Electronic Journal of Differential Equations, Vol. 2021 (2021), No. 32, pp. 1–12. ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu IMPROVED OSCILLATION CRITERIA FOR FIRST-ORDER DELAY DIFFERENTIAL EQUATIONS WITH VARIABLE DELAY JULIO G. DIX Abstract. This article concerns the oscillation of solutions to the delay dif- ferential equation x′(t) + p(t)x(τ(t)) = 0. Conditions for oscillation have been stated as lower bounds for the limit superior and limit inferior of ∫ t τ p. In this article we match the bound for the best case in [7], without using one of their hypotheses. Then assuming that hypothesis, we obtain a bound lower than the one in [12]. Then we apply our results to an equation with several delays. We employ iterated estimates of the solution. 1. Introduction In this article we improve existing conditions for the oscillation of all solutions to the delay differential equation x′(t) + p(t)x(τ(t)) = 0, t ≥ t0 , (1.1) where p, τ ∈ C([t0,∞), [0,∞)), τ is non-decreasing, τ(t) ≤ t for all ∈ [t0,∞), and limt→∞ τ(t) =∞. Let T0 = inf{τ(t) : t ≥ t0}. By a solution, we mean a function that is continuous for t ≥ T0, differentiable for t ≥ t0, and satisfies (1.1). Given an initial function φ defined on [T0, t0], we can obtain a unique solution by integrating (1.1) in successive intervals (a process known as the method of steps). A solution is called oscillatory if it has arbitrarily large zeros; otherwise it is called non-oscillatory. A solution x is called eventually positive if x(t) > 0 for all t sufficiently large. Throughout this article we use the following notation: τn+1(t) = τn(τ(t)) with τ1(t) = τ(t) and τ0(t) = t, α = lim inf t→∞ ∫ t τ(t) p(s) ds, β = lim sup t→∞ ∫ t τ(t) p(s) ds . (1.2) Moreover, we consider the equation λ = eαλ , where λ is a function of α. If α = 0, then λ = 1 is the only solution. If 0 < α < 1/e, then there are two solutions, λ1 < λ2; furthermore, λ1 is a continuous and increasing function of α. If α = 1/e, then λ = e is the only solution. If α > 1/e, there is 2010 Mathematics Subject Classification. 34K11, 34C10. Key words and phrases. Oscillation of solutions; first-order delay differential equation; eventually positive solution. ©2021 Texas State University. Submitted January 29, 2021. Published April 24, 2021. 1 2 J. G. DIX EJDE-2021/32 no solution. For 0 ≤ α ≤ 1/e. we denote the solutions to this equation by λ1, λ2, where λ1 ≤ λ2. It is well known [1, 3, 10] that if α > 1 e or β > 1 , (1.3) then every solution of (1.1) is oscillatory. On the other hand if ∫ t τ(t) p(s) ds ≤ 1/e holds sufficiently large t, then there is a non-oscillatory solution; see [3, Corollary 2.1.1] and [8]. From these statements, we see that if limt→∞ ∫ t τ(t) p(s) ds does not exists, then there is gap in the results. Many results have improved the above bounds, We just mention a few of them, and direct the reader to the references in this article. Lemma 1.1 ([7, Lemma 1]). Let 0 < α and x be an eventually positive solution of (1.1). Then 0 < α ≤ 1/e and λ1 ≤ lim inf t→∞ x(τ(t)) x(t) ≤ λ2 . (1.4) Lemma 1.2 ([6, Corollary 1]). Assume 0 < α ≤ 1/e, β < 1 and β > ln(λ1) + 1 λ1 − (1− α)− √ 1− 2α− α2 2 . (1.5) Then every solution of (1.1) is oscillatory. Lemma 1.3 ([7, Theorem]). Assume that 0 < α ≤ 1/e, β < 1, that there exists ω > 0 such that ∫ τ(t) τ(u) p(s) ds ≥ ω ∫ t u p(s) ds for τ(t) ≤ u ≤ t . (1.6) If β > ln(λ1) + 1 λ1 − (1− α)− √ (1− α)2 − 4A 2 , (1.7) where A = eαωλ1 − αωλ1 − 1 (ωλ1)2 , (1.8) then every solution of (1.1) is oscillatory. Lemma 1.4 ([12, Theoem]). Under the assumptions of Lemma 1.3, if β > ln(λ1) λ1 + −1 + √ 1 + 2ω − 2ωλ1B ωλ1 , (1.9) where B = ( 1−α− √ (1− α)2 − 4A)/2 and A is given by (1.8), then every solution of (1.1) is oscillatory. Note that the bounds for β depend on the value of α. In particular when α = 0 conditions (1.3), (1.5) and (1.7) become β > 1. A table of numerical values for these and other bounds can be found in [7, 12, 13]. Table 1 towards the end of this paper compares our results with the bounds from (1.7), (1.9). In this article, without assuming (1.6), we establish the bound β > 2α+ 2 λ1 − 1 EJDE-2021/32 IMPROVED OSCILLATION CRITERIA 3 which matches the bound given in [7] using assumption (1.6). In fact when (1.6) is assumed, we obtain a bound slightly lower than the one in (1.9). Our main tool is to integrate (1.1) which expresses x as an integral transform of itself. We then substitute the expression obtained into the integral. This substi- tution was done one time in the book by Erbe [3, Lemma 2.1.3]. Here we iterate this substitution several times as in [14, 15]. This process yields a multiple integral over an interval [τn(t), t∗]. Then we partition each interval [τk(t), τk−1(t)] and use Riemann sums to estimate a multiple integral. The idea of partitioning the domain comes from [15]. However our partition is different from theirs, and as t→∞ the resulting number of integrals we obtain can approach infinity. This is stark contrast to [15] where partitioning intervals depends of a positive parameter δ. However as n increases, δ becomes negative and the partitioning process stops. Equations with constant delay. A function f is called slowly varying at infinity [11] if for every s ∈ R, lim t→∞ f(t+ s)− f(t) = 0 . Garab et al [4, Theorem 4] showed the following result which is optimal for the constant delay case. Let τ(t) = t− τ0, p be a non-negative bounded and uniformly continuous function such that 0 < α, 1/e < β. Also let the mapping t 7→ ∫ t τ(t) p(s) ds be slowly varying at infinity. Then all solutions of (1.1) are oscillatory. This article is organized as follows. In Section 2, we study oscillation of solutions without assuming condition (1.6). In Section 3, we assume condition (1.6) for obtaining an oscillation criterion. Also we compare the bounds that we obtain with some bounds in the literature. In Section 4, we extend our results to equations with multiple delays. 2. Results without assuming (1.6) Lemma 2.1. For n ≥ 1 and t∗ > t, we have∫ t∗ t p(s1) ∫ s1 t p(s2) ∫ s2 t · · · ∫ sn−1 t p(sn) dsn . . . ds1 = ∫ t∗ t p(s1) ∫ t∗ s1 p(s2) ∫ t∗ s2 · · · ∫ t∗ sn−1 p(sn) dsn . . . ds1 = 1 n! (∫ t∗ t p(s) ds )n The above lemma can be proved by induction on the number of integrals, using integration by substitution. Lemma 2.2. Let 0 < α̂ < α and n ≥ 1. Then there exists t2, and for each t ≥ t2, there exists t∗ such that ∫ t∗ t p(s) ds = α̂, with τ(t∗) ≤ t < t∗, and ρn(t) := ∫ t∗ t p(s1) ∫ t τ(s1) p(s2) ∫ τ(t) τ(s2) p(s3)· · · ∫ τn−2(t) τ(sn−1) p(sn) dsn . . . ds1 ≥ α̂ n n! for all t ≥ t2, (2.1) Proof. From α̂ < α, we have ∫ t τ(t) p ≥ α̂ for all t large enough. For each one of those sufficiently large values of t, the continuity of the map u 7→ ∫ u t p(s) ds and 4 J. G. DIX EJDE-2021/32∫ t τ(t) p ≥ α̂, yield a t∗ such that ∫ t∗ t p(s) ds = α̂. The fact that t < t∗ follows from∫ t∗ t p > 0, and the fact that τ(t∗) ≤ t follows from ∫ t∗ τ(t∗) p ≥ α̂. We partition the interval [t, t∗] using the m+ 1 points t = u0,m < u0,m−1 < · · · < u0,0 = t∗ , so that ∫ u0,0 u0,k p(s) ds = α̂k m for k = 0, 1, . . . ,m . We partition the interval [τ(t), t] using the m+ 2 points τ(t) = u1,m+1 ≤ u1,m < · · · < u1,0 = t , so that ∫ u1,0 u1,k p(s) ds = ∫ u0,0 u0,k p(s) ds = α̂k m for k = 0, 1, . . . ,m . Then ∫ t τ(t) p(s) ds ≥ ∫ u1,0 u1,m p(s) ds = ∫ u0,0 u0,m p(s) ds = ∫ t∗ t p(s) ds . Note that we can not guarantee this inequality without the assumption ∫ t∗ t p = α̂. In a similarly way, we can partition the intervals [τ2(t), τ(t)], . . . , [τn−1(t), τn−2(t)]. Then ρn(t) ≥ m−1∑ k1=0 ∫ u0,k1 u0,k1+1 p(s1) k1−1∑ k2=0 ∫ u1,k2 u1,k2+1 p(s2) · · · kn−1−1∑ kn=0 ∫ un−1,kn un−1,kn+1 p(sn) dsn . . . ds1 . Since ∫ uj,0 uj,k p = ∫ u0,0 u0,k p and p is continuous, the expression above is a Riemann sum that approximates∫ t∗ t p(s1) ∫ t∗ s1 p(s2) ∫ t∗ s2 p(s3)· · · ∫ t∗ sn−1 p(sn) dsn . . . ds1 . Then by Lemma 2.1, this multiple multiple equals ( ∫ t∗ t p(s) ds )n /n! Taking the limit as n→∞ and using that ∫ t∗ t p(s) ds = α̂, we have the desired result. � Lemma 2.3. Let 0 < α̂ < α ≤ 1/e, and x be an eventually positive solution of (1.1). Then there exists t1 ≥ t0, so that for each t ≥ t1 there exists n = n(t) with limt→∞ n(t) =∞, and x(t) x(τ(t)) ≥ d(n, t, α̂) ∀t ≥ t1 , (2.2) where d(n, t, α̂) is the smaller root of quadratic equation d2 − (1− α̂)d+ fn(t, α̂) = 0 (2.3) and fn(t, α̂) = α̂2 2! + α̂3λ̂ 3! + · · ·+ α̂nλ̂n−2 n! , (2.4) where λ̂ is the smaller solution of λ = eα̂λ. EJDE-2021/32 IMPROVED OSCILLATION CRITERIA 5 Proof. Since 0 < α̂ < α ≤ 1/e, each one of the equations λ = eα̂λ and λ = eαλ has two solutions. Then by Lemma 1.1, λ̂ := λ̂1 < λ1 ≤ lim inf t→∞ x(τ(t)) x(t) ≤ λ2 < λ̂2 . From the fact that x is eventually positive and limt→∞ τ(t) = ∞, there is a t1 such that for all t ≥ t1 the following 4 conditions hold: 0 < x(t), 0 < x(τ(t)), α̂ ≤ ∫ t τ(t) p(s) ds (because α̂ < α which is a limit inferior), and λ̂ ≤ x(τ(t)/x(t) (because λ̂ < λ which is a limit inferior) . For each t > t1, we select n = n(t) as the largest integer for which τn(t) ≤ t1 ≤ τn−1(t) . (2.5) Then λ̂ ≤ x(τ j+1(t)) x(τ j(t)) for j = 0, 1, . . . , n− 1 . (2.6) Note that n is a non-decreasing function of t because τ is non-decreasing. Since τ is continuous and limt→∞ τ(t) =∞, we have, for each finite n, that lim t→∞ τn(t) = τ(. . . τ( lim t→∞ τ(t)) . . . ) =∞ . Now we claim that n → ∞ as t → ∞. To reach a contradiction, assume that n remains bounded as t→∞. Taking the limit in (2.5), ∞ = lim t→∞ τn(t) ≤ t1 which is a contradiction; therefore, n can not remain bounded as t→∞. By Lemma 2.2 there exists t2 ≥ t1, such that for each t ≥ t2 there exists t∗ such that ∫ t∗ t p = α̂. Integrating (1.1), we have inequalities of the form x(t) = x(t∗) + ∫ t∗ t p(s1)x(τ(s1)) ds1, (2.7) x(τ(s1)) = x(t) + ∫ t τ(s1) p(s2)x(τ(s2)) ds2, . . . , (2.8) x(τ(sn−1)) = x(τn−2(t)) + ∫ τn−2(t) τ(sn−1) p(sn)x(τ(sn)) dsn . (2.9) Since x(t) > 0, by (1.1), x′(t) ≤ 0 and x is non-increasing, and∫ τn−2(t) τ(sn−1) p(sn)x(τ(sn)) dsn ≥ x(τn−1(t)) ∫ τn−2(t) τ(sn−1) p(sn) dsn . Substituting (2.8)–(2.9) into (2.7), and using the definition of ρ, it follows by the above inequality, that x(t) ≥ x(t∗) + α̂x(t) + [ ρ2(t)x(τ(t)) + ρ3(t)x(τ2(t)) + · · ·+ ρn(t)x(τn−1(t)) ] . From (2.6), we have x(τ j+1(t)) ≥ λ̂jx(τ(t)) for j = 1, 2, . . . , n − 1. Then from Lemma 2.2, ∫ t∗ t = α̂, ∫ t τ(t) ≥ α̂, and the result in (2.1), we have x(t) ≥ x(t∗) + α̂x(t) + fn(t, α̂)x(τ(t)), ∀t ≥ t2 , where fn is defined by (2.4). Then (1− α̂)x(t) ≥ x(t∗) + fn(t, α̂)x(τ(t)) . (2.10) 6 J. G. DIX EJDE-2021/32 Since x and fn are positive, (1− α̂) > 0 which agrees with assumption 0 < α̂ < 1/e. Ignoring the term x(t∗), we have x(t) x(τ(t)) ≥ fn(t, α̂) 1− α̂ := d1 which is positive. Recalling that x is non-increasing and τ(t∗) ≤ t, we have x(t∗) ≥ d1x(τ(t∗)) ≥ d1x(t) . Using this inequality in (2.10) yields (1− α̂− d1)x(t) ≥ x(t∗) + fn(t, α̂)x(τ(t)) . Since x and fn are positive, (1− α̂− d1) > 0, which implies d1 < 1− α̂. Then x(t) x(τ(t)) ≥ fn(t, α̂) 1− α̂− d1 := d2 . Proceeding as above, (1 − α̂ − d2) > 0, which implies d2 < 1 − α̂. Also because d1 > 0, we have d1 = fn(t, α̂) 1− α̂ < fn(t, α̂) 1− α̂− d1 = d2 . As in [3, Lemma 2.1.3], repeating the above process, we have an increasing sequence {dk} that is bounded above by 1−α̂; therefore the sequence converges to the smaller solution of the quadratic equation d2 − (1− α̂)d+ fn(t, α̂) = 0. By (2.10), we have (1− α̂− d)x(t) ≥ x(t∗) + fn(t, α̂)x(τ(t)), and x(t) x(τ(t)) ≥ fn(t, α̂) 1− α̂− d = d . This completes the proof. � Lemma 2.4. Let 0 < α ≤ 1/e, and x be an eventually positive solution of (1.1). Then lim inf t→∞ x(t) x(τ(t)) ≥ 1− α− 1 λ1 . Proof. First in Lemma 2.3, for each value of t we select the largest possible n, and observe that n→∞ as t→∞. With the notation in Lemma 2.3, we have lim t→∞ fn(t, α̂) = 1 (λ̂)2 [ eα̂λ̂ − α̂λ̂− 1 ] = 1 (λ̂)2 [λ̂− α̂λ̂− 1] . Recall that the roots of a quadratic equation depend continuously on their coeffi- cients. Then, as n→∞, the roots of (2.3) approach the roots of d2 − (1− α̂)d+ 1 (λ̂)2 [λ̂− α̂λ̂− 1] = 0 . Then as λ̂→ λ1 and α̂→ α, the roots of the above equation approach the roots of d2 − (1− α)d+ 1 λ2 [λ1 − αλ1 − 1] = 0 , which are d = 1 − α − 1/λ1 and d = 1/λ1. For α ∈ [0, 1/e] and the corresponding lambda with λ = eαλ, the first root is smaller than the second. To complete the proof we compute the limits in (2.2) first as t→∞, and then as α̂→ α. � EJDE-2021/32 IMPROVED OSCILLATION CRITERIA 7 Theorem 2.5. Let 0 < α ≤ 1/e, and β > 2α+ 2 λ1 − 1 . (2.11) Then every solution of (1.1) is oscillatory. Proof. To obtain a contradiction, assume that x is an eventually positive solution of (1.1). Then by [6, Theorem 1], β ≤ ln(λ1) + 1 λ1 − lim inf t→∞ x(t) x(τ(t)) . Then by Lemma 2.4 β ≤ ln(λ1) + 1 λ1 − ( 1− α− 1 λ1 ) = 2α+ 2 λ1 − 1, which contradicts the assumption and completes the proof for eventually positive solutions. If a solution y is an eventually negative solution, we consider x = −y which is an eventually positive solution. � Note that the above theorem does not assume (1.6), and matches the best pos- sible case of (1.7), i.e. when ω = 1. Based on the example in [7], we build an example that satisfies the hypotheses in Theorem 2.5, but does not satisfy (1.6). Let τ(t) = t− 2 sin2(t)− 4/(e− 2) and p(t) = (e− 2)/(4e). Then∫ t τ(t) p(s) ds = e− 2 4e ( 2 sin2(t) + 4 e− 2 ) , so that lim inft→∞ ∫ t τ(t) p = 1/e and lim supt→∞ ∫ t τ(t) p = 1/2. Thus the assump- tions in Theorem 2.5 are satisfied. Condition (1.6) becomes e− 2 4e [ t− u− 2 sin2(t) + 2 sin2(u) ] ≥ ωe− 2 4e (t− u) . which is equivalent to sin2(t)− sin2(u) t− u ≥ 1− ω 2 . Because 0 < ω ≤ 1, we have (1 − ω)/2 ≤ 1/2. Meanwhile a linear approximation on the numerator of the left-hand side gives a term of the form sin(2u), so we can select t and u close to each other for which the above inequality is not satisfied. Therefore (1.6) does not hold in this example. 3. Bounds using condition (1.6) Lemma 3.1. If ω > 1 in (1.6), then (1.1) has a non-oscillatory solution. Proof. Let α̂ < α and β̂ > β. Then from the definition of the limit inferior and the limit superior, there exists t1 such that α̂ ≤ ∫ t τ(t) p(s) ds and ∫ t τ(t) p(s) ds ≤ β̂ ∀t ≥ t1 . For each t ≥ t1, let n = n(t) be the largest integer for which τn(t) ≥ t1. Then n→∞ as t→∞. By applying (1.6) repeatedly, we see that β̂ ≥ ∫ τn−1(t) τn(t) p(s) ds ≥ ωn−1 ∫ t τ(t) p(s) ds . 8 J. G. DIX EJDE-2021/32 As t → ∞, we see that n → ∞ and ωn−1 → ∞. Thus limt→∞ ∫ t τ(t) p(s) ds = 0. Then ∫ t τ(t) p(s) ds ≤ 1/e holds eventually, so (1.1) has a non-oscillatory solution [3, Corollary 2.1.1]. � Remark 3.2. In view of Lemma 3.1, we restrict our attention to 0 < ω ≤ 1. Using the series expansion of A in (1.8), we can show that A is an increasing function of ω. Therefore, A possesses its maximal value and the bound in (1.7) has its minimal value when ω = 1. Also the constant B has its maximal value, and the bound in (1.9) has its minimal value when ω = 1. Lemma 3.3. Assume (1.6) holds and τ(t) ≤ t∗ ≤ t. Then ρ̂n(t) := ∫ t t∗ p(s1) ∫ τ(t) τ(s1) p(s2) ∫ τ2(t) τ(s2) p(s3)· · · ∫ τn−1(t) τ(sn−1) p(sn) dsn . . . ds1 ≥ ω1+···+n−1 n! (∫ t t∗ p(s) ds )n . Proof. This is achieved by induction on the number of integrals. For the basic step n = 2, we have∫ t t∗ p(s1) ∫ τ(t) τ(s1) p(s2) ds2 ds1 ≥ ω ∫ t t∗ p(s1) ∫ t s1 p(s2) ds2 ds1 = ω 2! (∫ t t∗ p(s) ds )2 , where the equality follows from Lemma 2.1. For the induction step, we assume the inequality holds for n − 1 integrals, and show it holds for n integrals. Under this assumption and using (1.6), we see that ρ̂n(t) ≥ ∫ t t∗ p(s1) ω1+···+n−2 (n− 1)! (∫ τ(t) τ(s1) p(s) ds )n−1 ≥ ∫ τ(t) t∗ p(s1) ω1+···+n−1 (n− 1)! (∫ t s1 p(s) ds )n−1 . By the substitution method with u = ∫ t s1 p(s) ds, we have du = −p(s1) ds1, and∫ un−1 du = 1 nu n which yields the desired result. � Lemma 3.4. Assume 0 < α ≤ 1/e, (1.6), and that x is an eventually positive solution of (1.1). Then β ≤ α+ 1 ωλ1 ln ( 1 + 2ω − ωλ1 + αωλ1 ) . Proof. Since 0 < α ≤ 1/e, we have 1 < λ1, so there exists λ̂ ∈ (1, λ1). Then the conditions in Lemma 1.1 are satisfied, therefore (2.6) holds. As in [12, Lemma 3] we consider the function g(t) := x(τ(t))/x(t) which is continuous, g(τ(t)) = 1 < λ̂, and g(t) > λ̂. Then there exists t∗ ∈ (τ(t), t) such that x(τ(t)) x(t∗) = λ̂ . (3.1) From Lemma 2.4 for each d̂ < 1− α− 1 λ1 there exists t2 such that x(t) x(τ(t)) ≥ d̂ for all t ≥ t2 ; (3.2) EJDE-2021/32 IMPROVED OSCILLATION CRITERIA 9 If necessary we may increase t2 to make it greater than the t1 in (2.6). Dividing (1.1) by x(t) and then integrating, by (2.6), we have∫ t∗ τ(t) x′(s) x(s) ds = − ∫ t∗ τ(t) p(s) x(τ(s)) x(s) ds ≤ −λ̂ ∫ t∗ τ(t) p(s) ds . From this inequality and (3.1),∫ t∗ τ(t) p(s) ds ≤ ln(λ̂) λ̂ . (3.3) Now we estimate ∆ := ∫ t t∗ p(s) ds. Integrating (1.1) from t∗ to t, and proceeding as in (2.7), (2.9), we have x(t∗)− x(t) ≥ ∆x(τ(t)) + [ ρ̂2(t)x(τ2(t)) + ρ̂3(t)x(τ3(t)) + · · ·+ ρ̂n(t)x(τn(t)) ] . By (2.6), x(t∗)− x(t) ≥ ∆x(τ(t)) + [ ρ̂2(t)λ̂+ ρ̂3(t)λ̂2 + · · ·+ ρ̂n(t)λ̂n−1 ] x(τ(t)) . Dividing by x(τ(t)), using Lemma 3.3 and (3.2), we have ∆ + ∆2 2! ωλ̂+ ∆3 3! ω1+2λ̂2 + · · ·+ ∆n n! ω1+···+n−1λ̂n−1 ≤ x(t∗)) x(τ(t)) − x(t) x(τ(t)) ≤ 1 λ̂ − d̂ . (3.4) To solve the above inequality we define the polynomial Qn(∆) = ∆ + ∆2 2! ωλ̂+ ∆3 3! ω1+2λ̂2 + · · ·+ ∆n n! ω1+···+n−1λ̂n−1 − ( 1 λ̂ − d̂ ) . Note that all the coefficients of ∆ are positive and the independent term is negative, so by the Descartes’ rule of signs, Qn has at most one positive root. Since Qn(0) < 0 and lim∆→∞Qn(∆) = ∞, it follows that Qn has exactly one positive root. To satisfy (3.4), ∆ must be less than or equal to the positive root of Qn. When t → ∞, it follows that by definition t∗ → ∞ and n → ∞. Therefore we can increase n, which provides more accurate estimates for ∆. There are formulas for obtaining the roots when n = 1, 2, 3, 4, but not for n ≥ 5. Sficas et al [12] solved this equation when n = 2, by using its positive root as an estimate for the solution of (3.4). Our approach is to use the solution of lim n→∞ Qn(∆) = 0 as an estimate for the solution of (3.4). When ω = 1 we need to find the positive solution of ∆ + e∆λ̂ −∆λ̂− 1 λ̂ − ( 1 λ̂ − d̂ ) = 0 . In this case (3.4) is satisfied if ∆ ≤ 1 λ̂ ln ( 2− λ̂d̂ ) , (3.5) which corresponds to (3.7), below, with ω = 1. For 0 < ω < 1, we define a new polynomial Q̂n(∆) = ∆ + ∆2 2! ωλ̂+ ∆3 3! ω2λ̂2 + · · ·+ ∆n n! ωn−1λ̂n−1 − ( 1 λ̂ − d̂ ) . (3.6) 10 J. G. DIX EJDE-2021/32 Note that Q̂2 ≡ Q2, and Q2(∆) < Qn(∆) < Q̂n(∆) for n > 2, because 0 < ω < 1. Therefore the positive root of Q̂n is less than the positive root of Qn. The equation limn→∞ Q̂n(∆) = 0 is ∆ + e∆ωλ̂ −∆ωλ̂− 1 ωλ̂ − ( 1 λ̂ − d̂ ) = 0 . Therefore, (3.4) is satisfied if ∆ ≤ 1 ωλ̂ ln ( 1 + ω − ωλ̂d̂ ) . (3.7) Note that the right-hand side of this inequality is less than the right-hand side of (3.5), because they correspond to the roots of Q̂n and of Qn, respectively. Adding (3.3) and (3.7), and then computing the limit as λ̂ → λ1 and d̂ → 1− α− 1 λ1 yields lim sup t→∞ ∫ t τ(t) p(s) ds ≤ ln(λ1) λ1 + 1 ωλ1 ln ( 1 + ω − ωλ1(1− α− 1 λ1 ) ) = α+ 1 ωλ1 ln ( 1 + 2ω − ωλ1 + αωλ1) . This completes the proof. � Theorem 3.5. Assume (1.6), 0 < α ≤ 1/e, and β > α+ 1 ωλ1 ln ( 1 + 2ω − λ1ω + αλ1ω ) . (3.8) Then every solution of (1.1) is oscillatory. Proof. For the sake of contradiction, assume that there is an eventually positive solution. Then by Lemma 3.4 we have a contradiction to (3.8). On the other hand if y is an eventually negative solution of (1.1), we may consider x = −y which is an eventually positive solution. � Remark 3.6. For ω = 1, the bound in Theorem 3.5 is slightly lower than the one in [12], see Table 1. For an example of an equation that satisfies the assumptions in Theorem 3.5, we refer the reader to [12]. Table 1. Oscillation criteria when ω = 1 cond. α = 1/e, λ1 = e α = 2 ln(e/2)/e, λ1 = e/2 α = 0, λ1 = 1 (1.7) β > 0.471518 β > 0.923057 β > 1 (1.9) β > 0.459987 β > 0.741974 β > √ 3− 1 ≈ 0.732750 (3.8) β > 0.459188 β > 0.716267 β > ln(2) ≈ 0.693147 EJDE-2021/32 IMPROVED OSCILLATION CRITERIA 11 4. Equations with multiple delays As an application of the above results, we present a condition for the oscillation of solutions to the equation x′(t) + m∑ i=1 pi(t)x(τi(t)) = 0 , (4.1) where pi, τi ∈ C([t0,∞), [0,∞)), τi(t) ≤ t, and limt→∞ τi(t) = ∞. In this section we do not require τi to be monotonic, instead we redefine τ as the non-decreasing function τ(t) = max t0≤s≤t { max 1≤i≤m {τi(s)} } . If x is an eventually positive solution of (4.1), then x′(t) = − ∑m i=1 pi(t)x(τi(t)) ≤ 0 so x is non-increasing, and x′(t) + ( m∑ i=1 pi(t) ) x(τ(t)) ≤ 0 . (4.2) In this section the summation ∑m i=1 pi plays the role of p in the previous sections, while τ plays the same role as before. We redefine the constants α = lim inf to→∞ ∫ t τ(t) m∑ i=1 pi(s) ds, β = lim sup to→∞ ∫ t τ(t) m∑ i=1 pi(s) ds; while λ1 ≤ λ2 remain as the roots of λ = eαλ. In Sections 2 and 3 we replace p(·) by∑m i=1 pi(·), and replace the sign = by ≤, in (2.7)–(2.9). The rest of the inequalities remain valid, so we only restate the main results. Theorem 4.1. Let 0 < α ≤ 1/e, and β > 2α+ 2 λ1 − 1 . (4.3) Then every solution of (4.1) is oscillatory. Theorem 4.2. Assume (1.6), 0 < α ≤ 1/e, and β > α+ 1 ωλ1 ln ( 1 + 2ω − λ1ω + αλ1ω ) . (4.4) Then every solution of (4.1) is oscillatory. Our conditions for the oscillation of solutions of equations with multiple delays are rather basic. For alternative oscillation criteria, we refer the reader to [10, sec. 2.6], [2, 9]. We conclude this article by stating that the optimal bound β > 1/e has not been reached yet; so there is room for improvement. Acknowledgments. I want to thank editor Zhaosheng Feng for handling my sub- mission, also Professor Stewart Welsh and the anonymous referee for their sugges- tions. 12 J. G. DIX EJDE-2021/32 References [1] R. P. Agarwal, L. Berezansky, E. Braverman, A. Domoshnitsky; Nonoscillation Theory of Functional Differential Equations with Applications, Springer, 2012. MR 2908263 [2] J. P. Dix, J. G. Dix; Oscillation of solutions to nonlinear first-order delay differential equa- tions, Involve, 9:3 (2016), 465-482. [3] L. H. Erbe, Qingkai Kong, B. G. Zhang; Oscillation Theory for Functional Differential Equations, Marcel Dekker, New York, 1995. [4] A. Garab, M. Pituk, I. P. Stavroulakis; A sharp oscillation criterion for a linear delay differential equation, Appl. Math. 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Pituk; Oscillation of a linear delay differential equation with slowly varying coefficient, Appl. Math. Let., 73 (2017) 29-36. [12] Y. G. Sficas, I. P. Stavroulakis; Oscillation criteria for first-order delay equations, Bull. London Math. Soc., 35 (2003), 239-246. [13] I. P. Stavroulakis, Zh. Kh. Zhunussova, S. Sh. Ixanov, Belal S. H. Rababah; Optimal Os- cillation conditions for delay differential equations, Appl. Math. Inf. Sci., 33 (2019) no. 3, 417-425. [14] H. Wu, C. Chen, R. Zhuang; Oscillation criterion for first-order delay differential equations with sign-changing coefficients, Electronic J. Differential Equations, 2017 (2017) no. 126, 1–9. [15] Z. Zhuang, Q. Wang, H. Wu; A new oscillation criterion for first-order delay differential equatons by iteration, Appl. Math. Comp., 390 (2021), 125632. 5. Addendum posted October 13, 2022 The proof of Lemma 2.2 is incorrect. It uses that∫ u0,k u0,k p = ∫ u1,k u1,k p implies τ(u10,k) = τ(u10,k), which is not necessarily true. Therefore Lemma 2.2 needs a new proof. One possible solution is to assume that ∫ τ(v) τ(u) p ≥ ∫ v u p, i.e. (1.6) holds for ω = 1. However this assumption causes some difficulties. If ω = 1 and u ≤ v, then ∫ v τ(v) p = ∫ u τ(u) p+ ∫ v u p− ∫ τ(v) τ(u) p ≥ ∫ u τ(u) p. Therefore the mapping t 7→ ∫ t τ(t) p is non-decreasing which implies α = lim inf t→∞ ∫ t τ(t) p = lim sup t→∞ ∫ t τ(t) p = β . The equality α = β does not allow condition (4.3) to be satisfied in the typical case α = 1/e and λ1 = e. EJDE-2021/32 IMPROVED OSCILLATION CRITERIA 13 Another consequence of the above equality is that in Remark 3.6 and Table 1 we cannot use ω = 1, we can only use approximations such as ω = 0.9999. Julio G. Dix Department of Mathematics, Texas State University, 601 University Drive, San Marcos, TX 78666, USA Email address: jd01@txstate.edu 1. Introduction Equations with constant delay 2. Results without assuming (??) 3. Bounds using condition (??) 4. Equations with multiple delays Acknowledgments References 5. Addendum posted October 13, 2022