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Oscillation and nonoscillation of a class of neutral equations B. G. Zhang * and K. Gopalsamy Abstract. A new sufficient condition for the oscillation of the odd order neutral equation (x(t) - x(t - r))(n) + Q(t)x(t -σ)=0 is obtained. For η = 1, the condition obtained is sharp. Also, a necessary and sufficient condition for the existence of a positive solution of the odd order neutral equation τη (x(t) - x(t - τ ) ) ( n ) + Qi(t)x{t - a) = 0 i=l is established. 1991 Mathematics Subject Classification: 34K40, 34K15. 1. Introduction Recently, the

On solutions of a neutral equation with an oscillatory coefficient John R. Graef* and Paul W. Spikes* Abstract. The authors obtain results on the asymptotic behavior of solutions of the nonlinear neutral delay differential equation [y(t) + P(t)y(g(t))}{n) ~ QW (y(h(t))) = 0. Of special interest Me the cases where P(t) oscillates about —1 or P(t) has arbitrarily large zeros. 1991 Mathematics Subject Classification: Primary 34K40, 34K15; Secondary 34C11, 34C15. 1. Introduction In this paper we examine the asymptotic behavior of solutions of the nonlinear

Convergence in neutral equations with infinite delay arising from active compartmental systems Jianhong Wu* Abstract. We establish a convergence result for a system of integrodifferential equations of neutral type which was proposed as a model for the transmission dynamics of a bio- logical compartmental system. Our approach is based on an extension of an asymptotic stability theorem of Dancer and Hess to a class of set-condensing and quasi-strongly order-preserving semiflows. 1991 Mathematics Subject Classification: 34K40, 92B05. was proposed as a model

DEMONSTRATIO MATHEMATICA Vol. XXI No 2 198« Marek Barszcz ROOTS LOCALIZATION FOR NEUTRAL EQUATIONS AND THEIR SENSITIVITY FOR A CHANGE OF DELAY 1. In t roduc t ion In t h i s paper quasi-polynomials of the fol lowing form k (1.1) H(z) = J ] i=0 where (1 .2) h ^ z ) - Σ exp(yz) , βμ 1»ίΟ, j a ^ ^ o o V i , where M̂ c [o„m]cR are countable s e t s , such tha t V i 3 j > i t h a t sup Uj = max Mj? sup M^, V i 3 j ¿ i tha t inf Mj = min Mj s in f Μ±, (1.3) are considered. The l o c a l i z a t i o n of r o o t s w i l l be considered f o r suoh

asymptotic behavior of higher order neutral equations with variable coefficients”, Chin. Ann. of Math., Vol. 9B, (1988), pp. 322–338. [9] K. Gopalsamy, B.S. Lalli and B.G. Zhang: “Oscillation of odd order neutral differential equations”, Czech. Math. J., Vol. 42, (1992), pp. 313–323. [10] I. Győri and G. Ladas: Theory of Delay Differential Equations with Applications, Clarendon Press, Oxford, (1991). [11] J. Hale: Theory of functional differential equations, Springer-Verlag, New York, 1977. [12] N. Parhi and P.K. Mohanty: “Oscillation of neutral differential equations

. Key words and phrases: Integrodifferential equation, neutral equation, unbounded delay, asymptotic behavior, stability. 1. Introduction And Statement Of The Main Results During the past four decades, the theories of Volterra integral equations and Volterra integrodifferential equations have undergone rapid developments. Vari- ous classical problems in the theory of differential equations (ordinary or partial) lead to integral or integrodifferential equations and, in many cases, can be dealt with in a more satisfactory manner using these (integral or


The relationship between boundedness and oscillation of solutions of the third order neutral differential equations are presented.


Neutral differential equations are one of the most important extensions of classical ordinary differential equations and aim to give a better explanation for modeling phenomena where ordinary differential equations are insufficient. Naturally, all the questions studied in the scope of ordinary differential equations attracted the attention also for neutral differential equations. In this paper we study the oscillatory properties of second order half-linear neutral differential equations. We present oscillation criteria derived using a new approach. This approach allows us to reduce common restrictions on the deviations in arguments which are present in the currently known results of this type.


In this work, we present a new concept of Stepanov weighted pseudo almost periodic and automorphic functions which is more generale than the classical one, and we obtain a new existence result of μ-pseudo almost periodic and μ-pseudo almost automorphic mild solutions for some nonautonomous evolution equations with Stepanov μ-pseudo almost periodic terms. An example is shown to illustrate our results.

[1] LU, S.— REN, J.— GE, W.: Problems of periodic solutions for a kind of second order neutral functional differential equation, Appl. Anal. 82 (2003), 411–426. [2] ZHU, Y.— LU, S.: Periodic solutions for p-Laplacian neutral functional differential equation with multiple deviating arguments, J. Math. Anal. Appl. 336 (2007), 1357–1367. [3] DU, B.— GUO, L.— GE, W.— LU, S.: Periodic solutions for generalized Liénard neutral equation with variable parameter, Nonlinear Anal. 70