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On certain inequalities and their applications in the oscillation theory

Blanka Baculfkova and Jozef Dzurina*

"Correspondence: jozef.dzurina@tuke.sk Department of Mathematics, Faculty of ElectricalEngineering and Informatics, TechnicalUniversity of Kosice, Letna 9, Kosice, 042 00, Slovakia

Abstract

In the paper, we offer a set of inequalities involving delayed argument and offer their application for higher-order differential equations of the form

to be oscillatory. The conditions obtained essentially improve many other known results.

MSC: 34K11; 34C10

Keywords: higher order differential equations; delay argument; oscillation

x(n)(t) + q(t)x(r (t)) = 0

ft Spri

1 Introduction

The paper is organized as follows. In the first part we consider only properties of functions and their derivatives, and later we connect the estimate obtained with properties of solutions of differential equations. We shall investigate the properties of a couple of functions r (t) e C(I) and x(t) e Cl(I), I = [t0, to).

Lemma 1 Assume that l is a positive integer such that

x(t)>0, x'(t) > 0, ..., x(l)(t)>0, x(£+1)(t) < 0, (Q)

eventually. Then for any constant X e (0,1) and for every i = 1,2,..., l,

tix(i)(t) 1/A ..

<7 . x(t), (1.1)

ringer

i! X\ if eventually.

Proof Assume that (Q) holds for t > t0. Using the monotonicity of x(£)(t), it is easy to see that for any k e (0,1),

x(l-1)(t) > itx(£)(s)ds > (t - t0)xw(t) > ktx(l)(t), (1.2)

© 2013 Baculikova and Dzurina; licensee Springer. This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the originalwork is properly cited.

eventually, let us say, for t > ti > to. We define a sequence of functions [pi(t)}\ as follows:

Pi(t)=x(i-1)(t)-/ctx(i)(t), P2(t) = 2x(i-2)(t)-ktx(i-1)(t),

Pi(t) = ix(t)-ktx'(t). It follows from (1.2) that pi(t) > 0. An integration of this from ti to t yields

x(i-2)(t)[1 + k] - ktx(i-1)(t) > c = k(x(i-2)(ti) - tix(i-1)(ti)). On the other hand, since x(i-2)(t) ^ro as t we see that

x(i-2)(t)[i- k] > c. Combining the last two inequalities, we conclude that P2 (t) > 0.

Proceeding as above, we verify that pi(t) > 0, eventually, for all i = i, 2,..., i -1. Therefore,

ix(t) > ktx'(t), (i - 1)x (t) > ktx"(t),

x(i-1\t)>ktx(i\t),

or in other words

tx'(t) < 1 ix(t), k

t2x"(t)<^ri(i -1)x(t),

Lemma 2 Assume that t (t) < t and that i is a positive integer such that (Q) holds. Then, for any constant X e (0,1),

tlx(i)(t) < ^ji\x(t).

Setting X = ki, the last inequalities imply (1.1) and the proof is complete.

eventually.

Proof Taylor's theorem implies that

Employing (1.1), we have

The proof is complete.

The obtained estimates can be used, e.g., in the theory of functional equations. In the paper, we present their application in discussing oscillatory and asymptotic properties of higher-order delay differential equations.

2 Main results

We consider higher-order delay differential equation

(Hi) q(t)>0, t(t) < t.

Denote by N the set of all nonoscillatory solutions of (E). It follows from the classical lemma of Kiguradze [1] that the set N has the following decomposition:

N = No U N2 U ••• U Nn-i, n-odd, N = N1 U N3 U ••• U Nn-1, n-even,

where the nonoscillatory solution x(t), let us say positive, satisfies

A nonoscillatory solution x(t) of (E) is said to be of degree i if x(t) e N1. Following Kondratiev and Kiguradze, we say that (E) has property (A) provided that

N = N0, n-odd, N = 0, n-even.

The investigation of oscillatory properties of the second- and higher-order linear differential equations started with the Sturm comparison theorem [2]. Later Mahfoud [3]

x(n)(t) + q(t)x( t (t)) =0,

x(t) e Ni

x(i)(t) > 0; i = 0,1,...,i, (-1)i-ix%)>0; i = i,...,n -1.

essentially contributed to the subject and presented a very useful comparison technique for studying the properties of a delay differential equation from those of a differential equation without delay. A new impetus to investigation in this direction was given by papers of Chanturia and Kiguradze [4], Kusano and Naito [5] and Koplatadze etal. [6, 7]. See also [1-20]. In the paper, we employ Lemma 2 to establish new criteria for oscillation of (E). It is interesting to note that the condition

r n-1(t)q(t)dt = to (2.1)

is necessary for property (A) of (E). This fact has been observed in [6] and [7].

Theorem 1 Assume that (E) has a solution of degree l >0, then for any X e (0,1) so does the ordinary equation

x(n)(t) + X^p) q(t)x(t) = 0. (Ei)

Proof Assume that (E) possesses a nonoscillatory solution x(t) e Nl. We may assume that x(t) is positive. Then condition (1.3) of Lemma 1 implies that x(t) is a positive solution of the differential inequality

x(n)(t) + ^r(t^ q(t)x(t) < 0.

On the other hand, it follows from Theorem 2 of [5] that the corresponding equation (El) has also a solution of degree l. The proof is complete. □

So, if we eliminate solutions of degree l of equations (El), we get property (A) of studied equation (E). To do it, we recall the following comparison result which is due to Chanturia [4].

Theorem 2 Assume that

n to n to

/ q(s)ds > p(s)ds. (2.2)

If the differential equation x(n)(t) + p(t)x(t) = 0

has no solution ofdegree l, neither does the equation

x(n) (t) + q(t)x(t) = 0.

In view of Theorem 1, we apply this comparison theorem to equations (El) andthe Euler equation

x(n)(t) + ^ = 0 (2.3)

P(k)= -k(k-1 )(k-2)(k-3)(k-4)

Figure 1 Graph of function P(k).

to obtain new criteria for property (A) of (E). Properties of (2.3) are connected with properties of the polynomial Pn(k) = -k(k -1) ••• (k - n + 1). Let us denote

Mj = max {Pn(k)},

where j = 2,4,..., n - 1 for n odd, while j = 1,3,..., n - 1 for n even. In other words, Mj represents all local maxima of the polynomial Pn(k) (see Figure 1). Then it is easy to verify (see also [15]) that the following criterion for the to be empty holds true.

Lemma 3 Let I > 0. If

a > Mi, (.4)

where I = 1,3,..., n -1 for n odd and I = 1,3,..., n -1 for n even, then (2.3) has no solution of degree I.

Employing Theorem 2 to (2.3) and (Ei), in view of Theorem 1, one gets the following theorem.

Theorem 3 Assume that

liminftn-1 f 7Kf) V-1,(s)ds > M Jt \ s ) (n -

Then (E) has property (A).

Proof Assume that n is odd. Observing that r(t) < t and Mn-1 > Ml for every l = 2,4,..., n - 3, it follows from (P) that for every l = 2,4,..., n -1,

liminff-1 f 7lq(s)ds > -M-. (Pi)

t—TO Jt \ s / (n -1)

On the other hand, (Pl) implies that there exists a couple of constants X e (0,1) and a > Ml such that

Xl 1 f) lq(s)ds >pd1^ P5)

Since a > Ml, Euler equation (2.3) has no solution of degree l. On the other hand, taking (2.5) into account, Theorem 2 ensures that (El) has no solution of degree l. Finally, Theorem 1 guarantees that (E) has property (A). The proof is complete. □

For r(t) = at, 0 < a < 1, the previous result simplifies to the following.

liminf if1-1 I q(s)ds . ((

Corollary 1 Assume that

Jt q(s)ds> an-1(n -1)' Then the delay differential equation

x(n) (t) + q(t)x(at) = 0, 0 < a < 1, (E*)

has property (A).

Theorem 4 Let n be odd. Assume that (E) has property (A). Then every nonoscillatory solution x(t) of (E) satisfies

lim x(t) = 0.

t—>to

Proof First note that property (A) of (E) implies (2.1). Moreover, it follows from the definition of property (A) that every nonoscillatory solution x(t) e N0, which implies that there exists limt—TO x(t) = c > 0. We claim that c = 0. If not, then x( r(t)) > c >0. An integration of (E) from t to to yields

x(n-1)(t) > / q(s)x(r(s}) ds > c / q(s)ds. Jt Jt

Having repeated this procedure, we are led to c p to

x(ti) > (nZTi)? Jt (s - ti)n-1q(s)ds, which contradicts (2.1) and we conclude that c = 0. □

We support our results with the following illustrative example.

Example 1 Consider the fifth-order delay differential equation

x(5) (t) + ax(at) = 0, a > 0,0 < a <1. (Exl)

The graph of the polynomial P(k) = -k(k - 1)(k - 2)(k - 3)(k - 4) that corresponds to the fifth-order equation is presented in Figure 1. Employing Matlab, we easily evaluate that

M4 = 3.6314.

Consequently, criterion (P*) for property (A) of (E) reduces for (Ex1) to

aa4 > 3.6314. 3 Comparison

Theorem 2 essentially improves Chanturia's test [4] that guarantees property (A) of

x(n)(t) + q(t)x(t) = 0, (E1)

provided that

Kiguradze's test [1] that for property (A) of (E) requires

I sn-1-eq(s)ds = to for some e >0, J to

and Koplatadze's test [7] for property (A) of (E) that claims the condition

limsupi r (t) i tn-2(s)q(s)ds + i tn-1(s)q(s)ds

t-^TO ( Jt (t) Jt (t)

rx (t)

„«-2,

limsup M sn 2q(s)ds >(n -1)!.

t—TO Jt

+ t^/ stn-1(s)q(s)dsj >(n -1)!, (3.1)

where t (t) is nondecreasing. We provide details while comparing those criteria with our one.

Example 2 Consider once more the fifth-order delay differential equation (Exi). It is easy to see that Chanturia's test can be applied only when a = 1 and requires

for property (A) of (Ex1). Kiguradze's test fails. On the other hand, Koplatadze's test simplifies for a = 1 and a = 0.8 to

a >6 and a > 23.9522,

respectively, while our criterion needs only

a > 3.6314 and a > 8.8658,

respectively.

Competing interests

The authors declare that they have no competing interests. Authors' contributions

The authors have made the same contribution. Allauthors read and approved the finalmanuscript. Acknowledgements

This work was supported by the Slovak Research and Development Agency under the contract No. APVV-0008-10.

Received: 1 March 2013 Accepted: 23 May 2013 Published: 11 June 2013

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doi:10.1186/1687-1847-2013-165

Cite this article as: Baculikova and Dzurina: On certain inequalities and their applications in the oscillation theory.

Advances in Difference Equations 2013 2013:165.