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In this article, we are concerned with the question of the asymptotic decay of the oscillatory solutions to a class of first order linear unstable type delay differential equations with variable coefficients and constant delays. We establish a sufficient condition for all oscillatory solutions to tend to zero at ∞. This condition is substantially weaker than the one given by Ladas and the third author [Hiroshima Math. J. 18 (1988), 351-359].

Second-order linear (non-autonomous as well as autonomous) delay differential equations of unstable type are considered. In the non-autonomous case, sufficient conditions are given in order that all oscillatory solutions are bounded or all oscillatory solutions tend to zero at $\infty$. In the case where the equations are autonomous, necessary and sufficient conditions are established for all oscillatory solutions to be bounded or all oscillatory solutions to tend to zero at $\infty$.

Some new results are given concerning the behavior of the oscillatory solutions of first or second order delay differential equations. These results establish that all oscillatory solutions x of a first or second order delay differential equation satisfy x (t) = 0 (v(t)) as t --> infinity, where v is a nonoscillatory solution of a corresponding first or second order linear delay differential equation. Some applications of the results obtained are also presented. (C) 2003 Elsevier Inc. All rights reserved. Journal of Mathematical Analysis and Applications

This paper is concerned with the oscillatory behaviour of first-order delay differential equations of the form \begin{equation} x^{\prime}(t) + p(t)x(\tau(t)) = 0, \quad t \geqslant t_0, \end{equation} where be defined by \[ k = \mathop{{\lim\inf}}_{{t\rightarrow\infty}} \int\nolimits^t_{\tau(t)} p(s)ds \quad\hbox{and}\quad L = \mathop{\lim\sup}_{{t\rightarrow\infty}} \int\nolimits^t_{\tau(t)} p(s)ds. \]It is proved here that when all solutions of equation (1) oscillate in several cases in which the condition \[ L > \frac{\ln\lambda_1 - 1 + \sqrt{5 - 2\lambda_1 + 2k\lambda_1}}{\lambda_1} \] holds, where .

A new oscillation criterion is given for the delay differential equation , where and the function T defined by is increasing and such that . This criterion concerns the case where .

This paper is concerned with the existence of solutions of a boundary value problem for second-order functional differential equations. Specifically the following problem is considered: x(n)(t) = f(t, x(t), x'(t)), t is an element of [O, T], alpha(0)x(0) - alpha(1)x'(O) = phi is an element of C-r beta(0)x(T) + beta(1)x'(T) = A is an element of R(n). The results are based on a nonlinear alternative of Granas and the use of a priori bounds on solutions. Some examples are also discussed to illustrate how these results may be used to yield the existence of solutions of specific boundary value problems. (C) 1996 Academic Press, Inc. Journal of Mathematical Analysis and Applications

Existence results for initial value problems for neutral functional differential equations are given. The results are based on the Leray-Schauder alternative and rely on a priori bounds on solutions. Applications to certain integro-differential equations are also given.

Consider the n-th order (n ≥ 1 ) neutral differential equation where σ1 < σ 2 < ∞ and μ and η are increasing real-valued functions on [Ƭ1, Ƭ2] and [σ1, σ2] respectively. The function μ is assumed to be not constant on [Ƭ1, Ƭ2] and [Ƭ1, Ƭ2] for every Ƭ ∈ (Ƭ1, Ƭ2) similarly, for each σ ∈ (σ1, σ2), it is supposed that r\ is not constant on [σ1 , σ] and [σ, σ2]. Under some mild restrictions on Ƭ1,- and σ1, (ι = 1,2), it is proved that all solutions of (E) are oscillatory if and only if the characteristic equation of (E) has no real roots.

Applying an analytical method and several limiting equations arguments, some sufficient conditions are provided for the existence of a unique positive equilibriumK for the delay differential equationx=−γx+D(xt), which is the general form of many population models. The results are concerned with the global attractivity, uniform stability, and uniform asymptotic stability ofK. Application of the results to some known population models, which shows the effectiveness of the methods applied here, is also presented.

IN THIS paper we are interested in providing sufficient conditions guaranteeing the fact that all positive solutions of the discrete delay difference equation A II+1 = AA, + F(A,_,) (E) converge as n + 00. In (E), A is a constant real number with 0 0 sufficiently small. Thus, formally, equation (E) comes from x(t + h) x(t) h = -yx(t) + D(x(t r)) for small h. If we set ~,z(f) := x(ht), then the preceding equation for t = n and r/h = m becomes yh@ + 1) = (1 ?‘h)Y, (n) + hD(Y,(n m)), which is of the form (E), where A := 1 yh and F(u) := hD(u), u 2 0. Clearly h > 0 is small if and only if A E (0, 1). 1069 @h)

We obtain a necessary and sufficient condition for the oscillation of all solutions of the neutral delay differential equation: (1) \[ d d t [ x ( t ) + p x ( t − τ ) ] + Q ( t ) x ( t − σ ) = 0 , \tfrac {d}{{dt}}[x(t) + px(t - \tau )] + Q(t)x(t - \sigma ) = 0, \] where p ∈ R , Q ∈ C [ [ 0 , ∞ ) , R + ] , Q p \in {\mathbf {R}},Q \in C[[0,\infty ),{{\mathbf {R}}^ + }],Q is ω \omega -periodic with ω > 0 , Q ( t ) [ u n k ] 0 \omega > 0,Q(t)[unk]0 for t ≧ 0 t \geqq 0 , and there exist positive integers n 1 {n_1} and n 2 {n_2} such that τ = n 1 ω \tau = {n_1}\omega and σ = n 2 ω \sigma = {n_2}\omega . More precisely we show that every solution of (1) oscillates if and only if every solution of an associated neutral equation with constant coefficients oscillates.

We established a comparison result for the oscillation of all solutions of the linear delay equation with positive and negative coefficients $$\dot x(t) + P(t)x(t - \tau ) - Q(t)x(t - \sigma ) = 0,t \geqslant t_0$$ in terms of the oscillation of the « limiting » equation $$\dot y(t) + py(t - \tau ) - qy(t - \sigma ) = 0$$ (*) where p= lim infP(t) and q=lim supQ(t). t→∞ t→∞

This paper is concerned with the oscillatory behavior of first-order delay differential equations of the form (1) x'(t) + p(t)x(tau(t)) = 0, t greater than or equal to t(0), where p, tau is an element of C([t(0), infinity), R+), R+ = [0, infinity), tau(t) is non-decreasing, tau(t) < t for t greater than or equal to t(0) and lim(t-->infinity) tau(t) = infinity. Let the numbers k and L be defined by [GRAPHICS] It is proved here that when L < 1 and 0 < k less than or equal to 1/e all solutions of Eq. (1) oscillate in several cases in which the condition L > 2k + 2/lambda(1) -1 holds, where lambda(1) is the smaller root of the equation lambda = e(k lambda). Proceedings of the American Mathematical Society

We consider the (nonlinear) difference equation x n = a + ∑ k=1 m b k x n −k , n = 0,1,2… where a and bk (k = 1, 2, …, m) are nonnegative numbers with B ≡ ∑mk = 1bk > 0, and we are interested in whether all positive solutions are attracted by the positive equilibrium L = ( a 2 ) + ( a 2 ) 2 + B .

This paper deals with some asymptotic properties of nonoscillatory solutions of a class of n-th order (n < 1) differential equations with deviationg arguments involving the so called n-th order r-derivative of the unknown function x defined bywhere ri (i = 0,1…n) are positive continous functions on [t0, ∞). The fundamental purpose of this paper is to find for any integer m, 0 < m < n – 1, a necessary and sufficient condition (depending on m) in order that three exists at least one (nonoscillatory) solution x so that the exists in R – {0} The results obtained extend some recent ones due to Philos (1978a) and they prove, in a general setting, the validity of a conjecture made by Kusano and Onose (1975).

We established sufficient conditions for the global attractivity of the positive equilibrium of the delay differential equation N(t)=−mu;N(t)+∑i=1m pi exp[−γN(t−τi)], t⩾0, m⩾1. For m = 1, equation (1) was used by Wazewska-Czyzewska and Lasota as a model for the survival of red-blood cells in an animal.

is studied. The method used there in proving the main result (Theorem 1) requires a strong enough integral condition (Condition (iii)). So, results of the same nature with Theorem 1 in [2] concerning the unforced case cannot be included in it. The importance of an appropriate improvement of this condition is also remarked by Kartsatos at the end of [2]. The purpose of the present paper is to improve Theorem 1 in [2] in order to become a natural generalization of other known results and in the same time to give some interesting applications of it. The proof given here remains the same in the cases of ordinary, functional or differential equations with perturbed arguments. We are going to deal with the last case in order to derive, as a corollary, a recent result due to MarGiak [4] concerning retarded differential equations. Let now consider the lplth order differential equation with perturbed arguments

The oscillatory behavior of the solutions of the neutral delay differential equationwhere p, τ, and a are positive constants and Q ∊ C([t0, ∞), ℝ+), are studied.

Consider the n th n{\text {th}} order delay differential equation (1) \[ x ( n ) ( t ) + ( − 1 ) n + 1 ∑ i = 0 k p i x ( t − τ i ) = 0 , t ≥ t 0 , {x^{(n)}}(t) + {( - 1)^{n + 1}}\sum \limits _{i = 0}^k {{p_i}x(t - {\tau _i}) = 0, \qquad t \geq {t_0}}, \] where the coefficients and the delays are constants such that 0 = τ 0 > τ 1 > ⋯ > τ k ; p 0 ≥ 0 , p i > 0 , i = 1 , 2 , … , k ; k ≥ 1 0 = {\tau _0} > {\tau _{1}}\, > \cdots > {\tau _k};{p_0}\, \geq 0,{p_i} > 0,i = 1,2,\ldots ,k;k \geq 1 and n ≥ 1 n \geq 1 . The characteristic equation of (1) is (2) \[ λ n + ( − 1 ) n + 1 ∑ i = 0 k p i e − λ τ i = 0 . {\lambda ^n} + {( - 1)^{n + 1}}\;\sum \limits _{i = 0}^k {{p_i}{e^{ - \lambda {\tau _i}}} = 0}. \] We prove the following theorem. Theorem. (i) For n n odd every solution of (1) oscillates if and only if (2) has no real roots. (ii) For n n even every bounded solution of (1) oscillates if and only if (2) has no real roots in ( − ∞ , 0 ] ( - \infty ,0] . The above results have straightforward extensions for advanced differential equations.

Let us consider the retarded differential equation1for which the following assumptions are made:(i) p: [t0, ∞) → [0, ∞) is continuous and not identically zero for all large t.(ii) σ [t0, ∞)→ ℝ is continuous, strictly increasing,(iii) φ: ℝ → ℝ is continuous, non-decreasing and

On considere une equation differentielle fonctionnelle de type neutre {x(t)−g(t,x t )}'=f(t,x t ) ou f et g sont des fonctions continues de J×C r →R n , J=[t o ,t 0 +A]

Consider the neutral delay differential equation d2dt2 [y(t) + Py(t − τ)] + qy(t − σ) = 0, (1) where p, q ϵ R and τ, σ ϵ R+. We proved that Eq. (1) oscillates if and only if the characteristic equation λ2 + λ2pe−λτ + qe−λσ = 0 has no real roots.

In this paper we obtain sufficient conditions under which every solution of the retarded differential equation \[ ( 1 ) x ′ ( t ) + p ( t ) x ( t − τ ) = 0 , t ⩾ t 0 , (1)\quad x’(t) + p(t)x(t - \tau ) = 0,\quad t \geqslant {t_0}, \] , where τ \tau is a nonnegative constant, and p ( t ) > 0 p(t) > 0 , is a continuous function, tends to zero as t → ∞ t \to \infty . Also, under milder conditions, we prove that every oscillatory solution of (1) tends to zero as t → ∞ t \to \infty . More precisely the following theorems have been established. Theorem 1. Assume that ∫ t 0 ∞ p ( t ) d t = + ∞ \int _{{t_0}}^\infty {p(t)dt = + \infty } and lim t → ∞ ∫ t − τ t p ( s ) d s > π / 2 {\lim _{t \to \infty }}\int _{t - \tau }^t {p(s)ds > \pi /2} or lim sup t → ∞ ∫ t − τ t p ( s ) d s > 1 \lim {\sup _{t \to \infty }}\int _{t - \tau }^t {p(s)ds > 1} . Then every solution of (1) tends to zero as t → ∞ t \to \infty . Theorem 2. Assume that lim sup t → ∞ ∫ t → τ t p ( s ) d s > 1 \lim {\sup _{t \to \infty }}\int _{t \to \tau }^t {p(s)ds > 1} . Then every oscillatory solution of (1) tends to zero as t → ∞ t \to \infty .

Sufficient conditions are given for the occurrence of various types of asymptotic behaviour in the solution of a class of n th order neutral delay differential equations. The conditions are in the form of certain inequalities amongst the constants involved in the definition of the differential equations, and specify either oscillatory behavior, or asymptotic divergence, or solutions which converge to zero.

SynopsisThis paper is concerned with the existence of solutions of a two point boundary value problem for neutral functional differential equations. We consider the problemwhere M and N are n × n matrices. This is examined by using the “shooting method”. Also, an example is given to illustrate how our result can be applied to yield the existence of solutions of a periodic boundary value problem.

Consider the neutral equation with several positive and negative coefficients dd[tx(t)+∑Ipix(t−τi)−∑Jrjx(t−ϱj)]+∑Kqkx(t−σk)=0, (1) where I, J, and K are initial segments of natural numbers, pi, τi, rj, ϱj, qk are positive numbers, and σk ⩾ 0 for i ϵ I, j ϵ J, and k ϵ K. Then a necessary and sufficient condition for the oscillation of all solutions of (1) is that its characteristic equation λ+λ∑Ipie−λτi−λ∑Irje−λϱj+∑Kqke−λσk=0 has no real roots.

Consider the delay differential equation \[ (*)\quad y'(t) + py(t - \tau ) - qy(t - \sigma ) = 0\] where p, q, $\tau $, and $\sigma $ are positive constants.THEOREM. Assume that$\sigma \leqq \tau $, $q 1$. Then every solution of$(*)$oscillates.The above result was extended to equations with several delays.Finally we obtained sufficient conditions for the oscillation of delay differential equations with oscillating coefficients.

Consider the second order neutral differential equation1where the coefficients p and q and the deviating arguments τ and σ are real numbers. The characteristic equation of Eq. (1) is2The main result in this paper is the following necessary and sufficient condition for all solutions of Eq. (1) to oscillate.THEOREM. The following statements are equivalent:(a) Every solution of Eq. (1) oscillates.(b) Equation (2) has no real roots.

This paper is concerned with the oscillatory and asymptotic behavior of the solutions of the differential equation \[ ({\text{E}},\delta ) \qquad \left[ {r(t)x^{(n - 1)} (t)} \right]^\prime + \delta \sum_{i = 1}^N {p_i (t)f_i \left( {x\left[ {\tau _1 (t)} \right], \cdots ,x\left[ {\tau _m (t)} \right]} \right)} = 0 \] where $\delta = \pm 1$. Conditions which insure that every solution of $({\text{E}}, 1)$ is oscillatory or tending monotonically to zero are given. A classification of all solutions of $({\text{E}}, - 1)$ with respect to their behavior as $t \to \infty $ and to their oscillatory character is also obtained. Finally a comparison with the oscillatory behavior of second order equations is presented. The obtained results unify, extend and improve recent ones by Lovelady and by the authors.

C”) C w for some t, E prr w ? In 1947, Waiewski [15], in order to study this problem in the case where f is continuous and the solutions of (E) are uniquely determined with respect to their initial values, introduced its well-known topological method. In 1971 Jackson and Kladsen [3] proved the theorem of Waiewski without the assump- tion of uniqueness. Earlier, in 1965, Staikos [lo, 111, in order to avoid the assumption of uniqueness, introduced an analytical method which has the additional advantage that it does not require the existence of any retraction, as the topological method does. This will be referred as the “Anti-Lyapunov” method and it is based on the characterization of the boundary points of w as points of egress, by some Lyapunov-like functions. The purpose here is to formulate the last method for differential equations of Caratheodory type. The main difficulty arises from the fact that the derivative along the solutions of (E) of the Lyapunov-like functions used does not exist in all points of &. This difficulty has been surmounted by introducing the new concept of semiregularity of the set w (Definition 2.3) and by generalizing

In this paper we are dealing with the oscillatory and asymptotic behaviour of the nth order, (n> 1), retarded differential equationwheregis differentiable on the half-line (t0,∞) with (I)g(t) ≤ tfor everyt ≥ t0(II)g'(t)≥ 0 for everyt≥t0(III)= ∞.