Research Article | Open Access
On Properties of Third-Order Differential Equations via Comparison Principles
The objective of this paper is to offer sufficient conditions for certain asymptotic properties of the third-order functional differential equation , where studied equation is in a canonical form, that is, . 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.
We are concerned with the oscillatory and asymptotic behavior of all solutions of the third-order functional differential equations: In the sequel, we will assume and is the ratio of two positive odd integers,, , .
By a solution of we mean a function ,, which has the property and satisfies on . We consider only those solutions of which satisfy for all . We assume that possesses such a solution. A solution of is called oscillatory if it has arbitrarily large zeros on and otherwise it is called to be nonoscillatory. Equation is said to be oscillatory if all its solutions are oscillatory.
Recently, and its particular cases (see enclosed references) have been intensively studied. We establish new comparison theorems that permit to study properties of via properties of the second-order differential equations, in the sense that the oscillation of the second-order equations yields desired properties of .
Remark 1.1. All functional inequalities considered in this paper are assumed to hold eventually; that is, they are satisfied for all large enough.
2. Main Results
To simplify formulation of our main results, we recall the following definition:
Definition 2.2. We say that enjoys property (A) if all its positive solutions satisfy case (I) of Lemma 2.1.
Property (A) of has been studied by various authors; see enclosed references. We offer new technique for investigation property (A) of based on comparison theorems and Trench theory of canonical operators.
Remark 2.3. It is known that condition implies property (A) of . Consequently, in the sequel, we may assume that the integral on the left side of (2.1) is convergent.
Proof. Assuming the contrary, let be a solution of satisfying the Case (II) of Lemma 2.1. Using the monotonicity of , we see that Then evaluating and then integrating from to , we are lead to Setting to , we get Integrating to , we see that satisfies Let us denote the right hand side of (2.5) by . Then holds and moreover, Consequently, is a solution of the differential inequality , which contradicts our assumption.
Since is in noncanonical form, we apply Trench theory  to transform it to canonical form, which is more suitable for investigation.
Proof. Let be a positive solution of , such that holds. By direct computation, we can verify Trench result that the operator is equivalent to Therefore the differential inequality can be written in the form Applying the substitution , we can see that is a positive solution of the differential inequality Moreover, since our inequality is in canonical form, but according to Theorem 2 of , we get that the corresponding differential equation has also a positive solution. A contradiction. The proof is complete.
Remark 2.7. We do not stipulate whether or not is a delayed or advanced argument. Using any oscillatory condition for , we obtain criteria for property (A) of third-order equation . We offer several such results.
Proof. By Theorem 2.6, it is sufficient to prove that is oscillatory. Assume the contrary, that is, let be a positive solution of . Then An integration of from to leads to where . Integrating again from to , we obtain Let us denote Then Multiplying the previous inequality with and then integrating from to , we have Letting be , we get a contradiction with (2.13).
Now, we are prepared to provide another criterion for property (A) based on the Riccati transformation.
Let us denote
Proof. By Theorem 2.6, it is sufficient to prove that is oscillatory. Assume the contrary, that is, let be a positive solution of . Then satisfies (2.14). Since is decreasing, then there exists that We claim that . If not, then it is easy to see that Setting the last inequality to , we get where is arbitrary. Integrating the previous inequality from to , one gets Letting be , we get a contradiction with (2.22) and we conclude that On the other hand, it follows from that for any constant , we have We choose a positive constant , such that . It follows from (2.28) that eventually. Integrating from to , we obtain or equivalently We set Then and which in view of (2.29) implies It follows from (2.14) that Therefore where we have used (2.32). Applying the inequality , we are led to Integrating from to , we obtain in view of (2.23) as . A contradiction. The proof is complete now.
Example 2.12. Consider once more the third-order nonlinear delay differential equation with , and . It is easy to check that condition (2.22) is fulfilled and the condition (2.22) reduces to Choosing the condition (2.40) holds true and then Theorem 2.11 implies that enjoys property (A).
In this paper, we have presented new comparison theorems for deducing property (A) of from the oscillation of the suitable second-order delay differential equation. Our results here generalize those presented for linear differential equations [19, 25].
Research is in this paper supported by S.G.A. KEGA 019-025TUKE-4/2010.
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