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Gopalsamy, K.

Ladas, G.

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University of Rhode IslandDigitalCommons@URIMathematics Faculty Publications Mathematics1991Oscillations of delay differential equationsK. GopalsamyG. LadasUniversity of Rhode IslandFollow this and additional works at: https://digitalcommons.uri.edu/math_facpubsTerms of UseAll rights reserved under copyright.This Article is brought to you for free and open access by the Mathematics at DigitalCommons@URI. It has been accepted for inclusion inMathematics Faculty Publications by an authorized administrator of DigitalCommons@URI. For more information, please contactdigitalcommons@etal.uri.edu.Citation/Publisher AttributionGopalsamy, K., & Ladas, G. (1991). Oscillations of delay differential equations. The Journal of the Australian Mathematical Society. SeriesB. Applied Mathematics, 32(4), 377-381. doi:10.1017/S0334270000008493Available at: https://doi.org/10.1017/S0334270000008493/ Austral. Math. Soc. Ser. B 32(1991), 377-381OSCILLATIONS OF DELAY DIFFERENTIAL EQUATIONSK. GOPALSAMY'and G. LADAS2(Received 30 July 1989; revised 28 February 1990)AbstractSufficient conditions are established for all solutions of the linear systemto be oscillatory, where qtj 6 ( - o o , oo) , t^ 6 ( 0 , oo ) , i , j = 1,2,... , n.1. IntroductionConsider the system of delay differential equations* # + £v,('-'«>-0. 1-1.2a t *-*! lJ J IJ(l)where the coefficients are real numbers and the delays are positive real num-bers. We say that a solutiony(t) = (2)of (1) oscillates if for some i € (1 , 2, . . . , n), yt(t) has arbitrarily largezeros. A solution y(t) of (1) is said to be nonoscillatory if there exists at0 > 0 such that for each i = 1, 2 , . . . ,n, yt(t) ^ 0 for t > t0. The aimof this brief paper is to derive a set of sufficient conditions for all solutions'School of Mathematics, Flinders University, Bedford Park, S. A. 5042.2Department of Mathematics, University of Rhode Island, Kingston, R. I., U.S.A.© Copyright Australian Mathematical Society 1991, Serial-fee code 0334-2700/91377terms of use, available at https://www.cambridge.org/core/terms. https://doi.org/10.1017/S0334270000008493Downloaded from https://www.cambridge.org/core. University of Rhode Island Library, on 27 Nov 2018 at 20:04:44, subject to the Cambridge Core378 K. Gopalsamy and G. Ladas [2]of (1) to oscillate. Our result is an extension of a result of Gopalsamy in [2],where only bounded solutions of systems like (1) have been considered. Forreferences concerning the oscillation of systems, the reader is referred to thereferences in [2].2. Sufficient conditions for oscillationThe following lemma will be useful in the proof of our theorem below.LEMMA 1. Assume that (1) has a nonoscillatory solution (2). Then there arenumbersSte{-\, 1} /or i = 1 , 2 , . . . , / !such that the systemwherepij = -8qij M i , j = l , 2 , . . . , n (4)has a nonoscillatory solution [z^t), z2(t), . . . , zn(t)]T with eventually posi-tive components zt{t), i = 1, 2 , . . . , « .PROOF. The components y{t) of (2) are positive or negative eventually. Thatis, there exists a T > 0 such that yt{t) ^ 0 for t > T and / = 1, 2 , . . . , « .Set 8i = sign^^f)], i = 1, 2, . . . , n and t > T. It is now easy to see thatAt) = [Sxy{{t),d2y2{t),..., 5nyn{t)]T (5)satisfies (3) and <5,y,(0 > 0 for / = 1, 2, . . . , n and t > T.The next result is concerned with the asymptotic behaviour of nonoscilla-tory solutions of (1).LEMMA 2. Consider the system (1) and suppose that the constant coefficientsof (I) satisfynq = minn qu - ] T \qjt\ > 0. (6)Then every nonoscillatory solution y{t) = {yx,y2, ... ,yn) satisfiesterms of use, available at https://www.cambridge.org/core/terms. https://doi.org/10.1017/S0334270000008493Downloaded from https://www.cambridge.org/core. University of Rhode Island Library, on 27 Nov 2018 at 20:04:44, subject to the Cambridge Core[3] Delay Differential Equations 379PROOF. Clearly (6) is also satisfied with the qtj replaced by the respective p1;-of (4). From this and (5) it suffices to prove the lemma for nonoscillatorysolutions of (2) with eventually positive components. Let us assume thatthere is a tQ > 0 such that yt{t) > 0 for t > t0, i = 1, 2, . . . , n . If we letthen1=1 j=\or1=1 j=\It follows from (8) thatdw(t) y ,1=1An integration of both sides of (9) leads tortc i = lwhere x = max1<; < n T . . . A consequence of (10) is that w is bounded andyl; G Lx{tQ + x, oo) for / = 1, 2 , . . . , « . From the boundedness of w onecan conclude that of yt since w(t) = £"_, yt(t) and yt{t) > 0 eventually.It will now follow from (1) that yt is bounded for t > x, and thereforey( is uniformly continuous on [0, oo). The uniform continuity of yi on[0, oo), the eventual positivity of yt and the integrability of y( on a half-line together with a lemma of Barbalat [1], will imply that iiTnt_>0Oyj(t) =0, i = 1, 2, ... , n and this completes the proof.THEOREM. Let qu e (-oo, oo), x^ 6 (0, oo), i, j = 1, 2, . . . , n. IfqxM > - where q = mine i<i<nthen every s o l u t i o n of (I) oscillates.terms of use, available at https://www.cambridge.org/core/terms. https://doi.org/10.1017/S0334270000008493Downloaded from https://www.cambridge.org/core. University of Rhode Island Library, on 27 Nov 2018 at 20:04:44, subject to the Cambridge Core380 K. Gopalsamy and G. Ladas [4]PROOF. Assume for the sake of a contradiction that (1) has a nonoscillatorysolution (2). In view of Lemma 1 we can assume that the components ofyt(t) are eventually positive for i = 1, 2, . . . , n. We have directly from (1)thatj=\ ;=1which satisfies^ \dyt{t)1=1 ffl JWe have from (12) that w(t) = J2"=i ^,-(0 satisfiesdt i=\y,(t -T , , . )<O.Integrating both sides of (13) over (t, oo) and using the factw(t) -»• 0 as t -> oo (since ^ , (0 - » 0 , / = 1, 2 , . . . , « )we derive thatand this leads to/oo n^Zyjis-r1=1It is found from (15) that/•OOtw(0 > # / 1^(5)^5F{t)= f w(s)dsJt-x.orNow we let(12)(13)(14)(15)(16)(17)(18)terms of use, available at https://www.cambridge.org/core/terms. https://doi.org/10.1017/S0334270000008493Downloaded from https://www.cambridge.org/core. University of Rhode Island Library, on 27 Nov 2018 at 20:04:44, subject to the Cambridge Core[5] Delay Differential Equations 381and derive from (17) and (18) that<-qF{t-x.)\ «>2T.. (19)It follows from (19) that F is an eventually positive solution ofT . ) < O ; , > 2 V (20)But it is well known (from Ladas and Stavroulakis [3]) that when (11) holds,(20) cannot have an eventually positive solution and this contradiction com-pletes the proof.References[1] I. Barbalat, "Systemes d'equations differentielles d'oscillations nonlineaires," Rev.Roumain. Math. PuresAppl. 4 (1959) 267-270.[2] K. Gopalsamy, "Oscillatory properties of systems of first order delay differential inequal-ities," Pacific J. Math. 128 (1987) 299-305.[3] G. Ladas and I. P. Stavroulakis, "On delay differential inequalities of first order," Funk-cialaj Ekvacioj. 25 (1982) 105-113.terms of use, available at https://www.cambridge.org/core/terms. https://doi.org/10.1017/S0334270000008493Downloaded from https://www.cambridge.org/core. University of Rhode Island Library, on 27 Nov 2018 at 20:04:44, subject to the Cambridge Core

Oscillations of delay differential equations

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Oscillations of delay differential equations

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