General Permanence Conditions for Nonlinear Difference Equations of Higher Order

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1 JOURNAL OF MATHEMATICAL ANALYSIS AND APPLICATIONS 213, ARTICLE NO. AY General Peranence Condtons for Nonlnear Dfference Equatons of Hgher Order Hassan Sedaghat* Departent of Matheatcal Scences, Vrgna Coonwealth Un ersty, Rchond, Vrgna Subtted by Gerry Ladas Receved Deceber 2, INTRODUCTION Consder the th order nonlnear dfference equaton xn fž x n 1, x n 2,..., x n., n 1,2,3,..., Ž 1. where f: 0,. 0,. and the ntal values x,..., x 0, In studyng the global behavor of the solutons of Eq. Ž. 1, we often need to establsh that Ž. 1 s peranent,.e., t has the property that every one of ts solutons s eventually confned wthn a fxed copact nterval regardless of the ntal values chosen. Peranence s needed drectly or ndrectly n establshng other propertes such as persstent bounded oscllatons or the global attractvty of fxed ponts. For exaple, n 6 peranence of the equaton xn xn 1gŽ x n 1,..., xn. Ž 2. under certan condtons on the functon g s used drectly n the dervaton of global attractvty results. More recently, the oscllatons results n 9 requre the exstence of nontrval bounded solutons to Ž. 1. Other nstances where peranence s appled are noted n the sequel. In ths paper we obtan general suffcent condtons that ply Eq. Ž. 1 s peranent. Our approach here utlzes the exstence of lnear bounds on nally restrcted functons f Ž.e., when f s sublnear n the large.. *E-al address: hsedaghat@ruby.vcu.edu X 97 $25.00 Copyrght 1997 by Acadec Press All rghts of reproducton n any for reserved. 496

2 GENERAL PERMANENCE CONDITIONS 497 Thus we wll generally not requre that f be contnuous or onotonc n soe of ts arguents; nor s t necessary to assue that Ž. 1 possesses a splfed fxed pont structure as s coonly done when dealng wth global stablty or wth oscllatons. One consequence of ths generalzaton s that the results n ths paper extend and unfy the exstng peranence and boundedness results n the lterature. On the other hand, the restrctve hypotheses needed for stablty or for oscllatons but not assued here, soetes nvolve paraeter ranges that go beyond what s allowed for f here. Hence, the results of ths paper do not entrely subsue the exstng peranence results. In addton to the unfyng aspect entoned above, we also obtan copletely new results concernng broad classes of dfference equatons of arbtrary order. Many of these results edately yeld new bounded oscllaton theores when coupled wth the results n PRELIMINARIES DEFINITION 1. Equaton Ž. 1 s sad to be peranent f there exst constants L, M 0,., such that for each soluton x 4 of Ž. n 1, there s a postve nteger n n Ž x,..., x. such that x L, M n for all n n 0. Any copact nterval havng ths property ay be called an absorbng nter al for Ž. 1. Rearks. Ž. 1 Note that each absorbng nterval ust contan all attractng ponts and lt sets, so peranence puts restrctons on trajectores that ere boundedness does not. Indeed, boundedness of all solutons of Ž. 1 does not ply peranence Ž see below.. Ž. 2 If the functon f n Ž. 1 s bounded, then Ž. 1 s trvally peranent. Ž. 3 In 6 t s also requred that L 0. We shall not assue ths condton, but when t does hold we ay refer to Ž. 1 as post ely peranent. In any case, t s portant to note that constants L, M do not depend on the ntal condtons; further, n the absence of sharpness requreents or other consderatons, we ay let L 0. DEFINITION 2. Bold-faced letters denote vectors n the -densonal space R or the cone 0,., unless otherwse noted; we now lst three of

3 498 HASSAN SEDAGHAT the ost falar nors on R for reference: 4 u ax u,..., u 1 the ax-nor, sup-nor, or l -nor u 2 Ž u1 u. 2 Ž the Eucldean or l nor. u u u 1 Ž the su or l nor An eleentary fact about fnte densonal spaces s that all nors on the are equvalent Žsee, e.g., 7, p In partcular, ths eans that for any nor, there are constants C, C 0 such that C u u C u for all u R. In the sequel we refer to C Ž and C. as the upper Žand lower, respect ely. ax-nor coeffcents of. Clearly C 1 for the ax-nor tself, and t s easly verfed that C for the su nor and that C ' for the Eucldean nor. For both of the last two nors, C 1. We use the notaton u v to defne the falar scalar product n R : Ý u v u. The followng varaton on a rather falar result Žsee, e.g., 8, p. 20. s needed n the sequel. LEMMA 1. Let f, g: 0,. 0,., and assue that for all u 0,., fž u. gž u.. If g s nondecreasng n each of ts arguents and the equaton zn gž z n 1,..., zn. Ž 3. s peranent, then Eq. Ž. 1 s peranent. Also, f all solutons of Ž. 3 are bounded, then so are all solutons of Ž. 1. Proof. We ay assue that 0, M s an absorbng nterval for Ž. 3 where M s soe postve real nuber. Suppose that x 0,..., x1 s an arbtrary set of ntal values for Ž. 1 and defne z x for 1,...,0. Note that x1 fž x 0,..., x1. gž z 0,..., z1. z1 and by nducton x f x,..., x g z,..., z z. n n 1 n n 1 n n

4 GENERAL PERMANENCE CONDITIONS 499 Snce there s an nteger n0 1 such that zn M for all n n 0,t follows that 0, M s also an absorbng nterval for Ž. 1. The fnal asserton about boundedness s slarly proved. 3. THE MAIN RESULTS.. LEMMA 2. Let a 0, 1 and b 0,. Then x b 1 a s the unque fxed pont of x aax x,..., x 4 b Ž 4. n n 1 n and x s globally asyptotcally stable. Proof. Snce x s the only soluton of the equaton x a axx,..., x4 b, t follows that x s the unque fxed pont of Ž. 4 and also that ax b x. To show that x s globally attractng, we consder two cases: Case 1. x x for all k 1,..., 0. Defne k M ax x 0,..., x 1 4. Note that M x, whch ples that M am b ax b x. Snce x am b, t follows that M x x. Therefore, x aax x, x,..., x 4 b až am b. b am b1 a and so that M am b až am b. b ax b M x x x. 1 2 Proceedng n ths anner, we fnd nductvely that n 1 n xn am b Ý 0 s a nondecreasng sequence whch clearly approaches x as n. Case 2. xk x for soe k 1,..., 0. Defnng M as above, we see that M x so that a M am b x.

5 500 HASSAN SEDAGHAT Thus x x1 am b M plyng ax1 b x2 am b. Therefore, x x2 x1 M. Snce x 0, x1 are both present n axx,..., x 4 n 1 n for n 1,...,, t s evdent that for these values of n, x xn am b M Ž 1 n.. However, x 1 aax x, x 1,..., x14 b až am b. b am b and x s present n axx,..., x 4 n 1 n for n 1,...,2. It follows that 2 x xn am b1 a am b M 1 n 2. Contnung nductvely, for j 1, 2, 3,..., we have j 1 j x xn am bý a Ž Ž j 1. 1 n j.. 0 Once agan, t s clear that xn x as n. Fnally, the onotonc nature of the convergence of xn to x n both of the above cases ples that x s n fact stable, thus copletng the proof. Rearks. Ž. 1 When a 1, Eq. Ž. 4 s not peranent. In ths case, when b 0, Ž. 4 has the trval soluton xn M, n 1, 2, 3,... wth M as defned n the proof of Lea 2. Clearly all such solutons are bounded, although the dependence of M on the ntal values that defne t eans that Ž. 4 s not peranent. When a 1 and b 0, then t s easly seen that solutons of Ž. 4 are all of the unbounded varety xn M bn. Ž. 2 It ay be worth notng that n Lea 2, x s not a hyperbolc or lnearly stable fxed pont when 2 and a 0, snce the partal dervatves of the ax functon fž u 1,...,u. aaxu 1,...,u4 b do not exst at any pont Ž x,..., x. for x Ž 0,..

6 GENERAL PERMANENCE CONDITIONS 501 LEMMA 3. Let f: 0,. 0,., and assue that there exst a 0, 1. and b 0,. such that fž u. a u b. Ž. for all u 0,. Then Eq. 1 s peranent. Proof. If we defne gž u. a u b, then g s clearly nondecreasng n each coordnate of u Ž u,...,u. 1. Hence, an applcaton of Leas 1 and 2 copletes the proof. THEOREM 1. Let f: 0,. 0,.. If ether of the followng two sublnearty condtons hold, then Ž. 1 s peranent. Ž.. I f s bounded on the copact subsets of 0, and f Ž u. 1 l sup, u C u where C s the upper ax-nor coeffcent. Ž II. There s a 0,. wth a 1 1 and b 0,. such that for all u 0,., Frst, we show that Ž. I ples Eq. Ž. 1 s peranent. The func- Proof. ton f Ž u. b a u. Ž 5. f Ž u. h x sup : u x u ½ 5 s nonncreasng and l hž x. 1 C. Thus there s a 0, 1. x and r 0 such that hr a C. It follows that whch ples that f Ž u. a u C for all u r r fž u. a u for all u, C where C s the lower ax-nor coeffcent. Now, u r C f and only f axu,...,u 4 r C f and only f u 0, r C. Defne 1 4 b sup f u : u 0, r C and note that for all u 0,. t s true that fž u. a u b. Lea 3 now establshes the peranence of Ž. 1.

7 502 HASSAN SEDAGHAT Now suppose that condton II holds. Then fž u. b a u b Ý a axu,...,u 4 b a 1 u so that once agan Lea 3 apples. Rearks. 1 Ž. Ž. 1 Theore 1 I s false f 1 C s replaced by 1 for nors other than the ax-nor. Indeed, the functon u fž u,. b 2 results n a non-peranent second order equaton, as ay easly be checked. Note that f has the property that fž u,. b 1 1 Ž u,. 1 u 2 2 for all Ž u,. Ž 0,. 2. But when u Ž u,. 1 s large enough, b Žu Ž. 2 Theore 1 ay be false f f cannot be properly defned on the boundary of the cone 0,.. For exaple, consder the frst order equaton 1 x, Ž 6. n p x n 1 p where p, x 0. Here, f u u s contnuous on Ž 0,. 0 and strctly decreasng, approachng 0 as u. The general soluton of Ž. 6 s gven by x x Ž p. n. n 0 For 0 p 1, the unque fxed pont x 1 s globally asyptotcally stable, so n partcular, Ž. 6 s peranent. However, for p 1, whle every soluton s bounded Ž and of perod 2 f x 1., Ž 6. 0 s no longer peranent; and f p 1, then every soluton of Ž. 6 wth x 1 s unbounded APPLICATIONS Wth the ad of the results of the precedng secton, we obtan n ths secton general peranence and boundedness theores for varous classes of nonlnear dfference equatons. The frst result concerns a generalzaton of Eq. Ž. 2.

8 GENERAL PERMANENCE CONDITIONS 503 COROLLARY 1. Let g: 0,. 0,. be bounded and assue further that Then the equaton s peranent. Proof. Defnng we see edately that l sup g Ž u. 1. u xn xn kgž x n 1,..., x n., 1 k Ž 7. fž u,...,u. u gž u,...,u. Ž 8. 1 k 1 f Ž u. l sup l sup g Ž u. 1 u u u so the proof concludes by applyng Theore 1. Rearks. Ž. 1 Wth regard to Eq. Ž. 2, or when k 1 n Ž. 7, the sple condtons posed on g n Corollary 1 are qute dfferent fro hypotheses Ž H. Ž H. stated n 6, p , although there s a sgnfcant overlap. The latter hypotheses ply that Ž. 2 s peranent wthout requrng that g be bounded. The sae hypotheses, however, also put several restrctons on g Že.g., that g be contnuous and nonncreasng n u,..., u, that Ž 2. 2 have a unque postve fxed pont x, etc.. that we do not assue n Corollary 1. Ž. 2 Do the aforeentoned Ž H.Ž H. 1 4 ply the hypotheses of Theore 1Ž. I? It turns out that they do ply the boundedness of the functon fn Ž. 8 on the copact sets and the fnteness of the lt supreu n Theore 1; however, the sad lt supreu ay exceed 1 under the ax-nor snce the followng Ž fnte. quantty 1 Ž x,. 0, x 4 l sup g u : u x ay exceed 1 under Ž H. Ž H Evdently, utlzng the partcular product for of Ž. 2 as s done n Ž 6.e., ore substantally than n Corollary 1. s requred for provng peranence under Ž H. Ž H Ž. 3 The Bauol Wolff odel of productvty growth 1, p. 355, f generalzed by ntroducng addtonal lags or delays nto the basc frst order odel, represents an nterestng econoc applcaton of Eq. Ž. 2.

9 504 HASSAN SEDAGHAT The hypotheses n Corollary 1 as well as those n 6 are both general enough to be adssble; hence, the attractvty results for Ž. 2 n 6 and the oscllaton results n 9 provde edate nsghts nto the global behavor of the extended Bauol Wolff equatons. COROLLARY 2. Let functons f : 0,. 0,., 1,...,, be g en and assue that there exst constants a 0, 1. and b 0,. such that f Ž x. ax b for all x 0. Then the equaton s peranent. 4 x ax f Ž x.,..., f Ž x. n 1 n 1 n Proof. Let a a,...,a, b b,...,b, and defne 1 1 Note that 4 fž u,...,u. ax f Ž u.,..., f Ž u fž u. a axu,...,u 4 b a u b. 1 Now Lea 3 ay be appled to conclude the proof. COROLLARY 3. Let functons f : 0,. 0,., 1,...,, be g en and assue that for soe j 1, 2,..., 4, there are constants a 0, 1. and b 0,. such that f Ž x. ax b for all x 0. Then the equaton j 4 x n f Ž x.,..., f Ž x. n 1 n 1 n s peranent. Proof. Defne fž u,...,u. n f Ž u.,..., f Ž u.4 and note that fž u 1,...,u. fjž uj. auj b. The proof now concludes by applyng Theore 1 II. COROLLARY 4. Let functons f : 0,. 0,., 1,...,, satsfy the followng condtons: Ž. a There are nondecreasng functons g : 0,. 0,. such that f Ž x. g Ž x. for all x 0; 1 b l sup x Ł g Ž x. 1. Then the equaton s peranent. x Ł x f Ž x. Ž 9. n n

10 GENERAL PERMANENCE CONDITIONS 505 Proof. Defne f u f u,...,u Ł f Ž u. and note that 1 fž u. g Ž u. g Ž u., Ł Ł where the last nequalty holds because u u and g s nondecreasng for every 1,...,. Now condton Ž b. ples that f Ž u. l sup 1. u u Ž. Hence, by Theore 1, Eq. 9 s peranent. COROLLARY 5. Let functons f : 0,. 0,., 1,...,, satsfy the followng condtons: p 1; Ž. p a f x ax b for all x 0, where a 0, b 0, and 0 b Ether p Ý p 1, or p 1 and Ł a 1. Ž. Then Eq. 9 s peranent. p Proof. The functons ax b are ncreasng on 0,. and p p ax b x Ła Ž x., Ł where Ž x. conssts of a su of powers of x wth each power strctly less than p. Snce ether of the condtons n Ž b. ples condton Ž b. n Corollary 4, the proof s copleted by an applcaton of the latter boundary. Reark. Under sutable restrctve hypotheses, Corollary 5 ay hold even f soe of the exponents p are negatve whle others ay exceed 1; see Theore 4.1 n 2 for a second-order exaple. COROLLARY 6. Let f : 0,. 0,., 1,...,, be functons for whch there are constants a, b 0,., such that f Ž x. ax b on. 0,. If Ý a 1, then the equaton s peranent. Ý x f Ž x. n n

11 506 HASSAN SEDAGHAT Proof. Observe that Ý Ý Ý fž u,...,u. f Ž u. b au, b b 1 and apply Theore 1 to coplete the proof. The otted proof of the next corollary s slar to the proof of Corollary 6. COROLLARY 7. Let g: 0,. 0,. satsfy gž x. x on 0,. for constants, 0,.. The equaton n Ý n ž Ý n / x ax g b x, a, b 0,. Ž 10. s peranent f Ý a b 1. Rearks. Stablty of the fxed ponts of Eq. Ž 10. has been studed n 4, 5 Žor see. 6. Ths type of equaton appears n a odel of whale populatons 3. Peranence of Ž 10. s proved n 6 under several specalzed hypotheses, then used n establshng global stablty of a postve fxed pont x whose unque exstence s guaranteed by the sae hypotheses that are used n provng peranence. These latter hypotheses, n partcular, ply the followng: gž x. x wth sup gž x. 0 x x and 1 a, where a Ý a 1; also pled s Ý Ž a b. 1. The last equalty shows that under addtonal hypotheses on f, Corollary 7 holds even f the su Ý Ž a b. equals 1. On the other hand, when the last su s less than 1, Corollary 7 shows that peranence exsts wthout addtonal restrctve hypotheses. COROLLARY 8. Let g: Ž,. 0,. satsfy gž x. axx,04 on Ž,. for constants, 0,.. Then the equaton Ý x ax g bx cx, c, b, a 0,.,1,j,k n n n j n k Ž 11. s peranent f b Ý a 1.

12 GENERAL PERMANENCE CONDITIONS 507 Proof. Defne the functon Ž 1. Ý Ž j k. f u,...,u au g bu cu and observe that 4 fž u 1,...,u. Ý au ax buj cu k,0 bu Ý au. j Therefore, Theore 1 ples that 11 s peranent f b Ý a 1. Reark. Lke Corollary 7, n specal cases t s possble to prove the paraeter range n Corollary 8. For nstance, the second order equaton xn axn 1 gž xn 1 xn 2. Ž 12. s a specal case of Ž 11., so accordng to Corollary 8 peranence obtans for Ž 12. f, n partcular, a 1. However, t was shown n 10 that Ž 12. s peranent f only, a 1. Ths ncrease n paraeter range Žknown so far only for the second order case. cae at the prce of requrng g to be nondecreasng n addton to satsfyng the lnear bound of Corollary 8. Generalzatons of Ž 12. and varants of Ž 11. provde natural atheatcal settngs for dscussng the global behavor of accelerator-based odels of the acroeconoc busness cycle; see 9, 10. COROLLARY 9. Assue that f, g : 0,. 0,., 1,...,, are functons satsfyng fž x. b g Ž x. ax b.. on 0, for constants a, b 0, and b 0,. If Ýa 1 then the equaton s peranent. Ý fž xn. xn Ž 13. b Ý g Ž x. n

13 508 HASSAN SEDAGHAT Proof. Note that fž u,...,u Ý fž u.. b Ý g Ž u. 1 so that once agan Theore 1 apples. f Ž u. b au Ý Ý Ý b gž u. Reark. If the functons f, g are polynoals, then solutons of Ž 13. are called ratonal recursve sequences. In partcular, f the f, g are polynoals of degree 1 or 0, then Corollary 9 proves a theore of Ladas and Kocc 6, p. 61, who also use the ethodology presented n Lea 1 and Theore 1Ž II., though not at as general a level as we have done here. One advantage of the generalzaton s that the second part of the aforeentoned Kocc Ladas Theore need not be proved ndependently, as both parts of that theore are ncluded n Corollary 9. We close wth an applcaton of the results obtaned here to the oscllaton theory. For convenence, we quote the an result n 9 as a lea here. We state for reference that a bounded soluton x 4 of Ž. n 1 s sad to oscllate persstently f the sequence x 4 n has at least two dstnct lt ponts... LEMMA 4. Assue that the functon f: 0, 0, s contnuous and satsfes the followng condtons: Ž. a The equaton fž x,..., x. x has a fnte nuber of solutons 0 x1 xk ; Ž b. For e ery j 1,...,k, fž x,..., x, x. j j xj f x x j; Ž. c For 1,...,, the partal der at es f x exst contnuously at x j,..., x j, and e ery root of the characterstc polynoal Ý f x x,..., x has odulus greater than 1 for each j 1,...,k. j j Then all bounded solutons of Ž. 1 except the tr al solutons x j, j 1,...,k oscllate persstently. The next corollary gves a suffcent condton for the nontrval applcablty of Lea 4. COROLLARY 10. In addton to the assuptons of Lea 4, suppose that f satsfes one of the sublnearty condtons n Theore 1. Then all nontr al solutons of Ž. 1 e entually oscllate persstently wthn a fxed absorbng nter al.

14 GENERAL PERMANENCE CONDITIONS 509 In the second order case, the lnearzed nstablty condton Ž. c n Lea 4 s soewhat easer to characterze. Once agan, usng a result fro 9, we have the followng.. 2. COROLLARY 11. Assue that f: 0, 0, s contnuous and satsfes the followng condtons: Ž. a The equaton fž x, x. x has a fnte nuber of solutons 0 x 1 xk ; Ž b. For e ery j 1,...,k, fž x, x. j xj f x x j; Ž. c Both f x and f y exst contnuously at Ž x, x. j j for all j 1,...,k wth f f f Ž x j, x j. 1, Ž x j, xj. 1 Ž x j, x j.. y y x Ž d. Ether there are constants a, b, c 0 wth a b 1 such that fž x,y. ax by c for all x, y 0, or fž x, y. 1 l sup. x y x y 2 Then the second order nonlnear equaton: xn fž x n 1, x n 2., n 1, 2,3,..., x 1, x0 0,. s peranent and all of ts nontr al solutons e entually oscllate persstently n a fxed absorbng nter al. REFERENCES 1. W. J. Bauol and E. N. Wolff, Feedback between R & D and productvty growth: A chaos odel, n Cycles and Chaos n Econoc Equlbru Ž J. Benhabb, Ed.., Prnceton Unv. Press, Prnceton, E. Caouzs, E. A. Grove, V. L. Kocc, and G. Ladas, Monotone unstable solutons of dfference equatons and condtons for boundedness, J. Dfference Equatons Appl. 1 Ž 1995., C. W. Clark, A delayed recrutent odel of populaton dynacs wth an applcaton to baleen whale populatons, J. Math. Bol. 3 Ž 1976., M. E. Fsher, Stablty of a class of delay dfference equatons, Nonlnear Anal. 8 Ž 1984.,

15 510 HASSAN SEDAGHAT 5. M. E. Fsher and B. S. Goh, Stablty results for delayed recrutent n populaton dynacs, J. Math. Bol. 19 Ž 1984., V. L. Kocc and G. Ladas, Global Behavor of Nonlnear Dfference Equatons of Hgher Order wth Applcatons, Kluwer Acadec, Boston, E. Kreyszg, Introductory Functonal Analyss wth Applcatons, Wley, New York, V. Lakshkantha and D. Trgante, Theory of Dfference Equatons: Nuercal Methods and Applcatons, Acadec Press, San Dego, H. Sedaghat, Bounded oscllatons n the Hcks busness cycle odel and other delay equatons, J. Dfference Equatons Appl., n press. 10. H. Sedaghat, A class of nonlnear second order dfference equatons fro acroeconocs, J. Nonlnear Anal., n press.