T × [ t 1 , ) T .

Integrating both sides of the above inequality from s to t, we have

x(t) s t a 1 α (u) ( u s ) 1 α L 2 1 α x(s)Δu,(t,s) [ s , ) T × [ t 1 , ) T

or

x(t)A(t,s) L 2 1 α x(s),(t,s) [ s , ) T × [ t 1 , ) T ,

where A is defined as in (1.6). Then

x [ k ( τ ) ] A ( k ( τ ) , k ( t ) ) L 2 1 α x [ k ( t ) ] , ( k ( t ) , τ , t ) [ τ , ) T × [ t , ) T × [ t 1 , ) T .

Using the above inequality and (1.5) in (1.2), we have

L 3 x(τ)p(τ)h ( x [ k ( τ ) ] ) p(τ)h ( A ( k ( τ ) , k ( t ) ) ) h ( L 2 1 α x [ k ( t ) ] ) .

Integrating both sides of the above inequality from t to k(t), we get

L 2 x [ k ( t ) ] h ( L 2 1 α x [ k ( t ) ] ) t k ( t ) p(τ)h ( A ( k ( τ ) , k ( t ) ) ) Δτ.

Taking the lim sup of both sides of the above inequality as t, we obtain a contradiction to (3.1).

  1. (II)

    Assume that L 1 x(t)>0 and L 2 x(t)<0 for t [ t 2 , ) T . Then we have

    s t L 2 x(u)Δu= L 1 x(t) L 1 x(s),(t,s) [ s , ) T × [ t 1 , ) T .

Since L 2 x is nondecreasing, we have

x Δ (s) a 1 α (s) ( t s ) 1 α L 2 1 α x(t),(t,s) [ s , ) T × [ t 1 , ) T .

Integrating both sides of the above inequality from t 1 to t, we have

x(t)B(t, t 1 ) L 2 1 α x(t),t [ t 1 , ) T ,

where B is defined as in (1.7). This implies that there exists a t 2 [ t 1 , ) T such that

x [ g ( t ) ] B ( g ( t ) , t 1 ) L 2 1 α x [ g ( t ) ] ,g(t) [ t 1 , ) T .
(3.4)

Using the above inequality and (1.4) in (1.2), we have

L 3 x(t)q(t)f ( x [ g ( t ) ] ) q(t)f ( B ( g ( t ) , t 1 ) ) f ( L 2 1 α x [ g ( t ) ] ) .

Integrating both sides of the above inequality from g(t) to t, we find

L 2 x [ g ( t ) ] f ( L 2 1 α x [ g ( t ) ] ) g ( t ) t q(s)f ( B ( g ( s ) , t 1 ) ) Δs.

Taking the lim sup of both sides of the above inequality as t, we obtain a contradiction to (2.1). The proof is complete. □

Theorem 3.2 Assume that

h 1 α ( u ) u b 1 >0,
(3.5)

and

f 1 α ( u ) u c 1 >0,
(3.6)

for u0 and constants b 1 and c 1 . If

lim sup t t k ( t ) ( a ( s ) t s t u p ( r ) Δ r Δ u ) 1 α Δs> 1 b 1
(3.7)

and

lim sup t B ( g ( t ) , t 0 ) ( t q ( s ) Δ s ) 1 α > 1 c 1 ,
(3.8)

where B is defined as in (1.7), respectively, then (1.2) is oscillatory.

Proof Let x be an eventually positive solution of (1.2). As in the proof of Theorem 3.1 we have two cases to consider.

  1. (I)

    Assume that L 1 x(t)>0 and L 2 x(t)>0 for t [ t 1 , ) T . Integrating (1.2) from t to s 1 yields

    L 2 x( s 1 )h ( x [ k ( t ) ] ) t s 1 p(s)Δs.

Integrating the above inequality from t to s 2 [ s 1 , ) T gives

x Δ ( s 2 ) h 1 α ( x [ k ( t ) ] ) [ a ( s 2 ) ] 1 α ( t s 2 t s 1 p ( s ) Δ s Δ s 1 ) 1 α .

Again integrating the above inequality from t to k(t) we find

x ( k ( t ) ) h 1 α ( x [ k ( t ) ] ) t k ( t ) ( a ( s 2 ) t s 2 t s 1 p ( s ) Δ s Δ s 1 ) 1 α Δ s 2 .

Finally, dividing the above inequality by h 1 α (x(k(t))) and taking the lim sup of both sides of the above inequality as t, we obtain a contradiction to (3.5).

  1. (II)

    Assume that L 1 x(t)>0 and L 2 x(t)<0 for t [ t 1 , ) T . Integrating

    L 2 Δ x(t)q(t)f ( x [ g ( t ) ] )

from t to ∞ we have

L 2 x(t)f ( x [ g ( t ) ] ) t q(s)Δs.

Using (3.4) along with the above inequality, we have

x [ g ( t ) ] B ( g ( t ) , t 1 ) ( f ( x [ g ( t ) ] ) t q ( s ) Δ s ) 1 α .

Dividing the above inequality by f 1 α (x[g(t)]) and taking the lim sup of both sides of the resulting inequality as t, we obtain a contradiction to (3.6). The proof is complete. □

4 Examples

In this section we give examples to illustrate two of our main results. Recall

Theorem 4.1 ([[1], Theorem 1.75])

If f C rd and t T κ , then

t σ ( t ) f(τ)Δτ=μ(t)f(t).

And

Theorem 4.2 ([[1], Theorem 1.79 (ii)])

If [a,b] consists of only isolated points and a<b, then

a b f(t)Δt= t [ a , b ) μ(t)f(t).

Our first example illustrates Theorem 2.1.

Example 4.3 Consider the third-order delay dynamic equation

( 1 a ( t ) ( x Δ ( t ) ) α ) Δ Δ +q(t) x α ( t 3 ) =0,
(4.1)

where tT= 3 N 0 . Here α= 1 3 , q(t)=3+ ( 1 ) ln t ln 3 , g(t)= t 3 , and a(u)=f(u)= u α . Observe that if tT, then

q(t)=3+ ( 1 ) n ={ 2 , n  odd , 4 , n  even .

First we show that (1.3) holds. If s= 3 m and t= 3 n , m,n N 0 , we have

1 a 1 α (s)Δs= lim t 1 t a 1 α (s)Δs= lim n s = 1 ρ ( 3 n ) s(3ss)= lim n s = 1 3 n 1 2 s 2 =.

It is clear that f belongs to C(R,R), is nondecreasing for x0, and satisfies xf(x)>0 for x0 and (1.4). Also, (2.1) holds since

f 1 α ( u ) u = u u =1=c>0,u0.

Observe that if u= 3 k and s= 3 m for k,m N 0 , then

A ( g ( s ) , 1 ) =2 u = 1 ρ ( 3 m 1 ) u 2 ( u 1 ) 3 =2 k = 0 m 2 3 2 k ( 3 k 1 ) 3 =2 k = 1 m 2 3 2 k ( 3 k 1 ) 3 .

Note 3 k 1>1 for kN. Hence

A ( g ( s ) , 1 ) >2 k = 1 m 2 3 2 k >2 3 2 ( m 2 ) ,

and since f is nondecreasing and q(t)2 on , we obtain

q(s)f ( ( A ( g ( s ) , 1 ) ) ) 2 4 3 3 2 3 ( m 2 ) .

It follows that if t [ 1 , ) T

lim sup t g ( t ) t q ( s ) f ( A ( g ( s ) , t 0 ) ) Δ s lim sup n s = t 3 3 n 1 2 4 3 3 2 3 ( n 2 ) 2 s = 2 7 3 lim sup n m = n 1 n 1 3 2 3 ( n 2 ) 3 m > 1 ,

and so (2.2) holds. It remains to show that (2.3) holds for t [ 1 , ) T . This requires that we determine B(g(t),g(s)). Using the above representations of u, s, t, we have

B ( g ( t ) , g ( s ) ) =2 k = m 1 n 2 3 2 k ( 3 n 1 3 k ) 3 >2 3 2 ( n 2 ) ( 3 n 1 3 n 2 ) 3 .

The monotonicity of f and the fact that q(t)2 on yield

q(s)f ( B ( g ( t ) , g ( s ) ) ) > 2 4 3 3 2 3 ( n 2 ) ( 3 n 1 3 n 2 ) = 2 7 3 3 5 3 ( n 2 ) .

Therefore

lim sup t t 3 t q ( s ) f ( B ( g ( t ) , g ( s ) ) ) Δ s > lim sup n 2 7 3 3 5 3 ( n 2 ) s = 3 n 1 3 n 1 μ ( s ) = lim sup n 2 10 3 3 5 3 ( n 2 ) m = n 1 n 1 3 m > 1 ,

which shows that (2.3) holds. By Theorem 2.1, (4.1) is oscillatory.

Our second example illustrates Theorem 3.2.

Example 4.4 Consider the third-order advanced dynamic equation

( 1 a ( t ) ( x Δ ( t ) ) α ) Δ Δ =q(t) x α ( t q ) +p(t) x α ( q 3 t ) ,
(4.2)

where α is the ratio of two positive odd integers and tT= q N 0 , q>1. Here a(t)= ( q ( q 1 ) t ) α , q(t)= q q 1 + 1 + ( 1 ) ln t ln q ( q 1 ) t , p(t)= 1 t 2 q α ( q 1 ) 2 , g(t)= t q , k(t)= q 3 t, and h(u)=f(u)= u α . Then

h 1 α ( u ) u = f 1 α ( u ) u = u u =1= b 1 = c 1 >0,u0,

and so (3.5) and (3.6) hold. Next we show that (3.7) holds. For t [ 1 , ) T , we have

lim sup t t k ( t ) ( a ( s ) t s t u p ( r ) Δ r Δ u ) 1 α Δ s = lim sup t [ t σ ( t ) ( a ( s ) t s t u p ( r ) Δ r Δ u ) 1 α Δ s + σ ( t ) σ 2 ( t ) ( a ( s ) t s t u p ( r ) Δ r Δ u ) 1 α Δ s + σ 2 ( t ) σ 3 ( t ) ( a ( s ) t s t u p ( r ) Δ r Δ u ) 1 α Δ s ] = lim sup t [ μ ( t ) ( a ( t ) t t t u p ( r ) Δ r Δ u ) 1 α + μ ( σ ( t ) ) ( a ( σ ( t ) ) t σ ( t ) t u p ( r ) Δ r Δ u ) 1 α + μ ( σ 2 ( t ) ) ( a ( σ 2 ( t ) ) t σ 2 ( t ) t u p ( r ) Δ r Δ u ) 1 α ] = lim sup t [ μ ( σ ( t ) ) ( a ( σ ( t ) ) μ ( t ) t t p ( r ) Δ r ) 1 α + μ ( σ 2 ( t ) ) ( a ( σ 2 ( t ) ) t σ 2 ( t ) t u p ( r ) Δ r Δ u ) 1 α ] = lim sup t [ μ ( σ 2 ( t ) ) ( a ( σ 2 ( t ) ) ) 1 α ( μ ( t ) t t p ( r ) Δ r + μ ( σ ( t ) ) t σ ( t ) p ( r ) Δ r ) 1 α ] = lim sup t [ μ ( σ 2 ( t ) ) ( a ( σ 2 ( t ) ) ) 1 α ( μ ( σ ( t ) ) μ ( t ) p ( t ) ) 1 α ] = lim sup t [ ( q 1 ) q 2 t ( q 1 ) 2 α ( q t 2 ) 1 α q ( q 1 ) q 2 t 1 t 2 α q ( q 1 ) 2 α ] = lim sup t [ q 1 α ] = q 1 α > 1 .

Since q(t)>0 for all tT, we have

t q(s)Δs t σ ( t ) q(s)Δs=μ(t)q(t)= q q 1 +1+ ( 1 ) ln t ln q > q q 1 .

This implies

lim sup t B ( g ( t ) , t 0 ) ( t q ( s ) Δ s ) 1 α ( q q 1 ) 1 α lim sup t 1 t q a 1 α ( u ) ( t q u ) 1 α Δ u = ( q q 1 ) 1 α lim sup n k = 0 n 2 q ( q 1 ) q k ( q n q q k ) 1 α q k ( q 1 ) = q ( q q 1 ) 1 α lim sup n k = 0 n 2 ( q n q q k ) 1 α > 1 .

Thus (3.8) holds. By Theorem 3.2, (4.2) is oscillatory.

5 Discussion

While we were able to unify most results for (1.1) given in [3] and [4], the comparison result

Theorem Let (1.3)-(1.4) hold. If the first-order delay dynamic equations

y Δ (t)+q(t)f ( A ( g ( t ) , t 0 ) ) f ( y 1 α [ g ( t ) ] ) =0
(5.1)

and

z Δ (t)+q(t)f ( B ( t + g ( t ) 2 , g ( t ) ) ) f ( z 1 α [ t + g ( t ) 2 ] ) =0
(5.2)

are oscillatory, then (1.1) is oscillatory.

cannot be extended since t + g ( t ) 2 T is satisfied for few time scales. While the result holds for T=R and T=Z, this condition is not satisfied on q N ,q>1. Being aware that time scales are not generally closed under addition, we were able to prove the following.

Theorem 5.1 Let (1.3)-(1.4) hold. Furthermore, assume the delay function g:TT is a bijection. If the first-order delay dynamic equations

y Δ (t)+q(t)f ( A ( g ( t ) , t 0 ) ) f ( y 1 α [ g ( t ) ] ) =0
(5.3)

and

z Δ (t)+q(t)f ( B ( t + g ( t ) 2 , g ( t ) ) ) f ( z 1 α [ t + g ( t ) 2 ] ) =0
(5.4)

are oscillatory, where (t+g(t))/2T, then (1.1) is oscillatory.

In order to prove Theorem 5.1, it is necessary to define the function F: [ t 0 , ) T ×[0,)[0,) to be a nondecreasing function with respect to its second argument and with the property that F(,z()) C rd ( [ t 0 , ) T ,[0,)) for any function z C rd ( [ t 0 , ) T ,[0,)). It is also necessary to assume that τ:TT is a bijection with τ(t)<t for all tT and lim t τ(t)=, and to use the following definition and theorem.

Definition 5.2 Let t 1 [ t 0 , ) T . By a solution of the dynamic inequality

y Δ (t)+F ( t , y [ τ ( t ) ] ) 0
(5.5)

on an interval [ t 1 , ) T , we mean a rd-continuous function y defined on the interval [ τ ( t 1 ) , ) T , which is rd-continuously differentiable on [ t 1 , ) T and satisfies (5.5) for all t [ t 1 , ) T . A solution y of (5.5) is said to be positive if y(t)>0 for every t [ τ ( t 1 ) , ) T .

Theorem 5.3 Let y be a positive solution on an interval [ t 1 , ) T , t 1 t 0 of the delay dynamic inequality (5.5). Moreover, we assume that F is positive on any set of the form [ t ˆ , τ 1 ( t ˆ ) ] T ×(0,), t ˆ [ t 1 , ) T . Then there exists a positive solution x on [ τ 1 ( t 1 ) , ) T of the delay dynamic equation

x Δ (t)+F ( t , x [ τ ( t ) ] ) =0
(5.6)

such that

lim t x(t)=0

and

x(t)y(t)for every t [ t 1 , ) T .

Proof Let y be a positive solution (5.5). From (5.5) we obtain for all t ˜ , tT with t ˜ t t 1

y(t)y( t ˜ )+ t t ˜ F ( s , y [ τ ( s ) ] ) Δs> t t ˜ F ( s , y ( τ ( s ) ) ) Δs.
(5.7)

Hence, letting t ˜ we get

y(t) t F ( s , y [ τ ( s ) ] ) Δsfor every t [ t 1 , ) T .
(5.8)

Let X be the set of all nonnegative continuous functions x on the interval [ t 1 , ) T with x(t)y(t) for every t t 1 . Then by using (5.8) we can easily verify that for any function x in X the formula

(Sx)(t)={ t F ( s , x [ τ ( s ) ] ) Δ s , t [ τ 1 ( t 1 ) , ) T , τ 1 ( t 1 ) F ( s , x [ τ ( s ) ] ) Δ s + t τ 1 ( t 1 ) F ( s , y [ τ ( s ) ] ) Δ s , t [ t 1 , τ 1 ( t 1 ) ) T

defines an operator S:XX. If x 1 , x 2 X and x 1 (t) x 2 (t) for t [ t 1 , ) T , then we also have (S x 1 )(t)(S x 2 )(t) for t [ t 1 , ) T since F is nondecreasing with respect to its second argument. Thus, the operator S is monotone. Next, we put

x 0 =y,and x ν =S x ν 1 ,ν=1,2,

and observe that { x ν } ν = 0 , 1 , 2 is a decreasing sequence of functions in X. Furthermore, define

x= lim ν x ν  pointwise on  [ t 1 , ) T .

Then by applying the Lebesgue dominated convergence theorem, we obtain x=Sx. That is

x(t)={ t F ( s , x [ τ ( s ) ] ) Δ s if  t [ τ 1 ( t 1 ) , ) T , τ 1 ( t 1 ) F ( s , x [ τ ( s ) ] ) Δ s + t τ 1 ( t 1 ) F ( s , y [ τ ( s ) ] ) Δ s if  t [ t 1 , τ 1 ( t 1 ) ) T .
(5.9)

From (5.9) it follows that

x Δ (t)+F ( t , x [ τ ( t ) ] ) =0for all t [ τ 1 ( t 1 ) , ) T

and hence x is a solution of (5.6) on the interval t [ τ 1 ( t 1 ) , ) T . Also (5.9) yields

lim t x(t)=0.

Moreover, it is clear that x(t)y(t) for t [ t 1 , ) T . It remains to prove that x is positive on the interval [ t 1 , ) T . Taking into account that y is positive on the interval [ τ ( t 1 ) , ) T and the positivity of F on [ t 1 , τ 1 ( t 1 ) ] T ×(0,) we have

x(t) t τ 1 ( t 1 ) F ( s , y [ τ ( s ) ] ) Δs>0

for each t [ t 1 , τ 1 ( t 1 ) ) T . So, x is positive on an interval [ t 1 , τ 1 ( t 1 ) ) T . Next we will show that x is also positive on [ τ 1 ( t 1 ) , ) T . Assume that t ˆ is the first zero of x in [ τ 1 ( t 1 ) , ) T . Then x(t)>0 for t [ τ 1 ( t 1 ) , t ˆ ) T and x( t ˆ )=0. Then (5.9) yields

0=x( t ˆ )= t ˆ F ( s , x [ τ ( s ) ] ) Δs

and consequently

F ( s , x [ τ ( s ) ] ) =0for all s [ t ˆ , ) T .

That is, we can choose a t [ t ˆ , τ 1 ( t ˆ ) ) T such that

F ( t , x [ τ ( t ) ] ) =0.

On the other hand, taking into account that x(t)>0 for t [ τ 1 ( t 1 ) , t ˆ ) T and the positivity of F on [ t ˆ , τ 1 ( t ˆ ) ] T ×(0,) we get

F ( t , x [ τ ( t ) ] ) >0.

This leads to a contradiction and the proof is complete. □

Note that Theorem 5.3 holds for any unbounded time scale . Now we present the proof of Theorem 5.1.

Proof Let x be an eventually positive solution of (1.1). We consider two cases as we did in the proof of Theorem 2.1.

  1. (I)

    Assume that L 1 x(t)>0 and L 2 x(t)>0 for t [ t 1 , ) T . Then (2.4) holds. Using this and (1.4) in (1.1), we obtain

    L 2 Δ x(t)=q(t)f ( x [ g ( t ) ] ) q(t)f ( A ( g ( t ) , t 1 ) ) f ( L 2 1 α x [ g ( t ) ] ) ,t [ t 2 , ) T

or

y Δ (t)+q(t)f ( A ( g ( t ) , t 1 ) ) f ( y 1 α [ g ( t ) ] ) 0,t [ t 2 , ) T ,

where y(t):= L 2 x(t) for t [ t 2 , ) T . Since L 2 x(t)>0 for all t[ t 2 ,) and A(g(t), t 1 )>0 for all t[ t 1 ,), by Theorem 5.3, there exists a positive solution z of (5.3) such that lim t z(t)=0, which contradicts the hypothesis that (5.3) is oscillatory.

  1. (II)

    Assume that L 1 x(t)<0 and L 2 x(t)>0 for t t 1 . As in the proof of Theorem 2.1, we have (2.5). Then for t [ t 1 , ) T , we have

    x [ g ( t ) ] B ( t + g ( t ) 2 , g ( t ) ) L 2 1 α x ( t + g ( t ) 2 ) .

Using this inequality and (1.4) in (1.1) for t [ t 1 , ) T yields

L 2 Δ x(t)=q(t)f ( x [ g ( t ) ] ) q(t)f ( B ( t + g ( t ) 2 , g ( t ) ) ) f ( L 2 1 α x [ t + g ( t ) 2 ] )

or

z Δ (t)q(t)f ( B ( t + g ( t ) 2 , g ( t ) ) ) f ( z 1 α [ t + g ( t ) 2 ] ) ,

where z(t):= L 2 x(t) for t [ t 1 , ) T . Similar to Case (I) above, by Theorem 5.3, there exists a positive solution y of (5.4) such that lim t y(t)=0, which contradicts the fact that (5.4) is oscillatory. □

In [3] and [4], the authors prove a comparison result for (1.2) similar to the one given at the beginning of this section. That result involved t + k ( t ) 2 . Again, since time scales are not generally closed under addition, this result cannot be extended to a general time scale .

References

  1. Bohner M, Peterson A: Dynamic Equations on Time Scales: An Introduction with Applications. Birkhäuser Boston, Boston; 2001.

    Book  MATH  Google Scholar 

  2. Bohner M, Peterson A (Eds): Advances in Dynamic Equations on Time Scales. Birkhäuser Boston, Boston; 2003.

    MATH  Google Scholar 

  3. Agarwal RP, Grace SR, Wong PJY: On the oscillation of third order nonlinear difference equations. J. Appl. Math. Comput. 2010,32(1):189-203. 10.1007/s12190-009-0243-8

    Article  MathSciNet  MATH  Google Scholar 

  4. Agarwal RP, Grace SR, Wong PJY: Oscillation of certain third order nonlinear functional differential equations. Adv. Dyn. Syst. Appl. 2007,2(1):13-30.

    MathSciNet  MATH  Google Scholar 

  5. Erbe L, Peterson A, Saker SH: Oscillation and asymptotic behavior of a third-order nonlinear dynamic equation. Can. Appl. Math. Q. 2006,14(2):129-147.

    MathSciNet  MATH  Google Scholar 

  6. Erbe L, Hassan TS, Peterson A: Oscillation of third order functional dynamic equations with mixed arguments on time scales. J. Appl. Math. Comput. 2010,34(1-2):353-371. 10.1007/s12190-009-0326-6

    Article  MathSciNet  MATH  Google Scholar 

  7. Hassan TS: Oscillation of third order nonlinear delay dynamic equations on time scales. Math. Comput. Model. 2009,49(7-8):1573-1586. 10.1016/j.mcm.2008.12.011

    Article  MathSciNet  MATH  Google Scholar 

  8. Han Z, Li T, Sun S, Cao F: Oscillation criteria for third order nonlinear delay dynamic equations on time scales. Ann. Pol. Math. 2010,99(2):143-156. 10.4064/ap99-2-3

    Article  MathSciNet  MATH  Google Scholar 

  9. Agarwal R, Akın-Bohner E, Sun S: Oscillation criteria for fourth-order nonlinear dynamic equations. Commun. Appl. Nonlinear Anal. 2011,18(3):1-16.

    MathSciNet  MATH  Google Scholar 

  10. Grace SR, Bohner M, Sun S: Oscillation of fourth-order dynamic equations. Hacet. J. Math. Stat. 2010,39(4):545-553.

    MathSciNet  MATH  Google Scholar 

  11. Agarwal RP, Bohner M, Grace SR, O’Regan D: Discrete Oscillation Theory. Hindawi Publishing Corporation, New York; 2005. xiv+961

    Book  MATH  Google Scholar 

  12. Agarwal RP, Grace SR, O’Regan D: Oscillation Theory for Difference and Functional Differential Equations. Kluwer Academic, Dordrecht; 2000. viii+337

    Book  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

Authors

Corresponding author

Correspondence to Elvan Akın.

Additional information

Competing interests

The authors declare that they have no competing interests.

Authors’ contributions

All authors read and approved this submitted manuscript.

Rights and permissions

Open Access This article is distributed under the terms of the Creative Commons Attribution 2.0 International License ( https://creativecommons.org/licenses/by/2.0 ), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Reprints and permissions

About this article

Cite this article

Adıvar, M., Akın, E. & Higgins, R. Oscillatory behavior of solutions of third-order delay and advanced dynamic equations. J Inequal Appl 2014, 95 (2014). https://doi.org/10.1186/1029-242X-2014-95

Download citation

  • Received:

  • Accepted:

  • Published:

  • DOI: https://doi.org/10.1186/1029-242X-2014-95

Keywords