International Journal of Mathematics and Mathematical Sciences

International Journal of Mathematics and Mathematical Sciences / 2012 / Article

Research Article | Open Access

Volume 2012 |Article ID 975298 | 10 pages | https://doi.org/10.1155/2012/975298

On Properties of Third-Order Differential Equations via Comparison Principles

Academic Editor: Christian Corda
Received21 Mar 2012
Revised18 Jun 2012
Accepted19 Jun 2012
Published26 Jul 2012

Abstract

The objective of this paper is to offer sufficient conditions for certain asymptotic properties of the third-order functional differential equation [ π‘Ÿ ( 𝑑 ) [ π‘₯ β€² ( 𝑑 ) ] 𝛾 ] ξ…ž ξ…ž + 𝑝 ( 𝑑 ) π‘₯ ( 𝜏 ( 𝑑 ) ) = 0 , where studied equation is in a canonical form, that is, ∫ ∞ π‘Ÿ βˆ’ 1 / 𝛾 ( 𝑠 ) d 𝑠 = ∞ . Employing Trench theory of canonical operators, we deduce properties of the studied equations via new comparison theorems. The results obtained essentially improve and complement earlier ones.

1. Introduction

We are concerned with the oscillatory and asymptotic behavior of all solutions of the third-order functional differential equations: ξ€Ί π‘Ÿ ξ€Ί π‘₯ ( 𝑑 ) ξ…ž ξ€» ( 𝑑 ) 𝛾 ξ€» ξ…ž ξ…ž + 𝑝 ( 𝑑 ) π‘₯ ( 𝜏 ( 𝑑 ) ) = 0 . ( 𝐸 ) In the sequel, we will assume π‘Ÿ , 𝜏 , 𝑝 ∈ 𝐢 ( [ 𝑑 0 , ∞ ) ) and ( H 1 ) 𝛾 is the ratio of two positive odd integers, ( H 2 ) π‘Ÿ ( 𝑑 ) > 0 , 𝑝 ( 𝑑 ) > 0 , l i m 𝑑 β†’ ∞ 𝜏 ( 𝑑 ) = ∞ .

Moreover, we assume that ( 𝐸 ) is in a canonical form, that is, ξ€œ ∞ 𝑑 0 π‘Ÿ βˆ’ 1 / 𝛾 ( 𝑠 ) d 𝑠 = ∞ . ( 1 . 1 )

By a solution of ( 𝐸 ) we mean a function π‘₯ ( 𝑑 ) ∈ 𝐢 1 [ 𝑇 π‘₯ , ∞ ) , 𝑇 π‘₯ β‰₯ 𝑑 0 , which has the property π‘Ÿ ( 𝑑 ) ( π‘₯ β€² ( 𝑑 ) ) 𝛾 ∈ 𝐢 2 ( [ 𝑇 π‘₯ , ∞ ) ) and satisfies ( 𝐸 ) on [ 𝑇 π‘₯ , ∞ ) . We consider only those solutions π‘₯ ( 𝑑 ) of ( 𝐸 ) which satisfy s u p { | π‘₯ ( 𝑑 ) | ∢ 𝑑 β‰₯ 𝑇 } > 0 for all 𝑇 β‰₯ 𝑇 π‘₯ . We assume that ( 𝐸 ) possesses such a solution. A solution of ( 𝐸 ) is called oscillatory if it has arbitrarily large zeros on [ 𝑇 π‘₯ , ∞ ) and otherwise it is called to be nonoscillatory. Equation ( 𝐸 ) is said to be oscillatory if all its solutions are oscillatory.

Recently, ( 𝐸 ) and its particular cases (see enclosed references) have been intensively studied. We establish new comparison theorems that permit to study properties of ( 𝐸 ) via properties of the second-order differential equations, in the sense that the oscillation of the second-order equations yields desired properties of ( 𝐸 ) .

Our results complement and extend earlier ones presented in [1–23].

Remark 1.1. All functional inequalities considered in this paper are assumed to hold eventually; that is, they are satisfied for all 𝑑 large enough.

Remark 1.2. It is sufficient to deal only with positive solutions of ( 𝐸 ) .

2. Main Results

We begin with the classification of the possible nonoscillatory solutions of ( 𝐸 ) .

Lemma 2.1. Let π‘₯ ( 𝑑 ) be a positive solution of ( 𝐸 ) . Then π‘₯ ( 𝑑 ) satisfies, eventually, one of the following conditions:(I) π‘₯ β€² ( 𝑑 ) < 0 , [ π‘Ÿ ( 𝑑 ) [ π‘₯ β€² ( 𝑑 ) ] 𝛾 ] ξ…ž > 0 , [ π‘Ÿ ( 𝑑 ) [ π‘₯ β€² ( 𝑑 ) ] 𝛾 ] ξ…ž ξ…ž < 0 ;(II) π‘₯ β€² ( 𝑑 ) > 0 , [ π‘Ÿ ( 𝑑 ) [ π‘₯ β€² ( 𝑑 ) ] 𝛾 ] ξ…ž > 0 , [ π‘Ÿ ( 𝑑 ) [ π‘₯ β€² ( 𝑑 ) ] 𝛾 ] ξ…ž ξ…ž < 0 .

Proof. The proof follows immediately from the canonical form of ( 𝐸 ) .

To simplify formulation of our main results, we recall the following definition:

Definition 2.2. We say that ( 𝐸 ) enjoys property (A) if all its positive solutions satisfy case (I) of Lemma 2.1.

Property (A) of ( 𝐸 ) has been studied by various authors; see enclosed references. We offer new technique for investigation property (A) of ( 𝐸 ) based on comparison theorems and Trench theory of canonical operators.

Remark 2.3. It is known that condition ξ€œ ∞ 𝑑 0 𝑝 ( 𝑠 ) d 𝑠 = ∞ , ( 2 . 1 ) implies property (A) of ( 𝐸 ) . Consequently, in the sequel, we may assume that the integral on the left side of (2.1) is convergent.

Now, we offer a comparison result in which we reduce property (A) of ( 𝐸 ) to the absence of certain positive solution of the suitable second-order inequality.

Theorem 2.4. If the second-order differential inequality ξ‚΅ 1 𝑧 𝑝 ( 𝑑 ) ξ…ž ξ‚Ά ( 𝑑 ) ξ…ž + ξ€· 𝜏 ( 𝑑 ) βˆ’ 𝑑 1 ξ€Έ 1 / 𝛾 π‘Ÿ 1 / 𝛾 𝜏 ( 𝜏 ( 𝑑 ) ) ξ…ž ( 𝑑 ) 𝑧 1 / 𝛾 ξ€· 𝐸 ( 𝜏 ( 𝑑 ) ) ≀ 0 1 ξ€Έ has not any solution satisfying 𝑧 ( 𝑑 ) > 0 , 𝑧 ξ…ž ξ‚΅ 1 ( 𝑑 ) < 0 , 𝑧 𝑝 ( 𝑑 ) ξ…ž ξ‚Ά ( 𝑑 ) ξ…ž ξ€· 𝑃 < 0 , 1 ξ€Έ then ( 𝐸 ) has property (A).

Proof. Assuming the contrary, let π‘₯ ( 𝑑 ) be a solution of ( 𝐸 ) satisfying the Case (II) of Lemma 2.1. Using the monotonicity of [ π‘Ÿ ( 𝑑 ) [ π‘₯ β€² ( 𝑑 ) ] 𝛾 ] ξ…ž , we see that ξ€Ί π‘₯ π‘Ÿ ( 𝑑 ) ξ…ž ( ξ€» 𝑑 ) 𝛾 β‰₯ ξ€œ 𝑑 𝑑 1 ξ€Ί ξ€Ί π‘₯ π‘Ÿ ( 𝑠 ) ξ…ž ( ξ€» 𝑠 ) 𝛾 ξ€» ξ…ž ξ€Ί ξ€Ί π‘₯ d 𝑠 β‰₯ π‘Ÿ ( 𝑑 ) ξ…ž ( ξ€» 𝑑 ) 𝛾 ξ€» ξ…ž ξ€· 𝑑 βˆ’ 𝑑 1 ξ€Έ . ( 2 . 2 ) Then evaluating π‘₯ β€² ( 𝑑 ) and then integrating from 𝑑 1 to 𝑑 , we are lead to ξ€œ π‘₯ ( 𝑑 ) β‰₯ 𝑑 𝑑 1 ξ€· 𝑠 βˆ’ 𝑑 1 ξ€Έ 1 / 𝛾 π‘Ÿ 1 / 𝛾 ξ‚€ ξ€Ί ξ€Ί π‘₯ ( 𝑠 ) π‘Ÿ ( 𝑠 ) ξ…ž ξ€» ( 𝑠 ) 𝛾 ξ€» ξ…ž  1 / 𝛾 d 𝑠 . ( 2 . 3 ) Setting to ( 𝐸 ) , we get ξ€Ί ξ€Ί π‘₯ π‘Ÿ ( 𝑑 ) ξ…ž ξ€» ( 𝑑 ) 𝛾 ξ€» ξ…ž ξ…ž ξ€œ + 𝑝 ( 𝑑 ) 𝑑 𝜏 ( 𝑑 ) 1 ξ€· 𝑠 βˆ’ 𝑑 1 ξ€Έ 1 / 𝛾 π‘Ÿ 1 / 𝛾 ξ‚€ ξ€Ί ξ€Ί π‘₯ ( 𝑠 ) π‘Ÿ ( 𝑠 ) ξ…ž ξ€» ( 𝑠 ) 𝛾 ξ€» ξ…ž  1 / 𝛾 d 𝑠 ≀ 0 . ( 2 . 4 ) Integrating 𝑑 to ∞ , we see that 𝑦 ( 𝑑 ) = [ π‘Ÿ ( 𝑠 ) [ π‘₯ β€² ( 𝑠 ) ] 𝛾 ] ξ…ž satisfies ξ€œ 𝑦 ( 𝑑 ) β‰₯ ∞ 𝑑 ξ€œ 𝑝 ( 𝑠 ) 𝑑 𝜏 ( 𝑠 ) 1 ξ€· 𝑒 βˆ’ 𝑑 1 ξ€Έ 1 / 𝛾 π‘Ÿ 1 / 𝛾 𝑦 ( 𝑒 ) 1 / 𝛾 ( 𝑒 ) d 𝑒 d 𝑠 . ( 2 . 5 ) Let us denote the right hand side of (2.5) by 𝑧 ( 𝑑 ) . Then ξ€· 𝑃 1 ξ€Έ holds and moreover, ξ‚΅ 1 𝑧 𝑝 ( 𝑑 ) ξ…ž ξ‚Ά ( 𝑑 ) ξ…ž + ξ€· 𝜏 ( 𝑑 ) βˆ’ 𝑑 1 ξ€Έ 1 / 𝛾 π‘Ÿ 1 / 𝛾 𝜏 ( 𝜏 ( 𝑑 ) ) ξ…ž ( 𝑑 ) 𝑦 1 / 𝛾 ( 𝜏 ( 𝑑 ) ) = 0 . ( 2 . 6 ) Consequently, 𝑧 ( 𝑑 ) is a solution of the differential inequality ξ€· 𝐸 1 ξ€Έ , which contradicts our assumption.

Since ξ€· 𝐸 1 ξ€Έ is in noncanonical form, we apply Trench theory [24] to transform it to canonical form, which is more suitable for investigation.

Denote ξ€œ 𝜚 ( 𝑑 ) = ∞ 𝑑 𝑝 ( 𝑠 ) d 𝑠 . ( 2 . 7 )

Theorem 2.5. If the differential equation ξ‚΅ 𝜚 2 ( 𝑑 ) ξ‚Ά 𝑝 ( 𝑑 ) 𝑦 β€² ( 𝑑 ) ξ…ž + ξ€· 𝜏 ( 𝑑 ) βˆ’ 𝑑 1 ξ€Έ 1 / 𝛾 π‘Ÿ 1 / 𝛾 𝜏 ( 𝜏 ( 𝑑 ) ) ξ…ž ( 𝑑 ) 𝜚 ( 𝑑 ) 𝜚 1 / 𝛾 ( 𝜏 ( 𝑑 ) ) 𝑦 1 / 𝛾 ξ€· 𝐸 ( 𝜏 ( 𝑑 ) ) = 0 2 ξ€Έ is oscillatory, then ξ€· 𝐸 1 ξ€Έ has not any solution satisfying ξ€· 𝑃 1 ξ€Έ .

Proof. Let 𝑧 ( 𝑑 ) be a positive solution of ξ€· 𝐸 1 ξ€Έ , such that ξ€· 𝑃 1 ξ€Έ holds. By direct computation, we can verify Trench result that the operator ξ‚΅ 1 L 𝑧 = 𝑧 𝑝 ( 𝑑 ) ξ…ž ξ‚Ά ( 𝑑 ) ξ…ž ( 2 . 8 ) is equivalent to 1 𝐿 𝑧 = ξ‚΅ 𝜚 𝜚 ( 𝑑 ) 2 ( 𝑑 ) ξ‚΅ 𝑝 ( 𝑑 ) 𝑧 ( 𝑑 ) ξ‚Ά 𝜚 ( 𝑑 ) ξ…ž ξ‚Ά ξ…ž . ( 2 . 9 ) Therefore the differential inequality ξ€· 𝐸 1 ξ€Έ can be written in the form ξ‚΅ 𝜚 2 ( 𝑑 ) ξ‚΅ 𝑝 ( 𝑑 ) 𝑧 ( 𝑑 ) ξ‚Ά 𝜚 ( 𝑑 ) ξ…ž ξ‚Ά ξ…ž + ξ€· 𝜏 ( 𝑑 ) βˆ’ 𝑑 1 ξ€Έ 1 / 𝛾 π‘Ÿ 1 / 𝛾 𝜏 ( 𝜏 ( 𝑑 ) ) ξ…ž ( 𝑑 ) 𝜚 ( 𝑑 ) 𝑧 1 / 𝛾 ( 𝜏 ( 𝑑 ) ) ≀ 0 . ( 2 . 1 0 ) Applying the substitution 𝑦 = 𝑧 / 𝜚 , we can see that 𝑦 is a positive solution of the differential inequality ξ‚΅ 𝜚 2 ( 𝑑 ) ξ‚Ά 𝑝 ( 𝑑 ) 𝑦 β€² ( 𝑑 ) ξ…ž + ξ€· 𝜏 ( 𝑑 ) βˆ’ 𝑑 1 ξ€Έ 1 / 𝛾 π‘Ÿ 1 / 𝛾 𝜏 ( 𝜏 ( 𝑑 ) ) ξ…ž ( 𝑑 ) 𝜚 ( 𝑑 ) 𝜚 1 / 𝛾 ( 𝜏 ( 𝑑 ) ) 𝑦 1 / 𝛾 ( 𝜏 ( 𝑑 ) ) ≀ 0 . ( 2 . 1 1 ) Moreover, since ξ€œ ∞ 𝑝 ( 𝑠 ) 𝜚 2 ( 𝑠 ) d 𝑠 = ∞ , ( 2 . 1 2 ) our inequality is in canonical form, but according to Theorem 2 of [18], we get that the corresponding differential equation ξ€· 𝐸 2 ξ€Έ has also a positive solution. A contradiction. The proof is complete.

Combining Theorems 2.4 and 2.5, we get the following criterion for property (A) of ( 𝐸 ) .

Theorem 2.6. If the second-order differential equation ξ€· 𝐸 2 ξ€Έ is oscillatory, then ( 𝐸 ) has property (A).

Remark 2.7. We do not stipulate whether or not 𝜏 is a delayed or advanced argument. Using any oscillatory condition for ξ€· 𝐸 2 ξ€Έ , we obtain criteria for property (A) of third-order equation ( 𝐸 ) . We offer several such results.

Theorem 2.8. Let 𝛾 > 1 and 𝜏 ( 𝑑 ) ≀ 𝑑 . If ξ€œ ∞ 𝑑 0 𝜏 1 / 𝛾 ( 𝑠 ) π‘Ÿ 1 / 𝛾 𝜏 ( 𝜏 ( 𝑠 ) ) ξ…ž ( 𝑠 ) 𝜚 ( 𝑠 ) d 𝑠 = ∞ , ( 2 . 1 3 ) then ( 𝐸 ) has property (A).

Proof. By Theorem 2.6, it is sufficient to prove that ξ€· 𝐸 2 ξ€Έ is oscillatory. Assume the contrary, that is, let 𝑦 ( 𝑑 ) be a positive solution of ξ€· 𝐸 2 ξ€Έ . Then 𝑦 ξ…ž ξ‚΅ 𝜚 ( 𝑑 ) > 0 2 ( 𝑑 ) ξ‚Ά 𝑝 ( 𝑑 ) 𝑦 β€² ( 𝑑 ) ξ…ž < 0 . ( 2 . 1 4 ) An integration of ξ€· 𝐸 2 ξ€Έ from 𝑑 to ∞ leads to 𝜚 2 ( 𝑑 ) 𝑦 ξ…ž ( 𝑑 ) β‰₯ ξ€œ 𝑝 ( 𝑑 ) ∞ 𝑑 ξ€· 𝜏 ( 𝑠 ) βˆ’ 𝑑 1 ξ€Έ 1 / 𝛾 π‘Ÿ 1 / 𝛾 𝜏 ( 𝜏 ( 𝑠 ) ) ξ…ž ( 𝑠 ) 𝜚 ( 𝑠 ) 𝜚 1 / 𝛾 ( 𝜏 ( 𝑠 ) ) 𝑦 1 / 𝛾 ξ€œ ( 𝜏 ( 𝑠 ) ) d 𝑠 β‰₯ 𝑐 ∞ 𝑑 𝜏 1 / 𝛾 ( 𝑠 ) π‘Ÿ 1 / 𝛾 𝜏 ( 𝜏 ( 𝑠 ) ) ξ…ž ( 𝑠 ) 𝜚 ( 𝑠 ) 𝜚 1 / 𝛾 ( 𝜏 ( 𝑠 ) ) 𝑦 1 / 𝛾 ( 𝜏 ( 𝑠 ) ) d 𝑠 , ( 2 . 1 5 ) where 𝑐 ∈ ( 0 , 1 ) . Integrating again from 𝑑 1 to 𝜏 ( 𝑑 ) , we obtain ξ€œ 𝑦 ( 𝜏 ( 𝑑 ) ) β‰₯ 𝑐 𝑑 𝜏 ( 𝑑 ) 1 𝑝 ( 𝑣 ) 𝜚 2 ξ€œ ( 𝑣 ) ∞ 𝑣 𝜏 1 / 𝛾 ( 𝑠 ) π‘Ÿ 1 / 𝛾 𝜏 ( 𝜏 ( 𝑠 ) ) ξ…ž ( 𝑠 ) 𝜚 ( 𝑠 ) 𝜚 1 / 𝛾 ( 𝜏 ( 𝑠 ) ) 𝑦 1 / 𝛾 ξ€œ ( 𝜏 ( 𝑠 ) ) d 𝑠 d 𝑣 β‰₯ 𝑐 𝑑 𝜏 ( 𝑑 ) 1 𝑝 ( 𝑣 ) 𝜚 2 ξ€œ ( 𝑣 ) ∞ 𝑑 𝜏 1 / 𝛾 ( 𝑠 ) π‘Ÿ 1 / 𝛾 𝜏 ( 𝜏 ( 𝑠 ) ) ξ…ž ( 𝑠 ) 𝜚 ( 𝑠 ) 𝜚 1 / 𝛾 ( 𝜏 ( 𝑠 ) ) 𝑦 1 / 𝛾 ( ξ€œ 𝜏 ( 𝑠 ) ) d 𝑠 d 𝑣 = 𝑐 𝑑 𝜏 ( 𝑑 ) 1 𝑝 ( 𝑠 ) 𝜚 2 ξ€œ ( 𝑠 ) d 𝑠 ∞ 𝑑 𝜏 1 / 𝛾 ( 𝑠 ) π‘Ÿ 1 / 𝛾 𝜏 ( 𝜏 ( 𝑠 ) ) ξ…ž ( 𝑠 ) 𝜚 ( 𝑠 ) 𝜚 1 / 𝛾 ( 𝜏 ( 𝑠 ) ) 𝑦 1 / 𝛾 ( 𝜏 ( 𝑠 ) ) d 𝑠 . ( 2 . 1 6 ) Let us denote ξ€œ 𝐹 ( 𝑑 ) = ∞ 𝑑 𝜏 1 / 𝛾 ( 𝑠 ) π‘Ÿ 1 / 𝛾 𝜏 ( 𝜏 ( 𝑠 ) ) ξ…ž ( 𝑠 ) 𝜚 ( 𝑠 ) 𝜚 1 / 𝛾 ( 𝜏 ( 𝑠 ) ) 𝑦 1 / 𝛾 ( 𝜏 ( 𝑠 ) ) d 𝑠 . ( 2 . 1 7 ) Then 𝑦 1 / 𝛾 ( 𝜏 ( 𝑑 ) ) 𝐹 1 / 𝛾 ( β‰₯ ξ‚Έ 𝑐 ξ€œ 𝑑 ) 𝑑 𝜏 ( 𝑑 ) 1 𝑝 ( 𝑠 ) 𝜚 2 ξ‚Ή ( 𝑠 ) d 𝑠 1 / 𝛾 β‰₯ 𝑐 1 + 1 / 𝛾 𝜚 1 / 𝛾 . ( 𝜏 ( 𝑑 ) ) ( 2 . 1 8 ) Multiplying the previous inequality with 𝜏 1 / 𝛾 ( 𝑑 ) π‘Ÿ βˆ’ 1 / 𝛾 ( 𝜏 ( 𝑑 ) ) 𝜏 β€² ( 𝑑 ) 𝜚 ( 𝑑 ) 𝜚 1 / 𝛾 ( 𝜏 ( 𝑑 ) ) and then integrating from 𝑑 1 to 𝑑 , we have 𝑐 1 + 1 / 𝛾 ξ€œ 𝑑 𝑑 1 𝜏 1 / 𝛾 ( 𝑠 ) π‘Ÿ 1 / 𝛾 𝜏 ( 𝜏 ( 𝑠 ) ) ξ…ž ξ€œ ( 𝑠 ) 𝜚 ( 𝑠 ) d 𝑠 ≀ 𝑑 𝑑 1 βˆ’ 𝐹 ξ…ž ( 𝑠 ) 𝐹 1 / 𝛾 𝐹 ( 𝑠 ) d 𝑠 ≀ 1 βˆ’ 1 / 𝛾 ξ€· 𝑑 1 ξ€Έ . 1 βˆ’ 1 / 𝛾 ( 2 . 1 9 ) Letting 𝑑 be ∞ , we get a contradiction with (2.13).

Example 2.9. Consider the third-order nonlinear delay differential equation ξ€· 𝑑 ( π‘₯ β€² ( 𝑑 ) ) 3 ξ€Έ ξ…ž ξ…ž + π‘Ž 𝑑 2 ξ€· 𝐸 π‘₯ ( πœ† 𝑑 ) = 0 , π‘₯ 1 ξ€Έ with π‘Ž > 0 , and 0 < πœ† < 1 . It is easy to check that condition (2.13) is fulfilled and then Theorem 2.8 implies that ξ€· 𝐸 π‘₯ 1 ξ€Έ enjoys property (A).

Our results are new even for 𝛾 = 1 . Employing a generalization of Hille’s criterion [10] for oscillation of ξ€· 𝐸 2 ξ€Έ with 𝛾 = 1 , we get in view of Theorem 2.6.

Theorem 2.10. Let 𝜏 ( 𝑑 ) ≀ 𝑑 and 𝜏 β€² ( 𝑑 ) > 0 . If l i m i n f 𝑑 β†’ ∞ 1 ξ€œ 𝜚 ( 𝜏 ( 𝑑 ) ) ∞ 𝑑 𝜏 ( 𝑠 ) 𝜏 ξ…ž ( 𝑠 ) 𝜚 ( 𝑠 ) 𝜚 ( 𝜏 ( 𝑠 ) ) 1 π‘Ÿ ( 𝜏 ( 𝑠 ) ) d 𝑠 > 4 , ( 2 . 2 0 ) then ( 𝐸 ) has property (A).

Now, we are prepared to provide another criterion for property (A) based on the Riccati transformation.

Let us denote 𝜏 𝑄 ( 𝑑 ) = 1 / 𝛾 ( 𝑑 ) π‘Ÿ 1 / 𝛾 𝜏 ( 𝜏 ( 𝑑 ) ) ξ…ž ( 𝑑 ) 𝜚 ( 𝑑 ) 𝜚 1 / 𝛾 ( 𝜏 ( 𝑑 ) ) . ( 2 . 2 1 )

Theorem 2.11. Let 𝛾 β‰₯ 1 and 𝜏 ( 𝑑 ) ≀ 𝑑 . If ξ€œ ∞ 𝑑 0 𝑄 ( 𝑠 ) 𝜚 1 / 𝛾 ( 𝜏 ( 𝑠 ) ) d 𝑠 = ∞ , ( 2 . 2 2 ) and for some 𝑐 > 0 ξ€œ ∞ 𝑑 0 ξ‚΅ 𝑄 ( 𝑠 ) 𝑝 𝜚 ( 𝑠 ) βˆ’ 𝑐 2 ( 𝑠 ) 𝜚 1 + 1 / 𝛾 ( 𝜏 ( 𝑠 ) ) 𝜚 3 ( 𝑠 ) 𝑝 ( 𝜏 ( 𝑠 ) ) 𝜏 ξ…ž ξ‚Ά ( 𝑠 ) d 𝑠 = ∞ , ( 2 . 2 3 ) then ( 𝐸 ) has property (A).

Proof. By Theorem 2.6, it is sufficient to prove that ξ€· 𝐸 2 ξ€Έ is oscillatory. Assume the contrary, that is, let 𝑦 ( 𝑑 ) be a positive solution of ξ€· 𝐸 2 ξ€Έ . Then 𝑦 ( 𝑑 ) satisfies (2.14). Since ( 𝜚 2 ( 𝑑 ) / 𝑝 ( 𝑑 ) ) 𝑦 β€² ( 𝑑 ) is decreasing, then there exists that l i m 𝑑 β†’ ∞ 𝜚 2 ( 𝑑 ) 𝑦 𝑝 ( 𝑑 ) ξ…ž ( 𝑑 ) = β„“ β‰₯ 0 . ( 2 . 2 4 ) We claim that β„“ = 0 . If not, then it is easy to see that ξ€œ 𝑦 ( 𝑑 ) β‰₯ 𝑑 𝑑 1 ξ‚΅ 𝜚 2 ( 𝑠 ) 𝑦 𝑝 ( 𝑠 ) ξ…ž ξ‚Ά ( 𝑠 ) 𝑝 ( 𝑠 ) 𝜚 2 ξ€œ ( 𝑠 ) d 𝑠 β‰₯ β„“ 𝑑 𝑑 1 𝑝 ( 𝑠 ) 𝜚 2 β„“ ( 𝑠 ) d 𝑠 > 2 1 . 𝜚 ( 𝑑 ) ( 2 . 2 5 ) Setting the last inequality to ( 𝐸 ) , we get ξ‚΅ 𝜚 0 β‰₯ 2 ( 𝑑 ) 𝑦 𝑝 ( 𝑑 ) ξ…ž ξ‚Ά ( 𝑑 ) ξ…ž + ξ€· 𝜏 ( 𝑑 ) βˆ’ 𝑑 1 ξ€Έ 1 / 𝛾 π‘Ÿ 1 / 𝛾 𝜏 ( 𝜏 ( 𝑑 ) ) ξ…ž ( 𝑑 ) 𝜚 ( 𝑑 ) 𝜚 1 / 𝛾 ( 𝜏 ( 𝑑 ) ) 𝑦 1 / 𝛾 β‰₯ ξ‚΅ 𝜚 ( 𝜏 ( 𝑑 ) ) 2 ( 𝑑 ) 𝑦 𝑝 ( 𝑑 ) ξ…ž ξ‚Ά ( 𝑑 ) ξ…ž ξ‚€ β„“ + π‘˜ 2  1 / 𝛾 𝑄 ( 𝑑 ) 𝜚 1 / 𝛾 , ( 𝜏 ( 𝑑 ) ) ( 2 . 2 6 ) where π‘˜ ∈ ( 0 , 1 ) is arbitrary. Integrating the previous inequality from 𝑑 1 to 𝑑 , one gets 𝜚 2 ξ€· 𝑑 1 ξ€Έ 𝑝 ξ€· 𝑑 1 ξ€Έ 𝑦 ξ…ž ξ€· 𝑑 1 ξ€Έ ξ‚€ β„“ β‰₯ π‘˜ 2  1 / 𝛾 ξ€œ 𝑑 𝑑 1 𝑄 ( 𝑠 ) 𝜚 1 / 𝛾 ( 𝜏 ( 𝑠 ) ) d 𝑠 . ( 2 . 2 7 ) Letting 𝑑 be ∞ , we get a contradiction with (2.22) and we conclude that l i m 𝑑 β†’ ∞ 𝜚 2 ( 𝑑 ) 𝑝 ( 𝑑 ) 𝑦 β€² ( 𝑑 ) = 0 . ( 2 . 2 8 ) On the other hand, it follows from ξ€· 𝐸 2 ξ€Έ that for any constant π‘˜ ∈ ( 0 , 1 ) , we have ξ‚΅ 𝜚 2 ( 𝑑 ) 𝑦 𝑝 ( 𝑑 ) ξ…ž ξ‚Ά ( 𝑑 ) ξ…ž + π‘˜ 𝑄 ( 𝑑 ) 𝑦 1 / 𝛾 ( 𝜏 ( 𝑑 ) ) ≀ 0 . ( 2 . 2 9 ) We choose a positive constant 𝑐 1 , such that 𝑐 = ( 𝛾 𝑐 1 1 βˆ’ 1 / 𝛾 ) / ( 4 π‘˜ ) . It follows from (2.28) that 𝜚 2 ( 𝑑 ) 𝑦 𝑝 ( 𝑑 ) ξ…ž 𝑐 ( 𝑑 ) ≀ 1 2 , ( 2 . 3 0 ) eventually. Integrating from 𝑑 1 to 𝑑 , we obtain ξ€· 𝑑 𝑦 ( 𝑑 ) ≀ 𝑦 1 ξ€Έ + 𝑐 1 2 ξ€œ 𝑑 𝑑 1 𝜚 2 ( 𝑠 ) 𝑐 𝑝 ( 𝑠 ) d 𝑠 ≀ 1 , 𝜚 ( 𝑑 ) ( 2 . 3 1 ) or equivalently 𝑦 1 / 𝛾 βˆ’ 1 ( 𝜏 ( 𝑑 ) ) β‰₯ 𝑐 1 1 / 𝛾 βˆ’ 1 𝜚 1 βˆ’ 1 / 𝛾 ( 𝜏 ( 𝑑 ) ) . ( 2 . 3 2 ) We set 𝑀 𝜚 ( 𝑑 ) = ( 1 / 𝜚 ( 𝑑 ) ) ξ€· ξ€· 2 ξ€Έ 𝑦 ( 𝑑 ) / 𝑝 ( 𝑑 ) ξ…ž ξ€Έ ( 𝑑 ) 𝑦 1 / 𝛾 . ( 𝜏 ( 𝑑 ) ) ( 2 . 3 3 ) Then 𝑀 ( 𝑑 ) > 0 and 𝑀 ξ…ž 𝑝 ( 𝑑 ) = ( 𝑑 ) ( 𝜚 𝜚 ( 𝑑 ) 𝑀 ( 𝑑 ) + 1 / 𝜚 ( 𝑑 ) ) ξ€· ξ€· 2 ( ξ€Έ 𝑦 𝑑 ) / 𝑝 ( 𝑑 ) ξ…ž ( ξ€Έ 𝑑 ) ξ…ž 𝑦 1 / 𝛾 βˆ’ 1 ( 𝜏 ( 𝑑 ) ) 𝛾 𝑦 ξ…ž ( 𝜏 ( 𝑑 ) ) 𝜏 ξ…ž ( 𝑑 ) 𝑦 ( 𝜏 ( 𝑑 ) ) 𝑀 ( 𝑑 ) , ( 2 . 3 4 ) which in view of (2.29) implies 𝑀 ξ…ž 𝑝 ( 𝑑 ) ≀ ( 𝑑 ) 𝜚 ( 𝑑 ) 𝑀 ( 𝑑 ) βˆ’ π‘˜ 𝑄 ( 𝑑 ) βˆ’ 1 𝜚 ( 𝑑 ) 𝛾 𝑦 ξ…ž ( 𝜏 ( 𝑑 ) ) 𝜏 ξ…ž ( 𝑑 ) 𝑦 ( 𝜏 ( 𝑑 ) ) 𝑀 ( 𝑑 ) . ( 2 . 3 5 ) It follows from (2.14) that 𝜚 2 ( 𝜏 ( 𝑑 ) ) 𝑦 𝑝 ( 𝜏 ( 𝑑 ) ) ξ…ž 𝜚 ( 𝜏 ( 𝑑 ) ) > 2 ( 𝑑 ) 𝑦 𝑝 ( 𝑑 ) ξ…ž ( 𝑑 ) . ( 2 . 3 6 ) Therefore 𝑀 ξ…ž ( 𝑑 ) ≀ βˆ’ π‘˜ 𝑄 ( 𝑑 ) + 𝜚 ( 𝑑 ) 𝑝 ( 𝑑 ) 𝜏 𝜚 ( 𝑑 ) 𝑀 ( 𝑑 ) βˆ’ ξ…ž ( 𝑑 ) 𝛾 𝜚 2 ( 𝑑 ) 𝜚 2 ( 𝜏 ( 𝑑 ) ) 𝑝 ( 𝜏 ( 𝑑 ) ) 𝑦 𝑝 ( 𝑑 ) ξ…ž ( 𝑑 ) 𝑦 ( 𝜏 ( 𝑑 ) ) 𝑀 ( 𝑑 ) = βˆ’ π‘˜ 𝑄 ( 𝑑 ) + 𝜚 ( 𝑑 ) 𝑝 ( 𝑑 ) 𝜏 𝜚 ( 𝑑 ) 𝑀 ( 𝑑 ) βˆ’ ξ…ž ( 𝑑 ) 𝛾 𝜚 ( 𝑑 ) 𝜚 2 ( 𝜏 ( 𝑑 ) ) 𝑝 ( 𝜏 ( 𝑑 ) ) 𝑦 1 / 𝛾 βˆ’ 1 ( 𝜏 ( 𝑑 ) ) 𝑀 2 ( 𝑑 ) ≀ βˆ’ π‘˜ 𝑄 ( 𝑑 ) + 𝜚 ( 𝑑 ) 𝑝 ( 𝑑 ) 𝑐 𝜚 ( 𝑑 ) 𝑀 ( 𝑑 ) βˆ’ 1 1 / 𝛾 βˆ’ 1 𝜏 ξ…ž ( 𝑑 ) 𝛾 𝜚 ( 𝑑 ) 𝜚 2 ( 𝜏 ( 𝑑 ) ) 𝑝 ( 𝜏 ( 𝑑 ) ) 𝜚 1 βˆ’ 1 / 𝛾 ( 𝜏 ( 𝑑 ) ) 𝑀 2 ( 𝑑 ) , ( 2 . 3 7 ) where we have used (2.32). Applying the inequality 𝐴 𝑀 βˆ’ 𝐡 𝑀 2 ≀ 𝐴 2 / ( 4 𝐡 ) , we are led to 𝑀 ξ…ž ( 𝑑 ) ≀ βˆ’ π‘˜ 𝑄 ( 𝑑 ) 𝑝 𝜚 ( 𝑑 ) + 𝑐 π‘˜ 2 ( 𝑠 ) 𝜚 1 + 1 / 𝛾 ( 𝜏 ( 𝑠 ) ) 𝜚 3 ( 𝑠 ) 𝑝 ( 𝜏 ( 𝑠 ) ) 𝜏 ξ…ž . ( 𝑠 ) ( 2 . 3 8 ) Integrating from 𝑑 1 to 𝑑 , we obtain in view of (2.23) ξ€· 𝑑 𝑀 ( 𝑑 ) ≀ 𝑀 1 ξ€Έ ξ€œ βˆ’ π‘˜ 𝑑 𝑑 0 ξ‚΅ 𝑄 ( 𝑠 ) 𝑝 𝜚 ( 𝑠 ) βˆ’ 𝑐 2 ( 𝑠 ) 𝜚 1 + 1 / 𝛾 ( 𝜏 ( 𝑠 ) ) 𝜚 3 ( 𝑠 ) 𝑝 ( 𝜏 ( 𝑠 ) ) 𝜏 ξ…ž ξ‚Ά ( 𝑠 ) d 𝑠 ⟢ βˆ’ ∞ ( 2 . 3 9 ) as 𝑑 β†’ ∞ . A contradiction. The proof is complete now.

Example 2.12. Consider once more the third-order nonlinear delay differential equation ξ€· 𝑑 ( π‘₯ β€² ( 𝑑 ) ) 3 ξ€Έ ξ…ž ξ…ž + π‘Ž 𝑑 2 ξ€· 𝐸 π‘₯ ( πœ† 𝑑 ) = 0 , π‘₯ 2 ξ€Έ with π‘Ž > 0 , and 0 < πœ† < 1 . It is easy to check that condition (2.22) is fulfilled and the condition (2.22) reduces to ξ€œ ∞ 𝑑 0 1 𝑠 1 / 3 ξ‚΅ π‘Ž 1 / 3 πœ† 2 / 3 1 βˆ’ 𝑐 π‘Ž 2 / 3 πœ† 1 / 3 ξ‚Ά d 𝑠 = ∞ . ( 2 . 4 0 ) Choosing 𝑐 = ( π‘Ž πœ† ) / 2 the condition (2.40) holds true and then Theorem 2.11 implies that ξ€· 𝐸 π‘₯ 2 ξ€Έ enjoys property (A).

In this paper, we have presented new comparison theorems for deducing property (A) of ( 𝐸 ) from the oscillation of the suitable second-order delay differential equation. Our results here generalize those presented for linear differential equations [19, 25].

Acknowledgment

Research is in this paper supported by S.G.A. KEGA 019-025TUKE-4/2010.

References

  1. R. P. Agarwal, S.-L. Shieh, and C.-C. Yeh, β€œOscillation criteria for second-order retarded differential equations,” Mathematical and Computer Modelling, vol. 26, no. 4, pp. 1–11, 1997. View at: Publisher Site | Google Scholar | Zentralblatt MATH
  2. R. P. Agarwal, S. R. Grace, and D. O'Regan, β€œOn the oscillation of certain functional differential equations via comparison methods,” Journal of Mathematical Analysis and Applications, vol. 286, no. 2, pp. 577–600, 2003. View at: Publisher Site | Google Scholar | Zentralblatt MATH
  3. R. P. Agarwal, S. R. Grace, and T. Smith, β€œOscillation of certain third order functional differential equations,” Advances in Mathematical Sciences and Applications, vol. 16, no. 1, pp. 69–94, 2006. View at: Google Scholar | Zentralblatt MATH
  4. R. P. Agarwal and S. R. Grace, β€œOscillation theorems for certain neutral functional-differential equations,” Computers & Mathematics with Applications, vol. 38, no. 11-12, pp. 1–11, 1999. View at: Publisher Site | Google Scholar | Zentralblatt MATH
  5. B. Baculíková and J. Džurina, β€œOscillation of third-order neutral differential equations,” Mathematical and Computer Modelling, vol. 52, no. 1-2, pp. 215–226, 2010. View at: Publisher Site | Google Scholar | Zentralblatt MATH
  6. B. Baculíková and J. Džurina, β€œOscillation of third-order nonlinear differential equations,” Applied Mathematics Letters, vol. 24, no. 4, pp. 466–470, 2011. View at: Publisher Site | Google Scholar | Zentralblatt MATH
  7. B. Baculíková and J. Džurina, β€œOscillation of third-order functional differential equations,” Electronic Journal of Qualitative Theory of Differential Equations, vol. 43, pp. 1–10, 2010. View at: Google Scholar | Zentralblatt MATH
  8. D. D. Baĭnov and D. P. Mishev, Oscillation Theory for Neutral Differential Equations with Delay, Adam Hilger, Bristol, UK, 1991.
  9. M. Cecchi, Z. Došlá, and M. Marini, β€œOn third order differential equations with property A and B,” Journal of Mathematical Analysis and Applications, vol. 231, no. 2, pp. 509–525, 1999. View at: Publisher Site | Google Scholar | Zentralblatt MATH
  10. J. Džurina, β€œAsymptotic properties of third order delay differential equations,” Czechoslovak Mathematical Journal, vol. 45, no. 120, pp. 443–448, 1995. View at: Google Scholar | Zentralblatt MATH
  11. J. Džurina, β€œComparison theorems for functional-differential equations with advanced argument,” Unione Matematica Italiana. Bollettino A, vol. 7, no. 3, pp. 461–470, 1993. View at: Google Scholar | Zentralblatt MATH
  12. L. H. Erbe, Q. Kong, and B. G. Zhang, Oscillation Theory for Functional-Differential Equations, vol. 190 of Monographs and Textbooks in Pure and Applied Mathematics, Marcel Dekker, New York, NY, USA, 1995.
  13. S. R. Grace, R. P. Agarwal, R. Pavani, and E. Thandapani, β€œOn the oscillation of certain third order nonlinear functional differential equations,” Applied Mathematics and Computation, vol. 202, no. 1, pp. 102–112, 2008. View at: Publisher Site | Google Scholar | Zentralblatt MATH
  14. M. Gregus, Third Order Linear Differential Equations, Reidel, Dordrecht, The Netherlands, 1982.
  15. I. Györi and G. Ladas, Oscillation Theory of Delay Differential Equations: With Applications, Oxford Mathematical Monographs, Oxford Science Publications, The Clarendon Press Oxford University Press, New York, NY, USA, 1991.
  16. T. S. Hassan, β€œOscillation of third order nonlinear delay dynamic equations on time scales,” Mathematical and Computer Modelling, vol. 49, no. 7-8, pp. 1573–1586, 2009. View at: Publisher Site | Google Scholar | Zentralblatt MATH
  17. P. Hartman and A. Wintner, β€œLinear differential and difference equations with monotone solutions,” American Journal of Mathematics, vol. 75, pp. 731–743, 1953. View at: Publisher Site | Google Scholar | Zentralblatt MATH
  18. T. Kusano and M. Naito, β€œComparison theorems for functional-differential equations with deviating arguments,” Journal of the Mathematical Society of Japan, vol. 33, no. 3, pp. 509–532, 1981. View at: Publisher Site | Google Scholar | Zentralblatt MATH
  19. T. Kusano, M. Naito, and K. Tanaka, β€œOscillatory and asymptotic behaviour of solutions of a class of linear ordinary differential equations,” Proceedings of the Royal Society of Edinburgh A, vol. 90, no. 1-2, pp. 25–40, 1981. View at: Publisher Site | Google Scholar | Zentralblatt MATH
  20. G. S. Ladde, V. Lakshmikantham, and B. G. Zhang, Oscillation Theory of Differential Equations with Deviating Arguments, vol. 110 of Monographs and Textbooks in Pure and Applied Mathematics, Marcel Dekker, New York, NY, USA, 1987.
  21. N. Parhi and S. Pardi, β€œOn oscillation and asymptotic property of a class of third order differential equations,” Czechoslovak Mathematical Journal, vol. 49, no. 124, pp. 21–33, 1999. View at: Publisher Site | Google Scholar | Zentralblatt MATH
  22. Ch. G. Philos, β€œOn the existence of nonoscillatory solutions tending to zero at for differential equations with positive delays,” Archiv der Mathematik, vol. 36, no. 2, pp. 168–178, 1981. View at: Publisher Site | Google Scholar | Zentralblatt MATH
  23. A. Tiryaki and M. F. Aktaş, β€œOscillation criteria of a certain class of third order nonlinear delay differential equations with damping,” Journal of Mathematical Analysis and Applications, vol. 325, no. 1, pp. 54–68, 2007. View at: Publisher Site | Google Scholar | Zentralblatt MATH
  24. W. F. Trench, β€œCanonical forms and principal systems for general disconjugate equations,” Transactions of the American Mathematical Society, vol. 189, pp. 319–327, 1974. View at: Google Scholar
  25. K. Tanaka, β€œAsymptotic analysis of odd order ordinary differential equations,” Hiroshima Mathematical Journal, vol. 10, no. 2, pp. 391–408, 1980. View at: Google Scholar | Zentralblatt MATH

Copyright © 2012 B. Baculíková and J. Džurina. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.


More related articles

531Β Views | 487Β Downloads | 0Β Citations
 PDF  Download Citation  Citation
 Download other formatsMore
 Order printed copiesOrder

Related articles

We are committed to sharing findings related to COVID-19 as quickly and safely as possible. Any author submitting a COVID-19 paper should notify us at help@hindawi.com to ensure their research is fast-tracked and made available on a preprint server as soon as possible. We will be providing unlimited waivers of publication charges for accepted articles related to COVID-19. Sign up here as a reviewer to help fast-track new submissions.