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Oscillation of fourth-order neutral differential equations with p-Laplacian like operators

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Abstract

We study oscillatory behavior of a class of fourth-order neutral differential equations with a p-Laplacian like operator using the Riccati transformation and integral averaging technique. A Kamenev-type oscillation criterion is presented assuming that the noncanonical case is satisfied. This new theorem complements and improves a number of results reported in the literature. An illustrative example is provided.

MSC:34C10, 34K11.

1 Introduction

In this paper, we are concerned with oscillation of a class of fourth-order neutral differential equations with a p-Laplacian like operator

( r ( t ) | z ( t ) | p 2 z ( t ) ) + i = 1 l q i (t) | x ( τ i ( t ) ) | p 2 x ( τ i ( t ) ) =0,
(1.1)

where

z(t):=x(t)+a(t)x ( σ ( t ) ) .

It is interesting to study equation (1.1) since the p-Laplace differential equations have applications in continuum mechanics as seen from [1]. Throughout, we assume that p>1 is a constant, I:=[ t 0 ,), r C 1 (I,(0,)), r (t)0, a,σ, q i , τ i C(I,R), 0a(t)<1, q i (t)0, i=1,2,,l, σ(t)t, lim t σ(t)=, there exists a function τ C 1 (I,R) such that τ(t) τ i (t) for i=1,2,,l, τ(t)t, τ (t)>0, and lim t τ(t)=.

We use the notation t 1 := min t [ t 0 , ) {σ(t), τ 1 (t), τ 2 (t),, τ l (t)}. By a solution of (1.1), we mean a function xC([ t 1 ,),R) which has the property r | z | p 2 z C 1 (I,R) and satisfies (1.1) on I. We consider only those solutions x of (1.1) which satisfy sup{|x(t)|:t t }>0 for all t t 0 and tacitly assume that (1.1) possesses such solutions. A solution x of (1.1) is called oscillatory if it has arbitrarily large zeros on I; otherwise, it is said to be nonoscillatory. Equation (1.1) is termed oscillatory if all its solutions oscillate.

Fourth-order differential equations naturally appear in models concerning physical, biological, and chemical phenomena; see [2]. In mechanical and engineering problems, questions related to the existence of oscillatory solutions play an important role. During the past few years, there has been constant interest in obtaining sufficient conditions for oscillatory and nonoscillatory properties of different classes of fourth-order differential equations. We refer the reader to [321] and the references cited therein. Parhi and Tripathy [12, 13] and Thandapani and Savitri [15] studied a fourth-order neutral differential equation

( r ( t ) ( x ( t ) + p ( t ) x ( σ ( t ) ) ) ) +q(t)x ( τ ( t ) ) =0.

Most oscillation results reported in [6, 7, 9, 18] for (1.1) and its particular cases have been obtained under the assumption that

R( t 0 )=,
(1.2)

where

R(t):= t d s r 1 / ( p 1 ) ( s ) .

The analogue for (1.1) in case a(t)=0 has been studied in [10, 16, 17, 1921] under the condition that

R( t 0 )<,
(1.3)

which is called a noncanonical case. Assuming (1.3), a question regarding the oscillation and asymptotic behavior of solutions to (1.1) in the case

p=2,l=1,0a(t) a 1 <1,and τ 1 (t)t
(1.4)

has been studied by Li et al. [11]. Note that [[11], Theorem 2.2] ensures that every solution x of the studied equation is either oscillatory or tends to zero as t and, unfortunately, cannot distinguish solutions with different behaviors.

It should be noted that research in this paper is strongly motivated by the recent paper [11]. The purpose of this paper is to establish a Kamenev-type theorem which guarantees that all solutions of equation (1.1) are oscillatory in the case where (1.3) holds and without requiring conditions (1.4). In the sequel, all functional inequalities are assumed to hold for all t large enough.

2 Main results

We begin with the following lemma.

Lemma 2.1 (See [14])

Let f C n (I, R + ). Assume that f ( n ) is eventually of one sign for all large t, and there exists a t 1 t 0 such that f ( n ) (t) f ( n 1 ) (t)0 for all t t 1 . Then, for every constant λ(0,1), there exist a t λ [ t 1 ,) and a constant M>0 such that

f(λt)M t n 1 | f ( n 1 ) ( t ) |

for all t[ t λ ,).

Lemma 2.2 (See [[4], Lemma 2.2.3])

Let f be as in Lemma 2.1. If lim t f(t)0, then, for every constant k(0,1), there exists a t k [ t 1 ,) such that

f(t) k ( n 1 ) ! t n 1 | f ( n 1 ) ( t ) |

for all t[ t k ,).

Theorem 2.3 Assume (1.3) and let one of the following conditions hold:

t 0 R(s)ds=
(2.1)

and

t 0 u R(s)dsdu=.
(2.2)

Suppose also that there exist functions ρ C 1 (I,(0,)), H,ϱC(D,R), where D={(t,s):ts t 0 } such that

H(t,t)=0,t t 0 ,H(t,s)>0,t>s t 0 ,

and H has a nonpositive continuous partial derivative H/s satisfying, for all sufficiently large T t 0 , for some constant λ(0,1), and for all constants M>0,

lim sup t 1 H ( t , T ) T t [ H ( t , s ) ρ ( s ) i = 1 l q i ( s ) ( 1 a ( τ i ( s ) ) ) p 1 1 p p r ( s ) ( ϱ + ( t , s ) ) p ( λ M τ ( s ) τ 2 ( s ) ρ ( s ) ) p 1 ] d s = ,
(2.3)

where

ϱ + (t,s):=max { 0 , ϱ ( t , s ) }

and

H ( t , s ) s + ρ ( s ) ρ ( s ) H(t,s)= ϱ ( t , s ) ρ ( s ) ( H ( t , s ) ) ( p 1 ) / p .

If there exist functions δ C 1 (I,(0,)), K,ξC(D,R) such that

K(t,t)=0,t t 0 ,K(t,s)>0,t>s t 0 ,

and K has a nonpositive continuous partial derivative K/s satisfying, for all sufficiently large T t 0 and for some constant k(0,1),

lim sup t T t [ K ( t , s ) δ ( s ) ( k τ 2 ( s ) 2 ) p 1 i = 1 l q i ( s ) ( 1 a ( τ i ( s ) ) ) p 1 r ( s ) ( ξ + ( t , s ) ) p p p δ p 1 ( s ) ] d s > 0 ,
(2.4)

where

ξ + (t,s):=max { 0 , ξ ( t , s ) }

and

K ( t , s ) s + δ ( s ) δ ( s ) K(t,s)= ξ ( t , s ) δ ( s ) ( K ( t , s ) ) ( p 1 ) / p ,

then equation (1.1) is oscillatory.

Proof Let x be a nonoscillatory solution of (1.1). Without loss of generality, we may assume that x is eventually positive. Equation (1.1) implies that there exists a t 1 t 0 such that the following three possible cases hold for all t t 1 :

  1. (1)

    z(t)>0, z (t)<0, z (t)>0, z (t)<0, ( r | z | p 2 z ) (t)0;

  2. (2)

    z(t)>0, z (t)>0, z (t)>0, z ( 4 ) (t)0, ( r | z | p 2 z ) (t)0;

  3. (3)

    z(t)>0, z (t)>0, z (t)>0, z (t)<0, ( r | z | p 2 z ) (t)0.

We consider each of these cases separately.

Case 1. Assume that (1) is satisfied. Noting that r ( z ) p 1 is nondecreasing, we have, for st t 1 ,

r 1 / ( p 1 ) (s) z (s) r 1 / ( p 1 ) (t) z (t).

Dividing the latter inequality by r 1 / ( p 1 ) (s) and integrating the resulting inequality from t to ι, ιt t 1 , we obtain

z (ι) z (t)+ r 1 / ( p 1 ) (t) z (t) t ι d s r 1 / ( p 1 ) ( s ) .

Passing to the limit as ι, we conclude that

z (t) r 1 / ( p 1 ) (t) z (t)R(t).

Hence, there exists a constant c>0 such that

z (t)cR(t).
(2.5)

Integrating (2.5) from t 1 to t, we have

z (t) z ( t 1 )c t 1 t R(s)ds.

This yields

z ( t 1 )c t 1 t R(s)ds,

which contradicts (2.1). Next, integrating (2.5) from t to ∞, we get

z (t)c t R(s)ds.

Integrating again from t 1 to t, we have

z(t)+z( t 1 )c t 1 t u R(s)dsdu.

This implies that

z( t 1 )c t 1 t u R(s)dsdu,

which contradicts (2.2).

Case 2. Assume that (2) is satisfied and let λ(0,1) be an arbitrary constant. Then, there exists a t λ t 1 such that, for all t t λ , z(λτ(t))>0. For t t λ , define

ω(t):=ρ(t) r ( t ) ( z ( t ) ) p 1 z p 1 ( λ τ ( t ) ) .
(2.6)

Then ω(t)>0 for all t t λ , and

ω ( t ) = ρ ( t ) r ( t ) ( z ( t ) ) p 1 z p 1 ( λ τ ( t ) ) + ρ ( t ) ( r ( t ) ( z ( t ) ) p 1 ) z p 1 ( λ τ ( t ) ) ( p 1 ) λ ρ ( t ) τ ( t ) z p 2 ( λ τ ( t ) ) z ( λ τ ( t ) ) r ( t ) ( z ( t ) ) p 1 z 2 ( p 1 ) ( λ τ ( t ) ) .
(2.7)

By virtue of Lemma 2.1, we have, for some constant M>0 and for all sufficiently large t,

z ( λ τ ( t ) ) M τ 2 (t) z ( τ ( t ) ) M τ 2 (t) z (t).
(2.8)

Combining (2.7) and (2.8), we get

ω ( t ) ρ ( t ) r ( t ) ( z ( t ) ) p 1 z p 1 ( λ τ ( t ) ) + ρ ( t ) ( r ( t ) ( z ( t ) ) p 1 ) z p 1 ( λ τ ( t ) ) ( p 1 ) λ M τ 2 ( t ) τ ( t ) ρ ( t ) r ( t ) ( z ( t ) ) p z p ( λ τ ( t ) ) .
(2.9)

Recalling that z >0 and σ(t)t, we have

x(t)=z(t)a(t)x ( σ ( t ) ) z(t)a(t)z ( σ ( t ) ) ( 1 a ( t ) ) z(t).
(2.10)

Then it follows from (1.1), (2.6), (2.9), and (2.10) that there exists a t 3 t λ such that, for all t t 3 ,

ω ( t ) ρ ( t ) i = 1 l q i ( t ) ( 1 a ( τ i ( t ) ) ) p 1 + ρ ( t ) ρ ( t ) ω ( t ) ( p 1 ) λ M τ 2 ( t ) τ ( t ) ( r ( t ) ρ ( t ) ) 1 / ( p 1 ) ω p / ( p 1 ) ( t ) .

Multiplying the latter inequality by H(t,s) and integrating the resulting inequality from t 3 to t, we obtain

t 3 t H ( t , s ) ρ ( s ) i = 1 l q i ( s ) ( 1 a ( τ i ( s ) ) ) p 1 d s H ( t , t 3 ) ω ( t 3 ) + t 3 t [ H ( t , s ) s + ρ ( s ) ρ ( s ) H ( t , s ) ] ω ( s ) d s t 3 t ( p 1 ) λ M τ 2 ( s ) τ ( s ) ( r ( s ) ρ ( s ) ) 1 / ( p 1 ) H ( t , s ) ω p / ( p 1 ) ( s ) d s H ( t , t 3 ) ω ( t 3 ) + t 3 t ϱ + ( t , s ) ρ ( s ) ( H ( t , s ) ) ( p 1 ) / p ω ( s ) d s t 3 t ( p 1 ) λ M τ 2 ( s ) τ ( s ) ( r ( s ) ρ ( s ) ) 1 / ( p 1 ) H ( t , s ) ω p / ( p 1 ) ( s ) d s .
(2.11)

Now set

A p / ( p 1 ) := ( p 1 ) λ M τ 2 ( s ) τ ( s ) ( r ( s ) ρ ( s ) ) 1 / ( p 1 ) H(t,s) ω p / ( p 1 ) (s)

and

B 1 / ( p 1 ) := ( p 1 ) 1 / p ϱ + ( t , s ) ( r ( s ) ρ ( s ) ) 1 / p p ρ ( s ) ( λ M τ 2 ( s ) τ ( s ) ) ( p 1 ) / p .

Letting θ:=p/(p1) and using the inequality (see [22])

θA B θ 1 A θ (θ1) B θ ,θ>1,A0,B0,
(2.12)

we have

ϱ + ( t , s ) ρ ( s ) ( H ( t , s ) ) ( p 1 ) / p ω ( s ) ( p 1 ) λ M τ 2 ( s ) τ ( s ) ( r ( s ) ρ ( s ) ) 1 / ( p 1 ) H ( t , s ) ω p / ( p 1 ) ( s ) 1 p p r ( s ) ( ϱ + ( t , s ) ) p ( λ M τ ( s ) τ 2 ( s ) ρ ( s ) ) p 1 .

Hence, we conclude by (2.11) that, for all sufficiently large t,

1 H ( t , t 3 ) t 3 t [ H ( t , s ) ρ ( s ) i = 1 l q i ( s ) ( 1 a ( τ i ( s ) ) ) p 1 1 p p r ( s ) ( ϱ + ( t , s ) ) p ( λ M τ ( s ) τ 2 ( s ) ρ ( s ) ) p 1 ] d s ω ( t 3 ) ,

which contradicts (2.3).

Case 3. Assume that (3) is satisfied. We also have (2.10). By virtue of Lemma 2.2, we conclude that, for every constant k(0,1), there exists a t k t 1 such that, for all t t k ,

z(t) k 2 t 2 z (t).
(2.13)

Now define

ϕ(t):=δ(t) r ( t ) ( z ( t ) ) p 1 ( z ( t ) ) p 1 ,t t 1 .
(2.14)

Then ϕ(t)<0 for all t t 1 . It follows from (1.1), (2.10), (2.13), and (2.14) that there exists a t 4 t k such that, for all t t 4 ,

ϕ ( t ) = δ ( t ) i = 1 l q i ( t ) ( 1 a ( τ i ( t ) ) ) p 1 z p 1 ( τ ( t ) ) ( z ( τ ( t ) ) ) p 1 ( z ( τ ( t ) ) ) p 1 ( z ( t ) ) p 1 + δ ( t ) δ ( t ) ϕ ( t ) ( p 1 ) ( ϕ ( t ) ) p / ( p 1 ) ( r ( t ) δ ( t ) ) 1 / ( p 1 ) δ ( t ) ( k τ 2 ( t ) 2 ) p 1 i = 1 l q i ( t ) ( 1 a ( τ i ( t ) ) ) p 1 + δ ( t ) δ ( t ) ϕ ( t ) ( p 1 ) ( ϕ ( t ) ) p / ( p 1 ) ( r ( t ) δ ( t ) ) 1 / ( p 1 ) .
(2.15)

Multiplying (2.15) by K(t,s) and integrating the resulting inequality from t 4 to t, we obtain

t 4 t K ( t , s ) δ ( s ) ( k τ 2 ( s ) 2 ) p 1 i = 1 l q i ( s ) ( 1 a ( τ i ( s ) ) ) p 1 d s K ( t , t 4 ) ϕ ( t 4 ) + t 4 t [ K ( t , s ) s + δ ( s ) δ ( s ) K ( t , s ) ] ϕ ( s ) d s ( p 1 ) t 4 t K ( t , s ) ( ϕ ( s ) ) p / ( p 1 ) ( r ( s ) δ ( s ) ) 1 / ( p 1 ) d s K ( t , t 4 ) ϕ ( t 4 ) t 4 t ξ + ( t , s ) δ ( s ) ( K ( t , s ) ) ( p 1 ) / p ϕ ( s ) d s ( p 1 ) t 4 t K ( t , s ) ( ϕ ( s ) ) p / ( p 1 ) ( r ( s ) δ ( s ) ) 1 / ( p 1 ) d s .
(2.16)

Set

A p / ( p 1 ) :=(p1)K(t,s) ( ϕ ( s ) ) p / ( p 1 ) ( r ( s ) δ ( s ) ) 1 / ( p 1 )

and

B 1 / ( p 1 ) := ( p 1 ) 1 / p ( r ( s ) δ ( s ) ) 1 / p ξ + ( t , s ) p δ ( s ) .

Letting θ:=p/(p1) and using inequality (2.12), we have by (2.16) that, for all sufficiently large t,

t 4 t [ K ( t , s ) δ ( s ) ( k τ 2 ( s ) 2 ) p 1 i = 1 l q i ( s ) ( 1 a ( τ i ( s ) ) ) p 1 r ( s ) ( ξ + ( t , s ) ) p p p δ p 1 ( s ) ] d s K ( t , t 4 ) ϕ ( t 4 ) < 0 ,

which contradicts (2.4). This completes the proof. □

Remark 2.4 Choosing different combinations of functions H, ρ, K, and δ, one can derive from Theorem 2.3 a variety of efficient tests for oscillation of equation (1.1) and its particular cases.

3 Example and discussion

The following example illustrates applications of Theorem 2.3.

Example 3.1 For t1 and 0 a 0 <1, consider the fourth-order neutral differential equation

( t 2 ( x ( t ) + a 0 x ( t 2 π ) ) ) +(1+ a 0 ) t 2 x(t3π)+2(1+ a 0 )tx ( t + π 2 ) =0.
(3.1)

Let p=2, τ(t)=t3π, ρ(t)=δ(t)=1, and H(t,s)=K(t,s)= ( t s ) 2 . It is not difficult to verify that all assumptions of Theorem 2.3 are satisfied, and hence equation (3.1) is oscillatory. As a matter of fact, one such solution is x(t)=sint.

Remark 3.2 Oscillation theorem established in this paper for equation (1.1) complements, on one hand, results reported by Baculíková and Džurina [6], Karpuz [7], and Li et al. [9] because we use assumption (1.3) rather than (1.2) and, on the other hand, those by Li et al. [10] and Zhang et al. [16, 17, 1921] since our theorem can be applied to the case where a(t)0.

Remark 3.3 We point out that, contrary to [[11], Theorem 2.2], Theorem 2.3 does not need restrictive conditions (1.4) and can ensure that all solutions of equation (1.1) oscillate, which, in a certain sense, is a significant improvement compared to [[11], Theorem 2.2] for fourth-order neutral differential equations.

Remark 3.4 It would be of interest to study equation (1.1) in the case where

t 0 u R(s)dsdu<

for future research.

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Acknowledgements

The authors express their sincere gratitude to the anonymous referees for the careful reading of the original manuscript and useful comments that helped to improve the presentation of the results and accentuate important details. This research is supported by the National Key Basic Research Program of P.R. China (2013CB035604) and NNSF of P.R. China (Grant Nos. 61034007, 51277116, 51107069).

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Correspondence to Tongxing Li.

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Li, T., Baculíková, B., Džurina, J. et al. Oscillation of fourth-order neutral differential equations with p-Laplacian like operators. Bound Value Probl 2014, 56 (2014) doi:10.1186/1687-2770-2014-56

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Keywords

  • oscillation
  • fourth-order neutral differential equation
  • p-Laplace differential equation
  • noncanonical operator