Note on the paper of Džurina and Stavroulakis
Introduction
We are here concerned with the oscillatory behavior of the retarded functional differential equation of the formwhere 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 equationunder the following assumptionIn 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 assumption
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)Then Eq. (1) is oscillatory. Theorem B Let 0 < α < 1 and (2) holds. Assume thatwhereThen Eq. (1) is oscillatory.
It is evident that (6) implies that , 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 andThen Eq. (1) is oscillatory.
Section snippets
Main results
We need the following lemma. Lemma If A and B are nonnegative constants, thenand the equality holds if and only if A = B.See [6]
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 thatTherefore, we have that
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