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Oscillations of Delay Differential Equations with Variable Coefficients
Abstract
Consider the delay differential equationx′(t) + p(t)x(t − τ) = 0, wherep(t) ∈ C([t0, ∞), R + ) and τ is a positive constant. We obtain a sharp sufficient condition for the oscillation of this equation, which improves previously known results.
... which is also the critical point for the solution possibly to cause oscillations for the linear delay ODE [36] (1. 8) ...
... where k 1 and k 2 are arbitrarily given constants. Lemma 3.8 (see [2,8,13,15,33,36]). (i) When 0 < k 2 < k 1 , then, for any time-delay r > 0, it holds that ...
... The proof is standard. Here, (i)-(iii) are immediately derived from the textbooks [13,15,33], (iv) is originally from the stability analysis for linear delay differential equations by Boese [2] (see also the summery in the textbook [13]), and the oscillation part (v) is a simple corollary of [36] (see also the textbook [8]). ...
This paper is concerned with Nicholson's blowflies equation, a kind of time-delayed reaction-diffusion equation. It is known that when the ratio of birth rate coefficient and death rate coefficient satisfies 1 < p/d <= e, the equation is monotone and possesses monotone traveling wavefronts, which have been intensively studied in previous research. However, when p/d > e, the equation losses its monotonicity, and its traveling waves are oscillatory when the time-delay r or the wave speed c is large, which causes the study of stability of these nonmonotone traveling waves to be challenging. In this paper, we use the technical weighted energy method to prove that when e < p/d <= e(2), all noncritical traveling waves phi(x + ct) with c > c(*) > 0 are exponentially stable, where c(*) > 0 is the minimum wave speed. Here, we allow the traveling wave to be either monotone or nonmonotone with any speed c > c(*) and any size of the time-delay r > 0; however, when p/d > e(2) with a small time-delay r < [pi-arctan root ln p/d(ln p/d -2)]/d root ln p/d(ln p/d -2), all noncritical traveling waves phi(x + ct) with c > c(*) > 0 are exponentially stable, too. As a corollary, we also prove the uniqueness of traveling waves in the case of p/d > e(2), which to the best of our knowledge was open. Finally, some numerical simulations are carried out. When e < p/d <= e(2), we demonstrate numerically that after a long time the solution behaves like a monotone traveling wave for a small time-delay, and behaves like an oscillatory traveling wave for a big time-delay. When p/d > e(2), if the time-delay is small, then the solution numerically behaves like a monotone/nonmonotone traveling wave, but if the time-delay is big, then the solution is numerically demonstrated to be chaotically oscillatory but not an oscillatory traveling wave. These either confirm and support our theoretical results or open up some new phenomena for future research.
... Then, from (2.5) and (2.7), one can see that y(t) is positive solution of the delay differential inequality does not exist has been recently investigated by several authors [10], [18], [17], [9], [21], [24] and [22]. In view of the respective works presented in [10], [18], [17], [9], [21], [24] and [22] and the fact that every solution of equation ( Proof. ...
... Then, from (2.5) and (2.7), one can see that y(t) is positive solution of the delay differential inequality does not exist has been recently investigated by several authors [10], [18], [17], [9], [21], [24] and [22]. In view of the respective works presented in [10], [18], [17], [9], [21], [24] and [22] and the fact that every solution of equation ( Proof. Without loss of generality, we assume that equation (1.1) has an eventually positive solution x(t). ...
... (2-34) and then from (2.6), (2.34) and the fact that T > a, y(t) satisfies the inequality [10], [18], [17], [9], [21], [24] and [22] and the fact that every solution of (1. as T-+oo. ...
We extend some results in the literature concerning sufficient conditions for oscillation of all solutions of a class of first-order neutral delay differential equations with variable coefficients and delays.
... Now a days, peoples' concentration has been directed towards the oscillation criteria for linear delay differential equation. Among numerous papers dealing with the subject, we refer [2,3,4,5,6,7,9,10] cited there in. The above observations are motivated our interest in the study of new oscillation and nonoscillation linear delay differential equation which can be extended to neutral delay differential equation. ...
... which is equivalent to (24) in Theorem (10). Therefore every solution of (1) oscillates. ...
In this article the authors established sufficient condition for the first order delay differential equation in the form , ( ) where , = and is a non negative piecewise continuous function. Some interesting examples are provided to illustrate the results.
... to possibly occur oscillations [30,33]. That is, for b (v + ) < 0, the solution of the delayed ODE (1.14) is monotone for 0 < r < r and it may be oscillatory for r > r (cf. ...
... That is, for b (v + ) < 0, the solution of the delayed ODE (1.14) is monotone for 0 < r < r and it may be oscillatory for r > r (cf. [30]). However, if d < |b (v + )|, there exists a Hopf-bifurcation point to (1.14) [2,3] ...
... to possibly occur oscillations [30,33]. That is, for b (v + ) < 0, the solution of the delayed ODE (1.14) is monotone for 0 < r < r and it may be oscillatory for r > r (cf. ...
... That is, for b (v + ) < 0, the solution of the delayed ODE (1.14) is monotone for 0 < r < r and it may be oscillatory for r > r (cf. [30]). However, if d < |b (v + )|, there exists a Hopf-bifurcation point to (1.14) [2,3] ...
... to possibly occur oscillations [30,33]. That is, for b (v + ) < 0, the solution of the delayed ODE (1.14) is monotone for 0 < r < r and it may be oscillatory for r > r (cf. ...
... That is, for b (v + ) < 0, the solution of the delayed ODE (1.14) is monotone for 0 < r < r and it may be oscillatory for r > r (cf. [30]). However, if d < |b (v + )|, there exists a Hopf-bifurcation point to (1.14) [2,3] ...
This paper is concerned with the stability of critical traveling waves for a kind of non-monotone time-delayed reaction–diffusion equations including Nicholson's blowflies equation which models the population dynamics of a single species with maturation delay. Such delayed reaction–diffusion equations possess monotone or oscillatory traveling waves. The latter occurs when the birth rate function is non-monotone and the time-delay is big. It has been shown that such traveling waves exist for all and are exponentially stable for all wave speed [13], where is called the critical wave speed. In this paper, we prove that the critical traveling waves (monotone or oscillatory) are also time-asymptotically stable, when the initial perturbations are small in a certain weighted Sobolev norm. The adopted method is the technical weighted-energy method with some new flavors to handle the critical oscillatory waves. Finally, numerical simulations for various cases are carried out to support our theoretical results.
... (1.1) and Eq.(1.2). Tang and Shen in [16] obtained an other sharper sufficient condition for the oscillation of all solutions of Eq. (1.1) which improves previously known results. Moreover, Agwo in [2], extended the technique used in [16] to be suitable for delay equations with several delays. ...
... Tang and Shen in [16] obtained an other sharper sufficient condition for the oscillation of all solutions of Eq. (1.1) which improves previously known results. Moreover, Agwo in [2], extended the technique used in [16] to be suitable for delay equations with several delays. ...
New oscillation criteria are given for first order super-linear and sub-linear differential equations with deviating arguments with variable coefficients. Mathematics subject classification (2000): 34C15, 34K11, 34K15.
... In the recent literature, there have been numerous results on the oscillation for delay differential equation with discrete delays; see, for example, papers [1,4,[8][9][10][11][12][13][14][15][17][18][19][20][21][22][23], books [5,7,16] and references therein. But only a relatively few publications on oscillation for delay differential equation with distributed delay are found; see [2,3] and references cited ✩ This work was supported by NNSF of China. ...
... Integral conditions like (1.4) and (1.5) have been employed extensively in the study of the oscillatory properties of various functional differential equations. When the limit lim t →∞ t t −τ p(s) ds does not exist, there is an obvious gap between 1/e and 1; in the recent literature [9,11,[16][17][18][19][20][21] ...
In this paper, we obtain some oscillation criteria for the first order delay differential equation with distributed delay [GRAPHICS] By applying these results, we also establish some integral conditions for oscillation of the higher order delay differential equations.
... In particular, the linear nonautonomous difference equation (1) has been discussed extensively in the literature, where {Pn} is a sequence of nonnegative numbers and k is a positive integer, A denotes the forward difference operator Axn = Xn+l -Xn. See, for example, [1][2][3][4][5][6][7]. In 1989, Erbe and Zhang [1] first investigated the oscillatory behavior of (1), and proved the following theorem. ...
... See Example 2 in [6]. Though some sufficient conditions for oscillation of (1) which allows Pn to oscillate around kk/(k + 1) k+l have existed in [3][4][5][6]. However, these oscillation criteria are still less effective to handle equation (1) in the critical case. ...
First we establish an equivalence of the oscillation of the following two difference equations: where pn ≥ 0 and k is a positive integer. And then, we obtain some “sharp” oscillation and nonoscillation criteria for equation.
... In a recent paper [8], Tang showed that if n-1 ( k ~k+l Z p(s) > ~,~-~/ , for n > no for some no > 0, ...
... In a different approach, Chen and Yu [2] proved that every solution of equation (1) is oscillatory provided that liminf Z p(s)<_ (8) 72-'~OO 8~n--4 and ° 1( ) ...
We introduce some new techniques to analyze generalized characteristic equations to obtain several infinite sum sharp conditions for oscillation of the following delay difference equation: , where pi(n) ≥ 0 and ki are positive integers.
... But for a delay differential equations one can easily identify a solution. The idea of delayed-advanced interactions appeared atleast as early in the works of [1][2][3][4][5][6], [8][9][10][11][12][13][14][15]. These observations are motivated the authors to study of oscillatory behavior of solutions of delay differential equations (1) Where is a positive constant. ...
In this article the authors established sufficient condition for the oscillation of the first order linear delay differential equation ( * ) where is a positive constant, with several variable delays. Some interesting examples are provided to illustrate the results
... Li [14] and Tang and Shen [10] established another new criteria for the oscillation of (1.1) without condition (1.5). These results further improve many known results in literature. ...
... In simulation, we choose φ being the numerical solution to (3.4) corresponding to critical speed c * , and ε = 1, γ = 1. The results in [11,31] shows that when d (v + ) < |b (v + )|, the traveling wave exist for 0 < r < r, and no traveling waves exist for r ≥ r, where ...
This paper is concerned with the global stability of non-critical/critical traveling waves with oscillations for time-delayed nonlocal dispersion equations. We first theoretically prove that all traveling waves, especially the critical oscillatory traveling waves, are globally stable in a certain weighted space, where the convergence rates to the non-critical oscillatory traveling waves are time-exponential, and the convergence to the critical oscillatory traveling waves are time-algebraic. Both of the rates are optimal. The approach adopted is the weighted energy method with the fundamental solution theory for time-delayed equations. Secondly, we carry out numerical computations in different cases, which also confirm our theoretical results. Because of oscillations of the solutions and nonlocality of the equation, the numerical results obtained by the regular finite difference scheme are not stable, even worse to be blow-up. In order to overcome these obstacles, we propose a new finite difference scheme by adding artificial viscosities to both sides of the equation, and obtain the desired numerical results.
... where the functions p, τ ∈ C [t 0, ∞), R + (here R + = [0, ∞)), τ (t) is nondecreasing, τ (t) < t for t ≥ t 0 and lim t→∞ τ (t) = ∞, has been the subject of many investigations. See, for example, [11], [15], [17], [21]- [26], [28], [29]- [32], [33]- [42], [44], [47]- [52], [54], [55], [59], [60], [66], [73]- [80], [82]- [84], [90] and the references cited therein. By a solution of the equation (1) we understand a continuously differentiable function defined on τ (T 0 ), ∞ for some T 0 ≥ t 0 and such that the equation (1) is satisfied for t ≥ T 0 . ...
... where {p,, } and {p~(n)} are sequences of nonnegative real numbers, k and k~ are positive integers, has I)een the subject of many investigations. See, for example, [1][2][3][4][5][6][7][8][9][10][11][12][13][14][15][16] and tile references cited therein. By a solution of equation ( ...
This paper is concerned with the oscillation of all solutions of the delay difference equations x(n+1) - x(n) + p(n)x(n-k) = 0, n = 0, 1, 2.... (*) and x(n+1) - x(n) + Sigma (m)(i=1) pi(n)x(n-ki) = 0, n = 0, 1, 2,..., (**) where {p(n)} and {p(i)(n)} are sequences of nonnegative real numbers and k and k(i) are positive integers. New oscillation criteria of the forms lim(n --> infinity) sup p(n) > alpha + C(alpha) for equation (*) and lim(n --> infinity) sup Sigma (m)(i=1) Sigma (n+ki)(s=n) p(i) (s) > 1 for equation (**) are obtained, where alpha = lim inf(n --> infinity) (1/k) Sigma (n-1)(i=n-k) p(i) and C(alpha) is "the best possible" function of a in some sense.
... Moreover, such equations may exhibit several real world phenomena, such as rhythmical beating, merging of solutions, and noncontinuity of solutions. In the recent years, there is increasing interest on the oscillation/nonoscillation of impulsive delay differential equations, and numerous papers have been published on this class of equations and good results have been obtained (see [1,2,45678 etc. and the references therein). For example, in [4], Luo researched the equation ...
In this paper, we investigate the oscillation of second-order self-conjugate differential equation with impulses(1)(a (t) (x (t) + p (t) x (t - τ))′)′ + q (t) x (t - σ) = 0, t ≠ tk, t ≥ t0,(2)x (tk+) = (1 + bk) x (tk), k = 1, 2, ...,(3)x′ (tk+) = (1 + bk) x′ (tk), k = 1, 2, ...,where a, p, q are continuous functions in [t0, + ∞), q (t) ≥ 0, a (t) > 0, ∫t0∞ (1 / a (s)) d s = ∞, τ > 0, σ > 0, bk > - 1,0 < t0 < t1< t2 < ⋯ < tk < ⋯ and limk → ∞ tk = ∞. We get some sufficient conditions for the oscillation of solutions of Eqs. (1)-(3).
... (3.7) has been established when neither (3.11) nor (3.12) is satisfied. In this direction by employing the results in Tang [35], we see that if ...
The objective of this paper is to systematically study the qualitative properties of the solutions of the discrete nonlinear delay survival red blood cells model x(n + 1) − x(n) = −(n)x(n) + p(n)e −q(n)x(n−) , n= 1, 2,. .. , where (n), p(n) and q(n) are positive periodic sequences of period. First, by using the continuation theorem in coincidence degree theory, we prove that the equation has a positive periodic solution x(n) with strictly positive components. Second, we prove that the solutions are permanent and establish some sufficient conditions for oscillation of the positive solutions about x(n). Finally, we give an estimation of the lower and upper bounds of the oscillatory solution and establish some sufficient conditions for global attractivity of x(n). From applications point of view permanence guarantees the long term survival of mature cells, oscillation implies the prevalence of the mature cells around the periodic solution and the convergence implies the absence of any dynamical diseases in the population. Our results in the special case when the coefficients are positive constants involve and improve the oscillation and global attractivity results on the literature.
... x(t − π ) (n) + (n − 1)!(1 + cos t) e(t − π ) n−1 x(t − π ) ln[e + x 2 (t − π )] = 0, t ≥ t 0 > π , n is even. It is easy to show that (see [12]) ...
In this paper, some sufficient conditions are obtained for the oscillation of all solutions of even-order nonlinear neutral differential equations with variable coefficients. Our results improve and generalize known results. In particular, the results are new even when n=2.
... Several authors including Tang et al. [1], Li [2,3], Elbert and Stavroulakis [4], Tang and Shen [5], and Domshlak and Stavroulakis [6] have studied the oscillatory behavior of the first order differential equation in a critical state and obtained a lot of interesting results. In spite of these works, there is a need for further study of oscillation and non-oscillatory behavior of certain first order delay differential equations in a critical state. ...
In this work, some new oscillation criteria are obtained for the first order delay differential equation x′(t)+1τext−τ+f(t,xt−τ1,…,xt−τs)=0,t≥0, where τ,τ1,τ2,…,τs are nonnegative numbers, f is a function. Our results improve the known results from the literature, and an example of an application is given at the end.
We use an improved technique to establish new sufficient criteria of product type for the oscillation of the delay differential equation x′(t)+∑l=1mbl(t)x(σl(t))=0,t≥t0,\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document} $$\begin{aligned} x'(t)+\sum_{l=1}^{m} b_{l}(t)x\bigl(\sigma _{l}(t)\bigr)= 0,\quad t\geq t_{0}, \end{aligned}$$ \end{document} with bl,σl∈C([t0,∞),[0,∞))\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$b_{l},\sigma _{l}\in C([t_{0},\infty ),[0,\infty ))$\end{document} such that σl(t)≤t\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$\sigma _{l}(t)\leq t$\end{document} and limt→∞σl(t)=∞\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$\lim_{t \rightarrow \infty} \sigma _{l}(t)=\infty $\end{document}, l=1,2,…,m\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$l=1,2,\ldots,m$\end{document}. The obtained results are applicable for the nonmonotone delay case. Their strength is supported by a detailed practical example.
This paper is concerned with time-delayed reaction–diffusion equations on half space
with boundary effect. When the birth rate function is non-monotone, the solution of the delayed equation subjected to appropriate boundary condition is proved to converge time-exponentially to a certain (monotone or non-monotone) traveling wave profile with wave speed
, where
is the minimum wave speed, when the initial data is a small perturbation around the wave, while the wave is shifted far away from the boundary so that the boundary layer is sufficiently small. The adopted method is the technical weighted-energy method with some new flavors to handle the boundary terms. However, when the birth rate function is monotone under consideration, then, for all traveling waves with
, no matter what size of the boundary layers is, these monotone traveling waves are always globally stable. The proof approach is the monotone technique and squeeze theorem but with some new development.
Consider the first-order linear delay differential equation x ' (t)+p(t)x(τ(t))=0,t≥t 0 ,(1) where p,τ∈C([t 0 ,∞),ℝ + ),τ(t) is nondecreasing, τ(t)<t for t≥t 0 ,lim t→∞ τ(t)=∞, and the (discrete analogue) difference equation Δx(n)+p(n)x(τ(n))=0n=0,1,2,⋯,(1) ' where Δx(n)=(n+1)-x(n),p(n) is a sequence of nonnegative real numbers and τ(n) is a nondecreasing sequence of integers such that τ(n)≤n-1 for all n≥0 and lim n→∞ τ(n)=∞. Optimal conditions for the oscillation of all solutions to the above equations are presented.
First, in this paper, we establish the equivalence of the oscillation of the differential equations with several delays of the form: x′ (t) + ∑mi = 1 pi (t) x (t - τi) 0 for t ≥ t0 and the second-order differential equations without delay of the form: (equation presented) where pi (t) ∈ C ([t0, ∞), ℝ+) and τi > 0 for i = 1, ..., m. Next, we obtain some "sharp" conditions for oscillation and nonoscillation of the first equation.
Some new oscillation criteria for the first-order delay differential equationare established in the case whereAn open problem by A. Elbert and I. P. Stavroulakis (1995, Proc. Amer. Math. Soc.123, 1503–1510) is solved.
Asymptotic relations between the solutions of a linear autonomous functional differential equation and the solutions of the corresponding perturbed equation are established. In the scalar case, it is shown that the existence of a nonoscillatory solution of the perturbed equation often implies the existence of a real eigenvalue of the limiting equation. The proofs are based on a recent Perron type theorem for functional differential equations.
Consider the first-order linear delay differential equation xʹ(t) + p(t)x(τ(t)) = 0, t≥ t<sub>0</sub>, (1) where p, τ ∈ C ([t<sub>0</sub>,∞, ℝ<sup>+</sup>, τ(t) is nondecreasing, τ(t) < t for t ≥ t<sup>0</sup> and lim<sub>t→∞</sub> τ(t) = ∞, and the (discrete analogue) difference equation Δx(n) + p(n)x(τ(n)) = 0, n= 0, 1, 2,…, (1)ʹ where Δx(n) = x(n + 1) − x(n), p(n) is a sequence of nonnegative real numbers and τ(n) is a nondecreasing sequence of integers such that τ(n) ≤ n − 1 for all n ≥ 0 and lim<sub>n→∞</sub> τ(n) = ∞. Optimal conditions for the oscillation of all solutions to the above equations are presented.
We obtain some multiple integral average conditions for oscillation and nonoscillation of the first order nonautonomous delay equations. Our theorems improve several previous well-known results.
In this article we study oscillation and nonoscillation of the solutions of linear delay differential equations. We define generic oscillation and generic nonoscillation as a topological assessment of the extent of oscillatory or nonoscillatory solutions of a delay differential equation. We provide sufficient conditions for both phenomena based on the roots of the characteristic equation and concrete conditions for some special cases.
This paper is concerned with the nonoscillation of first order delay differential equations of the form x'(t)+p(t)x(tau (t))=0, tgreater than or equal to t(0), where p, tau is an element of C ([t(0), infinity), R(+)), R(+)=[0, infinity), tau(t) is nondecreasing, tau(t) = t(0) and lim(t ->infinity)(t)=oc. New nonoscillation conditions are obtained. These conditions concern the case when the well-known nonoscillation condition integral(t)(tau)(t) p(s) dsless than or equal to1/e is not satisfied. (C) 2003 Published by Elsevier Inc.
First, two comparison theorems are established between the first order delay differential equation x ' (t)+p(t)x(tau (t))= 0, t greater than or equal to t(0) (*) and two related second order ordinary differential equations in the case when f(tau (t))(t)p(s) ds greater than or equal to 1/e. (**) Next, by using these comparison theorems some new oscillation and non-oscillation criteria are given for (C) under condition (**). As a consequence, an open problem by Elbert and Stavroulakis is completely solved.
Some almost sharp sufficient conditions of oscillation and nonoscillation are obtained for the superlinear
delay differential equation
In this paper, we study the oscillatory behavior of first order nonlinear neutral delay differential equations. Several new
sufficient conditions which ensure that all solutions are oscillatory are given. The obtained results extend and improve several
known results in the literature. Some examples are considered to illustrate the main results.
Some new oscillation and nonoscillation criteria are given for linear delay or advanced differential equations with variable coefficients and not (necessarily) constant delays or advanced arguments. Moreover, some new results on the existence and the nonexistence of positive solutions for linear integrodifferential equations are obtained.
The oscillation theory of functional differential equations (FDEs) is surveyed including (I) formulation of oscillation problems; The oscillation theory of functional differential equations (FDEs) is surveyed including (I) formulation of oscillation problems;
(II) a brief history of the oscillation theory of FDEs; (III) main topics in the oscillation theory of FDEs; (IV) open problems (II) a brief history of the oscillation theory of FDEs; (III) main topics in the oscillation theory of FDEs; (IV) open problems
in the oscillation theory of FDEs. in the oscillation theory of FDEs.
In this paper, some sufficient conditions for oscillation of a first order delay differential equation with oscillating coefficients
of the form x′(t)+p(t)x(t−τ)=0 are established, which improve and generalize some of the known results in the literature.
In this paper, we investigate the oscillation of Third-order difference equation with impulses. Some sufficient conditions for the oscillatory behavior of the solutions of Third-order impulsive difference equations are obtained.
In this work, some sufficient conditions are obtained for oscillation of all solutions of the even order neutral differential equations with variable coefficients and delays. Our results improve and generalize the known results. In particular, the results are new even when n=2.
We obtain new sufficient conditions for the oscillation of all solutions of the first-order linear delay differential equation x′(t) + p(t)x(τ(t)) = 0, with p(t) ≥ 0, τ(t) < t. Our results improve the known results in the literature which include those obtained recently in papers by Elbert and Stavroulakis, Li, and Yu et al.
A new linearized oscillation criterion is established for a nonautonomous scalar delay differential equation. The assumptions include a weak recurrence property of a partial derivative of the nonlinearity. The importance of the recurrence assumption is shown by an example.
The objective of this paper is to systematically study the qualitative properties of the solutions of the discrete nonlinear delay survival red blood cells model where δ(n), p(n) and q(n) are positive periodic sequences of period ω. First, by using the continuation theorem in coincidence degree theory, we prove that the equation has a positive periodic solution with strictly positive components. Second, we prove that the solutions are permanent and establish some sufficient conditions for oscillation of the positive solutions about . Finally, we give an estimation of the lower and upper bounds of the oscillatory solution and establish some sufficient conditions for global attractivity of . From applications point of view permanence guarantees the long term survival of mature cells, oscillation implies the prevalence of the mature cells around the periodic solution and the convergence implies the absence of any dynamical diseases in the population. Our results in the special case when the coefficients are positive constants involve and improve the oscillation and global attractivity results on the literature.
The objective of this paper is to systematically study the qualitative behavior of solutions of the nonlinear delay population model where p(n) and q(n) are positive periodic sequences of period ω,m, and ω are positive integers and ω>1. First, by using the continuation theorem in conincidence degree theory, we establish a sufficient condition for the existence of a positive ω-periodic solution with strictly positive components. Second, we establish some sufficient conditions for oscillation of the positive solutions about a periodic solution. Finally, we give an estimation of the lower and upper bounds of the oscillatory solutions and establish some sufficient conditions for the global attractivity of . Some illustrative examples are included to demonstrate the validity and applicability of the results.
dIn this paper we shall consider an odd higher-order neutral delay differential equation with variable coefficients. Several finite and infinite sufficient conditions for oscillation of all solutions are obtained. These conditions are almost sharp which extend and improve some of the well known results in the literature, and can be extended to the more general equation with several coefficients and several delays. Some examples are considered to illustrate our results.
Some new oscillation criteria of first order nonlinear neutral delay differential equations with variable coefficients are given. Our results extend and improve some of the well known previously results in the literature. Some examples are considered to illustrate our main results.
In this paper, a class of nonlinear delay functional differential equations with variable coefficients is linearized,and through analogizing the oscillation theory of linear functional differential equation, we obtain many oscillationcriteria of this class of equation by using the Schauder fixed point theorem.
Journal of Mathematics Research, Vol. 1, No. 1, March 2009
In this paper, some new oscillation criteria are obtained for the first order nonlinear delay dierential equation x 0ðt Þþ pðtÞ f ðxðtt1Þ; ... ; xðttmÞÞ ¼ 0 and the corresponding advanced dierential equation. Our results improve the known results in the literature. And an example is given to demonstrate the advantage of our results.
We obtain new sufficient conditions for the oscillation of all solutions of first-order delay dynamic equations on arbitrary time scales, hence combining and extending results for corresponding differential and difference equations. Examples, some of which coincide with well-known results on particular time scales, are provided to illustrate the applicability of our results.
Oscillation criteria are obtained for all solutions of first-order nonlinear neutral delay differential equations. Our results extend and improve some results well known in the literature. Some examples are considered to illustrate our main results.
Consider the first-order linear delay differential equation $$ x'(t)+p(t)x(au (t))=0,quad tgeq t_{0}, $$ and the second-order linear delay equation $$ x''(t)+p(t)x(au (t))=0,quad tgeq t_{0}, $$ where $p$ and $au $ are continuous functions on $[t_{0},infty )$, $p(t)>0$, $au (t)$ is non-decreasing, $au (t)leq t$ for $tgeq t_{0}$ and $lim_{to infty }au (t)=infty $. Several oscillation criteria are presented for the first-order equation when $$ 0<liminf_{to infty }int_{au (t)}^{t}p(s)dsleq frac{1}{e} quad hbox{and}quad limsup_{to infty }int_{au (t)}^{t}p(s)ds<1, $$ and for the second-order equation when $$ liminf_{to infty }int_{au (t)}^{t}au (s)p(s)ds leq frac{1}{e}quad hbox{and}quad limsup_{to infty }int_{au (t)}^{t}au (s)p(s)ds<1,. $$
In this article we present infinite-integral conditions for the oscillation of all solutions of first-order delay differential equations with positive and negative coefficients.
We obtain sufficient conditions for the oscillation of all solutions of the neutral differential equation d/dt[y(t)+Py(t-τ)]+Q(t)y(t-σ)=0, where P∈ℝ, τ∈(0,∞), σ∈[0,∞) and Q∈C[[t 0 ,∞),ℝ + ]. Our theorems extend and improve some of the known results in the literature.
Consider the delay differential equation ẋ(t) + p(t)x(t − τ) = 0, where p(t) ∈ C[[t0, ∞), R+] and τ is a positive constant. We show that every solution of this equation oscillates if ∫tt−τp(t) dt ≥ 1/e for sufficiently large t and ∫∞t0+τp(t)[exp(∫tt−τp(s) ds − 1/e) − 1] dt = ∞.
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.
We obtain sufficient conditions for the oscillation of all solutions of the linear delay differential equation with positive and negative coefficients
where
Extensions to neutral differential equations and some applications to the global asymptotic stability of the trivial solution are also given.
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.
We obtain sufficient conditions for the oscillation of all solutions of the higher order neutral differential equation dn/dm[y(t) + P(t) y(t - μ)] + Q(t) y(t −σ) = 0, t ≧ t0 where n ≧ 1, P ϵ C[t0, ∞), R], Q ϵ C[t0, ∞), R] and τ, μ ϵ R+.
Our results extend and improve several known results in the literature.
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