The Equationx′(t) =ax(t) +bx(t− τ) With “Small” Delay

Stability and Oscillations in Delay Differential Equations of Population Dynamics

“…[x(t)e-I'tj = b 1 _ [4>(0) + be-I'Tj o e-1'84>(s)dS] being approached exponentially.IT Tj and T are real constants such that 0 ~ Tj ~ T, j = 1,2, ... , n then generalize the above result assuming n T L bjexp( -a + l/T)Tj < 1 j=1 for a system of the form (seeDriver et al [1973]) 6. Prove the asymptotic stability of the trivial solution ofdx(t) --;[t = -ax(t -T),if the positive constants a and T satisfy aT < 1j can you prove the same result if aT < 7[/2 or aT < If aj,Tj (j = 1,2, .. ·,n) are positive constants such that Ei=1 ajTj < 1, then prove that the trivial solution of is asymptotically stable; can you prove the same result, if n 3 "a"T"<7r/2or-?…”

A result on the behavior of the solutions for scalar first order linear autonomous neutral delay differential equations

“…Closely related are the results given by Driver [3], Driver, Sasser and Slater [6], Graef and Qian [8], Kordonis, Niyianni and Philos [12], Philos [13] and Philos and Purnaras [14,15]. Recently, Frasson and Verduyn Lunel published an excellent paper [7] concerning the large time behavior of linear functional differential equations.…”

The MID property for a second-order neutral time-delay differential equation

“…It is well-known that for any given initial function φ, there exists a unique solution of the initial value problem (7); see [12]. Namely, given the solution X(φ) of the initial value problem (7), we define the solution operator T (t) : C → C by the relation…”

“…Remark 2. Note that item 3 of the theorem is obtained as a corollary of the MID property, unlike (12) in Frasson [13] where dominancy is assumed.…”