Solutions of Delay Differential Equations Oscillate with Positive Coefficients and Negative Coefficients

Abstract

In this paper we introduced, the first order linear delay differential equation with positive and negative coefficients

                r′(t) + A(t) r (t – τ) – B(t) r (t – σ) = 0,                                                                          (1.1)

where A, B are continuous functions with positive and negative real coefficients and τ, σ are non-negative constants.

The standard form of the above equation is

                r′(t) +   = 0                                                        (1.2)

where ,  ([, ∞), R) and  [0, ∞), for i = 1 ,2,3,……,k. We consider a function

r(t)([ for some t ≥  and t₁ is hold for conditions (1.1) or (1.2), where r(t) is a continuous function and  = max{ τ_i , σ_i } and 1 ≤ i ≤ k .