Asymptotics of a nonlinear delay differential equation

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1 Asymptotics of a oliear delay differetial equatio Brads, J.J.A.M. Published: 01/01/1979 Documet Versio Publisher s PDF, also kow as Versio of Record (icludes fial page, issue ad volume umbers) Please check the documet versio of this publicatio: A submitted mauscript is the author's versio of the article upo submissio ad before peer-review. There ca be importat differeces betwee the submitted versio ad the official published versio of record. People iterested i the research are advised to cotact the author for the fial versio of the publicatio, or visit the DOI to the publisher's website. The fial author versio ad the galley proof are versios of the publicatio after peer review. The fial published versio features the fial layout of the paper icludig the volume, issue ad page umbers. Lik to publicatio Citatio for published versio (APA): Brads, J. J. A. M. (1979). Asymptotics of a oliear delay differetial equatio. (Eidhove Uiversity of Techology : Dept of Mathematics : memoradum; Vol. 7914). Eidhove: Techische Hogeschool Eidhove. Geeral rights Copyright ad moral rights for the publicatios made accessible i the public portal are retaied by the authors ad/or other copyright owers ad it is a coditio of accessig publicatios that users recogise ad abide by the legal requiremets associated with these rights. Users may dowload ad prit oe copy of ay publicatio from the public portal for the purpose of private study or research. You may ot further distribute the material or use it for ay profit-makig activity or commercial gai You may freely distribute the URL idetifyig the publicatio i the public portal? Take dow policy If you believe that this documet breaches copyright please cotact us providig details, ad we will remove access to the work immediately ad ivestigate your claim. Dowload date: 8. Nov. 018

EINDHOVEN UNIVERSITY OF TECHNOLOGY Departmet of Mathematics Memoradum 1979-14 November 1979 "Asymptotics of a oliear delay

2 EINDHOVEN UNIVERSITY OF TECHNOLOGY Departmet of Mathematics Memoradum November 1979 "Asymptotics of a oliear delay differetial equatio by J. J.1> M. Brads Techological Uiversity Departmet of Mathematics PO Box 513, Eidhove The Netherlads

ASYMPTOTICS OF A NONLINEAR DELAY DIFFERENTIAL EQUATION by J.J.A.M. Brads ABSTRAcr Sufficiet coditios will be give for the existece of the limit of f(x) for x ~ 00 if f is a solutio of w(x)f' (x) = g(f(x-l) - f(x».

3 ASYMPTOTICS OF A NONLINEAR DELAY DIFFERENTIAL EQUATION by J.J.A.M. Brads ABSTRAcr Sufficiet coditios will be give for the existece of the limit of f(x) for x ~ 00 if f is a solutio of w(x)f' (x) = g(f(x-l) - f(x». 1. INTRODUCTION I this ote we cosider the delay differetial equatio (1. 1) w(x)f'(x) = g(f(x-l) - f(x», x ~ 1, where (1. ) w is a positive cotiuous fuctio o [1,00), ad g is a odd real valued cotiuously differetiable fuctio o R such that g(x) > 0 if x > 0 We say that f is a solutio of (1.1) if f is a real valued fuctio, cotiuous o [O,~), differetiable o [1,00), satisfyig (1.1) o [1,00). Every give cotiuous fuctio ~ o [O,lJ ca be exteded to a solutio of (1.1), ad this extesio is uique. We just solve (1.1) as a ordiary differetial equatio o [1,J with iitial value f(l) = ~(1), ad repeat this process for the itervals [,+1J, =,3,.. N.G. de Bruij [1949,1950J ad J.J.A.M. Brads [197J treated the liear case of equatio (1.1) (i.e. g(x) = x) for a large class of fuctios w (cotaiig e.g. all fuctios w(x) = x-a, areal). Amog other thigs, they proved that uder some coditios for w (which for the specializatio w(x) = x-a reduce to the coditio a ~ ~) every solutio has a limit. J.L. Kapla, M. Sorg ad J.A. Yorke [1979J proved for a typical autoomous equatio (with a so-called order relatio as righthad side)

4 - - that every solutio has a limit. The autoomous case of equatios (1.1) (i.e. w(x) = costat> 0) is a specializatio of their equatio.. RESULTS We preset the results of this ote i three theorems all of which have the followig form: THEOREM.k. If w ad g satisfy coditio (1.) ad, i additio, the coditio (.k), the fop all solutios f of (1.1) lim f(x) exists. x-+oo Theorems.1,., ad.3 are obtaied by substitutio of k = 1,, ad 3. Thus, we obtai theorem.1 if the additioal coditio (.1) is satisfied, etc. The additioal coditios are: (. 1) (.) X+l -1 lim if (w(t» dt = 0 x f x w is cotiuously differetiable o [1,(0), w has a positive lower boud, say w(x) ~ L > 0, I x (w' (t» dt = 0'(X) (x-+<»), g' is positive ad odecpeasig o (0,00). (.3) w is cotiuously differetiable o [ 1,(0) with w' odecreasig w(x) -+0 (x-+<»), w' (x) = o(w(x» (x-+<»), g' is positive ad odecreasig, glg'is odecreasig o some iterval [O,aJ with a > 0 (or, equivaletly, xh I (x) is odecpeasig o some iterval [O,bJ with b > 0), J; [w(x)h ' (w(x»j dx = 00, where h is the iverse fuctio of g. The very simple proof of theorem.1 is give i sectio 3 as a applicatio of lemma 3.1. The proofs of theorems. ad.3 are preseted i sectios 4 ad 5, with proofs obtaied by adaptio of the oe i Brads [197J.

5 - 3 - I order to give a idea of that method we preset a short demostratio for a special case, viz. X-~f' (x) + f(x) = f(x-1) Squarig, itegratig from 1 to, ad itegratig I 1 x -~ f(x)f ' (x)dx by parts, we get 1 J x - 1 (f 1 (x) ) dx = J (f(x)) - 1 o J ( f (x) ) dx + ( f (1 ) ) -1 We coclude that J7 x -1 (f' (x)) dx < 00, exists. This method 1 1 J x- 3 / (f(x))dx from which we ca derive that lim f(x) ca be modified so as to be applicable, ot to (1.1) itself, but to the equatio obtaied by differetiatio. REMARK. All coditios o 'w i (.) ad (.3) ca be weakeed by requirig these coditios o a sub-iterval [l+b,oo) oly. This is easy to sea by applicatio of theorem. or.3 to f(x+b). EXAMPLE. Cosider the equatio -a I I 18 ' x, f (x) = f(x-1) - f(x) sg (f(x-1)-f(x)), with 8 ~ 1. If a ~ 8/ the (1.) ad (.3) are satisfied, hece lim f(x) exists for a solutio f, which, for 8 = 1 ad w(x) = x-a, is i agreemet with results of N.G. de Bruij [1950J ad J.J.A.M. Brads [197J. If 8 > 1 ad a > 8/ the the asymptotic behaviour is ot kow (The case 8 = l,a> 1/ is treated i N.G. de Bruij [1949J). 3. PRELIMINARIES Let coditios {1.) be satisfied ad let f be a solutio of (1.1). We defie fuctios M, m, 0 ad A by M (x) : = max {f (t) I x ~ t ~ x + 1}, m (x) : = mi {f (t) I x ~ t ~ x + 1}, A(x) := maxf ' (t) II x ~ t ~ x + 1}, o(x) = M(x) - m(x), for x ~ 1.

6 - 4 - LEMMA 3.1. The solutio f is bouded. Moreover~ the fuctios M ad - m are oicreasig. Furthermore~ f' has at least oe zero i every iterval [x,x+1], x ~ 1. If W is cotiuously differetiable~ the f" exists ad is cotiuous o [, 00) ~ ad JX+1 I f" (s) I ds ~ A (x) for x ~. Also x A(x) ~ fx+11 f' (s) I ds ~ o(x) for x ~ 1. x PROOF OF LEMMA 3.1. The proof of the statemet about M ad -m is obtaied by obvious modificatios from the oe i Brads [197]. The other statemets i lemma 3.1 are eve simpler. A obvious cosequece of lemma 3.1 is COROLLARY 3.1. If lim o(x) = 0 the lim f(x) exists. x-+oo X-KO PROOF OF THEOREM.1. Sice g(f(x-1) - f(x)) is bouded, we have x+1 x+1 o(x) ~ Ix If'(s)1 ds = 0'( J x (w(t))-1dt) (x ~ 1). Sice 0 is mootoic, the the0rem follows. We metio several simple statemets about the iverse fuctio h of g if g satisfies the extra coditio that g' is positive ad odecreasig. The proofs are easy ad therefore ommitted. h is a odd cotiuous fuctio o R, positive o (0,00), cotiuously differetiable o R\ {OJ. The derivative h' is positive o R\ {oj, oicreasig o (0,00), ad h' (x) = [g' (h(x)) ]-1 if x ~ O. If g' (0) f. 0, the the exclusio of x = 0 i the foregoig statemets about h ca be ommitted. LEMMA 3.. Suppose that~ i additio to (1.)~ the followig coditios are fulfilled: g' is positive ad odecreasig o (O,oo)~ w is cotiuously differetiable o [1,00)~ w(x) + 0 (x + 00), w' is egative ad odecreasig~ ad w(x) / w(x+1) has a upper boud o [l,oo)~ say W. The f' is bouded. PROOF OF LEMMA 3.. For a costat solutio f (i.e. f(x) = costat o [0,00)) lemma 3.. is obviously true. From ow o we suppose that f is a ocostat

7 - 5 - II solutio. The A() > 0 ( ~ 1), ad expressios h' (w(x )A(», appearig i the sequel of this proof, are defied. For every ~ is a umber x E [,+l] such that A() = I f' (x ) I. If x = the A() S A(-l). If < x S + 1 the f"(x ) sg (f' (x» ~ 0. Differetiatig (1.1) we get there (3.1) w (x) f" (x) + w' (x) f' (x) (f' (x-1) - f' (x»g' (h(w(x)f' (x») (x ~ ) It follows that [f' (x -1) sg (f' (x» - A()]g' (h(w(x )A(») - w' (x A(» ~ 0 ( ~ ) Hece we always have (3.) [1 + h' (w(x )A() )w' (x ) ]A() S A(-1) ( ~ ) Cosider the fuctio ~ [0,00) +~, defied by ~ (t) = [1 - o.h' (St)] t, where 0. ad S are positive umbers. Sice xh' (x) S hex) that o.th' (St) S o.s-1h (St). It follows that ~(t) (x > 0) we have + 0 if t + O. From the coditios o g we kow that h' (St) teds fo a fiite limit h' (00) ~ 0 if t If o.h' (00) < 1 the clearly ~(t) + 00 if t + 00, ad it follows that, give a positive umber y, there is a largest umber t such that ~(t) = y. Hece it is possible to defie a sequece B' ~ B =A, B [1 +w'(x )h'(w(x)b)] B O O -l O, as follows: ( ~ O)' where O is such that w' (O)h' (00) > - 1, ad where it is meat that, for > O' B is the largest umber satisfyig the equality. Trivially we have that A() S B < B+1 for ~ O Hece (3.3) B [1 + w'(x )h'(w(x)b )] $ B 1 O - where we have used the mootoicity of hi. ( > O)' Clearly,

8 ,... 6,.;.. B 00 is bouded if L = o w'(x )h'(w(x) B ) 1<00. We have O w'(x)h'(w(x)b) O ~ - w'()h'(w(+l) B ) O - ~ - w' ()h' (B W w(-l)) ~ O - w' (x)h' (BW-fw(x)) for - l' ~ x O ~, ~ O + 1. Hece, for N > O' Iw'(x )h'(w(x) B ) O - w' (x)h' (w(x) B W )dx ~ O This completes the proof of lemma 3.. LEMMA 3.3. Suppose that~ i additio to (1.)~ w is cotiuouszy differetiabze o [1,00). The y f [f' (x-i) - f' (x)] dx ~ C - where c is a positive costat ad G(t) := J y - G(f(x-l) - f(x)) (w(x)) w' (x)dx t J g(s)ds o PROOF OF LEMMA 3.3. Puttig u(x) := f(x-1) - f(x), we derive from (1.1) that (3.4) -1 u'(x) + (w(x)) g(u(x)) -1 (w(x-1)) g(u(x-1)) (x ~ ) Squarig both sides of (3.4), itegratig from to y, Y >, itegratig J(w(x)) y -1 g(u(x))u' (x)dx by parts ad rearragig terms, we fid

9 - 7 - (3.5) ((U, (x» dx = (U' (x» dx + (w (» -IG(u(» - (G (u(x» (w(x» -w ' (x 1 y f ( f' (x) ) dx - (w (y) )-1 G (u (y)). y-l Sice G(t) ~ 0 for all t E R the lemma follows. 4. PROOF OF THEOREM.. I the sequel symbols c 1 ' c ' etc. deote properly chose costats. Sice g(f(x)-1) - f(x)) is bouded ad w(x) ~ L > 0 o [1,~) it follows from (1.1) that f' is bouded o [1,~). Usig the boudedess of f, f' ad wf' we ifer from (3.1) (4.1) w(x) I fll(x) I ~ C 1 If' (x-l) - f' (x) 1+ C I w' (x) I (x ~ ). Squarig both sides of (4. 1), usig the iequality (a+b) ~ a + b, itegratig from to, >, we obtai (4.) I [W(X)fll(X)]dX~C3+C4 J (W,(x))dX + C5 f [f'(x-1) -f'(x)]dx. By Lemma 3.3, usig the boudedess of f ad l/w, ad usig the Schwarz's iequality, we have ) [f' (x-i) _ f' (x) j dx $ C 6 + c 7 ) I w' (x) I dx $ C 6 + c 7 ~[( I w' (x) 1d1 Moreover, by lemma 3.1 ad Schwarz's iequality f [w(x) fll (x)] dx ~ L (-) (o()) Sice ~~ I w' (x) r dx = C9'() ( -+ (0) we ca coclude that 0 () = 0'(1) ( -+ 00

- e - 5. PROOF OF THEOREM.3. From wl(x) = &(w(x» (x -+ 00) it follows that w(x) /w(x+l) has a upper boud W. Hece lemma 3. is applicable. Let K > 1 be a upper boud of I fl I We ifer from (3.1) (5.

10 - e - 5. PROOF OF THEOREM.3. From wl(x) = &(w(x» (x -+ 00) it follows that w(x) /w(x+l) has a upper boud W. Hece lemma 3. is applicable. Let K > 1 be a upper boud of I fl I We ifer from (3.1) (5.1) w(x)h l (KW(X)} I fll(x) 1$ If I (x-i) - fl (x) 1+ K I Wi (x) I hi (KW(x» Squarig both sides of (5.1), usig the iequality (a+b) $ a + b, itegratig from to, >, we obtai (5.) r J [w (x) h I (Kw (x) ) f" (x)] dx $ f [fl (x-l) - fl (x) ]dx + K I [ w I (x) h ( Kw (x) ) ] dx Applicatio of lemma 3.3, usig the boudedess of fl, ad itegratig by d parts, usig dt G(h(t» = th l (t), we ca write J [fl (x-l) - fl (x)]dx $ C e - J G(h(Kw(x») (w(x»- WI (x)dx = C g + (w(»-lg(h(kw(») - K f hi (Kw(x»w l (x)dx = C 10 + (w(»-lg(h(kw(») - Kh(Kw(» From G(x) $ xg(x) it follows that (w(»-lg(h(kw(») $ Kh(Kw(». We coclude that J~[fl (x-l) - fl (x)]dx is bouded. Sice hi is odecreasig we have that J [Wi (x)h l (Kw(x» ]dx $ I [w I (x) h I (w (x) ) ] dx Hece, a upper boud for the right had side of (5.)i5 C 11 + K J [ W I (x) h I (w (x) )] dx

- 9 - We ca fid a lower boud for the left had side of (5.) as follows: For ko sufficiecy large, the fuctio yw(x)h' (yw(x)) is oicreasig i x for x ~ k O ' ad odecreasig i y for 0 < y ~ K.

11 - 9 - We ca fid a lower boud for the left had side of (5.) as follows: For ko sufficiecy large, the fuctio yw(x)h' (yw(x)) is oicreasig i x for x ~ k O ' ad odecreasig i y for 0 < y ~ K. Let ko ~ k ~ - 1. The, usig Schwarz's iequality ad lemma 3.1 we have k+1 k+1 I k := f [w(x)h' (KW(x))f"(x)]dx ~ K- f [w(x)h' (w(x))f"(x)]dx ~ k K- [W(k+1)h' (w(k+1))] (0(k)) ~ K - (0 ()) J [w(x)h' (w(x)) ]dx Hece We coclude that f [w(x)h'(kw(x))f"(x)]dx ~ k k+1 k+ ko+1 K- (0()) f [w(x)h' (w(x))]dx. (0()) I [w(x)h' (w(x))]dx ~ C + K4 1 f [WI (x)h' (w(x)) Jdx ko+l Usig w' (x) = ef(w (x)) (x -+ 00) we easily ifer that 0 () = e'(1) ( -+ 00)

- 10 - REFERENCES N.G. DE BRUIJN [1949J [1950J The asymptotically periodic behaviour of some liear fuctioal equatios, Amer. J. Math. 71, 313-330. O some liear fuctioal equatios, Publ. Math. Debrece 1.

12 REFERENCES N.G. DE BRUIJN [1949J [1950J The asymptotically periodic behaviour of some liear fuctioal equatios, Amer. J. Math. 71, O some liear fuctioal equatios, Publ. Math. Debrece J.J.A.M. BRANDS [197J The asymptotic behaviour of the solutios of certai differece differetial equatios, Nederl. Akad. Wetesch. Proc. Ser A75 = Idag. Math. 34, J.L. KAPLAN, M. SORG ad J.A. YORKE [1979J Solutios of Xl (t) = f(x(t),x(t-l» have limits whe f is a order relatio, Noliear Aal., Theory) Methods Appl. 3,

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