Open Access

Forced oscillation of second-order differential equations with mixed nonlinearities

Journal of Inequalities and Applications20142014:520

https://doi.org/10.1186/1029-242X-2014-520

Received: 12 August 2014

Accepted: 18 November 2014

Published: 12 December 2014

Abstract

We study oscillatory behavior of a class of second-order forced differential equations with mixed nonlinearities. Some new oscillation theorems are presented that improve and complement those related results given in the literature. An example is provided to illustrate the main results.

MSC:34K11.

Keywords

oscillation forced differential equation second-order mixed nonlinearities

1 Introduction

This paper is concerned with the oscillation of solutions to a class of second-order forced differential equations with mixed nonlinearities
( r x ) ( t ) + q 0 ( t ) x ( τ 0 ( t ) ) + i = 1 n q i ( t ) | x ( τ i ( t ) ) | β i 1 x ( τ i ( t ) ) = e ( t ) sgn ( x ( t ) ) ,
(1.1)

where t t 0 > 0 , n 1 is a natural number, β i 1 ( i = 1 , 2 , , n ) are constants, r C 1 ( [ t 0 , ) , R ) , q j , τ j , e C ( [ t 0 , ) , R ) , r ( t ) > 0 , r ( t ) 0 , q j ( t ) 0 ( j = 0 , 1 , 2 , , n ), e ( t ) 0 . We also assume that there exists a function τ C 1 ( [ t 0 , ) , R ) such that τ ( t ) τ j ( t ) ( j = 0 , 1 , 2 , , n ), τ ( t ) t , lim t τ ( t ) = , and τ ( t ) > 0 .

We consider only those solutions x of equation (1.1) which satisfy condition sup { | x ( t ) | : t T } > 0 for all T t 0 . We assume that (1.1) possesses such solutions. As usual, a solution of (1.1) is called oscillatory if it has arbitrarily large zeros on the interval [ t 0 , ) ; otherwise, it is termed nonoscillatory. Equation (1.1) is said to be oscillatory if all its solutions are oscillatory.

Functional differential equations arise in many applied problems in natural sciences, technology, and automatic control; see, for instance, Hale [1]. In mechanical and engineering problems, questions related to the existence of oscillatory and nonoscillatory solutions play an important role. As a result, many theoretical studies have been undertaken during the past few years. We refer the reader to [212] and the references cited therein.

In what follows, we briefly comment on the related results that motivate our study. Li and Cheng [9] studied a differential equation
( r | x | α 1 x ) ( t ) + q ( t ) | x ( t ) | α 1 x ( t ) = e ( t ) .
Zheng et al. [11] considered the equation
( r | x | α 1 x ) ( t ) + q 0 ( t ) | x ( t ) | α 1 x ( t ) + i = 1 n q i ( t ) | x ( t ) | β i 1 x ( t ) = e ( t ) .

Equation (1.1) was studied by Zhong et al. [12] who established the following oscillation theorem.

Theorem 1.1 (see [[12], Theorem 3.1])

Assume that
t 0 r 1 ( t ) d t =
(1.2)
and there exists a function ρ C 1 ( [ t 0 , ) , ( 0 , ) ) such that
t 0 ( ρ ( t ) Q ( t ) r ( τ ( t ) ) ( ρ + ( t ) ) 2 4 ρ ( t ) τ ( t ) ) d t = ,
(1.3)
where
Q ( t ) : = q 0 ( t ) + i = 1 n β i [ n ( β i 1 ) ] ( 1 β i ) / β i ( q i ( t ) ) 1 / β i | e ( t ) | ( β i 1 ) / β i
and
ρ + ( t ) : = max { 0 , ρ ( t ) } .
(1.4)

Then equation (1.1) is oscillatory.

The purpose of this paper is to refine Theorem 1.1 in some cases and analyze the oscillatory behavior of solutions to (1.1) in the case when the integral in (1.2) is finite. This paper proceeds as follows: in Section 2, we present our main results; in Section 3, an example is provided to illustrate the results obtained.

2 Oscillation criteria

In what follows, all functional inequalities are tacitly assumed to hold eventually, that is, for all t large enough. Before stating the main results, we begin with the following lemma.

Lemma 2.1 (Bernoulli’s inequality)

For y 1 and γ 1 ,
( 1 + y ) γ 1 + γ y .
Theorem 2.2 Assume that condition (1.2) is satisfied, and let
i = 1 n ( 1 β i ) q i ( t ) e ( t ) 0 .
(2.1)
If there exists a function ρ C 1 ( [ t 0 , ) , ( 0 , ) ) such that, for all constants M > 0 ,
lim sup t t 0 t [ ρ ( s ) ( q 0 ( s ) + i = 1 n β i q i ( s ) + i = 1 n ( 1 β i ) q i ( s ) e ( s ) M τ ( s ) ) r ( τ ( s ) ) ( ρ + ( s ) ) 2 4 ρ ( s ) τ ( s ) ] d s = ,
(2.2)

where ρ + is defined as in (1.4), then equation (1.1) is oscillatory.

Proof Assume that (1.1) has a nonoscillatory solution x. Without loss of generality, we can assume that x is an eventually positive solution, i.e., there exists a t 1 t 0 such that x ( t ) > 0 for t t 1 . Equation (1.1) yields
( r x ) ( t ) = q 0 ( t ) x ( τ 0 ( t ) ) i = 1 n q i ( t ) x β i ( τ i ( t ) ) + e ( t ) 0 .
(2.3)
With a proof similar to that of [[12], Theorem 3.1], we conclude that
x ( t ) > 0 , x ( t ) > 0 , x ( t ) 0 , ( r x ) ( t ) 0 .
(2.4)
For t t 1 , define a function
u ( t ) : = ρ ( t ) r ( t ) x ( t ) x ( τ ( t ) ) .
(2.5)
Then u ( t ) > 0 for t t 1 . Differentiating (2.5), by virtue of (2.3) and (2.4), we have x ( τ ( t ) ) r ( t ) x ( t ) / r ( τ ( t ) ) , and so
u ( t ) = ρ ( t ) ρ ( t ) u ( t ) ρ ( t ) r ( t ) x ( t ) x 2 ( τ ( t ) ) x ( τ ( t ) ) τ ( t ) ρ ( t ) q 0 ( t ) x ( τ 0 ( t ) ) x ( τ ( t ) ) ρ ( t ) i = 1 n q i ( t ) x β i ( τ i ( t ) ) e ( t ) x ( τ ( t ) ) ρ + ( t ) ρ ( t ) u ( t ) τ ( t ) u 2 ( t ) ρ ( t ) r ( τ ( t ) ) ρ ( t ) [ q 0 ( t ) + i = 1 n q i ( t ) x β i ( τ i ( t ) ) e ( t ) x ( τ ( t ) ) ] .
(2.6)
Let y : = x ( τ i ( t ) ) 1 . It follows from Lemma 2.1 that
x β i ( τ i ( t ) ) β i x ( τ i ( t ) ) + 1 β i .
(2.7)
Hence, we deduce that
i = 1 n q i ( t ) x β i ( τ i ( t ) ) e ( t ) x ( τ ( t ) ) i = 1 n q i ( t ) [ β i x ( τ i ( t ) ) + ( 1 β i ) ] e ( t ) x ( τ ( t ) ) i = 1 n β i q i ( t ) + i = 1 n ( 1 β i ) q i ( t ) e ( t ) x ( τ ( t ) ) .
(2.8)
By virtue of (2.4), there exists a constant M > 0 such that
x ( t ) M t .
Thus, by (2.8), we obtain
i = 1 n q i ( t ) x β i ( τ i ( t ) ) e ( t ) x ( τ ( t ) ) i = 1 n β i q i ( t ) + i = 1 n ( 1 β i ) q i ( t ) e ( t ) M τ ( t ) .
(2.9)
Substitution of (2.9) into (2.6) implies that
u ( t ) ρ + ( t ) ρ ( t ) u ( t ) τ ( t ) u 2 ( t ) ρ ( t ) r ( τ ( t ) ) ρ ( t ) ( q 0 ( t ) + i = 1 n β i q i ( t ) + i = 1 n ( 1 β i ) q i ( t ) e ( t ) M τ ( t ) ) r ( τ ( t ) ) ( ρ + ( t ) ) 2 4 ρ ( t ) τ ( t ) ρ ( t ) ( q 0 ( t ) + i = 1 n β i q i ( t ) + i = 1 n ( 1 β i ) q i ( t ) e ( t ) M τ ( t ) ) .
Integrating the latter inequality from t 1 to t, we conclude that
t 1 t [ ρ ( s ) ( q 0 ( s ) + i = 1 n β i q i ( s ) + i = 1 n ( 1 β i ) q i ( s ) e ( s ) M τ ( s ) ) r ( τ ( s ) ) ( ρ + ( s ) ) 2 4 ρ ( s ) τ ( s ) ] d s u ( t 1 ) ,

which contradicts (2.2). This completes the proof. □

On the basis of Theorem 2.2, we can obtain the following results due to condition (2.1).

Corollary 2.3 Assume that conditions (1.2) and (2.1) are satisfied. If there exists a function ρ C 1 ( [ t 0 , ) , ( 0 , ) ) such that
lim sup t t 0 t [ ρ ( s ) ( q 0 ( s ) + i = 1 n β i q i ( s ) ) r ( τ ( s ) ) ( ρ + ( s ) ) 2 4 ρ ( s ) τ ( s ) ] d s = ,

where ρ + is defined as in (1.4), then equation (1.1) is oscillatory.

Using ρ ( t ) = t in Corollary 2.3, we can get the following criterion.

Corollary 2.4 Assume that conditions (1.2) and (2.1) are satisfied. If
lim sup t t 0 t [ s ( q 0 ( s ) + i = 1 n β i q i ( s ) ) r ( τ ( s ) ) 4 s τ ( s ) ] d s = ,

then equation (1.1) is oscillatory.

In what follows, we derive some oscillation criteria for (1.1) in the case where
t 0 r 1 ( t ) d t < .
(2.10)
Theorem 2.5 Assume that conditions (2.1) and (2.10) are satisfied, and let τ j ( t ) t ( j = 0 , 1 , 2 , , n ). Suppose also that there exists a function ρ C 1 ( [ t 0 , ) , ( 0 , ) ) such that (2.2) holds. If
lim sup t t 0 t [ δ ( s ) ( q 0 ( s ) + i = 1 n β i q i ( s ) + i = 1 n ( 1 β i ) q i ( s ) e ( s ) K ) 1 4 r ( s ) δ ( s ) ] d s =
(2.11)
holds for all constants K > 0 , where
δ ( t ) : = t r 1 ( s ) d s ,
(2.12)

then equation (1.1) is oscillatory.

Proof Assume that (1.1) has a nonoscillatory solution x. As above, we may assume that there is a t 1 t 0 such that x ( t ) > 0 for t t 1 . By virtue of (1.1), we have (2.3). Then there exist two possible cases, i.e., x ( t ) > 0 or
x ( t ) < 0 .
(2.13)
Assume first that x ( t ) > 0 . Then we obtain (2.4). Proceeding as in the proof of Theorem 2.2, we can obtain a contradiction to (2.2). Suppose now that (2.13) holds. Define a new function ω by
ω ( t ) : = r ( t ) x ( t ) x ( t ) , t t 1 .
(2.14)
Then ω ( t ) < 0 for t t 1 and
ω ( t ) = ( r x ) ( t ) x ( t ) r ( t ) x ( t ) x ( t ) x 2 ( t ) = q 0 ( t ) x ( τ 0 ( t ) ) i = 1 n q i ( t ) x β i ( τ i ( t ) ) + e ( t ) x ( t ) ω 2 ( t ) r ( t ) .
(2.15)
On the other hand, we have (2.7), and so
i = 1 n q i ( t ) x β i ( τ i ( t ) ) e ( t ) x ( t ) i = 1 n q i ( t ) [ β i x ( τ i ( t ) ) + ( 1 β i ) ] e ( t ) x ( t ) i = 1 n β i q i ( t ) + i = 1 n ( 1 β i ) q i ( t ) e ( t ) x ( t )
(2.16)
due to τ i ( t ) t ( i = 1 , 2 , , n ). By (2.13), there exists a constant K > 0 such that x ( t ) K . Hence, by virtue of (2.16), we conclude that
i = 1 n q i ( t ) x β i ( τ i ( t ) ) e ( t ) x ( t ) i = 1 n β i q i ( t ) + i = 1 n ( 1 β i ) q i ( t ) e ( t ) K .
(2.17)
It follows now from (2.15), (2.17), and τ 0 ( t ) t that
ω ( t ) q 0 ( t ) i = 1 n β i q i ( t ) i = 1 n ( 1 β i ) q i ( t ) e ( t ) K ω 2 ( t ) r ( t ) .
(2.18)
Using the condition ( r x ) ( t ) 0 , we have, for s t ,
x ( s ) r ( t ) x ( t ) r ( s ) .
Integrating the latter inequality from t to l, we deduce that
x ( l ) x ( t ) r ( t ) x ( t ) t l r 1 ( s ) d s .
Passing to the limit as l , we have
x ( t ) r ( t ) x ( t ) δ ( t ) ,
which yields
r ( t ) x ( t ) x ( t ) δ ( t ) 1 ,
i.e.,
ω ( t ) δ ( t ) 1 .
(2.19)
Multiplying (2.18) by δ ( t ) and integrating the resulting inequality from t 1 to t, we obtain
ω ( t ) δ ( t ) ω ( t 1 ) δ ( t 1 ) + t 1 t δ ( s ) ( q 0 ( s ) + i = 1 n β i q i ( s ) + i = 1 n ( 1 β i ) q i ( s ) e ( s ) K ) d s + t 1 t ω ( s ) r ( s ) d s + t 1 t ω 2 ( s ) δ ( s ) r ( s ) d s 0 .
Hence, we derive from (2.19) that
t 1 t [ δ ( s ) ( q 0 ( s ) + i = 1 n β i q i ( s ) + i = 1 n ( 1 β i ) q i ( s ) e ( s ) K ) 1 4 r ( s ) δ ( s ) ] d s 1 + ω ( t 1 ) δ ( t 1 ) ,

which contradicts (2.11). The proof is complete. □

Theorem 2.6 Assume that conditions (2.1) and (2.10) are satisfied, and let τ j ( t ) t ( j = 0 , 1 , 2 , , n ). Suppose further that there exists a function ρ C 1 ( [ t 0 , ) , ( 0 , ) ) such that (2.2) holds. If, for all constants K > 0 ,
lim sup t t 0 t [ δ ( s ) ( q 0 ( s ) δ ( τ 0 ( s ) ) δ ( s ) + i = 1 n β i q i ( s ) δ ( τ i ( s ) ) δ ( s ) ) + δ ( s ) i = 1 n ( 1 β i ) q i ( s ) e ( s ) K 1 4 r ( s ) δ ( s ) ] d s = ,
(2.20)

where δ is as in (2.12), then equation (1.1) is oscillatory.

Proof Assume again that there exists a t 1 t 0 such that x ( t ) > 0 for t t 1 . From (1.1), we have (2.3). Then there exist two possible cases, i.e., x ( t ) > 0 or (2.13). Suppose that x ( t ) > 0 . Following the same lines as in Theorem 2.2, we can obtain a contradiction to (2.2). Assume now that (2.13) is satisfied. Define the function ω by (2.14). We have ω ( t ) < 0 for t t 1 and (2.15). On the other hand, it has been established in Theorems 2.2 and 2.5 that (2.7) and (2.19) hold. By virtue of (2.19),
( x δ ) ( t ) 0 .
It follows from the latter inequality, τ j ( t ) t ( j = 0 , 1 , 2 , , n ), and (2.7) that
x ( τ 0 ( t ) ) x ( t ) δ ( τ 0 ( t ) ) δ ( t )
(2.21)
and
i = 1 n q i ( t ) x β i ( τ i ( t ) ) e ( t ) x ( t ) i = 1 n q i ( t ) [ β i x ( τ i ( t ) ) + ( 1 β i ) ] e ( t ) x ( t ) i = 1 n β i q i ( t ) δ ( τ i ( t ) ) δ ( t ) + i = 1 n ( 1 β i ) q i ( t ) e ( t ) x ( t ) .
(2.22)
Since x ( t ) < 0 , there exists a constant K > 0 such that x ( t ) K . Hence, by (2.22), we conclude that
i = 1 n q i ( t ) x β i ( τ i ( t ) ) e ( t ) x ( t ) i = 1 n β i q i ( t ) δ ( τ i ( t ) ) δ ( t ) + i = 1 n ( 1 β i ) q i ( t ) e ( t ) K .
(2.23)
Using (2.15), (2.21), and (2.23), we obtain
ω ( t ) q 0 ( t ) δ ( τ 0 ( t ) ) δ ( t ) i = 1 n β i q i ( t ) δ ( τ i ( t ) ) δ ( t ) i = 1 n ( 1 β i ) q i ( t ) e ( t ) K ω 2 ( t ) r ( t ) .
(2.24)

The remainder of the proof is similar to that of Theorem 2.5 and hence is omitted. This completes the proof. □

Remark 2.7 From the proof of Theorems 2.5 and 2.6, one can obtain oscillation results for equation (1.1) with delayed and advanced arguments. The details are left to the reader.

3 Example

The following example illustrates possible applications of the theoretical results presented in this paper.

Example 3.1 For t 1 , consider a second-order differential equation
x ( t ) + γ t 2 x ( t 2 ) + 1 t 2 | x ( t 5 ) | x ( t 5 ) + 1 t 2 x 3 ( t 40 ) = 3 t 2 sgn ( x ( t ) ) ,
(3.1)
where γ > 0 is a constant. Let n = 2 , r ( t ) = 1 , q 0 ( t ) = γ / t 2 , q 1 ( t ) = q 2 ( t ) = 1 / t 2 , τ 0 ( t ) = t / 2 , τ 1 ( t ) = t / 5 , τ ( t ) = τ 2 ( t ) = t / 40 , e ( t ) = 3 / t 2 , β 1 = 2 , and β 2 = 3 . Then i = 1 n ( 1 β i ) q i ( t ) e ( t ) = 0 and
lim sup t t 0 t [ s ( q 0 ( s ) + i = 1 n β i q i ( s ) ) r ( τ ( s ) ) 4 s τ ( s ) ] d s = ( γ 5 ) lim sup t 1 t d s s = , provided that  γ > 5 .

Hence, by Corollary 2.4, equation (3.1) is oscillatory for any γ > 5 .

Let Q be defined as in Theorem 1.1. Then
Q ( t ) = q 0 ( t ) + i = 1 n β i [ n ( β i 1 ) ] ( 1 β i ) / β i ( q i ( t ) ) 1 / β i | e ( t ) | ( β i 1 ) / β i = 1 t 2 [ γ + i = 1 2 β i ( 2 ( β i 1 ) 3 ) ( 1 β i ) / β i ] = 1 t 2 [ γ + 2 ( 3 2 ) 1 / 2 + 3 ( 3 4 ) 2 / 3 ] < 1 t 2 ( γ + 2.45 + 2.477 ) = 1 t 2 ( γ + 4.927 ) .

Using the latter inequality and ρ ( t ) = t in (1.3), we observe that Theorem 1.1 cannot ensure oscillation of (3.1) on the interval ( 5 , 5.073 ] . Therefore, Corollary 2.4 improves Theorem 1.1.

Remark 3.2 In this paper, several new oscillation criteria for equation (1.1) are obtained by using the Riccati substitution and Bernoulli’s inequality. Employing inequalities different from those exploited in [12], we improve Theorem 1.1; see Example 3.1. Furthermore, Theorems 2.5 and 2.6 complement those by Zhong et al. [12] since our results can be applied to the case where (2.10) holds.

Declarations

Acknowledgements

The authors express their sincere gratitude to the editors and 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 AMEP of Linyi University, P.R. China.

Authors’ Affiliations

(1)
School of Informatics, Linyi University, Linyi
(2)
LinDa Institute of Shandong Provincial Key Laboratory of Network Based Intelligent Computing, Linyi University
(3)
Ramanujan Institute for Advanced Study in Mathematics, University of Madras

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