Note on the paper of Džurina and Stavroulakis

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Abstract

In this paper we will establish some new oscillation criteria for the second-order retarded differential equation of the form(r(t)|u(t)|α-1u(t))+p(t)|u[τ(t)]|α-1u[τ(t)]=0.The results obtained essentially improve and extend those of Džurina and Stavroulakis [Oscillation criteria for second-order delay differential equations, Appl. Math. Comput., 140 (2003) 445–453]. An open problem is proposed at the end of this paper.

Introduction

We are here concerned with the oscillatory behavior of the retarded functional differential equation of the form(r(t)|u(t)|α-1u(t))+p(t)|u[τ(t)]|α-1u[τ(t)]=0,tt0,where α > 0 is a constant, r(t), τ(t)  C1[t0, ∞), r(t) > 0, τ(t)  t, τ′(t)  0 for t  t0, limt→∞τ(t) = ∞, p(t)  C[t0, ∞) and p(t)  0 for t  t0.

In what follows, we shall consider only the nonconstant solutions of (1) which are defined for all large t. The oscillatory behavior is considered in the usual sense, i.e., a solution of (1) is called oscillatory if it has arbitrarily large zeros, otherwise, it is called nonoscillatory. Eq. (1) is said to be oscillatory if all of its nonconstant solutions are oscillatory.

Recently, Mirzov [10], [11], [12], Elbert [4], [5], Kusano et al. [7], [8], [9], Chern et al. [2], Agarwal et al. [1], Džurina and Stavroulakis [3] have observed some similar properties between Eq. (1) and the corresponding linear equation(r(t)u)+q(t)u[τ(t)]=0under the following assumptionlimtR(t)=t0t1r1/α(s)ds=.In this paper, we shall further the investigation and offer some criteria for the oscillation of (1). Our results improve the main results of [1], [2], [3]. Examples are given to illustrate the sharpness of our results at the end of this paper. On the other hand, we shall also study the oscillatory and asymptotic behavior of (1) under the assumptionlimtR(t)=t0t1r1/α(s)ds<.

Now, let us list the main results of [3] as follows:

Theorem A

Let α  1 and (2) holds. Assume that for some k  (0, 1)Rα[τ(t)]p(t)-ατ(t)4kR[τ(t)]r1/α[τ(t)]dt=.Then Eq. (1) is oscillatory.

Theorem B

Let 0 < α < 1 and (2) holds. Assume thatRα[τ(t)]p(t)-ατ(t)4R2-α[τ(t)]r(2/α)-1[τ(t)]P¯(t)dt=,whereP¯(t)=1r[τ(t)]tp(s)ds(1-α)/α.Then Eq. (1) is oscillatory.

It is evident that (6) implies that t0p(t)dt<, so Theorem B cannot be applied to the more general case of Eq. (1). In Section 2, we will show that (6) is redundant. We give a perfect result for all α > 0 which simplifies and improves Theorem A, Theorem B. Our main result is as the following:

Theorem 2.1

Assume that (2) holds andRα[τ(t)]p(t)-αα+1α+1τ(t)R[τ(t)]r1/α[τ(t)]dt=.Then Eq. (1) is oscillatory.

Section snippets

Main results

We need the following lemma.

Lemma

See [6]

If A and B are nonnegative constants, thenAλ-λABλ-1+(λ-1)Bλ0,λ>1and the equality holds if and only if A = B.

Now, let us give the proof of Theorem 2.1.

Proof of Theorem 2.1

Suppose that Eq. (1) has a nonoscillatory solution u(t). Without loss of generality we may assume that u(t) > 0 for all large t. The case of u(t) < 0 can be considered by the same method. From (1), (2), we can easily obtain that there exists a t1 > t0 such thatu(t)>0,u(t)>0,[r(t)(u(t))α]0fortτ(t1).Therefore, we have thatr(

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