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Abstract
The oscillatory behavior of the solutions of the neutral delay differential equation
where p, τ, and a are positive constants and Q ∊ C([t 0 , ∞), ℝ ⁺ ), are studied.
... = 0, t^t 0 have been recently considered by several authors (Ladas and Sficas [5,6]], Wang [8], Zahariev and Bainov [10] and Zhang and Gopalsamy [12]). The purpose of this article is to discuss the asymptotic behavior of (1.1) when n is an odd positive integer. ...
... • Remark 4.L The condition (4.8) improves the condition of Theorem 3 in [5] since the parameter of the neutral term appears in (4.8) whereas such parametes do not appear in the condition used in Theorem 3 in [5]. ...
... • Remark 4.L The condition (4.8) improves the condition of Theorem 3 in [5] since the parameter of the neutral term appears in (4.8) whereas such parametes do not appear in the condition used in Theorem 3 in [5]. ...
... al. [10], Grove et. al. [11], $¥mathrm{J}¥mathrm{a}¥mathrm{r}¥mathrm{o}¥forall ¥mathrm{s}$ and Kusano [12], Ladas and Sficas [16], [17], Ruan [20], Sficas and Stavroulakis [21] and Zahariev and Bainov [22] and the references cited therein. However, much of the research on the subject has been restricted to first and second order linear equations with constant coefficients and constant deviations, and very little has appeared on higher order nonlinear equations. ...
... However, much of the research on the subject has been restricted to first and second order linear equations with constant coefficients and constant deviations, and very little has appeared on higher order nonlinear equations. For some results on higher order equations we refer to the papers [8], [12], [16], [17] and [22]. ...
... Using inequality (15) in equation (7) we get (16) $L_{n}z(t)+(-1)^{n+1}¥sum_{i=0}^{N}¥sum_{¥ell=1}^{m}¥sum_{j=1}^{k}q_{j}¥alpha_{¥ell}^{i}z[t-¥sigma_{j}-i¥tau_{¥ell}]¥leq 0$ , $t¥geq t_{1}$ , Inequality (16) has a bounded and eventually negative solution on [ $t_{1}$ , $¥infty)$ for $n$ even and a bounded and eventually positive solution on [ $t_{1}$ , $¥infty)$ for $n$ odd. From Corollary 1' in [19] it follows that equation $(E_{3})$ has a solution with the same properties as that of a solution of inequality (16). ...
... During the past few decades, neutral differential equations have been studied extensively and the oscillatory theory for these equations is well developed; see [1][2][3][4][5][6][7][8][9][10][11][12][13][14][15][16][17][18][19] and the references cited therein. In fact, the developments of oscillation theory for the neutral differential equations began in 1986 with the appearance of the paper of Ladas and Sficas [15]. ...
... During the past few decades, neutral differential equations have been studied extensively and the oscillatory theory for these equations is well developed; see [1][2][3][4][5][6][7][8][9][10][11][12][13][14][15][16][17][18][19] and the references cited therein. In fact, the developments of oscillation theory for the neutral differential equations began in 1986 with the appearance of the paper of Ladas and Sficas [15]. A survey of the most significant efforts in this theory can be found in the excellent monographs of Győri and Ladas [12] and Agarwal et al. [1]. ...
... In [5], some finite integral conditions for oscillation of all solutions of (1) when ( ) ≡ 1 are given under less restrictive hypothesis on . See also Grammatikopoulos et al. [10], Ladas and Sficas [15], and Al-Amri [4]. ...
We will consider a class of neutral functional differential equations. Some infinite integral conditions for the oscillation of all solutions are derived. Our results extend and improve some of the previous results in the literature.
... Delay differential equations (DDEs) have been applied widely in many fields, such as oscillation theory [1,2,3,4,5,6,9,13,14,18,19,20,21,28,32], stability theory [23,26,30,33], periodic solutions [24,25,27,29], population dynamics [10,18], dynamical behavior of delayed network systems [17,36] and so on. Theoretical studies on oscillation of solutions of DDEs have fundamental significance [15,16]. ...
... In the research article [34] was derived that if p ∈ (0, 1) and qσe > 1 − p, then all the solutions of (1.1) are oscillatory. The result improves the corresponding result in [20]. Afterward, for p ∈ (0, 1), many authors have involved to study this problem and have obtained various kinds of better results, see [11,22,31,35,37]. ...
... Afterward, for p ∈ (0, 1), many authors have involved to study this problem and have obtained various kinds of better results, see [11,22,31,35,37]. However, the negative side of all the conclusions reported in literature [11,20,22,31,34,35,37] are limited to sufficient conditions when 0 < p < 1. Therefore, the main aim of this paper is to establish a set of necessary and sufficient conditions for oscillation of all the solutions of (1.1) for the cases 0 < p < 1 and p > 1. ...
In this article, we concerned with oscillation of the neutral delay
differential equation $[x(t)-px(t-\tau)]'+qx(t-\sigma)=0$ with constant
coefficients. By constructing several suitable auxiliary functions,
we obtained some necessary and sufficient conditions for oscillation
of all the solutions of the aforementioned equation for the cases $0
... In the last two decades, there has been an increasing interest in obtaining sufficient conditions for the oscillation and/or nonoscillation of solutions of neutral delay differential equations. Particularly, we mention the papers by Ladas and Sficas [1], Chuanxi and Ladas [2], Ruan [3], Elabbasy and Saker [4], Kulenović et al. [5], and Karpuz andÖcalan [6] who investigated NDDEs with variable coefficients. To a large extent, this is due to its theoretical interest as well as to its importance in applications. ...
... It suffices to note that NDDEs appear in the study of networks containing lossless transmission lines (as in high-speed computers where the lossless transmission lines are used to interconnect switching circuits) in population dynamics and also in many applications in epidemics and infection diseases. We refer reader to [1][2][3][4][5][6][7][8][9][10][11][12][13][14][15][16][17][18] for relevant studies on this subject. ...
... Zhang [19], Ladas and Sficas [1], Grammatikopoulos et al. [10], and Yu et al. [8] considered (1) when 1 ≡ 0, and they obtained some sufficient conditions for oscillation of (1). The purpose of this work is to present some new sufficient conditions under which all solutions of (1) are oscillatory. ...
Some new sufficient conditions for oscillation of all solutions of the first-order linear neutral delay differential equations are obtained. Our new results improve many well-known results in the literature. Some examples are inserted to illustrate our results.
1. Introduction
A neutral delay differential equation (NDDE) is a differential equation in which the highest-order derivative of the unknown function is evaluated both at the present state at time and at the past state at time for some positive constant .
In the last two decades, there has been an increasing interest in obtaining sufficient conditions for the oscillation and/or nonoscillation of solutions of neutral delay differential equations. Particularly, we mention the papers by Ladas and Sficas [1], Chuanxi and Ladas [2], Ruan [3], Elabbasy and Saker [4], Kulenović et al. [5], and Karpuz and Öcalan [6] who investigated NDDEs with variable coefficients. To a large extent, this is due to its theoretical interest as well as to its importance in applications. It suffices to note that NDDEs appear in the study of networks containing lossless transmission lines (as in high-speed computers where the lossless transmission lines are used to interconnect switching circuits) in population dynamics and also in many applications in epidemics and infection diseases. We refer reader to [1–18] for relevant studies on this subject.
In this paper, we consider the linear first-order NDDE of the type where and . When and is a constant, Jaroš [9] established some new oscillation conditions for all solutions of (1), and his technique was based on the study of the characteristic equation
Zhang [19], Ladas and Sficas [1], Grammatikopoulos et al. [10], and Yu et al. [8] considered (1) when , and they obtained some sufficient conditions for oscillation of (1). The purpose of this work is to present some new sufficient conditions under which all solutions of (1) are oscillatory. In order to achieve this object, we are first concerned with NDDE (1) with constant coefficients (when is a constant). That is,
Some illustrating examples are given. In some sense, the established results extend and improve some previous investigations such as [1, 8–10, 19].
As usual, a solution of (1) is said to be oscillatory if it has arbitrarily large zeros and nonoscillatory if it is eventually positive or eventually negative. A function is called eventually positive (or negative) if there exists such that (or ) for all. Equation (1) is called oscillatory if all its solutions are oscillatory; otherwise, it is called nonoscillatory.
2. Main Results
In this section, we give some new sufficient conditions for the oscillation of all solutions of (1) and (3). This is done by using the following well-known lemmas which are from [11, 12].
Lemma 1. Consider the NDDE where , and .
Let be a positive solution of (4). Set
If , then is a positive and decreasing solution of (4); that is,
Lemma 2. Let and be positive constants. Let be an eventually positive solution of the delay differential inequality
Then for sufficiently large, where
Our main results can now be given as follows.
Theorem 3. Consider NDDE (3). Assume that (i), and(ii),
where is the unique real root of the equation
Then all solutions of (3) are oscillatory.
Proof. Assume, for the sake of a contradiction, that (3) has a nonoscillatory solution . Without loss of generality, assume that . Let So that is also a positive solution of (3).
That is, where
Set for
Thus it follows from Lemma 1 that is a positive and decreasing solution of and in particular (as implies that .), it follows that
But we have
This implies that
Applying Lemma 2 with (18) we get
Then is bounded.
Dividing (16) by and integrating from to , we get
Let .
Then, it follows from (20) that for and sufficiently small,
As is arbitrary, so we have
Let
Then
Let be the unique real root of the equation
Then
Hence
This contradicts condition (ii) and then completes the proof.
Example 4. Consider the NDDE
We note that
Then we have (i) , (ii) where is the unique real root of the equation Then all the hypotheses of Theorem 3 are satisfied, and therefore every solution of (28) oscillates. (Indeed is such a solution.)
Theorem 5. Consider the NDDE (1). Assume that (iii) , and is periodic with period , (iv) ,where is defined as in Theorem 3. Then all solutions of (1) are oscillatory.
Proof. Assume, for the sake of contradiction, that (1) has a nonoscillatory solution . Without loss of generality, assume that . Let which is oscillation invariant transformation. Then is a positive solution of the equation where is periodic with period .
Let
Then is decreasing positive solution of the equation
Set
This implies that , since .
Dividing both sides of (33) by and then integrating from to , we obtain that
Hence
Since is periodic with period , then we obtain
Substituting in (38) we find, for all ,
Now, we want to prove that is bounded.
Applying the assumption (iv), we can find such that where is similar as in the proof of Theorem 3.
Integrating (33) from to we obtain
Using Bonnet’s Theorem and in particular (as ), we get
Integrating (33) from to , we get
Using Bonnet’s Theorem and in particular (as ), we get
Combining (43) and (45), we conclude or
Then is bounded.
Now, let
But we have proved that is bounded; that is, is finite.
From (40), we obtain
Therefore, we get
Hence
This contradicts our assumption (iv) and then completes the proof.
Example 6. Consider the NDDE where
Then we have(1); (2) is periodic with period and satisfies
where is the unique real root of the equation
Therefore (52) satisfies all the hypotheses of Theorem 5. Hence every solution of this equation is oscillatory.
Theorem 7. Suppose that condition (iii) holds. If (v) ,
then every solution of (1) is oscillatory.
Proof. Proceeding as in the proof of Theorem 5, we get (49) which implies that
Hence
But this is a contradiction of assumption (v), and then the proof is complete.
Example 8. Consider the NDDE
Here we have
Note that is positive and periodic with period , and also(1), (2)
Then (58) satisfies hypotheses of Theorem 7, and so all its solutions are oscillatory.
Funding
This research has been completed with the support of these Grants: ukm-DLP-2011-049, DIP-2012-31 and FRGS/1/2012/SG04/ukm/01/1.
... In the special case where k = 1, Ladas and Sficas [21] studied the asymptotic and oscillatory behavior of the solutions of Eq. (1) when c e [-1, 0], and then Grammatikopoulos, Grove and Ladas [10] investigated the other possible cases c < -1 and c > 0. In this special case Sficas and Stavroulakis [26] obtained a necessary and sufficient condition for the oscillation of all solutions of Eq. (1) in terms of its characteristic equation, and then Grove, Ladas and Meimaridou [13] extended this result for Eq. (*). ...
... in the case of equations (2) and (4). Then the following lemma is easily established by using arguments similar to those in [10] and [21]. See also Lemma 1 in [19] and Lemma 2 in [12]. ...
... Combining the above results into a single statement we have the following. Observe that (21) λ + λce~λ r + £? =1 Pi e-λτi = 0 is the characteristic equation of (1) and (2), and (22) λ + λce-» + ΣU Pi*~λ τi + Σί=i 4^' = 0 ...
... We remark that the condition (2.2) is better than the corresponding con-ditions obtained by Zhang [25] and Ladas and Sficas [17]. For instance in the example i(l)-;+ 1)+&r-2)=0, (2.10) peo = 1-c = $ and the results of Zhang [25] and Ladas and Sficas [ 171 do not apply for (2.10). ...
... In a recent article Ladas and Sticas [17] have proved the following result: "If p, G, r are positive constants and Q is r-periodic such that QEC(Cb, co), R), a>~, p<l, and ...
... is decreasing, we can combine (2.20) and (2.21) so as to have Remark. We note that the existence of the limit in (2.13) has been assumed in [17]; such an assumption is not required in Theorem 2.2. Proof We shall show that the existence of a nonoscillatory solution of (2.30) leads to a contradiction. ...
Verifiable sufficient conditions are obtained respectively for the oscillation and nonoscillation of the neutral differential equation x(t) − cx(t − τ) + px(t − σ) = 0.
... In a recent paper, Ladas and Was [9] studied the oscillatory behavior of solutions of neutral delay differential equations (NDDE) of the form ; Cy(t)+py(t-~)l+Q(t)~(t-a)=O, In this paper, we investigate the behavior of the solutions of Eq. ...
... Although the oscillatory theory of delay differential equations has been extensively developed during the last few years (see, for example, [ 1, 7, 8, l&13]), there is hardly any work at this time (except for [9]) dealing with the oscillatory behavior of solutions of neutral delay differential equations. (For some results on second-order NDDE, see [ 141.) ...
... Proofs of extensions of these results are given in Section 3. A schematic summary of the oscillatory and asymptotic results for Eq. (2) including the results of Ladas and Sficas [9], is given at the end of this section. ...
Consider the neutral delay differential equation (∗) are positive constants, while pϵ (−∞, −1) ∪ (0, + ∞). (For the case pϵ [−1, 0] see Ladas and Sficas, Oscillations of neutral delay differential equations (to appear)). The following results are then proved. Theorem 1. Assume p < − 1. Then every nonoscillatory solution y(t) of Eq. (∗) tends to ± ∞ as t → ∞. Theorem 2. Assume p < − 1, τ > σ, and . Then every solution of Eq. (∗) oscillates. Theorems 3. Assume p > 0. Then every nonoscillatory solution y(t) of Eq. (∗) tends to zero as t → ∞. Theorem 4. Assume p > 0. Then a necessary condition for all solutions of Eq. (∗) to oscillate is that σ > τ. Theorem 5. Assume . Then every solution of Eq. (∗) oscillates. Extensions of these results to equations with variable coefficients are also obtained.
... Lemma 1 [18] Assume that h(t) and k(t) are nonnegative continuous functions for t t 0 , then for α 1 and all t t 0 , we have ...
... The proof procedure is easily obtained from Jensen inequality in [18], we omit the details. We outline one of the main stability results below. ...
A uniform stability analysis is developed for a type of neutral delays differential equations which depend on more general nonlinear integral inequalities. Many original investigations and results are obtained. Firstly, generations of two integral nonlinear inequalities are presented, which are very effective in dealing with the complicated integro-differential inequalities whose variable exponents are greater than zero. Compared with existed integral inequalities, those proposed here can be applied to more complicated differential equations, such as time-varying delay neutral differential equations. Secondly, the notions of (w, Ω) uniform stable and (w, Ω) uniform asymptotically stable, especially (c
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, c
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) uniform stable and (c
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, c
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) uniform asymptotically stable, are presented. Sufscient conditions on about (c
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, c
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) uniform stable and (c
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, c
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) uniform asymptotically stable of time-varying delay neutral differential equations are established by the proposed integral inequalities. Finally, a complex numerical example is presented to illustrate the main results effectively. The above work allows to provide further applications on the proposed stability analysis and control system design for different nonlinear systems.
... In fact, the paper of Zahariev and Baȋnov [8] seems to be the first work dealing with oscillation of neutral equations. A systematic development of oscillation theory of neutral equations was initiated by Ladas and Sficas [9]. ...
... which has been established in [9] (see, e.g., [1, 6, 12, 21]. ...
We study the oscillatory behaviour of all solutions of first-order neutral equations with variable coefficients. The obtained results extend and improve some of the well-known results in the literature. Some examples are given to show the evidence of our new results.
... 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. ...
... (8), it follows that ( ) is the maximum value of ( ) in (−∞, 0). By (10), we know that (19) is equivalent to ( ) < 0 for ∈ (−∞, 0). From Theorem 7, we obtain the following corollary that extends Theorem 1 in [25] for < . ...
We are concerned with oscillation of the first order neutral delay differential equation [ x (t) - p x (t - τ) ] ′ + q x (t - σ) = 0 with constant coefficients, and we obtain some necessary and sufficient conditions of oscillation for all the solutions in respective cases 0 < p < 1 and p > 1.
... During the past decade the oscillation theory of first order neutral delay differential equations has been extensively developed by many authors. Particularly, we mention the papers by Grove, Kulennovic and Ladas [2], and Ladas and Sficas [4,5], who investigated neutral delay differential equations with variable coefficients. ...
... U REMARK. Let q(t) = 0 in Theorem 1; then we get Theorem 7 of Ladas and Sficas [4]. Next we consider the following equation Then every solution of equation (14) is oscillatory. ...
In this paper, sufficient conditions for oscillations of the first order neutral differential equation with variable coefficients
are obtained, where c , τ, σ and µ are positive constants, p, q ∈ C ([ t 0 , ∞), R ⁺ ).
... Several results concerning the oscillation and asymptotic behavior of the solutions of (1), when k = 1, were recently obtained by Ladas and Sficas [5] and Grammatikopoulos, Grove and Ladas [2]. ...
... Using arguments similar to those in [2] and [5] the following lemma is easily established. ...
Consider the neutral delay differential equation
where p ∈ R , τ ≥ 0, q 1 > 0, σ 1 ≥ 0, for i = 1, 2, …, k . We prove the following result.
Theorem. A necessary and sufficient condition for the oscillation of all solutions of Eq. (1) is that the characteristic equation
has no real roots .
... Introduction. Recently, Ladas and Sficas [7] obtained sufficient conditions for the oscillation of all solutions of the neutral delay differential equation (NDDE) where 0 ~ p ~ 1, q > 0 and r, a~ 0.(2) Our aim in this paper is to extend some of the results obtained in [7] to systems of NDDEs of the formd m dt [x(t)-Px(t-r)] + L Qkx(t-ak) = 0,where P is an n x n diagonal matrix with diagonal entries P1, P2, ... , Pn such that O:Spi:S1 for i=1,2, ... ,n,the delays r and ak for k = 1, 2, ... , m, are nonnegative and for each k = 1, 2, ... , m the entries qg) of then x n matrix Qk are real numbers.For related results on systems of non-neutral delay differential equations, the reader is referred to [3] and [4]. Let p = max{ T, a 1, a2, ... , ak}. ...
... (2) Our aim in this paper is to extend some of the results obtained in [7] to systems of NDDEs of the form ...
We obtain sufficient conditions for the oscillation of all solutions of the system of neutral delay differential equations d dt[x(t)-Px(t-τ)]+∑ k=1 m Q k x(t-σ k )=0, where P is an n×n diagonal matrix with diagonal entries p 1 ,p 2 ,···,p n such that 0≤p i ≤1 for i=1,2,···,n, the delays τ and σ k for k=1,2,···,m are nonnegative and for each k=1,2,···,n the entries q ij (k) of the n×n matrix Q k are real numbers. Our results can be extended to systems with the Q k ' s continuous n×n matrices.
... tends to zero as t → ∞ if (1) holds and Q(t + τ/n) ≤ Q(t) for t ∈ [0, ∞) where n is any fixed positive integer. However, Ladas and Sficas [6] have shown that every solution of ...
... Let y(t) > 0 for t ≥ t 0 > 0. The case y(t) < 0 for t ≥ t 0 > 0 may be dealt with similarly. Setting z(t) and w(t) as in (5) we obtain (6). ...
In this paper, sufficient conditions have been obtained under which every solution of
$$\left[ {y(t) \pm y(t - \tau )]'} \right. \pm Q(t)G(y(t - \sigma )) = f(t), t \geqslant 0$$
, oscillates or tends to zero or to ±∞ → ∞. Usually these conditions are stronger than
$$\int\limits_0^\infty {Q(t)dt = \infty } $$
. An example is given to show that the condition (*) is not enough to arrive at the above conclusion. Existence of a positive (or negative) solution of
$$[y(t) - y(t - \tau )]' + Q(t)G(y(t - \sigma )) = f(t)$$
is considered.
... Note that a significant part of the research on the topic under study of the qualitative theory of differential equations/inclusions is contained in the following works; for second-order DFIs, qualitative properties were analyzed in [1,[3][4][5]7,8,10,13,31,33,37,38]. The works [9,28,34] considered oscillation criteria for neutral DFIs. [28] described the oscillatory and asymptotic behaviour of second-order neutral differential equations with several delays. ...
... Examples are provided to illustrate the results. Further none of the results in the papers [3][4][5][6][7][8] can be applied to Eq. (1.1) to yield any conclusion. ...
... Oscillatory behavior of [x (t) − rx (t − ρ)] ′ + px (t − τ ) = 0 (8) is investigated by many authors. We improve the result of [22] by G. Ladas and Y. G. Sficas holding ...
... In the case where = 1, ( ) ≡ 1, and and are constants, Karpuz andÖcalan [1] improved the result of Ladas and Sficas [2] holding 0 ≤ ≤ 1, ≥ 0, and ( − ) > (1/ )(1− ) conditions for oscillation. Also, the case including continuous functions as coefficients ...
A class of nonlinear neutral delay differential equations is considered. Some new oscillation criteria of all solutions are derived. The obtained results generalize and extend some of well known previous results in the literature.
... The literature on oscillation of solutions of neutral functional differ ential equations is growing day by day. We refer, in particular, to the papers of Grammatikopoulos, Grove and Ladas [2-4], Grammatikopoulos, Ladas an Meimaridou [5], Grammatikopoulos, Ladas and Sficas [6], Grove, Kulenovic and Ladas [7], Grove, Ladas and Meimaridou [8], Ivanov and Kusano [9], the present authors [10], Kulenovic, Ladas and Meimaridou [12], Ladas and Sficas [14], Ruan [19], Sficas and Stavroulakis [20] and Zahariev and Bainov [21][22]. Most of the papers mentioned above, however, concerns the first and second order equations. ...
... From a well-known inequality we see that for \i > 0, Hence (2.7) holds for all \i > 0, and £ = r, <JI ,..., cr n , and the theorem is proved. • Theorem 5.1 gives a sharp condition for oscillation in the sense that for the constant coefficient case it coincides with our recent result in [5] and it is better than the corresponding results in [2,6,7,8,13]. ...
We obtain a number of new conditions for oscillation of the first order neutral delay equation with nonconstant coefficients of the form
Comparison results are also given as well as conditions for the existence of nonoscillatory solutions.
... [t 0 ,oo) -» R, t 0 ^ 0 and /: R -> R are continuous; q(t) ^ 0 and is not identically zero on any ray of the form [t*, oo), t* ^ t 0 . The function g is such that lim g(t) = oo The oscillatory behavior of neutral equations of the type (l.i;6) with e(t) = 0 has been extensively studied by many authors, see, for example [1], [2], [5], [6], [11] and [12], and the reference cited therein. When p = 0 Kartsatos ([7], [8]) obteined some criteria for (1.3;6), however, for the case when py-0, very little is known. ...
... We note that the problem of existence of $¥ovalbox{¥tt¥small REJECT}_{¥iota}$ type solutions was first studied in [4]. 5. Concluding remaks I) A natural question arises as to the structure of the sets of nonoscillatory solutions $x(t)$ of the equations $d^{2n+1}$ (5. 1) $¥overline{dt^{2n+1}}[x(t)-h(t)x(¥tau(t))]+f(t, ¥mathrm{x}(g_{1}(t)), ¥ldots, ¥mathrm{x}(g_{N}(t)))=0$ , This equation has a nonosciUatory solution $x(t)=e^{-t}$ which satisfies $x(t)-$ $¥lambda x(t-¥tau)=(1-¥lambda e^{¥tau})e^{-t}<0$ for all $t$ . ...
... There exists also a variety of "easily verifiable" sufficient conditions which guarantee the oscillatory behaviour of (1) (cf. Gopalsamy [2], Grammatikopoulos, Grove and Ladas [3], Grove, Kulenovic and Ladas [4], Gyori and Ladas [6], Ladas and Sficas [10], Ruan [12] $)$ . However, the known criteria are not sharp enough and they fail to work, for example, if $¥sigma=0$ in the equation under consideration. ...
... In fact, Zahariev and Baȋnov [8] is the first work dealing with oscillation of neutral equations. A systematic development of oscillation theory of NDDEs was initiated by Ladas and Sficas [9]. For the oscillation of (1) when ( ) = 1 and ( ) and ( ) are constants, we refer the readers to the articles by Ladas and Schults [10], Sficas and Stavroulakis [11], Grammatikopoulos et al. [12], Zhang [13], and Gopalsamy and Zhang [14]. ...
Some new oscillation criteria are given for first order neutral delay differential equations with variable coefficients. Our results generalize and extend some of the well-known results in the literature. Some examples are considered to illustrate the main results.
... Such a solution is said to be oscillatory if it has arbitrarily large zeros in [t,, co) and it is said to be nonoscillatory otherwise. Surveying the rapidly expanding literature devoted to the study of oscillatory and asymptotic properties of neutral equations, one finds that most of the papers concerns linear equations with constant coefficients and constant deviations (see, for example, Gopalsamy [Z, 31, Grammatikopoulos , Grove, and Ladas [4,5], Grove, Kulenovic and Ladas [6], Grove, Ladas and Meimaridou [7], Gyori and Ladas [9], JaroS [ll], Kulenovic, Ladas, and Meimaridou [ 133, Ladas and Sficas [14], and Ruan [ 181). Results that apply to the neutral equations of form (A) (or (B)) with general h(t) and t(t) seem to be few. ...
... is investigated by many authors. We improve the result of[22]by G. Ladas and Y. G. Sficas holding 1 ≥ r ≥ 0, p ≥ 0, ...
In this article, we study delay differential equations having forms x' (t) + P (t)x(t �) Q(t)x(t �) = 0 (x(t) R(t)x(t �))' + P (t)x(t �) = 0
... (3.5) From z'(t) ~ 0 for t 2: T, as in [6], z(t) must be eventually positive, say z(t) > 0, t 2: T. We note that 0 < z(t) < y(t) fort 2: T. Hence, (3.1) becomes ...
Some new sufficient conditions for oscillation of first order linear delay differential equations, as well as neutral differential equations have been obtained. Forced oscillation for first order differential equations with deviating arguments has been studied also.
In this paper, we analyze robust stability of differential dynamical system with deviating argument in derivative part. By using inequality technique and stochastic analysis idea, we obtain the upper bounds of the interval length of deviating argument and the noise intensity, respectively. First, it is proved theoretically that for a given exponentially stable differential dynamical system (DDS), if the interval length of deviating argument is lower than the upper bound, DDS with deviating argument will still maintain exponentially stable. In addition, it is also proved that for a given exponentially stable DDS, if the interval length of deviating argument and noise intensity are lower than the upper bound, stochastic DDS (SDDS) with deviating argument will remain exponentially stable. Finally, theoretical findings are supported by two examples.
This paper mainly focuses on the pth(p≥2)-moment input-to-state stability (ISS) of neutral stochastic delay differential equations (NSDDEs) with Le´vy noise and Markovian switching. By using the generalized integral inequality and the Lyapunov function methodology, the input-to-state stability (ISS), integral input-to-state stability (iISS), and stochastic input-to-state stability (SISS) of such equations are obtained. When the input signal is a constant signal and a zero signal, the pth(p≥2)-moment ISS reduces to the pth(p≥2)-moment practical exponential stability and the pth(p≥2)-moment exponential stability, respectively. Finally, an example of the mass-spring-damping (MSD) model under the stochastic perturbation is given to verify the validity of the results.
A class of first order neutral delay differential equations are investigated, and sufficient condition is derived for all solutions to be oscillatory. This result solves an open problem in the literature.
In this paper, we discuss the relationship between three time lags (the consumption lag, the investment decision lag and the gestation lag) and oscillatory economic fluctuations in the Keynesian IS-LM system. We first confirm that in the absence of time lags, the monotone convergence to the unique equilibrium is observed. Next, we demonstrate that, in the Keynesian IS-LM system, in the existence of the investment decision lag and the gestation lag, oscillatory fluctuations are generated around the equilibrium if the gestation lag is relatively long, while in the existence of the consumption lag and the gestation lag, oscillatory behaviors may not occur. We then conclude that the existence of time lags may be one of the major causes of oscillatory fluctuations.
This paper considers the asymptotic behavior of the nonoscillating solutions for a class of nonautonomous neutral equations d/ dt[x(t)-∑i=1nPi(t)x(t - τ i)]+q(t)x(t)+∫0a(t) x(t - s) dr(t, s) = 0, where pi(t) (i = 1, 2, ⋯, n), q(t) are nonnegative functions, and the integral in the above equations is in the sense of Riemann-Stieltjes. The asymptotic behavior results of each nonoscillating solution for this kind of equations are obtained when α(t), r(t, s), pi(t) (i = 1,2, ⋯, n) and q(t) satisfy some adequate conditions. The results improve and generalize some known results in the related literature.
We discuss the oscillations of first order nonlinear differential equations of neutral type and obtain "sharp conditions" that ensure all solutions oscillate. These conditions are necessary and sufficient when the coefficient function reduces to a constant.
The neutral differential equation [x(t)+px(t-τ)] ' +∑ i=1 m Q i (t)x(Δ i (t,x(t)))=0(*) is considered, where p∈ℝ, τ>0, Q i ∈C(ℝ + ,ℝ), Δ i ∈C(ℝ + ×ℝ,ℝ), Δ * (t)≤Δ i (t,x)≤t, i=1,⋯,m and lim t→+∞ Δ * (t)=+∞. It is shown that equation (*) has a nonoscillatory solution, if p≠-1 and ∫ 0 ∞ ∑ i=1 m |Q i (t)|dt<+∞.
In this paper the oscillatory behaviors of the solutions of the neutral differential difference equation were studied. The results were generalized and the main results were proved.
Consider the neutral delay differential equation [display math001] where [display math002] We studied the asymptotic behavior of the nonoscillatory solutions of Eq. (1) and we obtained sufficient conditions for the oscillation of all solutions, all bounded solutions, and all unbounded solutions of Eq. (1)
Sufficient conditions are established that guarantee the convergence of the positive solutions of a neutral-type difference equation of the form
$$ \Delta [x(n) - q(n)x(\sigma (n))] + p(n)f(x({\tau_1}(n)), \ldots, x({\tau_k}(n))) = 0, $$
where σ(n); n = 1, 2, …, are retarded arguments and τ
j
(n) j = 1, …, k, are general deviated arguments.
New necessary and sufficient conditions and verifiable sufficient conditions are obtained for the oscillation of the neutral delay differential equations. These sufficient conditions include and are in many cases weaker than those known.
Oscillation criteria for the neutral delay differential equation are established. The results are applicable to superlinear and linear neutral delay differential equations.
The authors consider the first order nonlinear neutral delay differential equation (E) [y(t) + P(t) y(t − τ)]′ − Q(t) ƒ(y(t − σ)) = 0, where P, Q, and ƒ are continuous, if u ≠ 0. They give sufficient conditions for all nonoscillatory solutions of (E) to converge to zero as t → ∞. Two oscillation theorems for equation (E) are also proved.
This paper is devoted to the study of the oscillation of solutions of delay differential equations of the neutral and mixed types. Some general results are proved for certain general Volterra type neutral differential equations and many particular cases are discussed.
Received July 9, 1986 Consider the neutral delay differential equation $x(r)-P(f)x(f-r)]+Q(t)x(r-0)-O, tat,, (1) where P, Q E C[ [ I~, co), (w + ] and T, D E Iw +. We obtain sutlicient conditions for all solutions of Eq. (1) to oscillate. Our conditions are “sharp” in the sense than when the coefficients P and Q are constants the conditions are also necessary. We also obtain sufficient conditions for Eq. (1) to have a nonoscillatory solution when P(r) is a constant in the interval
The present paper deals with some oscillating and asymptotic properties of the functional differential equations of the form
where λ is an arbitrary positive constant and τ > 0 is a constant delay.
Consider the nth order ( n ≥ 1) delay differential inequalities and and the delay differential equation , where q ( t ) ≥ 0 is a continuous function and p , τ are positive constants. Under the condition pτe ≥ 1 we prove that when n is odd (1) has no eventually positive solutions, (2) has no eventually negative solutions, and (3) has only oscillatory solutions and when n is even (1) has no eventually negative bounded solutions, (2) has no eventually positive bounded solutions, and every bounded solution of (3) is oscillatory. The condition pτe > 1 is sharp. The above results, which generalize previous results by Ladas and by Ladas and Stavroulakis for first order delay differential inequalities, are caused by the retarded argument and do not hold when τ = 0.
In this paper the long-term behavior of solutions to the equation in the title are examined, where qi(t) and Ti(t) are positive. In particular, it is shown that if lim inft → ∝ ∑i = 1nTi(t) qi(t) > 1/e, all solutions oscillate about 0 infinitely often.
In this paper we study the oscillatory behavior of equations of the forms (*)y′(t) + ∑i = 1nPiy(t − τi) = 0 and (**) y′(t) − ∑i = 1npiy(t + τi) = 0, where piand τi, i = 1, 2,…, n, are positive constants. We prove that each one of the following conditions (1)piτi1/e for some i, i = 1, 2,ߪ, n, (2)(∑i = 1npi)τ>1/e,whereτ = min {τ1,τ2, …, τn}, (3)[Π i = 1nPi]1/n (∑i = 1n τi)>1/e, or (4)(1/n) (∑i = 1n (Piτi)1/2)21/e implies that every solution of (*) and (**) oscillates. A generalization in the case where the coefficientspi, i = 1, 2,…, n, are positive and continuous functions is also presented.
For ordinary differential equations it is well known that an equilibrium solution is stable if the linearized equation has only exponentially decaying solutions. The differential-difference equation u' (t+s) - u' (t) + au (t + )s) = g (t, u (t), u (t+ s)), a > O, s > O, when linearized, also has decaying exponentials exp (st), but the characteristic roots s approach the imaginary axis. The problem arises -- do solutions of this equation decay. An affirmative answer is obtained for this problem, the rate of decay depending upon the smoothness of the initial data and nonlinear term and the rate of approach of the characteristic roots to the imaginary axis. Then this result is applied to a transmission line problem. Also, an example is given of a homogeneous linear ordinary differential-difference equation with constant coefficients, whose characteristic roots all lie in the left half plane, having an unbounded solution. (Author)
AMS (MOS) subject classification (1970): Primary 34K15; Secondary 34K25, 34E10. A sufficient condition under which all solutions of the delay differential equation , where p(t≤0 and continuous and τ>0 and constant, are oscillatory is presented. It is explained that the condition is the best possible for oscillations. When the coefficient p(t) in equation (1) is a positive constant, p, then the condition becomes pτe>0 which is necessary and sufficient for all solutions of the DDE to be oscillatory.
Iosif Polterovich Département de mathématiques et de statistique Université de Montréal Pav. André Aisenstadt CP 6128, succ. Centre-ville
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Ren Zhu
Department of Mathematics
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Asymptotic Methods in the Theory of Nonlinear Oscillations, Kiev "Naukova Dumka
- V N Sevelo
- N V Vareh
V. N. Sevelo and N. V. Vareh, Asymptotic Methods in the Theory of Nonlinear Oscillations, Kiev
"Naukova Dumka," 1979 (Russian).