Oscillation Properties of a Logistic Equation with Several Delays

Transcription

1 Journal of Maheaical Analysis and Applicaions 247, Ž 2. doi:1.16 jaa.2.683, available online a hp: on Oscillaion Properies of a Logisic Equaion wih Several Delays Leonid Berezansy 1 Deparen of Maheaics and Copuer Science, Ben-Gurion Uni ersiy of he Nege, Beer-She a 8415, Israel E-ail: brznsy@cs.bgu.ac.il and Elena Braveran 1 Copuer Science Deparen, Technion Israel Insiue of Technology, Haifa 32, Israel E-ail: aelena@cs.echnion.ac.il Subied by Gerry Ladas Received Deceber 15, 1999 For a scalar delay logisic equaion ž y h y y Ý r 1, h, he oscillaion properies are esablished, which are well nown for a linear delay differenial equaion, such as coparison heores, explici nonoscillaion and oscillaion condiions, dependence of he soluion sign on he iniial funcion and he iniial value. 2 Acadeic Press The delay logisic equaion 1. INTRODUCTION ž y h y r y 1, h, Ž 1. 1 Suppored by Israel Minisry of Absorpion X $35. Copyrigh 2 by Acadeic Press All righs of reproducion in any for reserved. 11

2 OSCILLATION OF LOGISTIC EQUATION 111 is nown as Huchinson s equaion, if r and are posiive consans and h for a posiive consan. Huchinson s equaion has been invesigaed by several auhors; see, for exaple, 12, 13, 16, 19. The oscillaion of soluions of delay logisic equaion 1 was sudied by Gopalsay and Zhang 8, 21 who gave sufficien condiions for he oscillaion of 1. In order o inroduce several delays, Eq. 1 can be odified o he for 1 ž y r y 1 Ý a yž., Ž 2. where a and are nonnegaive consans. For he recen conribuions o he sudy of Eq. 2, we refer o 8, 11, 15. Oher generalizaions of Eq. 1 and he relevan references are presened in he recen onograph 7 and papers 9, 1, 14, 17, 18, 2. The paper 9 conains an ineresing ehod for he research of oscillaion of he uliplicaive delay logisic equaion ž y g y r y 1 Ł. Ž 3. The auhors connec he oscillaory properies of Eq. 3 wih he oscillaion of a cerain linear delay differenial equaion. Unlie he oher wors on he logisic equaion, in 9 funcions r and g are no assued o be coninuous. This allows o consider he logisic equaion wih piecewise consan delays. In our paper 1, we also consider he addiive delay logisic equaion y h y y Ý r 1 Ž 4 ž. wihou he assupion ha he funcions r and g are coninuous. We prove ha he oscillaion properies of 4 can be deduced fro he corresponding properies of he linear equaion x Ý r x h. Ž 5. In his paper, we coninue he invesigaion ha was begun in 1. For logisic equaion 4, we obain he properies which were forerly esablished only for a linear equaion: coparison of he oscillaion properies of wo equaions, coparison of soluions of a differenial equaion and differenial inequaliies, posiiveness condiion for a given soluion, ec.

3 112 BEREZANSY AND BRAVERMAN Here we apply he sae ehod which we used before for he sudy of oscillaion properies for various classes of linear funcional differenial and ipulsive equaions 2 6. Siilar o he above invesigaions, we prove ha he exisence of a nonoscillaory soluion for a differenial equaion is equivalen o he exisence of a nonnegaive soluion for soe explicily consruced nonlinear inegral inequaliy. Inegral equaions ha correspond o such inequaliies are widely used in he oscillaion heory Žsee, for exaple, 7, 8, 11.. However, for soe oscillaion probles, inegral inequaliies can be a ore useful ool han inegral equaions. All he resuls of his paper are based on such an inequaliy. The paper is organized as follows. In Secions 2 4, we consider an equaion, which is obained fro 4 by he following subsiuion y x 1. On he base of he resuls of he previous secions, in Secion 5 we invesigae delay logisic equaion 4 and soe generalized logisic equaion. 2. PRELIMINARIES Consider a scalar delay differenial equaion x Ý r x h 1 xž.,, Ž 6. under he following condiions: Ž a1. r, 1,...,, are Lebesgue easurable funcions essenially bounded in each finie inerval, b, r, Ž a2. h :,. R are Lebesgue easurable funcions, h, li h, 1,...,. Togeher wih 6, we consider for each an iniial value proble x Ý r x h 1 xž.,, Ž 7. x Ž.,, xž. x. Ž 8. We also assue ha he following hypohesis holds: Ž a3. : Ž,. R is a Borel easurable bounded funcion.

4 OSCILLATION OF LOGISTIC EQUATION 113 DEFINITION. An absoluely coninuous in each inerval, b funcion x: R R is called a soluion of proble 7, 8, if i saisfies Eq. 7 for alos all,. and equaliies Ž 8. for. We will presen here soe leas which will be used in he proof of he ain resuls. Consider he linear delay differenial equaion x Ý r x h,, Ž 9. and he differenial inequaliies x Ý r x h,, Ž 1. x Ý r x h,, Ž 11. LEMMA 1 Žsee. 2. Le Ž a1. Ž a3. hold. Then he following saeens are equi alen: 1. There exiss a nonoscillaory soluion of Eq There exiss an e enually posii e soluion of he inequaliy Ž There exiss an e enually negai e soluion of he inequaliy Ž There exiss such ha he inequaliy Ý ½H 5 h u r exp už s. ds,, Ž 12. where he su conains only such ers for which h, has a nonnega- i e locally inegrable in,. soluion. If x, y, z,, are posii e soluions of 9, Ž 1,. Ž 11,. respeci ely, x y z,, hen y x z,. Ž. LEMMA 2 see 11. Le a1 a3 hold. If Ý H li sup r s ds 1 e, 13 in h hen Eq. 9 has a nonoscillaory soluion. If in addiion x, x, hen he soluion of iniial alue proble 9, 8 is posii e.

5 114 BEREZANSY AND BRAVERMAN If Ý H li inf r s ds 1 e, 14 ax h hen all he soluions of Eq. 9 are oscillaory. 3. OSCILLATION CRITERIA Togeher wih Eq. 6, consider differenial inequaliies x Ý r x h 1 xž.,, Ž 15. x Ý r x h 1 xž.,. Ž 16. In his secion and he nex one, we assue ha Ž a1. Ž a3. hold and consider only such soluions of Ž 6., Ž 15., and Ž 16. for which he following condiion holds: 1 x. Ž 17. THEOREM 1. The following saeens are equi alen: 1. Inequaliy Ž 15. has an e enually posii e soluion. 2. There exis, : Ž,.,., and c such ha he inequaliy where ž ½ H 5 Ý u 1 c exp už s. ds Ž F u.ž., Ž r exp Hh už s. ds, if h, Ž Fu. 1 exp H už s. ds Ž h Ž.., if h, c has a nonnegai e locally inegrable on,. soluion. 3. Equaion 6 has an e enually posii e soluion. Proof Le x be a soluion of Ž 15. and x for 1. Then here exiss such ha h for. Denoe 1 1 x,, and c x.

6 OSCILLATION OF LOGISTIC EQUATION 115 Le u Ž x x.,. For soluion x of Ž 15., we have x,, consequenly, u. We can rewrie now x in he for ½ c exp H už s. ds 4,, x Ž 19. Ž.,. By subsiuing x in inequaliy Ž 15., we will obain inequaliy Ž Le u be a nonnegaive soluion of inequaliy Ž 18.. Denoe a sequence ž ½ 5 Ž n. Žn 1. H Ý Ž. u 1 c exp u s ds F u, u u. Ž 2. Ž1. Fro 18, we have u u, and by inducion, we can prove ha Ž n. Ž n 1. u u u. There exiss a poinwise lii of he nonincreasing posiive sequence Ž n. Ž n. u. If we le u li u 1 n, hen by he Lebesgue Convergence Theore we conclude ha he funcion u is locally inegrable and li Ž Fu Ž n.. Ž Fu 1.Ž.. n Hence, for u fro 2 we obain 1 ž 1 ½ H 5 Ý 1 u 1 c exp u Ž s. ds Ž F u.ž.. Siilarly, we can consruc a sequence u, such ha and obain ha n n ½ H n 1 5 Ý n ž u 1 c exp u Ž s. ds Ž F u. un un 1 už.. The sae arguen as before gives ha here exiss a poinwise lii u of he sequence u, and for his funcion we have n ž ½ H 5 Ý 1 u 1 c exp už s. ds Ž F u.ž.. Ž 21.

7 116 BEREZANSY AND BRAVERMAN Hence, a funcion x defined by equaliy Ž 19. is an evenually posiive soluion of Eq. 6. THEOREM 2. The following saeens are equi alen: 1. Inequaliy Ž 16. has an e enually negai e soluion. 2. There exis, : Ž,. Ž,, and c such ha inequaliy Ž 18. has a nonnegai e locally inegrable on,. soluion u and ½ H 5 1 c exp už s. ds. Ž Equaion 6 has an e enually negai e soluion. Proof Le x be a soluion of Ž 16. and x for 1. Then here exiss such ha h for. Denoe 1 1 x,, and c x. Le u Ž x x.,. For soluion x of Ž 16., we have x, ; herefore, u. We can rewrie now x in he for Ž 19.. By subsiuing x fro Ž 19. in inequaliy Ž 16., we will obain inequaliy Ž Le u be a nonnegaive soluion of inequaliy Ž 18.. Denoe a sequence ž ž ½ H 5 1 ½ H 5 Ý u 1 c exp u Ž s. ds Ž F u.ž., u 1 c exp u Ž s. ds Ž F u.ž.. n n 1 n 1 Fro Ž 18., we obain ha u u, and by inducion, u 1 n u u. Then here exiss he poinwise lii u n 1 of he sequence u n. Hence, u is a soluion of Eq. Ž 21.. Thus x defined by Ž 19. is an evenually negaive soluion of 6. COROLLARY 1. Suppose here exis and c Ž c. such ha he inequaliy Ý ž ½ H 5 Ý ½H 5 h u 1 c exp už s. ds r exp už s. ds has a nonnegai e locally inegrable soluion, where he su conains only such ers for which h. Then Eq. 6 has an e enually posii e Ž an e enually negai e. soluion.

8 OSCILLATION OF LOGISTIC EQUATION 117 The corollary follows fro saeen 2 of Theore 1 Ž Theore 2. if we assue. COROLLARY 2. Le H Ý r d. Then here exiss a nonoscillaory soluion of Eq. 6. Proof. There exiss such ha exp 2H Ý r Ž s. ds4 h 2,. We assue u 2Ý r. Then u is a soluion of he inequaliy Ý H 5 h u r exp už s. ds. ½ Corollary 1 iplies ha Eq. 6 has an evenually negaive soluion. 3 Siilarly, here exiss such ha exp3h Ý r Ž s. ds4 h 2,. We assue u 3Ý r. Then Ý u 2 r exp už s. ds ½ H 5 h ½ H 5 Ý ½H 5 h ž 1 exp u s ds r exp u s ds, hence, by Corollary 1, Eq. 6 also has an evenually posiive soluion. 4. COMPARISON THEOREMS Copare firs oscillaion properies of Eq. 6 and of x Ý b x g 1 xž.,, Ž 23. where for b, g, condiions Ž a1. Ž a2. hold. THEOREM 3. Suppose b rž., g hž., Ž 24. and Eq. 6 has a nonoscillaory soluion. Then Ž 23. has a nonoscillaory soluion. Suppose b r, g h, and all he soluions of Eq. 6 are oscillaory. Then all he soluions of Ž 23. are also oscillaory.

9 118 BEREZANSY AND BRAVERMAN Proof. Le Ž 24. hold and le Ž 6. have a nonoscillaory soluion. Theores 1 and 2 iply ha inequaliy Ž 18. has a nonnegaive soluion u. The inequaliy also holds if r and h are replaced by b and g. Hence, Eq. Ž 23. also has a nonoscillaory soluion. The second saeen of he heore is a consequence of he firs one. Now we will copare soluions of he differenial equaion and he differenial inequaliies. THEOREM 4. Le x be a soluion of proble 7, 8, y1 be a soluion of Ž 15., Ž 8., y be a soluion of Ž 16., Ž 8., and z be a soluion of Ž 9,. Ž Suppose y, z,, x. Then y x 1 1 z. 2. Suppose y, z,, x. Then z x 2 y. 2 Proof. Suppose 1 holds. By he proof of iplicaion 1 2 in Theore 1, we have he represenaion ½ y1 xž. exp H už s. ds 4,, Ž.,, where u is a nonnegaive soluion of inequaliy Ž 18.. Fro he proof of iplicaion 2 3 in Theore 1, i follows ha for soluion u of Ž 21., he inequaliy u u,, holds. Hence, for soluions of 7, 8 and Ž 15., Ž 8., we have ½ xž. exp H už s. ds 4,, x y1ž.. Ž.,, If x is a posiive soluion of 7, 8, hen x Ý r x h,. Lea 1 iplies x z. The proof of 2 is siilar. Suppose 7, 8 is nonoscillaory. Consider he dependence of he sign of he soluion on he iniial funcion and he iniial value. To his end, denoe h in h, a li sup H Ý r Ž s. ds. h

10 OSCILLATION OF LOGISTIC EQUATION THEOREM 5. Suppose a, x is a soluion of he proble e 7, If x 1, hen x ae,. 2. If x 1, hen x,. Proof. Le condiions of 1 hold. By Lea 2, he equaion Ž. Ý Ž. y 1 xž. r x h Ž 25. wih iniial condiions 8 has he soluion y,. Since y is a nonincreasing funcion, hen y x. Then y 1 y Ý r x h, i.e., y is a posiive soluion of inequaliy Ž 15.. Theore 4 iplies ha x y, where x is he soluion of 7, 8. Le now condiion 2 of he heore hold. Consider he proble y Ý r y h, yž. xž., Ž 26. y Ž.,. By Lea 2, one can obain ha for soluion y of his proble, we have y, hen y. Equaion Ž 26. iplies ha y is nondecreasing, hence y 1 y Ý r y h, and 1 y 1 y. Theore 4 iplies now ha x y, where x is he soluion of 7, 8. Now le us copare he oscillaion properies of Eq. 6 and a linear delay differenial equaion. THEOREM Le here exis a nonoscillaory soluion of he linear equaion y Ý r y h. Ž 27. Then here exiss a nonoscillaory soluion of Eq Le here exis an e enually posii e soluion of Eq. 6. Then here exiss a nonoscillaory soluion of linear equaion Ž 27..

11 12 BEREZANSY AND BRAVERMAN 3. Le, for e ery sufficienly sall, all he soluions of equaion y Ž 1. Ý r y h Ž 28. be oscillaory. Then all soluions of Eq. 6 are oscillaory. 4. Le all he soluions of 6 be oscillaory. Then all he soluions of Ž 27. are oscillaory. Proof. 1. Le Eq. Ž 27. have a nonoscillaory soluion. Firs, suppose H Ý r d. Corollary 2 of Theores 1 and 2 iplies ha here exiss a nonoscillaory soluion of 6. Second, suppose ha H Ý r d and y is a negaive soluion of Ž 27. for. Then li y. Hence, here exiss T such ha 1 y 1, T. Therefore, y 1 y Ý r y h, T. Theore 2 iplies ha here exiss an evenually negaive soluion of Suppose y,, is a posiive soluion of Eq. 6. Then y Ý r y h,. Lea 1 iplies ha Ž 27. has a nonoscillaory soluion. 3. Le Eq. 6 have a nonoscillaory soluion. If his soluion is evenually posiive, hen by saeen 2 of his heore, Eq. Ž 28. for also has an evenually posiive soluion. We have a conradicion. Suppose ha y,, is a negaive soluion of 6. Oscillaion of all soluions of 28 iplies ha H Ý r d. Then li y. Hence, here exiss T such ha y, T. Therefore, y is a negaive soluion of he inequaliy y Ž 1. Ý r y h, T. Hence, by Lea 1 here exiss a nonoscillaory soluion of Ž 28., which gives a conradicion. 4 follows fro 1.

12 OSCILLATION OF LOGISTIC EQUATION 121 and 5. MAIN RESULTS Consider now delay logisic equaion 4 and he differenial inequaliies y h y y Ý r 1, Ž 29 ž. y h y y Ý r 1, Ž 3 ž. where r, h saisfy condiions Ž a1. Ž a2.,, and iniial funcion saisfies Ž a3.. There exiss a unique soluion of Ž 4. wih he iniial condiion y Ž.,, yž. y. Ž 31. In his secion, we assue ha an addiional condiion Ž a4. y,,, holds. Then, as in case 1 8, 21, he soluion of Ž 4, 31. is posiive. A posiive soluion y of 4 is said o be oscillaory abou if here exiss a sequence,, such ha y n n n, n 1, 2,... ; y is said o be nonoscillaory abou if here exiss T such ha y for T. A soluion y is said o be evenually posiive Ževenually negaive. abou if y is evenually posiive Ž evenually negaive.. y Suppose y is a posiive soluion of 4 and define x as x 1. Then x is a soluion of 6 such ha 1 x. If y is a soluion of Ž 29., hen x is a soluion of Ž 15.. If y is a soluion of Ž 3., hen x is a soluion of Ž 16.. Hence, oscillaion Ž or nonoscillaion. of y abou is equivalen o oscillaion Ž nonoscillaion. of x. Besides, y for is equivalen o x for. By applying Theores 1 6, we obain he following resuls for Eq. 4. THEOREM 7. The following saeens are equi alen: 1. Inequaliy Ž 29. has an e enually posii e abou soluion. 2. There exis, : Ž,.,., and c such ha inequaliy Ž 18. has a nonnegai e locally inegrable on,. soluion. 3. Equaion 4 has an e enually posii e abou soluion. THEOREM 8. The following saeens are equi alen: 1. Inequaliy 3 has an e enually negai e abou soluion.

13 122 BEREZANSY AND BRAVERMAN 2. There exis, : Ž,. Ž,, and c such ha inequaliy Ž 18. has a nonnegai e locally inegrable on,. soluion u and ½ H 5 1 c exp u s ds. 3. Equaion 4 has an e enually negai e abou soluion. Copare now he oscillaion properies of Eq. 4 and where b, g saisfy Ž a1. Ž a2.. THEOREM 9. y g y y Ý b 1, Ž 32 ž. Suppose b rž., g hž., Ž 33. and Eq. 4 has a nonoscillaory abou soluion. Then Ž 32. has a nonoscillaory abou soluion. Suppose b r, g h, and all he soluions of Eq. 4 are oscillaory abou. Then all he soluions of Ž 32. are oscillaory abou. Now copare soluions of he differenial equaion and he differenial inequaliies. THEOREM 1. Le y be a soluion of he proble 4, Ž 31., y1 be a soluion of Ž 29., Ž 31., y be a soluion of Ž 3., Ž 31., and z of Ž 9., Ž Suppose y, z,, y. Then y 1 1 y z. 2. Suppose y, z,, y. Then z 2 y y. 2 Denoe h in h, a li sup H Ý r s ds. h 1 THEOREM 11. Le a, and le y be a soluion of he proble Ž 4, 31. e. 1. Suppose y. Then y ae,. 2. Suppose y. Then y,. The following resul copares he oscillaion properies of Eq. 4 wih he oscillaion properies of a linear delay differenial equaion.

14 OSCILLATION OF LOGISTIC EQUATION 123 THEOREM If here exiss a nonoscillaory soluion of linear equaion y Ý r yž h., Ž 34. hen here exiss a nonoscillaory abou soluion of Eq If here exiss an e enually posii e abou soluion of Eq. 4, hen here exiss a nonoscillaory soluion of linear equaion Ž If for e ery sufficienly sall, all he soluions of y Ž 1. Ý r y h Ž 35. are oscillaory, hen all he soluions of Eq. 4 are oscillaory abou. 4. Le all soluions of 4 be oscillaory abou. Then all he soluions of Ž 34. are oscillaory. By Theore 12 and Lea 2, one can obain explici condiions of oscillaion abou for logisic equaion 4. COROLLARY. Le H 1 Ý li sup rž s. ds. e in h Then here exiss a nonoscillaory abou soluion of Eq. 4. Le H 1 Ý li inf rž s. ds. e ax h Then all soluions of 4 are oscillaory abou. The ehod applied in his paper can be used for soe oher nonlinear funcional differenial equaions. Consider, for exaple, he following one, which has no been invesigaed before as far as we now, y h 1 y y y Ý r 1,, Ž 36 ž. where. Condiion Ž 31. iplies Ž as in he case 1. ha he soluion of Ž 36., Ž 31. is posiive. Hence, we can rewrie Ž 36. in he for y h Ž. y y Ý r 1,. Ž 37 ž.

15 124 BEREZANSY AND BRAVERMAN y Afer he subsiuion x 1 in Ž 37., we have he equaion 1 Ž.Ž. x Ý r x h 1 x,. 38 For his equaion, one can obain he sae resuls as for Eq. 6. As corollaries, we obain oscillaion resuls for Eq. Ž 36., in paricular, he following heore. THEOREM 13. Le 1 li suph Ý rž s. ds. e in h Then here exiss a nonoscillaory abou soluion of Eq. 36. Le 1 li infh Ý rž s. ds. e ax h Then all soluions of 36 are oscillaory abou. REFERENCES 1. L. Berezansy and E. Braveran, On non-oscillaion of a logisic equaion wih several delays, J. Cop. Appl. Mah. 113 Ž 2., L. Berezansy and E. Braveran, On nonoscillaion of a scalar delay differenial equaion, Dynaic Sys. Appl. 6 Ž 1997., L. Berezansy, E. Braveran, and H. Aça, On oscillaion of a linear delay inegro-differenial equaion, Dynaic Sys. Appl. 8 Ž 1999., L. Berezansy and E. Braveran, Soe oscillaion probles for a second order linear delay differenial equaion, J. Mah. Anal. Appl. 22 Ž 1998., L. Berezansy and E. Braveran, Oscillaion of a linear delay ipulsive differenial equaion, Co. Appl. Nonlinear Anal. 3 Ž 1996., L. Berezansy and E. Braveran, On oscillaion of a second order ipulsive linear delay differenial equaion, J. Mah. Anal. Appl. 233 Ž 1999., L. N. Erbe, Q. ong, and B. G. Zhang, Oscillaion Theory for Funcional Differenial Equaions, Marcel Deer, New Yor, Basel, Gopalsay, Sabiliy and Oscillaions in Delay Differenial Equaions of Populaion Dynaics, luwer Acadeic Publishers, Dordrech, Boson, London, S. R. Grace, I. Gyori, and B. S. Lalli, Necessary and sufficien condiions for he oscillaions of uliplicaive delay logisic equaion, Q. Appl. Mah. 53 Ž 1995., E. A. Grove, G. Ladas, and C. Qian, Global araciviy in a food-liied populaion odel, Dynaic Sys. Appl. 2 Ž 1993., I. Gyori and G. Ladas, Oscillaion Theory of Delay Differenial Equaions, Clarendon Press, Oxford, 1991.

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