一些脉冲泛函微分方程解的稳定性与周期性
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摘要
本文研究脉冲泛函微分方程的渐近稳定性及脉冲作用下种群模型的周期解。
在第二章，研究脉冲泛函微分方程的渐近稳定性，建立了脉冲泛函微分方程零解的渐近稳定性和一致渐近稳定性定理，得到的结果推广或改进了相关的结果。
借助重合度理论这一工具，在第三章，研究具有脉冲和无穷时滞的捕食-食饵系统的正周期解的存在性；在第四章，研究具有脉冲和时滞的Lotka-Volterra系统的正周期解的存在性和全局渐近稳定性；在第五章，研究脉冲中立型单种群模型
海南师范大学硕士学位论文
的正周期解的存在性，分别得到了一些充分条件.
In this paper,we study the asympotic stability for impulsive functional differential equation and periodic solutions of population models influenced by impulses.
In chapter two,we study the asympotic stability for impulsive functional differential equation:
and establish the asympotic stability theorems and the uniformly asympotic stability theorems of zero solutions of the impulsive functional differential equation, the theorems extend or improve the former results.
By using the method of coincidence degree,in chapter three,we study the existence of positive periodic solution of Predator-Prey system with impulses and infinite delay:
in chapter four,we study the existence and global asymptotic stability of the positive periodic solution of a Lotka-Volterra system with delays and impulses:
in chapter five,we study the existence of positive periodic solution of a impulsive neutral modal of single-species population growth:
and respectively obtain some new sufficient conditions.
在第二章，研究脉冲泛函微分方程的渐近稳定性，建立了脉冲泛函微分方程零解的渐近稳定性和一致渐近稳定性定理，得到的结果推广或改进了相关的结果。
借助重合度理论这一工具，在第三章，研究具有脉冲和无穷时滞的捕食-食饵系统的正周期解的存在性；在第四章，研究具有脉冲和时滞的Lotka-Volterra系统的正周期解的存在性和全局渐近稳定性；在第五章，研究脉冲中立型单种群模型
海南师范大学硕士学位论文
的正周期解的存在性，分别得到了一些充分条件.
In this paper,we study the asympotic stability for impulsive functional differential equation and periodic solutions of population models influenced by impulses.
In chapter two,we study the asympotic stability for impulsive functional differential equation:
and establish the asympotic stability theorems and the uniformly asympotic stability theorems of zero solutions of the impulsive functional differential equation, the theorems extend or improve the former results.
By using the method of coincidence degree,in chapter three,we study the existence of positive periodic solution of Predator-Prey system with impulses and infinite delay:
in chapter four,we study the existence and global asymptotic stability of the positive periodic solution of a Lotka-Volterra system with delays and impulses:
in chapter five,we study the existence of positive periodic solution of a impulsive neutral modal of single-species population growth:
and respectively obtain some new sufficient conditions.
引文
[1] Lakshmikantham V. , Bainov D. D. and Simeonov P. S. , Theory of impulsive differential equations[M]. Singapore: World Scientific, 1989.
[2] Bainov D. D. and Simeonov P. S. , Systems with impulse effect stability, theory and applications[M]. New York: Chichester, 1989.
[3] Anokhin A. V. , On linear impulsive systems for functional differential equations[J]. Soviet Math. Dokl. , 1986, 33: 220-223.
[4] Yu J. S. and Zhang B. G. , Stability theorems for delay differential equations with impulses[J]. J. Math. Anal. Appl. , 1996, 198(1): 285-297.
[5] Gopalsamy K. and Zhang B. , On delay differential equations with impulses[J]. J. Math. Anal. Appl. , 1989, 139(1): 110-122.
[6] Shen J. and Yan J. , Razumikin type stability theorems for impulsive functional differential equation[J]. Nonlinear Anal. , 1998, 33: 519-531.
[7] Yan J. and Shen J, Impulsive stabilization of functional differential equations by Lyapunov-Razumikhin functions[J]. Non. Anal. , 1999, 37: 245-255.
[8] Bainov D. D. and Hristova S. G. , Impulsive integral inequalities with a deviation of the argument[J]. Math. Nachr. , 1995, 171: 19-27.
[9] Ballinger G. and Liu X. , On boundedness of solutions of impulsive systems[J]. Nonlinear Studies, 1997, 4(1): 121-131.
[10] Yan J. , Comparison results for impulsive delay differential inequalities and equations[J]. Portugaliae Math. , 1997, 54: 255-261.
[11] Chen Y. and Feng W. , Oscillations of second order nonlinear ODE with impulses[J]. J. Math. Anal. Appl. , 1997, 210(1): 150-169.
[12] Ballinger G. and Liu X. , Existence, uniqueness and boundedness results for impulsive delay differential equations[J]. Appl. Anal. , 1993, 74: 71-93.
[13] Anokhin A. V. , Berezansky L. and Braverman E. , Exponential stability of linear delay impulsive differential equations[J]. J. Math. Anal. Appl. , 1995, 193(3): 923-941.
[14] Zhang Y. , Zhao A. and Yan J. , Oscillation criteria for impulsive delay differential equations[J]. J. Math. Anal. Appl. , 1997, 205(2): 461-470.
[15] Sheng J. , Luo Z. and Liu X., Impulsive stability of functional differential equations via Lyapunnov functionals[J]. J. Math. Anal. Appl. , 1999, 240(1): 1-15.
[16] Hatvani L. , On the asymptotic stability for nonautonomous functional dif-
ferential equations by Liapunov functionals[J]. Trans. Amer. Math. Soc. , 2002, 354: 3555-3571.
[17]马知恩，种群生态学的数学模型与研究[M]．合肥：安徽教育出版社，1996．
[18]靳祯，马知恩，环境污染中三维时变Volterra捕食-被捕食系统的持续生存[J]．生物数学学报，2000，13(2)：126-135．
[19]Pan J. , Jin Z. and Ma Z. , Threshold of survival for an n-dimensional Volterra mutualistic system in a polluted environment[J]. J. Math. Anal. Appl. , 2000, 252(2): 519-531.
[20]Chen L. , Lu Z. and Wang W. , The effect of delays on the permanence for Lotka Volterra systems[J]. Appl. Math. Letter. 1995. 8: 71-73.
[21]藤志东，陈兰荪，具有时滞的周期Lotka-Volterra型系统的全局渐近稳定性[J]．数学物理学报，2000，20(3)：296-303．
[22]Tineo A. , An iterative scheme for the n-competing species problem[J]. J. Math. Anal. Appl. , 1995, 196: 1-15.
[23]Eilbeck J. C. and Lopez-Gomez J. , On the periodic Lotka-Volterra competition model[J]. J. Math. Anal. Appl. , 1997, 210: 58-87.
[24]范猛，王克，具有偏差变元的捕食者．食饵系统的全局周期解的存在性[J]．应用数学学报，2000，23(4)1557-561．
[25]Huang X. K. and Xiang Z. G. , 2π-periodic solutions of duffing equations with a deviating argument[J]. Chinese Sci. Bull. , 1994, 39(3): 201-203.
[26]Li Y. , Periodic solutions for delay Lotka-Volterra competition systems [J]. J. Math. Anal. Appl. , 2000, 246: 230-244.
[27]Teng Z. and Yu Y. , Some new results of nonautonomous Lotka-Volterra competitive systems with delays[J]. J. Math. Anal. Appl. , 2000, 241: 254-275.
[28]Fan M. and Wang K. , Global existence of positive periodic solutions of periodic Predator-Prey system with infinite delay[J]. J. Math. Anal. Appl. , 2001, 262: 1-11.
[29]李玉昆，一类时滞微分方程周期正解的存在性和全局吸引性[J]．中国科学(A辑)，1998，28：108-118．
[30]蒋达清，魏俊杰，非自治时滞微分方程周期正解的存在性[J]．数学年刊，1999，20A(6)：715-720．
[31]Yu J. , On the stability of impulsive delay differential equations caused byimpulse[J]. Acta. Math. Sinia, 1997, 13: 45-52.
[32]Shen J. , Some asymptotic stability results of impulsive integro-differential equations[J]. Chinese Ann. Math. Ser. A, 1996, 17: 759-765.
[33]Grains R. E. and Mawhin J. L. , Coincidence and nonlinear differential equations[M]. Horwood: Springe Verlin, 1997.
[34]郭大钧，孙经先，刘兆理，非线性常微分方程泛函方法[M]．济南：山东科技出版社，1995．
[35]藤志东，Lotka-Volterra型种群竞争系统的持久性和稳定性[J]．数学学报，2002，45(2)：905-918．
[36]王志成，庾建设，中立型时滞微分方程的振动性[J]．数学学报，1994，37(1)：129-134．
[37]俞元洪，中立型时滞微分方程解的振动性[J]．应用数学学报，1991，14(3)：404-410．
[38]张玉珠，党新益，脉冲中立型时滞微分方程解的振动性[J]．数学学报，1998，41(1)：219-224．
[39]Fang L. and Li J. B. , On the existence of periodic solution of a neutral delay model of single-species population growth[J]. J. Math. Anal. Appl. , 2001, 259: 8-17.
[40]Li Y. K. , Positive periodic solution for neutral delay model[J]. Acta Math. sinica, 1996, 39: 789-795.
[41]Cao Y. , Multiexistence of slowly oscillating periodic solutions for differential delay equations[J]. SIAM J. Math. Anal. , 1995, 26: 436-445.
[42]李正荣，李继彬，哈密顿系统与时滞微分方程的周期解[M]．北京：科学技术出版社，1996．
[43]唐扬斌，一类中立型泛函微分方程的周期解[J]．应用数学学报，2000，23(3)：321-328．
[44]文贤章，多种群生态竞争-捕食时滞系统正周期解的全局吸引性[J]．数学学报，2000，45(1)：83-92．
[45]Erbe L. , Krawcewicz W. and Wu J. , A composite coincidence degree with applications to boundary value problems of neutral equations[J]. Trans. Amer. Math. Soc. , 1993, 335: 459-478.
[46]周厚高，朱军，闫新甫，中立型时滞模型周期正解的一个充分条件[J]．生物数学学报，2000，15(1)：15-20．
[47]陈福来，文贤章，具有脉冲和无穷时滞的捕食-食饵系统的正周期解．湖南师范大学自然科学学报，2003，26(1)：6-11．
[48]陈福来，文贤章，具有脉冲和时滞的Lotka-Volterra系统的正周期解的存在性和全局渐近稳定性．生物数学学报，2004，19(2)：1-8．
[49]Liu B. , Chen L. , The periodic competing Lokta-Volterra model with impulsive
effect[J]. (in press)
[2] Bainov D. D. and Simeonov P. S. , Systems with impulse effect stability, theory and applications[M]. New York: Chichester, 1989.
[3] Anokhin A. V. , On linear impulsive systems for functional differential equations[J]. Soviet Math. Dokl. , 1986, 33: 220-223.
[4] Yu J. S. and Zhang B. G. , Stability theorems for delay differential equations with impulses[J]. J. Math. Anal. Appl. , 1996, 198(1): 285-297.
[5] Gopalsamy K. and Zhang B. , On delay differential equations with impulses[J]. J. Math. Anal. Appl. , 1989, 139(1): 110-122.
[6] Shen J. and Yan J. , Razumikin type stability theorems for impulsive functional differential equation[J]. Nonlinear Anal. , 1998, 33: 519-531.
[7] Yan J. and Shen J, Impulsive stabilization of functional differential equations by Lyapunov-Razumikhin functions[J]. Non. Anal. , 1999, 37: 245-255.
[8] Bainov D. D. and Hristova S. G. , Impulsive integral inequalities with a deviation of the argument[J]. Math. Nachr. , 1995, 171: 19-27.
[9] Ballinger G. and Liu X. , On boundedness of solutions of impulsive systems[J]. Nonlinear Studies, 1997, 4(1): 121-131.
[10] Yan J. , Comparison results for impulsive delay differential inequalities and equations[J]. Portugaliae Math. , 1997, 54: 255-261.
[11] Chen Y. and Feng W. , Oscillations of second order nonlinear ODE with impulses[J]. J. Math. Anal. Appl. , 1997, 210(1): 150-169.
[12] Ballinger G. and Liu X. , Existence, uniqueness and boundedness results for impulsive delay differential equations[J]. Appl. Anal. , 1993, 74: 71-93.
[13] Anokhin A. V. , Berezansky L. and Braverman E. , Exponential stability of linear delay impulsive differential equations[J]. J. Math. Anal. Appl. , 1995, 193(3): 923-941.
[14] Zhang Y. , Zhao A. and Yan J. , Oscillation criteria for impulsive delay differential equations[J]. J. Math. Anal. Appl. , 1997, 205(2): 461-470.
[15] Sheng J. , Luo Z. and Liu X., Impulsive stability of functional differential equations via Lyapunnov functionals[J]. J. Math. Anal. Appl. , 1999, 240(1): 1-15.
[16] Hatvani L. , On the asymptotic stability for nonautonomous functional dif-
ferential equations by Liapunov functionals[J]. Trans. Amer. Math. Soc. , 2002, 354: 3555-3571.
[17]马知恩，种群生态学的数学模型与研究[M]．合肥：安徽教育出版社，1996．
[18]靳祯，马知恩，环境污染中三维时变Volterra捕食-被捕食系统的持续生存[J]．生物数学学报，2000，13(2)：126-135．
[19]Pan J. , Jin Z. and Ma Z. , Threshold of survival for an n-dimensional Volterra mutualistic system in a polluted environment[J]. J. Math. Anal. Appl. , 2000, 252(2): 519-531.
[20]Chen L. , Lu Z. and Wang W. , The effect of delays on the permanence for Lotka Volterra systems[J]. Appl. Math. Letter. 1995. 8: 71-73.
[21]藤志东，陈兰荪，具有时滞的周期Lotka-Volterra型系统的全局渐近稳定性[J]．数学物理学报，2000，20(3)：296-303．
[22]Tineo A. , An iterative scheme for the n-competing species problem[J]. J. Math. Anal. Appl. , 1995, 196: 1-15.
[23]Eilbeck J. C. and Lopez-Gomez J. , On the periodic Lotka-Volterra competition model[J]. J. Math. Anal. Appl. , 1997, 210: 58-87.
[24]范猛，王克，具有偏差变元的捕食者．食饵系统的全局周期解的存在性[J]．应用数学学报，2000，23(4)1557-561．
[25]Huang X. K. and Xiang Z. G. , 2π-periodic solutions of duffing equations with a deviating argument[J]. Chinese Sci. Bull. , 1994, 39(3): 201-203.
[26]Li Y. , Periodic solutions for delay Lotka-Volterra competition systems [J]. J. Math. Anal. Appl. , 2000, 246: 230-244.
[27]Teng Z. and Yu Y. , Some new results of nonautonomous Lotka-Volterra competitive systems with delays[J]. J. Math. Anal. Appl. , 2000, 241: 254-275.
[28]Fan M. and Wang K. , Global existence of positive periodic solutions of periodic Predator-Prey system with infinite delay[J]. J. Math. Anal. Appl. , 2001, 262: 1-11.
[29]李玉昆，一类时滞微分方程周期正解的存在性和全局吸引性[J]．中国科学(A辑)，1998，28：108-118．
[30]蒋达清，魏俊杰，非自治时滞微分方程周期正解的存在性[J]．数学年刊，1999，20A(6)：715-720．
[31]Yu J. , On the stability of impulsive delay differential equations caused byimpulse[J]. Acta. Math. Sinia, 1997, 13: 45-52.
[32]Shen J. , Some asymptotic stability results of impulsive integro-differential equations[J]. Chinese Ann. Math. Ser. A, 1996, 17: 759-765.
[33]Grains R. E. and Mawhin J. L. , Coincidence and nonlinear differential equations[M]. Horwood: Springe Verlin, 1997.
[34]郭大钧，孙经先，刘兆理，非线性常微分方程泛函方法[M]．济南：山东科技出版社，1995．
[35]藤志东，Lotka-Volterra型种群竞争系统的持久性和稳定性[J]．数学学报，2002，45(2)：905-918．
[36]王志成，庾建设，中立型时滞微分方程的振动性[J]．数学学报，1994，37(1)：129-134．
[37]俞元洪，中立型时滞微分方程解的振动性[J]．应用数学学报，1991，14(3)：404-410．
[38]张玉珠，党新益，脉冲中立型时滞微分方程解的振动性[J]．数学学报，1998，41(1)：219-224．
[39]Fang L. and Li J. B. , On the existence of periodic solution of a neutral delay model of single-species population growth[J]. J. Math. Anal. Appl. , 2001, 259: 8-17.
[40]Li Y. K. , Positive periodic solution for neutral delay model[J]. Acta Math. sinica, 1996, 39: 789-795.
[41]Cao Y. , Multiexistence of slowly oscillating periodic solutions for differential delay equations[J]. SIAM J. Math. Anal. , 1995, 26: 436-445.
[42]李正荣，李继彬，哈密顿系统与时滞微分方程的周期解[M]．北京：科学技术出版社，1996．
[43]唐扬斌，一类中立型泛函微分方程的周期解[J]．应用数学学报，2000，23(3)：321-328．
[44]文贤章，多种群生态竞争-捕食时滞系统正周期解的全局吸引性[J]．数学学报，2000，45(1)：83-92．
[45]Erbe L. , Krawcewicz W. and Wu J. , A composite coincidence degree with applications to boundary value problems of neutral equations[J]. Trans. Amer. Math. Soc. , 1993, 335: 459-478.
[46]周厚高，朱军，闫新甫，中立型时滞模型周期正解的一个充分条件[J]．生物数学学报，2000，15(1)：15-20．
[47]陈福来，文贤章，具有脉冲和无穷时滞的捕食-食饵系统的正周期解．湖南师范大学自然科学学报，2003，26(1)：6-11．
[48]陈福来，文贤章，具有脉冲和时滞的Lotka-Volterra系统的正周期解的存在性和全局渐近稳定性．生物数学学报，2004，19(2)：1-8．
[49]Liu B. , Chen L. , The periodic competing Lokta-Volterra model with impulsive
effect[J]. (in press)
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