scholarly journals Necessary and sufficient condition for oscillations of neutral differential equations

1987 ◽  
Vol 28 (3) ◽  
pp. 362-375 ◽  
Author(s):  
M. R. S. Kulenović ◽  
G. Ladas ◽  
A. Meimaridou
Keyword(s):  
Sufficient Condition ◽  
Necessary And Sufficient Condition ◽  
Neutral Delay ◽  
Neutral Differential Equations ◽  
Real Roots ◽  
Neutral Delay Differential Equation ◽  
All Solutions ◽  
Image Position ◽  
Delay Differential ◽  
Necessary And Sufficient

AbstractConsider the neutral delay differential equationwhere p ∈ R, τ ≥ 0, q1 > 0, σ1 ≥ 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 equationhas no real roots.

scholarly journals Oscillations of higher order neutral differential equations

1992 ◽  
Vol 52 (2) ◽  
pp. 261-284 ◽  
Author(s):  
S. J. Bilchev ◽  
M. K. Grammatikopoulos ◽  
I. P. Stavroulakis
Keyword(s):  
Differential Equation ◽  
Differential Equations ◽  
Sufficient Conditions ◽  
Sufficient Condition ◽  
Necessary And Sufficient Condition ◽  
Neutral Differential Equations ◽  
Real Roots ◽  
All Solutions ◽  
Image Position ◽  
Necessary And Sufficient

AbstractConsider the nth-order neutral differential equation where n ≥ 1, δ = ±1, I, K are initial segments of natural numbers, pi, τi, σk ∈ R and qk ≥ 0 for i ∈ I and k ∈ K. Then a necessary and sufficient condition for the oscillation of all solutions of (E) is that its characteristic equation has no real roots. The method of proof has the advantage that it results in easily verifiable sufficient conditions (in terms of the coefficients and the arguments only) for the oscillation of all solutionso of Equation (E).

Oscillation and Global Attractivity in a Periodic Delay Equation

1996 ◽  
Vol 39 (3) ◽  
pp. 275-283 ◽  
Author(s):  
J. R. Graef ◽  
C. Qian ◽  
P. W. Spikes
Keyword(s):  
Global Attractivity ◽  
Positive Periodic Solution ◽  
Continuous Functions ◽  
Sufficient Condition ◽  
Necessary And Sufficient Condition ◽  
All Solutions ◽  
Periodic Delay ◽  
Image Position ◽  
Delay Differential ◽  
Necessary And Sufficient

AbstractConsider the delay differential equationwhere α(t) and β(t) are positive, periodic, and continuous functions with period w > 0, and m is a nonnegative integer. We show that this equation has a positive periodic solution x*(t) with period w. We also establish a necessary and sufficient condition for every solution of the equation to oscillate about x*(t) and a sufficient condition for x*(t) to be a global attractor of all solutions of the equation.

scholarly journals Oscillation of neutral delay differential equations

1992 ◽  
Vol 45 (2) ◽  
pp. 195-200 ◽  
Author(s):  
Jianshe Yu ◽  
Zhicheng Wang ◽  
Chuanxi Qian
Keyword(s):  
Delay Differential Equations ◽  
Necessary Condition ◽  
Sufficient Condition ◽  
Neutral Delay ◽  
Open Problems ◽  
Neutral Delay Differential Equation ◽  
All Solutions ◽  
Neutral Delay Differential Equations ◽  
Image Position ◽  
Delay Differential

Consider the following neutral delay differential equationwhere P ∈ ℝ, T ∈ (0, ∞), σ ∈ (0, ∞) and Q ∈ C[(t0, ∞), [0, ∞)]. We obtain a sufficient condition for the oscillation of all solutions of Equation (*) with P = −1, which does not require thatBut, for the cases −1 < P < 0 and P < −1, we show that (**) is a necessary condition for the oscillation of all solutions of Equation (*). These new results solve some open problems in the literature.

scholarly journals Oscillation Criteria for Second Order Neutral Differential Equations

1996 ◽  
Vol 48 (4) ◽  
pp. 871-886 ◽  
Author(s):  
Horng-Jaan Li ◽  
Wei-Ling Liu
Keyword(s):  
Differential Equation ◽  
Differential Equations ◽  
Delay Differential Equations ◽  
Second Order ◽  
Neutral Delay ◽  
Oscillation Criteria ◽  
Neutral Differential Equations ◽  
Neutral Delay Differential Equation ◽  
Image Position ◽  
Delay Differential

AbstractSome oscillation criteria are given for the second order neutral delay differential equationwhere τ and σ are nonnegative constants, . These results generalize and improve some known results about both neutral and delay differential equations.

Oscillations of Second Order Neutral Differential Equations

1993 ◽  
Vol 36 (4) ◽  
pp. 485-496 ◽  
Author(s):  
Shigui Ruan
Keyword(s):  
Differential Equations ◽  
Delay Differential Equations ◽  
Sufficient Conditions ◽  
Oscillatory Behavior ◽  
Second Order ◽  
Neutral Delay ◽  
Neutral Differential Equations ◽  
Neutral Delay Differential Equation ◽  
Image Position ◽  
Delay Differential

AbstractIn this paper, we consider the oscillatory behavior of the second order neutral delay differential equationwhere t ≥ t0,T and σ are positive constants, a,p, q € C(t0, ∞), R),f ∊ C[R, R]. Some sufficient conditions are established such that the above equation is oscillatory. The obtained oscillation criteria generalize and improve a number of known results about both neutral and delay differential equations.

Oscillations of Second Order Neutral Equations

1988 ◽  
Vol 40 (6) ◽  
pp. 1301-1314 ◽  
Author(s):  
G. Ladas ◽  
E. C. Partheniadis ◽  
Y. G. Sficas
Keyword(s):  
Second Order ◽  
List Type ◽  
Necessary And Sufficient Condition ◽  
Real Numbers ◽  
Neutral Equations ◽  
Deviating Arguments ◽  
Real Roots ◽  
All Solutions ◽  
Image Position ◽  
Necessary And Sufficient

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.

scholarly journals Existence of nonoscillatory solutions of first order nonlinear neutral equations

1990 ◽  
Vol 32 (2) ◽  
pp. 180-192 ◽  
Author(s):  
Lu Wudu
Keyword(s):  
Nonoscillatory Solution ◽  
Sufficient Condition ◽  
Neutral Equation ◽  
Necessary And Sufficient Condition ◽  
Nonoscillatory Solutions ◽  
Neutral Equations ◽  
First Order ◽  
Image Position ◽  
Necessary And Sufficient

AbstractConsider the nonlinear neutral equationwhere pi(t), hi(t), gj(t), Q(t) Є C[t0, ∞), limt→∞hi(t) = ∞, limt→∞gj(t) = ∞ i Є Im = {1, 2, …, m}, j Є In = {1, 2, …, n}. We obtain a necessary and sufficient condition (2) for this equation to have a nonoscillatory solution x(t) with limt→∞ inf|x(t)| > 0 (Theorems 5 and 6) or to have a bounded nonoscillatory solution x(t) with limt→∞ inf|x(t)| > 0 (Theorem 7).

scholarly journals Oscillation of Nonlinear First Order Neutral Differential Equations

2007 ◽  
Vol 4 (3) ◽  
pp. 485-490
Author(s):  
Baghdad Science Journal
Keyword(s):  
Differential Equations ◽  
Delay Differential Equations ◽  
Integral Conditions ◽  
Neutral Delay ◽  
Neutral Differential Equations ◽  
First Order ◽  
All Solutions ◽  
Neutral Delay Differential Equations ◽  
Delay Differential

In this paper, the author established some new integral conditions for the oscillation of all solutions of nonlinear first order neutral delay differential equations. Examples are inserted to illustrate the results.

Prime Producing Quadratic Polynomials and Class-Numbers of Real Quadratic Fields

1990 ◽  
Vol 42 (2) ◽  
pp. 315-341 ◽  
Author(s):  
Stéphane Louboutin
Keyword(s):  
Class Number ◽  
Quadratic Field ◽  
Sufficient Condition ◽  
Necessary And Sufficient Condition ◽  
Quadratic Polynomials ◽  
Real Quadratic Fields ◽  
Consecutive Integers ◽  
Image Position ◽  
Necessary And Sufficient ◽  
Real Quadratic

Frobenius-Rabinowitsch's theorem provides us with a necessary and sufficient condition for the class-number of a complex quadratic field with negative discriminant D to be one in terms of the primality of the values taken by the quadratic polynomial with discriminant Don consecutive integers (See [1], [7]). M. D. Hendy extended Frobenius-Rabinowitsch's result to a necessary and sufficient condition for the class-number of a complex quadratic field with discriminant D to be two in terms of the primality of the values taken by the quadratic polynomials and with discriminant D (see [2], [7]).

On Global Inverse Theorems of Szász and Baskakov Operators

1979 ◽  
Vol 31 (2) ◽  
pp. 255-263 ◽  
Author(s):  
Z. Ditzian
Keyword(s):  
Rate Of Convergence ◽  
Exponential Growth ◽  
Polynomial Growth ◽  
Continuous Functions ◽  
Sufficient Condition ◽  
Necessary And Sufficient Condition ◽  
Baskakov Operators ◽  
Inverse Theorems ◽  
Image Position ◽  
Necessary And Sufficient

The Szász and Baskakov approximation operators are given by1.11.2respectively. For continuous functions on [0, ∞) with exponential growth (i.e. ‖ƒ‖A ≡ supx\ƒ(x)e–Ax\ < M) the modulus of continuity is defined by1.3where ƒ ∈ Lip* (∝, A) for some 0 < ∝ ≦ 2 if w2(ƒ, δ, A) ≦ Mδ∝ for all δ < 1. We shall find a necessary and sufficient condition on the rate of convergence of An(ƒ, x) (representing Sn(ƒ, x) or Vn(ƒ, x)) to ƒ(x) for ƒ(x) ∈ Lip* (∝, A). In a recent paper of M. Becker [1] such conditions were found for functions of polynomial growth (where (1 + \x\N)−1 replaced e–Ax in the above). M. Becker explained the difficulties in treating functions of exponential growth.

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