Acta Math. Univ. Comenianae Vol. LXVII, 2(1998), pp. 335-342 335 INVARIANT CONE OF SLOWLY OSCILLATING SOLUTION IN TWO DELAYS DIFFERENTIAL EQUATIONS N. YOUSFI and O. ARINO Abstract. Scalar delay differential equations with two delays are considered in this paper. Some monotonicity results permit to establish existence of non trivial slowly oscillating solutions. 1. Introduction Oscillations of delay differential equations have been considered recently by many authors (see [1], [2], [3], [6], [7]). In this work, we investigate the monotony properties to establish existence of slowly oscillating solution (s.o.) of retarded differential equations with two delays. This paper is organized in three sections. In the introduction, we present our model and give the definition of a s.o. solution. In the second section, we construct an invariant cone K ⊆ C ([−σ, 0], R) of s.o. solutions. Section 3 is devoted to an example. We consider the equation (1) ˙ x (t)= −f (x (t − τ )) − g (x (t − σ)) under the following general assumptions: f and g are smooth functions such that: 1. 0 ≤ f (x) x ≤ a for x = 0, 2. 0 ≤ g(x) x for x = 0, 3. τ<σ. Definition 1. A solution of (1) is slowly oscillating if the distance between two successive zeros of x(t) is not less than max(τ,σ). Received January 8, 1998; revised March 27, 1998. 1980 Mathematics Subject Classification (1991 Revision). Primary 34K15; Secondary 34C35.