On the oscillation of first‐order neutral delay differential equations with real coefficients

Abstract

We prove sufficient conditions for the oscillation of all solutions of a scalar first-order neutral delay differential equationẋ(t)− cẋ(t − τ)+ n i=1 p i x(t − σ i ) = 0 for all 0 < c < 1, τ,σ i > 0, and p i ∈ R, i = 1, 2,...,n.2000 Mathematics Subject Classification: 34C15, 34K40.1. Introduction. The theory of neutral delay differential equations presents complications and the results which are true for neutral differential equations may not be true for nonneutral differential equations. Besides its theoretical interest, the study of oscillatory behaviour of solutions of neutral delay differential equations has some importance in applications. Neutral delay differential equations appear in networks containing lossless transmission lines (as in high-speed computers where the lossless transmission lines are used to interconnect switching circuits), in the study of vibrating masses attached to an elastic bar and also in population dynamics (see Gopalsamy In fact, Zahariev and Baȋnov [11] seems to be the first paper dealing with oscillation of neutral equations. A systematic development of oscillation theory of neutral equations was initiated by Ladas and Sficas [10].Ladas and Schultz [9] obtained a necessary and sufficient condition for oscillation of all solutions of the neutral delay differential equatioṅ

Some Further Results on Oscillations for Neutral Delay Differential Equations with Variable Coefficients

“…For the sake of obtaining a contradiction, assume that there is an eventually positive solution ( ) of (1). Let ( ) be defined by (16). Proceeding as in the proof of Theorem 7, we again obtain (37), which guarantees that eventually Then every solution of NDDE (1) oscillates.…”

“…For the sake of obtaining a contradiction, assume that there is an eventually positive solution ( ) of (1). Let ( ) be defined by (16). Then by Lemma 4, we obtain…”

“…which has been studied by several authors including Gopalsamy and Zhang [10], Chuanxi and Ladas [11,12], Choi and Koo [13], Greaf et al [14], Erbe et al [15], and Al-Amri [16].…”

Oscillations for Neutral Functional Differential Equations

“…In [ 5 ], some finite integral conditions for oscillation of all solutions of ( 1 ) when r ( t ) ≡ 1 are given under less restrictive hypothesis on p . See also Grammatikopoulos et al [ 10 ], Ladas and Sficas [ 15 ], and Al-Amri [ 4 ].…”

Oscillation Criteria for Linear Neutral Delay Differential Equations of First Order

“…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.…”