几类三阶时滞泛函微分方程周期解问题的研究
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摘要
泛函微分方程主要描述的是带有时滞现象的数学模型.带有周期时滞的泛函微分方程在生物学、经济学、生态学和人口动力系统等实际问题中有着广泛的应用,因此,对带有周期时滞的泛函微分方程周期解存在性的研究就更具有现实意义.
本文主要讨论几类三阶时滞泛函微分方程周期解的存在唯一性问题,全文共分为五章.
第一章介绍泛函微分方程周期解的背景知识和有关的研究动态,并介绍本文的主要结果.
第二章利用重合度理论,研究一类具有偏差变元的三阶时滞泛函微分方程的T-周期解问题,获得了上述方程周期解存在唯一性的新结果.
第三章利用重合度理论,在第二章的基础上,研究一类三阶多时滞泛函微分方程的T-周期解问题,获得了保证其周期解存在唯一性的充分性条件.
第四章采用不同于第三章的研究方法,利用K-集压缩算子抽象连续性定理和一些新的分析技巧,研究一类三阶多时滞泛函微分方程的T-周期解问题,获得了上述方程T-周期解存在和唯一性的若干新结果.
第五章采用重合度理论中的延拓定理,运用一些新的分析方法,讨论了如下一类三阶带分布时滞的p-Laplacian方程(φp((x(t)-cx(t-σ))"))'+f1(x(t))x'(t)+f2(x'(t))x"(t)+g(t,x(t),x(t-τ-(t)),(?),x(t+s)dm(s))=e(t)的T-周期解问题,得到了上述方程存在T-周期解的若干新结果.
Functional differential equation is a mathematical model describing the phe-nomenon with time delays.The functional differential equations with periodic delays represent a natural framework for mathematical modeling of many real world phe-nomena such as biology, economy, ecology, the population dynamic system and so on.Therefore, the researches on existence and uniqueness of periodic solutions for functional differential equations with periodic delays have practical significance.
This paper mainly discusses the problems on existence and uniqueness of peri-odic solutions for some classes of three order functional differential equations with delays. The full text is divided into five chapters.
In chapter Ⅰ, the background knowledge of the subjects relevant to this dis-sertation and the study of dynamic are introduced.Then, the main results of the thesis are introduced.
In Chapter Ⅱ,using the theory of coincide degree,a type of third order func-tional differential equation with delays is considered.Some new results on the existence and uniqueness of periodic solutions are obtained.
In Chapter Ⅲ,using the theory of coincide degree,on the based of the second chapter,a type of third order functional differential equation with more delays is considered.Some sufficient conditions that guarantee the existence and uniqueness of periodic solution of the equation are obtained.
In Chapter IV,comparing the fourth chapter,a different research methods is used to study this equation.Specifically,using the continuation theory for K-set con-tractive operator,a type of third order functional differential equation with more delays is considored.The sufficient conditions for the existance of unique T-periodic solution are obtained.
In Chapter V,by using continuation theorem in coincidence degree theory, a type of third order p-Laplacian Equation with a deviating argument ((φp((x(t)-cx(t-σ))"))'+f1(x(t))x'(t)+f2(x'(t))x"(t)+g(t, x(t), x(t-τ(t)),(?)x(t+s) dm(s))=e(t) is considered.The sufficient condition for the existance of T-periodic solution is obtained.
本文主要讨论几类三阶时滞泛函微分方程周期解的存在唯一性问题,全文共分为五章.
第一章介绍泛函微分方程周期解的背景知识和有关的研究动态,并介绍本文的主要结果.
第二章利用重合度理论,研究一类具有偏差变元的三阶时滞泛函微分方程的T-周期解问题,获得了上述方程周期解存在唯一性的新结果.
第三章利用重合度理论,在第二章的基础上,研究一类三阶多时滞泛函微分方程的T-周期解问题,获得了保证其周期解存在唯一性的充分性条件.
第四章采用不同于第三章的研究方法,利用K-集压缩算子抽象连续性定理和一些新的分析技巧,研究一类三阶多时滞泛函微分方程的T-周期解问题,获得了上述方程T-周期解存在和唯一性的若干新结果.
第五章采用重合度理论中的延拓定理,运用一些新的分析方法,讨论了如下一类三阶带分布时滞的p-Laplacian方程(φp((x(t)-cx(t-σ))"))'+f1(x(t))x'(t)+f2(x'(t))x"(t)+g(t,x(t),x(t-τ-(t)),(?),x(t+s)dm(s))=e(t)的T-周期解问题,得到了上述方程存在T-周期解的若干新结果.
Functional differential equation is a mathematical model describing the phe-nomenon with time delays.The functional differential equations with periodic delays represent a natural framework for mathematical modeling of many real world phe-nomena such as biology, economy, ecology, the population dynamic system and so on.Therefore, the researches on existence and uniqueness of periodic solutions for functional differential equations with periodic delays have practical significance.
This paper mainly discusses the problems on existence and uniqueness of peri-odic solutions for some classes of three order functional differential equations with delays. The full text is divided into five chapters.
In chapter Ⅰ, the background knowledge of the subjects relevant to this dis-sertation and the study of dynamic are introduced.Then, the main results of the thesis are introduced.
In Chapter Ⅱ,using the theory of coincide degree,a type of third order func-tional differential equation with delays is considered.Some new results on the existence and uniqueness of periodic solutions are obtained.
In Chapter Ⅲ,using the theory of coincide degree,on the based of the second chapter,a type of third order functional differential equation with more delays is considered.Some sufficient conditions that guarantee the existence and uniqueness of periodic solution of the equation are obtained.
In Chapter IV,comparing the fourth chapter,a different research methods is used to study this equation.Specifically,using the continuation theory for K-set con-tractive operator,a type of third order functional differential equation with more delays is considored.The sufficient conditions for the existance of unique T-periodic solution are obtained.
In Chapter V,by using continuation theorem in coincidence degree theory, a type of third order p-Laplacian Equation with a deviating argument ((φp((x(t)-cx(t-σ))"))'+f1(x(t))x'(t)+f2(x'(t))x"(t)+g(t, x(t), x(t-τ(t)),(?)x(t+s) dm(s))=e(t) is considered.The sufficient condition for the existance of T-periodic solution is obtained.
引文
[1]李永祥.二阶非线性常微分方程的正周期解[J].数学学报,2002,45(3)：481-488.
[2]弥鲁芳,武延树.一类具时滞耗散型Duffing方程的周期解[J].数学的实践与认识,2008,38(19)：235-238.
[3]黄记洲.一类时滞Duffing方程周期解存在性的充分性定理[J].甘肃联合大学学报：自然科学版,2006,20(06)：29-31.
[4]汪娜,章家顺等.一类三阶具偏差变元微分方程的周期解[J].大学数学,2006,22(06)：41-47.
[5]Gaines R,Mawhin J.Coincide Degree and Nonlinear Differential Equation[M].Berlin: Springer-Verlag.1977.
[6]王根强,燕居让.二阶非线性中立型泛函微分方程周期解的存在性[J].数学学报,2004,47(2)；379-384.
[7]唐美兰,刘新歌等.一类时滞Duffing型方程周期解的存在性[J].上饶师范学院学报,2004,24(6)：12-15.
[8]汪娜,鲁世平.一类三阶具时滞微分方程的周期解[J].安徽师范大学学报,2006,29(1)：17-22.
[9]章家顺,沈佐民等.一类三阶泛函微分方程周期解的存在性[J].池州师专学报,2006,20(05)：4-6.
[10]汪一高,鲁世平.一类三阶微分方程周期解的存在唯一性[J].数学研究,2009,42(1)：58-62.
[11]LU Shi-ping,GE Wei-gao.Sufficient conditions for the existence of periodic solution-s to some second order different equations with a deviating argernent[J].Math Anal Appl,2005,308(2):393-419
[12]张莉,王全义.具有偏差变元的二阶泛函微分方程周期解的存在性[J].数学物理学报,2009,29A(2)：263-261.
[13]鲁世平,葛渭高,郑祖庥.具复杂变元的Rayleigh方程周期解的问题[J].数学学报,2004,47(2)：299-304.
[14]朱艳玲,鲁世平.一类二阶中立型泛函微分方程周期解的存在性[J].安徽师范大学学报：自然科学版,2006,29(1)：12-16.6(1996),181-186.
[15]刘锡平,贾梅,任睿.时滞Duffing型方程周期解的存在唯一性[J].数学物理学报,2007,27A(1)：37-44.
[16]Liu B W,Huang L H.Existence and uniqueness of periodic solutions for a kind of first order neutral functional differential equations[J].J Math Anal Appl,2006,327(03):121-132.
[17]Wang Z X,Qian L X,Lu S P.The existence and uniqueness of periodic solution-s for a kind of Duffing-type euqation with two deviating arguments [J].Nonlinear Analysis,2010,73(06):3034-3043.
[18]郭大钧.非线性泛函分析[M].济南：山东科学技术出版社.2001
[19]Petryshyn W V,Yu Z S.Existence theorems for higher order nonlinear periodic bound-ary value problems[J]. Nonlinear Anal,1982,9:943-969.
[20]Chenug.W.S.and Ren J.L. Periodic solutions for p-Laplacian Lienard equation with a deviating argument[J].Nonlinear Analysis,2004,59:107-120.
[21]Chenug.W.S.and Ren J.L. On the existence of Periodic solutions for p-Laplacian gen-eralized Lienard equation[J]. Nonlinear Analysis,2005,60:65-75.
[22]倪晋波,张建军.具多偏差变元的p-Laplacian方程周期解边值问题的可解性[J].南京大学学报半年刊,2009,26(1)：56-64.
[23]Zhu Y.L and Lu S.P,Periodic solutions for p-Laplacian neutral functional differential Equation with deviating arguments[J].J.Math.Anal.Appl,2007,325:377-385.
[24]彭世国.具有偏差变元的P-Laplacian中立型Lienard方程的周期解[J].数学年刊,2008,29A(05)：617-626.
[25]朱艳玲,汪凯.具有p-Laplacian中立型微分方程的周期解[J].系统科学与数学,2009,29(06)：808-817.
[26]曹凤娟,韩振来.具偏差变元P-Laplace微分方程周期解的存在性[J].济南大学学报(自然科学版),2010,24(1)：95-98.
[27]Gaines R,Mawhin J.Coincide Degree and Nonlinear Differential Equation[M].Berlin: Springer-Verlag.1977.
[28]Lu Shiping,Zheng Zuxiu.Periodic solutions to neutral differential Equation with devi-ating arguments[J].Appl,Math,comput,2004,15(2):17-27.
[29]Cheng Jun,Lu Shiping.Periodic solutions for a third order p-Laplacian equation[J]. Journal of Anhui Normal university (Natural Science),2009,34(4):311-315.
[30]I.Podlumbny. Fractional Differential Equations[M]. Academic Press,San Diego,New Y-ork,London,1999.
[31]S.Samko,A.Kilbas,O.Marichev. Fractional Integrals and Derivatives:Theory and Appli-cations[M]. Gordon and Breach Science Publisher,Yverdon,1993.
[32]Salem H.A.H. On the existence of continuous solutions for a singular sys-tem of non-linear fractional differential equations[J]. Applied Mathematics and Computation,2008,198(1):445-452.
[33]M.A.Krasnoselskii. Positive Solutions of Operator Equations[M]. Noordhoff, Gronin-gen,1984.
[34]Zhu Y.L and Lu S.P,Periodic solutions for p-Laplacian neutral functional differential Equation with deviating arguments[J].J.Math.Anal.Appl,2007,325:377-385.
[2]弥鲁芳,武延树.一类具时滞耗散型Duffing方程的周期解[J].数学的实践与认识,2008,38(19)：235-238.
[3]黄记洲.一类时滞Duffing方程周期解存在性的充分性定理[J].甘肃联合大学学报：自然科学版,2006,20(06)：29-31.
[4]汪娜,章家顺等.一类三阶具偏差变元微分方程的周期解[J].大学数学,2006,22(06)：41-47.
[5]Gaines R,Mawhin J.Coincide Degree and Nonlinear Differential Equation[M].Berlin: Springer-Verlag.1977.
[6]王根强,燕居让.二阶非线性中立型泛函微分方程周期解的存在性[J].数学学报,2004,47(2)；379-384.
[7]唐美兰,刘新歌等.一类时滞Duffing型方程周期解的存在性[J].上饶师范学院学报,2004,24(6)：12-15.
[8]汪娜,鲁世平.一类三阶具时滞微分方程的周期解[J].安徽师范大学学报,2006,29(1)：17-22.
[9]章家顺,沈佐民等.一类三阶泛函微分方程周期解的存在性[J].池州师专学报,2006,20(05)：4-6.
[10]汪一高,鲁世平.一类三阶微分方程周期解的存在唯一性[J].数学研究,2009,42(1)：58-62.
[11]LU Shi-ping,GE Wei-gao.Sufficient conditions for the existence of periodic solution-s to some second order different equations with a deviating argernent[J].Math Anal Appl,2005,308(2):393-419
[12]张莉,王全义.具有偏差变元的二阶泛函微分方程周期解的存在性[J].数学物理学报,2009,29A(2)：263-261.
[13]鲁世平,葛渭高,郑祖庥.具复杂变元的Rayleigh方程周期解的问题[J].数学学报,2004,47(2)：299-304.
[14]朱艳玲,鲁世平.一类二阶中立型泛函微分方程周期解的存在性[J].安徽师范大学学报：自然科学版,2006,29(1)：12-16.6(1996),181-186.
[15]刘锡平,贾梅,任睿.时滞Duffing型方程周期解的存在唯一性[J].数学物理学报,2007,27A(1)：37-44.
[16]Liu B W,Huang L H.Existence and uniqueness of periodic solutions for a kind of first order neutral functional differential equations[J].J Math Anal Appl,2006,327(03):121-132.
[17]Wang Z X,Qian L X,Lu S P.The existence and uniqueness of periodic solution-s for a kind of Duffing-type euqation with two deviating arguments [J].Nonlinear Analysis,2010,73(06):3034-3043.
[18]郭大钧.非线性泛函分析[M].济南：山东科学技术出版社.2001
[19]Petryshyn W V,Yu Z S.Existence theorems for higher order nonlinear periodic bound-ary value problems[J]. Nonlinear Anal,1982,9:943-969.
[20]Chenug.W.S.and Ren J.L. Periodic solutions for p-Laplacian Lienard equation with a deviating argument[J].Nonlinear Analysis,2004,59:107-120.
[21]Chenug.W.S.and Ren J.L. On the existence of Periodic solutions for p-Laplacian gen-eralized Lienard equation[J]. Nonlinear Analysis,2005,60:65-75.
[22]倪晋波,张建军.具多偏差变元的p-Laplacian方程周期解边值问题的可解性[J].南京大学学报半年刊,2009,26(1)：56-64.
[23]Zhu Y.L and Lu S.P,Periodic solutions for p-Laplacian neutral functional differential Equation with deviating arguments[J].J.Math.Anal.Appl,2007,325:377-385.
[24]彭世国.具有偏差变元的P-Laplacian中立型Lienard方程的周期解[J].数学年刊,2008,29A(05)：617-626.
[25]朱艳玲,汪凯.具有p-Laplacian中立型微分方程的周期解[J].系统科学与数学,2009,29(06)：808-817.
[26]曹凤娟,韩振来.具偏差变元P-Laplace微分方程周期解的存在性[J].济南大学学报(自然科学版),2010,24(1)：95-98.
[27]Gaines R,Mawhin J.Coincide Degree and Nonlinear Differential Equation[M].Berlin: Springer-Verlag.1977.
[28]Lu Shiping,Zheng Zuxiu.Periodic solutions to neutral differential Equation with devi-ating arguments[J].Appl,Math,comput,2004,15(2):17-27.
[29]Cheng Jun,Lu Shiping.Periodic solutions for a third order p-Laplacian equation[J]. Journal of Anhui Normal university (Natural Science),2009,34(4):311-315.
[30]I.Podlumbny. Fractional Differential Equations[M]. Academic Press,San Diego,New Y-ork,London,1999.
[31]S.Samko,A.Kilbas,O.Marichev. Fractional Integrals and Derivatives:Theory and Appli-cations[M]. Gordon and Breach Science Publisher,Yverdon,1993.
[32]Salem H.A.H. On the existence of continuous solutions for a singular sys-tem of non-linear fractional differential equations[J]. Applied Mathematics and Computation,2008,198(1):445-452.
[33]M.A.Krasnoselskii. Positive Solutions of Operator Equations[M]. Noordhoff, Gronin-gen,1984.
[34]Zhu Y.L and Lu S.P,Periodic solutions for p-Laplacian neutral functional differential Equation with deviating arguments[J].J.Math.Anal.Appl,2007,325:377-385.