Skip to main content

Oscillation theorems for second order nonlinear neutral difference equations

Abstract

This paper deals with the oscillation of second order neutral difference equations of the form

△ ( a n ( △ ( x n + p n x τ ( n ) ) ) α ) + q n x σ ( n ) β =0.
(E)

The oscillation of all solutions of this equation is established via comparison theorems. Examples are provided to illustrate the main results.

MSC:39A10.

1 Introduction

Consider the second order nonlinear neutral delay difference equation of the form

△ ( a n ( △ ( x n + p n x τ ( n ) ) ) α ) + q n x σ ( n ) β =0,n≥ n 0 ∈N,
(1.1)

where N={0,1,2,…} and △ is the forward difference operator defined by △ x n = x n + 1 − x n , subject to the following hypotheses:

  • (H1) { p n } and { q n } are nonnegative real sequences with { q n } not identically zero for infinitely many values of n;

  • (H2) { a n } is a positive sequence such that R n = ? s = n 0 n - 1 1 a s 1 a ?8 as n?8;

  • (H3) a and ß are ratio of odd positive integers;

  • (H4) {t(n)} and {s(n)} are nondecreasing sequences of integers such that lim n ? 8 t(n)= lim n ? 8 s(n)=8 and t°s=s°t;

  • (H5) there is a positive constant p such that 0= p n =p<8.

By a solution of equation (1.1) we mean a real sequence { x n } defined, and satisfying equation (1.1), for all n≥ n 0 . We consider only those solutions { x n } of equation (1.1) which satisfy sup{| x n |:n≥N}>0 for all N≥ n 0 . We assume that equation (1.1) possesses such a solution. A solution { x n } of equation (1.1) is oscillatory if it is neither eventually positive nor eventually negative, and nonoscillatory otherwise.

Since the second order equations have applications in various problems in physics, biology, and economics, there is a permanent interest in obtaining new sufficient conditions for the oscillation or nonoscillation of solutions of various types of second order equations; see for example [1–12], and the references cited therein.

In [3, 4] the authors proved that 0≤ p n <1 together with

∑ n = n 0 ∞ q n (1− p n − σ )=∞

guarantees the oscillation of all solutions of the neutral difference equation

△ 2 ( x n + p n x n − τ )+ q n x n − σ =0.
(1.2)

In [13], the author discussed the oscillatory behavior of all solutions of the difference equation

△ ( a n △ ( x n + p n x τ ( n ) ) ) + q n f( x σ ( n ) )=0
(1.3)

with

f ( x ) x ≥μ>0for x≠0,0≤ p n <1and ∑ n = n 0 ∞ 1 a n =∞.

In [14] and [15], the author considered the equation

△ ( a n ( △ ( x n + p n x τ ( n ) ) ) α ) +f(n, x σ ( n ) )=0
(1.4)

with f ( n , x ) x α ≥c q n for x≠0 and studied the oscillatory behavior under the conditions ∑ n = n 0 ∞ 1 a n 1 α =∞ and 0≤ p n <1. In [16], Saker considered the difference equation

△ ( a n ( △ ( x n + p n x n − τ ) ) α ) +f(n, x n − σ )=0
(1.5)

with

f(n,x)≥ q n x n α ,0≤ p n <1and ∑ n = n 0 ∞ 1 a n 1 α =∞

and presented oscillation criteria which improved the existing results for the equation.

In [17], the authors established some sufficient conditions for the oscillation of all solutions of equation (1.3) via comparison theorems. In [18], Sun and Saker considered equation (1.5) and obtained some new oscillation criteria, which improve and complement that given in [16].

Recently in [19, 20], the authors studied the oscillatory behavior of all solutions of equation (1.1) under the conditions 0≤ p n ≤p<1 and α>β or α<β or α=β.

Motivated by the above observation in this paper we shall investigate the oscillatory properties of equation (1.1) without assuming any usual restrictions on { p n }, {σ(n)}, and {Ï„(n)}. We shall provide new comparison theorems in which we compare the second order equation (1.1) with the first order difference equations in the sense that the oscillation nature of these first order difference equations yields the oscillation of equation (1.1). In Section 2, we present some basic lemmas, and in Section 3 we establish oscillation results. In Section 4, we provide several examples to illustrate the main results and we present our conclusion in Section 5.

2 Some basic lemmas

In this section we present some basic lemmas which will be used to prove the main results.

Lemma 2.1 Let A≥0, B≥0, and α≥1. Then

( A + B ) α ≤ 2 α − 1 ( A α + B α ) .
(2.1)

Proof If A=0 or B=0, then the result is true. Next, we assume that 0<A≤B. Consider the function f(x) defined by f(x)= x α . Then

f ″ (x)=α(α−1) x α − 2 ≥0

for x>0. Therefore, f(x) is convex and f( A + B 2 )≤ f ( A ) + f ( B ) 2 , which implies the result (2.1). □

Lemma 2.2 Let A≥0, B≥0, 0<α≤1. Then

( A + B ) α ≤ A α + B α .
(2.2)

Proof If A=0 or B=0, then the result is true. For A≠0, let x= B A . Then the result (2.2) takes the form ( 1 + x ) α ≤1+ x α , which is clearly true for all x>0. □

Next, we present the structure of positive solutions of equation (1.1) since the opposite case is similar.

Lemma 2.3 If { x n } is a positive solution of equation (1.1), then z n = x n + p n x Ï„ ( n ) satisfies

z n >0,△ z n >0,△ ( a n ( △ z n ) α ) <0
(2.3)

eventually.

Proof Assume that x n >0 is a solution of equation (1.1). Then equation (1.1) implies

△ ( a n ( △ z n ) α ) =− q n x σ ( n ) β <0.

Therefore, a n ( △ z n ) α is decreasing and thus either △ z n >0 or △ z n <0 eventually for n≥ n 1 ∈N. If △ z n <0, then there exists a negative constant c such that

△ z n ≤ c a n 1 α <0.

Summing the above inequality from n 1 to n−1, one obtains

z n ≤ z n 1 +c ∑ s = n 1 n − 1 1 a s 1 α →−∞as n→∞.

This contradiction proves (2.3). □

The proof of the following lemma is found in [21].

Lemma 2.4 Let γ>1 be a quotient of odd positive integers. Assume that k is a positive integer, { d n } is a positive sequence defined for all n≥ n 0 ∈N, and there exists λ> 1 k logγ such that

lim n → ∞ inf [ d n exp ( − e λ n ) ] >0,
(2.4)

then all the solutions of the difference equation

△ y n + d n y n − k γ =0

are oscillatory.

Next we state a result given in [22].

Lemma 2.5 Assume that { ϕ n } is a nonnegative real sequence for all n∈N and

lim n → ∞ inf ∑ j = n − l n − 1 ϕ j > ( l l + 1 ) l + 1 ,

where l is a positive integer, then

△ w n + ϕ n w n − l ≤0

has no eventually positive solution.

Lemma 2.6 Let γ be such that 0<γ≤1 be a quotient of odd positive integers and σ(n)=n−k where k is a positive integer. Assume that { d n } is a positive real sequence defined for all n≥ n 0 ∈N. If

lim n → ∞ inf ∑ s = n − k n − 1 d s > ( k k + 1 ) k + 1 ,
(2.5)

then every solution of the first order delay difference equation

△ y n + d n y n − k γ =0
(2.6)

is oscillatory.

Proof Assume that { y n } is a positive solution of equation (2.6). First, observe that condition (2.5) implies

∑ n = n 1 ∞ d n =∞.
(2.7)

Since { y n } is decreasing, there exists l such that lim n → ∞ y n =l≥0. If l>0, then summing the equation (2.6) from n 1 to n−1, we obtain

y n 1 ≥ ∑ s = n 1 n − 1 d s y s − k γ ≥ l γ ∑ s = n 1 n − 1 d s →∞as n→∞,

which is a contradiction, and therefore, we conclude that lim n → ∞ y n =0, and also 0< y n <1, eventually. Therefore, y n − k γ ≥ y n − k . Substituting this into the equation (2.6) we deduce that { y n } is a positive solution of the difference inequality

△ y n + d n y n − k ≤0.
(2.8)

But this contradicts Lemma 2.5 according to which condition (2.5) ensures that (2.8) has no positive solution. The proof is now complete. □

We conclude this section with the following lemma.

Lemma 2.7 Assume that the difference inequality

△ w n + ϕ n w n − k γ ≤0
(2.9)

has an eventually positive solution, where ϕ n ≥0, ϕ n is not identically zero, and γ>0 is a ratio of odd positive integers. Then the difference equation

△ w n + ϕ n w n − k γ =0
(2.10)

also has an eventually positive solution.

Proof Let { w n } be an eventually positive solution of (2.9). Define a set S={ u n :0≤ u n ≤ w n ,n≥N∈N}. Then define a mapping Γ on S as follows:

( Γ u ) n = { ∑ s = n ∞ ϕ s u s − k γ , n ≥ N , ( Γ u ) N + w n − w N , n 0 ≤ n ≤ N .

Define a sequence { u n ( k ) }, k=1,2,… , as follows:

u n ( 1 ) = w n , u n ( k + 1 ) = ( Γ u ( k ) ) n , k = 1 , 2 , … .

Since { w n } is a solution of (2.9), we obtain

△ w n + ϕ n w n − k γ ≤0.

Summing the last inequality from n to n 1 and letting n 1 →∞, we obtain

− w n + ∑ s = n ∞ ϕ s w s − k γ ≤0

or

u n ( 2 ) = ( Γ w ) n = ∑ s = n ∞ ϕ s w s − k γ ≤ w n = u n ( 1 ) .

By induction, we see that

0≤ u n ( k ) ≤ u n ( k − 1 ) ≤⋯≤ u n ( 1 ) = w n ,n≥ n 0 .

Hence, lim k → ∞ u n ( k ) = u n exists with 0≤ u n ≤ w n . Then we can apply the Lebesgue dominated convergence theorem to show that u=Γu, that is,

u n = ∑ s = n ∞ ϕ n u n − k γ ,n≥N.

Clearly, { u n } is an eventually positive solution of (2.10) for all n≥N. Since u n >0 for all n 0 ≤n≤N, it follows that u n >0 for all n≥ n 0 . Hence equation (2.10) has an eventually positive solution for all n≥ n 0 . This completes the proof. □

3 Oscillation results

In this section, we establish some new oscillation criteria for equation (1.1). To simplify our notation, let us denote

Q n =min{ q n , q τ ( n ) }, Q n ⋆ = Q n ( ∑ s = n 1 σ ( n ) − 1 1 a s 1 α ) β ,

where n 1 is sufficiently large. We first study the case α=β=1.

Theorem 3.1 Let α=β=1 in equation (1.1). Assume that the first order neutral difference inequality

△( y n +p y τ ( n ) )+ Q n ( R σ ( n ) − R n 1 ) y σ ( n ) ≤0
(3.1)

has no positive solution, then every solution of equation (1.1) is oscillatory.

Proof Assume that { x n } is a positive solution of equation (1.1). Then the corresponding function z n satisfies

z σ ( n ) = x σ ( n ) + p σ ( n ) x τ ( σ ( n ) ) ≤ x σ ( n ) +p x σ ( τ ( n ) ) ,
(3.2)

where we have used the hypotheses (H4) and (H5). From equation (1.1), we have

△( a n △ z n )+ q n x σ ( n ) =0
(3.3)

and

0=p△( a τ ( n ) △ z τ ( n ) )+p q τ ( n ) x τ ( σ ( n ) ) .
(3.4)

Combining (3.3) and (3.4), we are led to

△( a n △ z n +p a τ ( n ) △ z τ ( n ) )+ Q n z σ ( n ) ≤0.
(3.5)

It follows from Lemma 2.3 that y n = a n â–³ z n >0 is decreasing, and then

z n ≥ ∑ s = n 1 n − 1 1 a s ( a s △ z s )≥ y n ∑ s = n 1 n − 1 1 a s = y n ( R n − R n 1 ).
(3.6)

Therefore, (3.6) together with (3.5) ensures that { y n } is a positive solution of inequality (3.1). This contradicts our assumption and therefore the proof is complete. □

Remark 3.1 The condition τ∘σ=σ∘τ of the hypothesis (H4) is satisfied, for example τ(n)=n−τ and σ(n)=n−σ.

Remark 3.2 When studying the oscillatory properties of neutral type equations one usually assumes σ(n)≤τ(n), σ(n)≤n, Ï„(n)≤n, 0≤ p n <1, etc. In Theorem 3.1 no such constraint is involved and what is more we do not impose is Ï„(n) is delayed or advanced and accordingly σ(n) can be delayed or advanced. Hence, our result is of high generality and extends and complements the known ones.

Theorem 3.2 Let α=β=1 in equation (1.1), and assume that

τ(n)≥n.
(3.7)

If the first order difference inequality

△ w n + Q n ( 1 + p ) ( R σ ( n ) − R n 1 ) w σ ( n ) ≤0
(3.8)

has no positive solution, then every solution of equation (1.1) is oscillatory.

Proof We assume that { x n } is a positive solution of equation (1.1). It follows from Lemma 2.3 and the proof of Theorem 3.1 that y n = a n â–³ z n >0 is decreasing and it satisfies (3.1). Let us denote w n = y n +p y Ï„ ( n ) .

It follows from (3.7) that

w n ≤ y n (1+p).

Substituting this term into (3.1), we obtain { w n } is a positive solution of (3.8), a contradiction. This completes the proof. □

Adding the restriction that σ(n)=n−σ, and using suitable criterion for the absence of positive solutions of equation (3.8) (see e.g. Lemma 2.5), we obtain an easily verifiable oscillation result for equation (1.1).

Corollary 3.1 Assume that α=β=1, and (3.7) holds. If

σ(n)=n−σ,σ is a positive integer
(3.9)

and

lim n → ∞ inf ∑ s = n − σ n − 1 Q s R σ ( s ) >(1+p) ( σ σ + 1 ) σ + 1 ,
(3.10)

then every solution of equation (1.1) is oscillatory.

Proof It is easy to see that if (3.10) holds, then

lim n → ∞ inf ∑ s = n − σ n − 1 Q s ( 1 + p ) ( R σ ( s ) − R n 1 )> ( σ σ + 1 ) σ + 1 .

But this condition, according to Lemma 2.5, guarantees that (3.8) has no positive solution and the assertion now follows from Theorem 3.2. □

Now, we turn our attention to the case when τ(n) is delayed. We use the notation τ − 1 (n) for its inverse function.

Theorem 3.3 Let α=β=1 in equation (1.1), and assume that

τ(n)≤n.
(3.11)

If the difference inequality

△ w n + Q n ( 1 + p ) ( R σ ( n ) − R n 1 ) w τ − 1 σ ( n ) ≤0
(3.12)

has no positive solution, then every solution of equation (1.1) is oscillatory.

Proof Assume that { x n } is a positive solution of equation (1.1). Then y n = a n △ z n >0 is decreasing solution of (3.1). We denote w n = y n +p y τ ( n ) . Then by (3.11), we have w n ≤ y τ ( n ) (1+p). Substituting this into (3.1), we see that { w n } is a positive solution of inequality (3.12), a contradiction. This completes the proof. □

Corollary 3.2 Assume that α=β=1, τ(n)=n−τ, σ(n)=n−σ with

n−σ≤n−τ,σ and Ï„ are nonnegative integers
(3.13)

and

lim n → ∞ inf ∑ s = n + τ − σ n − 1 Q s R s >(1+p) ( σ − τ σ − τ + 1 ) σ − τ + 1 ,
(3.14)

then every solution of equation (1.1) is oscillatory.

Proof The proof is very similar to that of Corollary 3.1 and hence it is omitted. □

Next, we consider the case 0<β≤1, and σ(n)≤n in equation (1.1).

Theorem 3.4 Let 0<β≤1. If the difference inequality

△ ( w n + p β w τ ( n ) ) + Q n ⋆ w σ ( n ) β α ≤0
(3.15)

has no positive solution, then every solution of equation (1.1) is oscillatory.

Proof Let { x n } be a positive solution of equation (1.1). Then from equation (1.1), we have

0=△ ( a n ( △ z n ) α ) + q n x σ ( n ) β
(3.16)

and

0= p β △ ( a τ ( n ) ( △ z τ ( n ) ) α ) + p β q τ ( n ) x σ ( τ ( n ) ) .
(3.17)

Combining (3.16) and (3.17), we obtain

△ ( a n ( △ z n ) α ) + p β △ ( a τ ( n ) ( △ z τ ( n ) ) α ) + Q n ( x σ ( n ) β + p β x σ ( τ ( n ) ) β ) ≤0.
(3.18)

By Lemma 2.2, we have

z σ ( n ) β = ( x σ ( n ) + p σ ( n ) x σ ( τ ( n ) ) ) β ≤ x σ ( n ) β + p β x σ ( τ ( n ) ) β .
(3.19)

Using (3.19) in (3.18), we obtain

△ ( a n ( △ z n ) α ) + p β △ ( a τ ( n ) ( △ z τ ( n ) ) α ) + Q n z σ ( n ) β ≤0.
(3.20)

It follows from Lemma 2.3 that w n = a n ( â–³ z n ) α >0 is decreasing and so

z n ≥ ∑ s = n 1 n − 1 ( a s ( △ z s ) α ) 1 α a s − 1 α ≥ w n 1 α ∑ s = n 1 n − 1 1 a s 1 α .

Using the last inequality in (3.20), we see that { w n } is a positive solution of

△ ( w n + p β w τ ( n ) ) + Q n ∗ w σ ( n ) β α ≤0,

which is a contradiction. This completes the proof. □

Next, we shall deduce new sufficient conditions for inequality (3.15) to have no positive solutions, to obtain new oscillation criteria for equation (1.1). We shall discuss both the cases when Ï„(n) is delayed and advanced.

Theorem 3.5 Let 0<β≤1 and τ(n)≥n. If the difference equation

△ y n + Q n ⋆ ( 1 + p β ) β α y σ ( n ) β α =0
(3.21)

is oscillatory, then so is equation (1.1).

Proof We assume that { x n } is a positive solution of equation (1.1). Then it follows from the proof of Theorem 3.4 that w n = a n ( â–³ z n ) α >0 is decreasing and it satisfies (3.15). We denote

y n = w n + p β w τ ( n ) .
(3.22)

Then

y n ≤ w n ( 1 + p β ) .

Substituting this into (3.15), we see that y n is a positive solution of the difference inequality

△ y n + Q n ⋆ ( 1 + p β ) β α y σ ( n ) β α ≤0.

It follows from Lemma 2.7 that the associated difference equation (3.21) also has a positive solution, which contradicts the oscillatory nature of equation (3.21). □

Theorem 3.6 Let 0<β≤1 and σ(n)≤τ(n)≤n. If the difference equation

△ y n + Q n ⋆ ( 1 + p β ) β α y τ − 1 ( σ ( n ) ) β α =0
(3.23)

is oscillatory, then so is equation (1.1).

Proof Assume that { x n } is a positive solution of equation (1.1). Then it follows from (3.22) that

y n ≤ w τ ( n ) ( 1 + p β )

or

w σ ( n ) β α ≥ 1 ( 1 + p β ) β α y τ − 1 ( σ ( n ) ) β α .

Using the above inequality in (3.15), we see that { y n } is a positive solution of the difference inequality

△ y n + Q n ⋆ ( 1 + p β ) β α y τ − 1 ( σ ( n ) ) β α ≤0.

By Lemma 2.7, the associated difference equation (3.23) also has a positive solution, which is a contradiction. This completes the proof. □

Now, we derive a criterion for equations (3.21) and (3.23) to be oscillatory. Employing this criterion, one can easily verify sufficient conditions for the oscillation of all solutions of equation (1.1).

Applying condition (2.6) to equations (3.21) and (3.23) in view of Theorem 3.5 and Theorem 3.6, immediately we obtain the following oscillatory criteria for equation (1.1).

Corollary 3.3 Let the equation (1.1) with 0<β≤1, β≤α, τ(n)≥n, and (3.9). If

lim n → ∞ inf ∑ s = n − σ n − 1 Q s ⋆ > ( 1 + p β ) β α ( σ σ + 1 ) σ + 1 ,
(3.24)

then every solution of equation (1.1) is oscillatory.

Corollary 3.4 Let the equation (1.1) with 0<β≤1, β≤α, τ(n)=n−τ, and (3.9) with σ>τ>0. If

lim n → ∞ inf ∑ s = n + τ − σ n − 1 Q s ⋆ > ( 1 + p β ) β α ( σ − τ σ − τ + 1 ) σ − τ + 1 ,
(3.25)

then every solution of (1.1) is oscillatory.

In view of Lemma 2.4, and Theorems 3.5 and 3.6, we have the following oscillation criteria for equation (1.1).

Corollary 3.5 Let the equation (1.1) with 0<β≤1, β>α, τ(n)=n+τ with τ>0, and (3.9). Assume that there exists λ> 1 σ log β α such that

lim n → ∞ inf [ Q n ⋆ exp ( − e λ n ) ] >0,
(3.26)

then every solution of equation (1.1) is oscillatory.

Corollary 3.6 Let the equation (1.1) with 0<β≤1, β>α, τ(n)=n−τ, and (3.9) with σ>τ>0. Assume that there exists λ> 1 σ − τ log β α such that

lim n → ∞ inf [ Q n ⋆ exp ( − e λ n ) ] >0,
(3.27)

then every solution of equation (1.1) is oscillatory.

Next, we turn our attention to the case β≥1 and we rewrite our previous results to cover this case.

Theorem 3.7 Let β≥1 and σ(n)≤n. If the difference inequality

△ ( w n + p β w τ ( n ) ) + 2 1 − β Q n ⋆ w σ ( n ) β α ≤0
(3.28)

has no positive solution, then every solution of equation (1.1) is oscillatory.

Proof This theorem can be proved exactly as Theorem 3.4; we need to replace the inequality (3.19) by

z σ ( n ) β = ( x σ ( n ) + p σ ( n ) x τ ( σ ( n ) ) ) β ≤ 2 β − 1 ( x σ ( n ) β + p β x σ ( τ ( n ) ) β ) ,

which follows from Lemma 2.1. □

The following results are equivalent to Theorems 3.5 and 3.6 and the proofs are omitted.

Theorem 3.8 Let β≥1 and τ(n)≥n. If the difference equation

△ y n + Q n ⋆ ( 1 + p β ) β α 2 1 − β y σ ( n ) β α =0
(3.29)

is oscillatory, then every solution of equation (1.1) is oscillatory.

Theorem 3.9 Let β≥1 and σ(n)≤τ(n)≤n. If the difference equation

△ y n + Q n ⋆ ( 1 + p β ) β α 2 1 − β y τ − 1 ( σ ( n ) ) β α =0
(3.30)

is oscillatory, then solution of equation (1.1) is oscillatory.

Combining Lemma 2.6 with Theorems 3.8 and 3.9, we have the following oscillation criteria for equation (1.1).

Corollary 3.7 Let α≥β≥1, τ(n)≥n, and σ(n)=n−σ in equation (1.1). If

lim n → ∞ inf ∑ s = n − σ n − 1 Q s ⋆ > 2 β − 1 ( 1 + p β ) β α ( σ σ + 1 ) σ + 1 ,
(3.31)

then every solution of equation (1.1) is oscillatory.

Corollary 3.8 Let α≥β≥1, τ(n)=n−τ, σ(n)=n−σ with σ>τ in equation (1.1). If

lim n → ∞ inf ∑ s = n + τ − σ n − 1 Q s ⋆ > 2 β − 1 ( 1 + p β ) β α ( σ − τ σ − τ + 1 ) σ − τ + 1 ,
(3.32)

then every solution of equation (1.1) is oscillatory.

Combining Lemma 2.4 with Theorems 3.8 and 3.9, we have following results.

Corollary 3.9 Let β>α≥1, τ(n)≥n, and σ(n)=n−σ in equation (1.1). Assume that there exists λ> 1 σ log β α such that

lim n → ∞ inf [ Q n ⋆ exp ( − e λ n ) ] >0,

then every solution of equation (1.1) is oscillatory.

Corollary 3.10 Let β>α≥1, τ(n)=n−τ, σ(n)=n−σ with σ>τ in equation (1.1). Assume that there exists λ> 1 σ − τ log β α such that

lim n → ∞ inf [ Q n ⋆ exp ( − e λ n ) ] >0,

then every solution of equation (1.1) is oscillatory.

4 Examples

In this section, we present some examples to illustrate the main results.

Example 4.1 Consider the difference equation

△ ( 1 n △ ( x n + p x τ ( n ) ) ) + b n ( n + 1 ) x n − 2 =0,n≥1,
(4.1)

where 0<p<∞, τ(n)=an, a≥1 is an integer and b>0. Here R(n)= n ( n − 1 ) 2 , Q n = b a n ( a n + 1 ) . Condition (3.10) reduces to

27 b 8 a 2 >(1+p).
(4.2)

If the condition (4.2) is satisfied, then by Corollary 3.1, every solution of equation (4.1) is oscillatory.

If τ(n)=n−1, then condition (3.10) reduces to

2b>(1+p).
(4.3)

If the condition (4.3) holds, then by Corollary 3.2, every solution of equation (4.1) is oscillatory. Hence, we have covered the oscillation of equation (4.1) when Ï„(n) is delayed or advanced.

Example 4.2 Consider the difference equation

△ ( 1 n ( △ ( x n + p x τ ( n ) ) ) 1 3 ) + b n 2 3 ( n + 1 ) 2 3 x n − 2 1 3 =0,n≥1,
(4.4)

where 0<p<∞, b>0, and τ(n)=an, a≥1 is an integer, and α=β= 1 3 . Here R(n)= n ( n − 1 ) 2 , Q n = b ( a n ) 2 3 ( a n + 1 ) 2 3 . Then condition (3.24) reduces to

2 1 3 b a 4 3 > 8 27 ( 1 + p 1 3 ) .
(4.5)

Therefore, by Corollary 3.3, every solution of equation (4.4) is oscillatory if condition (4.5) holds.

If τ(n)=n−1, then condition (3.25) reduces to

2 4 3 b>1+ p 1 3 .
(4.6)

Therefore, by Corollary 3.4, every solution of equation (4.4) is oscillatory if condition (4.6) holds.

Example 4.3 Consider the difference equation

△ ( 1 n 1 5 ( △ ( x n + p x n + 1 ) ) 1 5 ) + b e e 2 n ( n + 1 ) 2 3 x n − 1 1 3 =0,n≥1,
(4.7)

where b>0, α= 1 5 , β= 1 3 , Ï„=1, σ=1. Choose λ=2, then condition (3.26) reduces to b 2 1 3 >0. Therefore, by Corollary 3.5 every solution of equation (4.7) is oscillatory.

Example 4.4 Consider the difference equation

△ ( 1 n 5 ( △ ( x n + p x τ ( n ) ) ) 5 ) + μ ( n + 1 ) 6 x n − 2 3 =0,n≥1,
(4.8)

where μ>0, 0<p<∞, α=5, β=3, τ(n)=an, a≥1 is an integer. Here Q n ⋆ = μ ( a n + 1 ) 6 ( ( n − 2 ) ( n − 3 ) 2 ) 3 . Then condition (3.31) reduces to

27 μ 128 a 6 > ( 1 + p 3 ) 3 5 .
(4.9)

Therefore, by Corollary 3.7, every solution of equation (4.8) is oscillatory if condition (4.9) holds.

Further, if τ(n)=n−1, then condition (3.32) reduces to

μ>8 ( 1 + p 3 ) 3 5 .
(4.10)

Therefore, by Corollary 3.8, every solution of equation (4.8) is oscillatory if condition (4.10) holds.

5 Conclusions

In this paper, we have introduced new comparison theorems for investigation of the oscillation of equation (1.1). The established comparison principles reduce the study of the oscillation of the second order neutral difference equations to a study of the oscillation properties of various types of first order difference inequalities, which clearly simplifies the investigation of the oscillation of equation (1.1). Further, the method used here permits us to relax the restrictions usually imposed on the coefficients of equation (1.1). So the results obtained here are of high generality and easily may be applicable, as illustrated with suitable examples.

References

  1. Agarwal RP: Difference Equations and Inequalities. 2nd edition. Dekker, New York; 2000.

    MATH  Google Scholar 

  2. Elizabeth S, Graef JR, Sundaram P, Thandapani E: Classifying nonoscillatory solutions and oscillation of neutral difference equations. J. Differ. Equ. Appl. 2005, 11: 605–618. 10.1080/10236190412331334491

    Article  MathSciNet  MATH  Google Scholar 

  3. Lalli BS, Grace SR: Oscillation theorems of second order delay and neutral difference equations. Util. Math. 1994, 45: 197–212.

    MathSciNet  MATH  Google Scholar 

  4. Li HJ, Yeh CC: Oscillation criteria for second order neutral difference equations. Comput. Math. Appl. 1998, 36: 123–132.

    Article  MathSciNet  MATH  Google Scholar 

  5. Thandapani E, Manuel MMS: Asymptotic and oscillatory behavior of second order neutral delay difference equations. Eng. Simul. 1998, 15: 423–430.

    Google Scholar 

  6. Thandapani E, Mohankumar P: Oscillation and nonoscillation of nonlinear neutral delay difference equations. Tamkang J. Math. 2007, 38: 323–333.

    MathSciNet  MATH  Google Scholar 

  7. Thandapani, E, Vijaya, M, Gyori, I: New oscillation criteria for forced superlinear neutral type difference equations. Fasc. Math. (to appear)

  8. Zhang BG, Saker SH: Kamenev-type oscillation criteria for nonlinear neutral delay difference equation. Indian J. Pure Appl. Math. 2003,34(11):1571–1584.

    MathSciNet  MATH  Google Scholar 

  9. Agarwal RP, Manuel MMS, Thandapani E: Oscillatory and nonoscillatory behavior of second order neutral delay difference equations. Math. Comput. Model. 1996, 24: 5–11. 10.1016/0895-7177(96)00076-3

    Article  MathSciNet  MATH  Google Scholar 

  10. Agarwal RP, Wong PJY: Advanced Topics in Difference Equations. Kluwer Academic, Dordrecht; 1997.

    Book  MATH  Google Scholar 

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

    Book  MATH  Google Scholar 

  12. Bainov DD, Mishev DP: Classification and existence of positive solutions of second order nonlinear neutral difference equations. Funkc. Ekvacioj 1997, 40: 371–393.

    Google Scholar 

  13. Jiaowan L: Oscillation criteria for second order neutral difference equations. Ann. Differ. Equ. 1998, 14: 262–266.

    MathSciNet  MATH  Google Scholar 

  14. Jiang J: Oscillatory criteria for second-order quasilinear neutral delay difference equations. Appl. Math. Comput. 2002, 125: 287–293. 10.1016/S0096-3003(00)00130-2

    Article  MathSciNet  MATH  Google Scholar 

  15. Jiang J: Oscillation of second order nonlinear neutral delay difference equations. Appl. Math. Comput. 2003, 146: 791–801. 10.1016/S0096-3003(02)00631-8

    Article  MathSciNet  MATH  Google Scholar 

  16. Saker SH: New oscillation criteria for second order nonlinear neutral delay difference equations. Appl. Math. Comput. 2003, 142: 99–111. 10.1016/S0096-3003(02)00286-2

    Article  MathSciNet  MATH  Google Scholar 

  17. Grace SR, El-Morshedy HA: Oscillation criteria of comparison type for second order difference equations. J. Appl. Anal. 2000, 6: 87–103.

    Article  MathSciNet  MATH  Google Scholar 

  18. Sun YG, Saker SH: Oscillation of second-order nonlinear neutral delay difference equations. Appl. Math. Comput. 2005, 163: 909–918. 10.1016/j.amc.2004.04.017

    Article  MathSciNet  MATH  Google Scholar 

  19. Jinfa C: Kamenev-type oscillation criteria for delay difference equations. Acta Math. Sci. Ser. B 2007,27(3):574–580. 10.1016/S0252-9602(07)60057-5

    Article  MathSciNet  MATH  Google Scholar 

  20. Thandapani, E, Vijaya, M: Oscillation of second order nonlinear neutral delay difference equations. Int. J. Differ. Equ. (to appear)

  21. Tang XH, Liu Y: Oscillation for nonlinear delay difference equations. Tamkang J. Math. 2001, 32: 275–280.

    MathSciNet  MATH  Google Scholar 

  22. Ladas G, Philos CG, Sficas YG: Sharp condition for the oscillation of delay difference equations. J. Appl. Math. Simul. 1989, 2: 101–112. 10.1155/S1048953389000080

    Article  MathSciNet  MATH  Google Scholar 

Download references

Acknowledgements

This paper was supported by CINAMIL - Centro de Investigação, Desenvolvimento e Inovação da Academia Militar.

Author information

Authors and Affiliations

Authors

Corresponding author

Correspondence to Sandra Pinelas.

Additional information

Competing interests

The authors declare that they have no competing interests.

Authors’ contributions

All authors contributed equally to the writing of this paper. All authors read and approved the final manuscript.

Rights and permissions

Open Access  This article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons licence, and indicate if changes were made.

The images or other third party material in this article are included in the article’s Creative Commons licence, unless indicated otherwise in a credit line to the material. If material is not included in the article’s Creative Commons licence and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder.

To view a copy of this licence, visit https://creativecommons.org/licenses/by/4.0/.

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Selvarangam, S., Thandapani, E. & Pinelas, S. Oscillation theorems for second order nonlinear neutral difference equations. J Inequal Appl 2014, 417 (2014). https://doi.org/10.1186/1029-242X-2014-417

Download citation

  • Received:

  • Accepted:

  • Published:

  • DOI: https://doi.org/10.1186/1029-242X-2014-417

Keywords