Necessary and Sufficient Conditions of Oscillation in First Order Neutral Delay Differential Equations

and Applied Analysis 3 If qσe ⩾ 1 − pe, we suppose furthermore that q/θ+qσ < 1−p (otherwise, all solutions of (1) are oscillatory by the above conclusion); that is, θ > q/(1 − p − qσ). Since qσe is a minimum value of the function (q/μ)eμσ at μ = 1/σ, we have that

In this paper, we consider a class of neutral DDEs [ () −  ( − )]  +  ( − ) = 0,  ⩾  0 , where  0 is a positive number and , , , and  are positive constants.Generally, a solution of (1) is called oscillatory if it is neither eventually positive nor eventually negative.Otherwise, it is nonoscillatory.It can be seen in the literature that the oscillation theory regarding solutions of (1) has been extensively developed in the recent years.
In [18], Zhang came to the following conclusion.
This result in Theorem I improves the corresponding result in [19].Afterward, many authors have been devoted to studying this problem and have obtained many better results.For details, Gopalsamy and Zhang [20] obtained the improved result shown in Theorem II.
Further, Zhou and Yu [21] proved the following theorem.
Continuing to improve the research work, Xiao and Li [22] obtained the following.
Finally, Lin [23] obtained the result shown in Theorem V.
Theorem V. Assume that  ∈ (0, 1) and  > 1 −  /(1−−) ; then all solutions of (1) are oscillatory.However, all the conclusions mentioned above are limited to sufficient conditions in the case 0 <  < 1.The aim of this paper is to establish systematically the necessary and sufficient conditions of oscillation for all solutions of (1) for the cases 0 <  < 1 and  > 1.

Main Results
It is well known [24] that all solutions of (1) are oscillatory if and only if the characteristic equation of ( 1) has no real roots.
Theorem 1. Assume that  ∈ (0, 1) and let Then all solutions of (1) are oscillatory if and only if where  is a unique zero of () in (0, 1/).
In addition, Thus, we get that function () has a unique zero  in (0, 1/).
From Theorem 1, we obtain immediately the following.
From Theorem 1, all solutions of (1) are oscillatory if and only if one of (H 1 ) or (H 2 ) holds.
Theorem 4. Assume that  ∈ (0, 1); then all solutions of (1) are oscillatory if one of the following conditions holds: where  is a unique zero of () in (0, 1/).
Proof.If / +  ⩾ 1 − , we have that From the proof of Theorem 1, all solutions of (1) are oscillatory.
If  ⩾ 1 −  /(1−−) , we suppose furthermore that / +  < 1 −  (otherwise, all solutions of (1) are oscillatory by the above conclusion); that is,  > /(1 −  − ).Since  is a minimum value of the function (/)  at  = 1/, we have that and the result follows.So far, for  ∈ (0, 1) we have discussed the necessary and sufficient conditions of oscillation for all solutions of (1).Our results have perfected the results in [23] (see Theorem 4).Next, we will discuss the behavior of oscillation of solutions of (1) in the case  > 1.
Lemma 5. Let  > 1; then all solutions of (1) are oscillatory if and only if the equation has no real roots in (− ln /, 0).
Proof.It is similar to the proof of Theorem 1; () is the maximum value of () for  ∈ (−∞, 0).This and Lemma 5 imply the result.
Theorem 7. Assume that  > 1 and  < ; then all solutions of (1) are oscillatory if and only if where  is a unique zero of (3) in (−∞, 0).
From Theorem 9, we obtain the following corollary immediately.
so that all the solutions of (28) are oscillatory from Theorem 9.