On the oscillation of nonlinear delay differential equations and their applications

Omar Bazighifan; Sameh Askar

doi:10.1515/phys-2021-0097

Article Open Access

On the oscillation of nonlinear delay differential equations and their applications

Omar Bazighifan and Sameh Askar

Published/Copyright: December 23, 2021

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From the journal Open Physics Volume 19 Issue 1

Abstract

The oscillation of nonlinear differential equations is used in many applications of mathematical physics, biological and medical physics, engineering, aviation, complex networks, sociophysics and econophysics. The goal of this study is to create some new oscillation criteria for fourth-order differential equations with delay and advanced terms

( a 1 ( x ) ( w ‴ ( x ) ) n ) ′ + ∑ j = 1 r β j ( x ) w k ( γ j ( x ) ) = 0 ,

and

( a 1 ( x ) ( w ‴ ( x ) ) n ) ′ + a 2 ( x ) h ( w ‴ ( x ) ) + β ( x ) f ( w ( γ ( x ) ) ) = 0 .

The method is based on the use of the comparison technique and Riccati method to obtain these criteria. These conditions complement and extend some of the results published on this topic. Two examples are provided to prove the efficiency of the main results.

Keywords: Riccati method; advanced term; oscillation; fourth-order; delay differential equations; damped

1 Introduction

Nonlinear differential equations with delay term contribute to some applications in applications of physics, acoustics, biological and medical physics, engineering, aviation, complex networks, sociophysics and econophysics, see refs [1,2].

Nowadays, the questions regarding the study of oscillation criteria of differential equations have become an important area of research [3,4].

The authors in refs [5,6,7] discuss some oscillation theories of differential equations of second-order and they used different techniques to obtain the qualitative properties of these equations. In refs [8,9] the authors discussed some oscillation criteria for higher-order differential equations and used comparison techniques with equations of different orders. Agarwal et al. [10], Chatzarakis and Li [11], and Bazighifan et al. [12] obtained many criteria for oscillation and qualitative properties of differential equations of different orders. Conversely, there has been a great research interest about the study of fractional differential equations and numerical solutions to these equations [13,14,15] and fractional-ordered differential systems, see refs [16,17,18].

Park et al. [19] establish some qualitative properties of the equation:

(1) ( a 1 ( x ) ( w ( n − 1 ) ( x ) ) n ) ′ + β ( x ) w k ( γ ( x ) ) = 0 ,

where k ≤ n and under condition

∫ x 0 ∞ a 1 − 1 / n ( s ) d s = ∞ .

In ref. [20], Zhang et al. obtained some oscillatory conditions for (1) under

(2) ∫ x 0 ∞ a 1 − 1 / n ( s ) d s < ∞ .

Baculikova et al. [21] used the comparison technique to find the parameters of the oscillation of the equation:

[ a 1 ( x ) ( z ( r − 1 ) ( x ) ) n ] ′ + β ( x ) f ( z ( γ ( x ) ) ) = 0 .

Moreover, Moaaz and Muhib [22,23] presented some oscillation theorems of (1). Also, Zhang et al. [24] establish oscillation conditions for (1), where n = k .

Zhang et al. [25] obtained some oscillatory theorems by the following equation:

[ a 1 ( x ) ( w ‴ ( x ) ) n ] ′ + β ( x ) f ( w ( γ ( x ) ) ) = 0 ,

where n = 4 . Bazighifan [26] obtained some oscillatory conditions by the following equation:

[ a 1 ( x ) ( w ‴ ( x ) ) n ] ′ + β ( x ) w k ( γ ( x ) ) = 0 .

The authors in ref. [27] use the comparison technique to find oscillation theorems of (4), where n = k = 1 and under the condition (6).

Motivated by the results we presented earlier, the oscillation conditions for the following equations are studied:

(3) ( a 1 ( x ) ( w ‴ ( x ) ) n ) ′ + ∑ j = 1 r β j ( x ) w k ( γ j ( x ) ) = 0

and

(4) ( a 1 ( x ) ( w ‴ ( x ) ) n ) ′ + a 2 ( x ) h ( w ‴ ( x ) ) + β ( x ) f ( w ( γ ( x ) ) ) = 0 .

Throughout, the following hypotheses have been assumed:

a 1 , a 2 , β ∈ C ( [ x 0 , ∞ ) , [ 0 , ∞ ) ) , γ j ( x ) ∈ C ( [ x 0 , ∞ ) , R ) , n and k are quotient of odd positive integers,
a 1 ′ ( x ) + a 2 ( x ) ≥ 0 , a 1 ( x ) > 0 , β j ( x ) > 0 , γ j ( x ) ≤ x , lim x → ∞ γ j ( x ) = ∞ , j = 1 , 2 , … , r ,
h , f ∈ ( R , R ) , h ( u ) ≥ b h u n > 0 , f ( u ) ≥ b f u n > 0 for u ≠ 0 and b h , b f are constants.

Moreover, equation (3) has been studied under the condition:

(5) ∫ x 0 ∞ 1 a 1 1 / n ( s ) d s = ∞ ,

and equation (4) has been studied under the conditions γ ( x ) ≥ x and

(6) ∫ x 0 ∞ 1 a 1 ( s ) exp − ∫ x 0 s a 2 ( u ) a 1 ( u ) d u 1 / n d s < ∞ .

Definition 1.1

[12] A solution of (3) and (4) is said to be non-oscillatory if it is positive or negative, ultimately; otherwise, it is said to be oscillatory.

Definition 1.2

[12] Equations (3) and (4) are said oscillatory if each of their solutions is oscillatory.

The motivation of studying this article is to obtain new oscillatory properties and to continue the previous works [20,27].

In this article, the Riccati method was used, which depends on reducing the order of the equation. The comparison technique with a first-order equation has also been relied upon, so these methods give more accurate criteria. These conditions expand and complement some of the previously published results.

2 Oscillation criteria

Here are some lemmas we need to prove the results:

Lemma 2.1

[28] Let w ∈ C m ( [ x 0 , ∞ ) , ( 0 , ∞ ) ) and w ( m ) ( x ) is of a fixed sign, on [ x 0 , ∞ ) such that, for all x ≥ x 1 ,

w ( m − 1 ) ( x ) w ( m ) ( x ) ≤ 0 .

If we have lim x → ∞ w ( x ) ≠ 0 and λ ∈ ( 0 , 1 ) , then

w ( x ) ≥ λ ( m − 1 ) ! x m − 1 ∣ w ( m − 1 ) ( x ) ∣ .

Lemma 2.2

[29] If w ( r ) ( x ) > 0 , r = 0 , 1 , … , r , and w ( r + 1 ) ( x ) < 0 , then

w ( x ) x r / r ! ≥ w ′ ( x ) x r − 1 / ( r − 1 ) ! .

Lemma 2.3

([3], Lemma 2.2.2) Let w ∈ C m ( [ x 0 , ∞ ) , ( 0 , ∞ ) ) and w ( m − 1 ) ( x ) w ( m ) ( x ) ≤ 0 , then

w ( θ x ) ≥ M t m − 1 w ( m − 1 ) ( x ) ,

for θ ∈ ( 0 , 1 ) , there exists M > 0 and for all sufficient large x .

Lemma 2.4

[30] Let F > 0 . Then

(7) E u − F u ( r + 1 ) / r ≤ r r ( r + 1 ) r + 1 E r + 1 F − r .

For convenience, we write the following notations:

R ( x ) ≔ ∫ x ∞ 1 a 1 ( x ) ∫ x ∞ ∑ j = 1 r β j ( s ) d s 1 / n d x , R ˜ ( x ) ≔ μ 2 k / n ∫ x ∞ 1 a 1 ( x ) ∫ x ∞ ∑ j = 1 r β j ( s ) γ j ( s ) s k d s 1 / n d x , ζ x 0 ( x ) ≔ exp ∫ x 0 x a 2 ( x ) a 1 ( x ) d x

and

R ^ ( x ) ≔ μ 2 k / n ∫ x ∞ 1 a 1 ( x ) ζ x 0 ( x ) ∫ x ∞ ζ x 0 ( x ) ∑ j = 1 r β j ( s ) γ j ( s ) s k d s 1 / n d x ,

where μ 2 ∈ ( 0 , 1 ) .

γ ( x 0 , x ) = exp ∫ x 0 x a 2 ( u ) a 1 ( u ) d u , δ ( x ) = ∫ x ∞ d s ( a 1 ( s ) γ ( x 0 , s ) ) 1 n , χ ( x ) = ζ ′ ( x ) ζ ( x ) − b h a 2 ( x ) a 1 ( x ) , σ ( x ) = 1 γ 1 n ( x 0 , x ) − δ ( x ) a 2 ( x ) a 1 1 − n / n ( x ) n

and

σ ˜ ( x ) = a 2 ( x ) a 1 ( x ) + n ( n + 1 ) ζ ( x ) σ n + 1 ( x ) γ ( x 0 , x ) δ ( x ) a 1 1 n ( x ) .

Lemma 2.5

Let w is an eventually positive solution of (3), then w ′ > 0 and w ‴ > 0 .

Theorem 2.6

(8) y ′ ( x ) + λ k 6 k ∑ j = 1 r β j ( x ) γ j 3 k ( x ) a 1 k / n ( γ j ( x ) ) y k / n ( γ j ( x ) ) = 0

is oscillatory, then (3) is oscillatory.

Proof

Suppose that (3) has a nonoscillatory solution in [ x 0 , ∞ ) . Then w ( x ) > 0 and w ( γ j ( x ) ) > 0 for x ≥ x 1 . Let

y ( x ) ≔ a 1 ( x ) ( w ‴ ( x ) ) n > 0 [by Lemma 2.5] .

Then from (3), we obtain

(9) y ′ ( x ) + ∑ j = 1 r β j ( x ) w k ( γ j ( x ) ) = 0 .

lim x → ∞ w ( x ) ≠ 0 . By Lemma 2.1, we see

(10) w k ( γ j ( x ) ) ≥ λ k 6 k γ j 3 k ( x ) ( w ‴ ( γ j ( x ) ) ) k ,

for all λ ∈ ( 0 , 1 ) . By (9) and (10), we see that

y ′ ( x ) + λ k 6 k ∑ j = 1 r β j ( x ) γ j 3 k ( x ) ( w ‴ ( γ j ( x ) ) ) k ≤ 0 .

So, we obtain y ( x ) > 0 and

y ′ ( x ) + λ k 6 k ∑ j = 1 r β j ( x ) γ j 3 k ( x ) a 1 k / n ( γ j ( x ) ) y k / n ( γ j ( x ) ) ≤ 0 .

From ([31], Theorem 1), equation (8) is nonoscillatory. This is a contradiction. Proof completed.□

Corollary 2.7

If n = k and

(11) lim inf x → ∞ ∫ γ ( x ) x λ k 6 k ∑ j = 1 r β j ( s ) γ j 3 k ( s ) a 1 k / n ( γ j ( s ) ) d s > 1 e ,

then (3) is oscillatory.

Lemma 2.8

(12) ∫ x 0 ∞ M k − n π ( x ) ∑ j = 1 r β j ( x ) γ j 3 n ( x ) x 3 n − 2 n ( n + 1 ) n + 1 a 1 ( x ) ( π ′ ( x ) ) n + 1 μ n x 2 n π n ( x ) d s = ∞ ,

then w ″ < 0 .

Proof

Let w ″ ( x ) > 0 . From Lemmas 2.1 and 2.2, we find

(13) w ( γ j ( x ) ) w ( x ) ≥ γ j 3 ( x ) x 3

and

(14) w ′ ( x ) ≥ μ 2 x 2 w ‴ ( x ) .

Let

(15) z 1 ( x ) ≔ π ( x ) a 1 ( x ) ( w ‴ ( x ) ) n w n ( x ) > 0 .

From (13)–(15), we find

(16) z 1 ′ ( x ) ≤ π ′ ( x ) π ( x ) z 2 ( x ) − π ( x ) ∑ j = 1 r β j ( x ) γ j 3 n ( x ) x 3 n w k − n ( γ j ( x ) ) − n μ 2 x 2 π 1 / n ( x ) a 1 1 / n ( x ) z 1 1 + 1 / n ( x ) .

Since w ′ ( x ) > 0 . From Lemma 2.4 with E = π ′ / π , F = n μ x 2 / ( 2 a 1 1 / n ( x ) π 1 / n ( x ) ) , and u = z 1 , we see that

z 1 ′ ( x ) ≤ − M k − n π ( x ) ∑ j = 1 r β j ( x ) γ j 3 n ( x ) x 3 n + 2 n ( n + 1 ) n + 1 a 1 ( x ) ( π ′ ( x ) ) n + 1 μ n x 2 n π n ( x ) .

This implies that

∫ x 1 x M k − n π ( x ) ∑ j = 1 r β j ( x ) γ j 3 n ( x ) x 3 n − 2 n ( n + 1 ) n + 1 a 1 ( x ) ( π ′ ( x ) ) n + 1 μ n x 2 n π n ( x ) d s ≤ z 1 ( x 1 ) ,

This is a contradiction (12). Proof completed.□

Theorem 2.9

(17) u ″ ( x ) + M k − n R ˜ ( x ) u ( x ) = 0

is oscillatory, then (3) is oscillatory.

Proof

From Theorem 2.6 and by Lemmas 2.1 and 2.5, we find

(18) w ′ ( x ) > 0 , w ″ ( x ) < 0 and w ‴ ( x ) > 0 .

Integrating (3), we find

(19) a 1 ( b ) ( w ‴ ( b ) ) n = a 1 ( x ) ( w ‴ ( x ) ) n − ∫ x b ∑ j = 1 r β j ( s ) w k ( γ j ( s ) ) d s .

By Lemma 3 in ref [30] with (18), we obtain

w ( γ j ( x ) ) w ( x ) ≥ λ γ j ( x ) x ,

which with (19) gives

a 1 ( b ) ( w ‴ ( b ) ) n − a 1 ( x ) ( w ‴ ( x ) ) n + λ k ∫ x b ∑ j = 1 r β j ( s ) γ j ( s ) s k w k ( s ) d s ≤ 0 .

By w ′ > 0 , we see

(20) a 1 ( b ) ( w ‴ ( b ) ) n − a 1 ( x ) ( w ‴ ( x ) ) n + λ k w k ( x ) ∫ x b ∑ j = 1 r β j ( s ) γ j ( s ) s k d s ≤ 0 .

Taking b → ∞ , we find

− a 1 ( x ) ( w ‴ ( x ) ) n + λ k w k ( x ) ∫ x ∞ ∑ j = 1 r β j ( s ) γ j ( s ) s k d s ≤ 0 ,

that is,

w ‴ ( x ) ≥ λ k / n a 1 1 / n ( x ) w k / n ( x ) ∫ x ∞ ∑ j = 1 r β j ( s ) γ j ( s ) s k d s 1 / n .

Integrating from x to ∞ , we see

− w ″ ( x ) ≥ λ k / n w k / n ( x ) ∫ x ∞ 1 a 1 ( x ) ∫ x ∞ ∑ j = 1 r β j ( s ) γ j ( s ) s k d s 1 / n d x .

Hence,

(21) w ″ ( x ) ≤ − R ˜ ( x ) w k / n ( x ) .

Let

z 2 ( x ) = w ′ ( x ) w ( x ) ,

then z 2 ( x ) > 0 for x ≥ x 1 , and

z 2 ′ ( x ) = w ″ ( x ) w ( x ) − w ′ ( x ) w ( x ) 2 .

From (21), we find

(22) z 2 ′ ( x ) ≤ − R ˜ ( x ) w k / n ( x ) w ( x ) − z 2 2 ( x ) .

Since w ′ ( x ) > 0 . Thus, (22) becomes

(23) z 2 ′ ( x ) + z 2 2 ( x ) + M k − n R ˜ ( x ) ≤ 0 ,

From [32], equation (17) is nonoscillatory, and this is a contradiction. Proof completed.□

Theorem 2.10

Suppose that k ≥ n and

(24) 1 γ j ′ ( x ) u ′ ( x ) ′ + M k / n − 1 R ( x ) u ( x ) = 0

is oscillatory, and then (3) is oscillatory.

Proof

Suppose that (12) and (19) hold. So, we see γ j ′ ( x ) ≥ 0 and w ′ ( x ) ≥ 0 .

(25) a 1 ( b ) ( w ‴ ( b ) ) n − a 1 ( x ) ( w ‴ ( x ) ) n + w k ( γ j ( x ) ) ∫ x b ∑ j = 1 r β j ( s ) d s ≤ 0 .

Thus, (18) becomes

(26) w ″ ( x ) ≤ − R ( x ) w k / n ( γ j ( x ) ) .

Let

(27) z 3 ( x ) = w ′ ( x ) w ( γ j ( x ) ) ,

then z 3 ( x ) > 0 for x ≥ x 1 , and

z 3 ′ ( x ) = w ″ ( x ) w ( γ j ( x ) ) − w ′ ( x ) w 2 ( γ j ( x ) ) w ′ ( γ j ( x ) ) γ j ′ ( x ) ≤ w ″ ( x ) w ( γ j ( x ) ) − γ j ′ ( x ) w ′ ( x ) w ( γ j ( x ) ) 2 .

From (26) and (27), we find

(28) z 3 ′ ( x ) + M k / n − 1 R ( x ) + γ j ′ ( x ) z 3 2 ( x ) ≤ 0 .

From [32], (24) is nonoscillatory, and this is a contradiction. This completes the proof.□

Lemma 2.11

([24], Theorem 2.1) Suppose that w is an eventually positive solution of (4). Then, there exist two possible cases:

( N 1 ) w ( x ) > 0 , w ′ ( x ) > 0 , w ‴ ( x ) > 0 , w ( 4 ) ( x ) < 0 ; ( N 2 ) w ( x ) > 0 , w ″ ( x ) > 0 , w ‴ ( x ) < 0 .

for x ≥ x 1 .

Lemma 2.12

Let w ( x ) > 0 and ( N 1 ) holds. If

(29) z 4 ( x ) = ζ ( x ) a 1 ( x ) ( w ‴ ) n ( x ) w n ( x / 2 ) > 0 ,

where ζ ∈ C 1 ( [ x 0 , ∞ ) , ( 0 , ∞ ) ) and M > 0 is a constant, then

(30) z 4 ′ ( x ) ≤ − b f ζ ( x ) β ( x ) + χ ( x ) z 4 ( x ) − n M t 2 2 ( a 1 ( x ) ζ ( x ) ) 1 / n z 4 n + 1 n ( x ) .

Proof

Suppose that w ( x ) > 0 and using Lemma 2.11, we get ( N 1 ) holds. From Lemma 2.3, we see

(31) w ′ ( x / 2 ) ≥ M t 2 w ‴ ( x ) .

From z 4 ( x ) , we obtain

z 4 ′ ( x ) = ζ ′ ( x ) a 1 ( x ) ( w ‴ ) n ( x ) w n ( x / 2 ) + ζ ( x ) ( a 1 ( w ‴ ) n ) ′ ( x ) w n ( x / 2 ) − n ζ ( x ) w ′ ( x / 2 ) a 1 ( x ) ( w ‴ ) n ( x ) 2 w n + 1 ( x / 2 ) .

Using (29) and (31), we find

z 4 ′ ( x ) ≤ ζ ′ ( x ) ζ ( x ) z 4 ( x ) + ζ ( x ) ( a 1 ( w ‴ ) n ) ′ ( x ) w n ( x / 2 ) − n M t 2 ζ ( x ) a 1 ( x ) ( w ‴ ) n + 1 ( x ) 2 w n + 1 ( x / 2 ) .

From (4), we see

z 4 ′ ( x ) ≤ ζ ′ ( x ) ζ ( x ) z 4 ( x ) − b h a 2 ( x ) z 4 ( x ) a 1 ( x ) − b f ζ ( x ) β ( x ) w n ( γ ( x ) ) w n ( x / 2 ) − n M t 2 z 4 n + 1 n ( x ) 2 ( ζ ( x ) a 1 ( x ) ) 1 / n ≤ − b f ζ ( x ) β ( x ) + ζ ′ ( x ) ζ ( x ) − b h a 2 ( x ) a 1 ( x ) z 4 ( x ) − n M t 2 z 4 n + 1 n ( x ) 2 ( ζ ( x ) a 1 ( x ) ) 1 / n .

Hence, we find

z 4 ′ ( x ) ≤ − b f ζ ( x ) β ( x ) + χ ( x ) z 4 ( x ) − n M t 2 z 4 n + 1 n ( x ) 2 ( ζ ( x ) a 1 ( x ) ) 1 / n .

The proof is complete.□

Lemma 2.13

Let b f > 1 is a constant and ( N 2 ) holds. If

(32) z 5 ( x ) = − a 1 ( x ) ( − w ‴ ) n ( x ) ( w ″ ) n ( x ) < 0 ,

then

(33) z 5 ′ ( x ) ≤ b h a 2 ( x ) a 1 ( x ) δ n ( x ) γ ( x 0 , x ) − b f β ( x ) μ 2 γ 2 ( x ) n − n z 5 n + 1 n ( x ) a 1 1 n ( x ) .

Proof

Let that w ( x ) > 0 and ( N 2 ) holds. Since

( − a 1 ( x ) ( − w ‴ ( x ) ) n γ ( x 0 , x ) ) ′ = ( − a 1 ( x ) ( − w ‴ ( x ) ) n ) ′ γ ( x 0 , x ) + ( − a 1 ( x ) ( − w ‴ ( x ) ) n ) γ ( x 0 , x ) a 2 ( x ) a 1 ( x ) = ( − 1 ) n + 1 ( − a 2 ( x ) h ( w ‴ ( x ) ) − β ( x ) f ( w ( γ ( x ) ) ) ) γ ( x 0 , x ) − a 2 ( x ) ( − w ‴ ( x ) ) n γ ( x 0 , x ) ≤ ( − 1 ) n + 1 ( − b h a 2 ( x ) ( w ‴ ( x ) ) n − b f β ( x ) w n ( γ ( x ) ) ) γ ( x 0 , x ) − a 2 ( x ) ( − w ‴ ( x ) ) n γ ( x 0 , x ) = ( − a 2 ( x ) ( − w ‴ ( x ) ) n ( 1 − b h ) + b f β ( x ) ( − w n ( γ ( x ) ) ) ) γ ( x 0 , x ) = ( − 1 ) n ( − a 2 ( x ) ( w ‴ ( x ) ) n ( 1 − b h ) + b f β ( x ) ( w n ( γ ( x ) ) ) ) γ ( x 0 , x ) ≤ − b f β ( x ) w n ( γ ( x ) ) γ ( x 0 , x ) < 0 ,

we deduce that − a 1 ( x ) ( − w ‴ ( x ) ) n γ ( x 0 , x ) is decreasing. Thus, for s ≥ x ≥ x 1 ,

(34) ( a 1 ( s ) γ ( x 0 , s ) ) 1 / n w ‴ ( s ) ≤ ( a 1 ( x ) γ ( x 0 , x ) ) 1 / n w ‴ ( x ) .

Dividing both sides of (34) by ( a 1 ( s ) γ ( x 0 , s ) ) 1 / n and integrating from x to b , we get

w ″ ( b ) ≤ w ″ ( x ) + ( a 1 ( s ) γ ( x 0 , s ) ) 1 / n w ‴ ( x ) ∫ x b d s ( a 1 ( s ) γ ( x 0 , s ) ) 1 / n .

Letting u → ∞ , we arrive that

0 ≤ w ″ ( x ) + ( a 1 ( x ) γ ( x 0 , x ) ) 1 / n w ‴ ( x ) δ ( x ) ,

which yields

− w ‴ ( x ) w ″ ( x ) δ ( x ) ( a 1 ( x ) γ ( x 0 , x ) ) 1 / n ≤ 1 .

Hence,

a 1 ( x ) ( w ‴ ( x ) ) n ( w ″ ( x ) ) n ≥ − 1 δ n ( x ) γ ( x 0 , x ) .

From (32), we find

(35) z 5 ( x ) ≥ − 1 δ n ( x ) γ ( x 0 , x ) .

and

z 5 ′ ( x ) = ( − a 1 ( x ) ( − w ‴ ( x ) ) n ) ′ ( w ″ ( x ) ) n − n − a 1 ( x ) ( − w ‴ ( x ) ) n + 1 ( w ″ ( x ) ) n + 1 .

From (4) and (32), we obtain

(36) z 5 ′ ( x ) = − b h a 2 ( x ) a 1 ( x ) z 5 ( x ) − b f β ( x ) w n ( γ ( x ) ) ( w ″ ( x ) ) n − n z 5 n + 1 n ( x ) a 1 1 n ( x ) . = − b h a 2 ( x ) a 1 ( x ) z 5 ( x ) − b f β ( x ) w n ( γ ( x ) ) ( w ″ ( γ ( x ) ) ) n ( w ″ ( γ ( x ) ) ) n ( w ″ ( x ) ) n − n z 5 n + 1 n ( x ) a 1 1 n ( x ) .

By Lemma 2.1, we find

(37) w ( x ) ≥ μ 2 x 2 w ″ ( x ) .

Thus, from (35) and (37), we see

z 5 ′ ( x ) ≤ b h a 2 ( x ) a 1 ( x ) δ n ( x ) γ ( x 0 , x ) − b f β ( x ) μ 2 γ 2 ( x ) n − n z 5 n + 1 n ( x ) a 1 1 n ( x ) .

The proof is complete.□

Theorem 2.14

Let the functions ζ , ϑ ∈ C 1 ( [ x 0 , ∞ ) , ( 0 , ∞ ) ) such that

(38) lim sup x → ∞ ∫ x 0 x b f ζ ( s ) β ( s ) − 2 M s 2 n a 1 ( s ) ζ ( s ) ( χ ( s ) ) n + 1 ( n + 1 ) n + 1 d s = ∞ .

(39) ϑ ( x ) δ ( x ) ( a 1 ( x ) γ ( x 0 , x ) ) 1 / n + ϑ ′ ( x ) ≤ 0

and, for M > 0 and μ ∈ ( 0 , 1 ) ,

(40) lim sup x → ∞ ∫ x 0 x b f β ( s ) μ γ 2 ( s ) 2 ϑ ( γ ( s ) ) ϑ ( s ) δ ( s ) n γ ( x 0 , s ) − σ ˜ ( s ) d s = ∞ ,

then every solution of (4) is oscillatory.

Proof

Proceeding as in the proof of Theorem 2.6.

For case ( N 1 ) . By Lemma 2.12, we find (30) holds. From Lemma 2.4, we set

F = χ ( x ) , E = n M t 2 / ( 2 ( a 1 ( x ) ζ ( x ) ) 1 / n ) and u = z 4 ,

we have

(41) z 4 ′ ( x ) ≤ − b f ζ ( x ) β ( x ) + 2 M t 2 n a 1 ( x ) ζ ( x ) ( χ ( x ) ) n + 1 ( n + 1 ) n + 1 .

Integrating from x 1 to x , we obtain

∫ x 1 x b f ζ ( s ) β ( s ) − 2 M s 2 n a 1 ( s ) ζ ( s ) ( χ ( s ) ) n + 1 ( n + 1 ) n + 1 d s ≤ z 4 ( x 1 ) ,

which contradicts (38).

For case ( N 2 ) . From the proof of Lemma 2.13, we see

w ‴ ( x ) w ″ ( x ) ≥ − 1 δ ( x ) ( a 1 ( x ) γ ( x 0 , x ) ) 1 / n .

From (39), we obtain

w ″ ( x ) ϑ ( x ) ′ = w ‴ ( x ) ϑ ( x ) − w ″ ( x ) ϑ ′ ( x ) ϑ 2 ( x ) ≥ w ″ ( x ) ϑ 2 ( x ) ϑ ( x ) δ ( x ) ( a 1 ( x ) γ ( x 0 , x ) ) 1 / n + ϑ ′ ( x ) ≥ 0 ,

and this means that w ″ ( x ) / ϑ ( x ) is nondecreasing. So, it follows from γ ( x ) ≥ x that

w ″ ( γ ( x ) ) w ″ ( x ) ≥ ϑ ( γ ( x ) ) ϑ ( x ) .

Thus, by (36) and (37), we get

(42) z 5 ′ ( x ) ≤ b h a 2 ( x ) a 1 ( x ) δ n ( x ) γ ( x 0 , x ) − b f β ( x ) μ 2 γ 2 ( x ) n ϑ ( γ ( x ) ) ϑ ( x ) n − n z 5 n + 1 n ( x ) a 1 1 n ( x ) .

Multiplying (42) by δ n ( x ) γ ( x 0 , x ) and integrating from x 1 to x , we get

δ n ( x ) γ ( x 0 , x ) z 5 ( x ) − δ n ( x 1 ) γ ( x 0 , x 1 ) z 5 ( x 1 ) − ∫ x 1 x a 2 ( s ) a 1 ( s ) d s + n ∫ x 1 x a 1 − 1 n ( s ) δ n − 1 ( s ) γ ( x 0 , s ) σ ( s ) z 5 ( s ) d s + ∫ x 1 x b f β ( s ) μ 2 γ 2 ( s ) n ϑ ( γ ( s ) ) ϑ ( s ) n δ n ( s ) γ ( x 0 , s ) d s + n ∫ x 1 x z 5 n + 1 n ( s ) a 1 1 n ( s ) δ n ( s ) γ ( x 0 , s ) d s ≤ 0 .

Using Lemma 2.4, we set

E = δ n ( s ) γ ( x 0 , s ) / a 1 1 n ( s ) , F = ∫ x 1 x a 1 − 1 n ( s ) δ n − 1 ( s ) γ ( x 0 , s ) σ ( s ) , u = z 5 ( x ) .

Thus, we get

δ n ( x ) γ ( x 0 , x ) z 5 ( x ) − δ n ( x 1 ) γ ( x 0 , x 1 ) z 5 ( x 1 ) − ∫ x 1 x a 2 ( s ) a 1 ( s ) d s + ∫ x 1 x b f β ( s ) μ 2 γ 2 ( s ) n ϑ ( γ ( s ) ) ϑ ( s ) n δ n ( s ) γ ( x 0 , s ) d s + ∫ x 1 x n ( n + 1 ) ζ ( s ) σ n + 1 ( s ) γ ( x 0 , s ) δ ( s ) a 1 1 n ( x ) d s ≤ 0 .

Hence, by (35), we obtain

∫ x 1 x b f β ( s ) μ γ 2 ( s ) 2 ϑ ( γ ( s ) ) ϑ ( s ) δ ( s ) n γ ( x 0 , s ) − σ ˜ ( s ) d s ≤ δ n ( x ) γ ( x 0 , x ) z 5 ( x 1 ) + 1 ,

which contradicts (40).

Theorem 2.14 is proved.□

Example 2.15

Let the equation:

(43) ( x 3 ( w ‴ ( x ) ) 3 ) ′ + β 0 x 7 w 3 ( γ x ) = 0 ,

where x ≥ 1 , γ ∈ ( 0 , 1 ] and β 0 > 0 . Let n = k = 3 , γ j ( x ) = γ x , a 1 ( x ) = x 3 , and β ( x ) = β 0 / x 7 . So, we obtain

R ˜ ( x ) = λ β 0 6 1 / 3 γ 1 2 x 2 .

By Corollary 2.7, we find (43) is oscillatory if

β 0 > 6 3 e ln 1 γ γ 6 , β 0 > 3 4 2 1 γ 9 ,

and

β 0 > 6 1 4 γ 3 .

So, equation (43) is oscillatory if

(44) β 0 > max 3 4 2 1 γ 9 , 6 1 4 γ 3 = 3 4 2 1 γ 9 .

Example 2.16

Let the equation

(45) ( x 2 ( w ‴ ( x ) ) ) ′ + x 2 w ‴ ( x ) + β 0 x 2 w ( 2 x ) = 0 ,

where β 0 > 0 is a constant. Let n = 1 , x 0 = 1 , a 1 ( x ) = x 2 , a 2 ( x ) = x / 2 , β ( x ) = x , and γ ( x ) = 2 x , we now set ζ ( x ) = x , b h = b f = 1 , then

γ ( x 0 , x ) = exp ∫ x 0 x a 2 ( u ) a 1 ( u ) d u = x 1 / 2 , δ ( x ) = ∫ x ∞ d s ( a 1 ( s ) γ ( x 0 , s ) ) 1 n = 2 x − 3 / 2 3 , ϑ ( x ) = 2 x − 3 / 2 3 , σ ( x ) = 1 γ 1 n ( x 0 , x ) − δ ( x ) a 2 ( x ) a 1 1 − n / n ( x ) n = 2 x − 1 / 2 3 , χ ( x ) = − 1 2 x ,

and

σ ˜ ( x ) = a 2 ( x ) a 1 ( x ) + n ( n + 1 ) ζ ( x ) σ n + 1 ( x ) γ ( x 0 , x ) δ ( x ) a 1 1 n ( x ) = 2 x − 1 / 3 3 .

Thus, we get

lim sup x → ∞ ∫ x 0 x b f ζ ( s ) β ( s ) − 2 M s 2 n a 1 ( s ) ζ ( s ) ( χ ( s ) ) n + 1 ( n + 1 ) n + 1 d s = ∞

and

ϑ ( x ) δ ( x ) ( a 1 ( x ) γ ( x 0 , x ) ) 1 / n + ϑ ′ ( x ) = 0 .

Also,

lim sup x → ∞ ∫ x 0 x b f β ( s ) μ γ 2 ( s ) 2 ϑ ( γ ( s ) ) ϑ ( s ) δ ( s ) n γ ( x 0 , s ) − σ ˜ ( s ) d s = ∞ .

By Theorem 2.14, we find (45) is oscillatory.

3 Conclusion

In this article, by using the Riccati method and the comparison technique, some new oscillation conditions for fourth-order differential equations are established with delay and advanced terms. Conditions complement and extend some of the results published on this topic. Two examples are discussed to illustrate the efficiency of the main results. Furthermore, nonlinear equations contribute in many applications of mathematical physics, biological and medical physics, engineering, complex networks, aviation, sociophysics and econophysics. The future work is to further study the oscillatory properties of neutral differential equations with p -Laplacian:

( a 1 ( x ) ( w ‴ ( x ) ) p − 1 ) ′ + ∑ j = 1 r β j ( x ) w p − 1 ( γ j ( x ) ) = 0

and

( a 1 ( x ) ( w ‴ ( x ) ) p − 1 ) ′ + a 2 ( x ) h ( w ‴ ( x ) ) + β ( x ) f ( w ( γ ( x ) ) ) = 0 .

Under the assumptions:

∫ x 0 ∞ 1 a 1 1 / p − 1 ( s ) d s < ∞

and

∫ x 0 ∞ 1 a 1 ( s ) exp − ∫ x 0 s a 2 ( u ) a 1 ( u ) d u 1 / p − 1 d s = ∞ ,

where p > 1 .

Funding information: Research supporting Project number (RSP-2021/167), King Saud University, Riyadh, Saudi Arabia.
Author contributions: All authors have accepted responsibility for the entire content of this manuscript and approved its submission.
Conflict of interest: The authors state no conflict of interest.

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Received: 2021-10-23

Revised: 2021-11-28

Accepted: 2021-11-30

Published Online: 2021-12-23

This work is licensed under the Creative Commons Attribution 4.0 International License.

Articles in the same Issue

https://doi.org/10.1515/phys-2021-0097

Keywords for this article

Riccati method; advanced term; oscillation; fourth-order; delay differential equations; damped

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