几类泛函微分方程的周期解
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摘要
本论文主要讨论了一类无限时滞中立型Volterra型积分微分方程周期解的存在性与唯一性、一类中立型Duffing型微分方程的周期解、一类泛函微分方程正周期解的存在性与多解性.全文共分为四章.
第一章简述了泛函微分方程的周期解存在性、唯一性的历史与研究现状,以及本文的主要工作.
第二章讨论了一类具有无限时滞中立型积分微分系统的周期解.利用线性系统的指数型二分性理论,Schauder不动点定理,得到了这一类方程周期解的存在性与唯一性理论.
第三章讨论了一类中立型Duffing型微分方程aχ"(t)+cχ'(t-τ)+bχ(t)+g(χ(t-τ)))=p(t)的周期解,利用迭合度方法得到了这一类方程周期解存在的充分条件.
第四章讨论了一类脉冲泛函微分方程χ'(t)=A(t,χ(t))χ(t)-λf(t-τ(t)),t≠τk,k∈Nχ(τk+)=χ(τk)+Ek(χ(τk)),t=τk的正周期解的存在性与多解性.利用Krasnoselskii不动点定理,获得了判断这一类方程正周期解的存在性与多解性的一些结论.
This thesis is composed of four chapters, which mainly studied the ex-istence and uniqueness of periodic solutions for one kind of neutral Volterra integro-differential equations with infinite delay、periodic solution for one kind of neutral Duffing equations and existence and multiplicity of positive periodic solutions for one kind of functional differential equations with impulses.
As the introductions, in Chapter 1, the background and history of the existence and uniqueness or multiplicity of Periodicity solutions problems, and the main work of this paper are given.
In Chapter 2, we studied the existence, uniqueness and stability of neutral Integro-differential equations By using the theory of exponential dichotomies of linear system and Schauder fixed point theorem, we obtain the existence and uniqueness theorem of peri-odic solutions for this kind of functional differential equations.
In Chapter 3, we study a class of neutral Duffing equation, which have periodicity solutions ax"(t)+cx'(t-τ)+bx(t)+g(x(t-τ)))= p(t). By using the theory of coincidence degree, we obtain a sufficient theorem is obtained for the existence of a periodic solution of this kind of equations.
In the last Chapter, we study the existence of multiple positive periodic solutions for functional differential equations with impulsesχ(t)= A(t,x(t))x(t)-λf(t-τ(t)),t≠Tk,k∈N x(τk+)=χ(Tk)+Ek(x(Tk)), t= Tk By using Krasnoselskii fixed point theorem, we obtain the existence of multiple positive periodic solutions for this kind of functional differential equations with impulses. Our results improve some known results or are new.
第一章简述了泛函微分方程的周期解存在性、唯一性的历史与研究现状,以及本文的主要工作.
第二章讨论了一类具有无限时滞中立型积分微分系统的周期解.利用线性系统的指数型二分性理论,Schauder不动点定理,得到了这一类方程周期解的存在性与唯一性理论.
第三章讨论了一类中立型Duffing型微分方程aχ"(t)+cχ'(t-τ)+bχ(t)+g(χ(t-τ)))=p(t)的周期解,利用迭合度方法得到了这一类方程周期解存在的充分条件.
第四章讨论了一类脉冲泛函微分方程χ'(t)=A(t,χ(t))χ(t)-λf(t-τ(t)),t≠τk,k∈Nχ(τk+)=χ(τk)+Ek(χ(τk)),t=τk的正周期解的存在性与多解性.利用Krasnoselskii不动点定理,获得了判断这一类方程正周期解的存在性与多解性的一些结论.
This thesis is composed of four chapters, which mainly studied the ex-istence and uniqueness of periodic solutions for one kind of neutral Volterra integro-differential equations with infinite delay、periodic solution for one kind of neutral Duffing equations and existence and multiplicity of positive periodic solutions for one kind of functional differential equations with impulses.
As the introductions, in Chapter 1, the background and history of the existence and uniqueness or multiplicity of Periodicity solutions problems, and the main work of this paper are given.
In Chapter 2, we studied the existence, uniqueness and stability of neutral Integro-differential equations By using the theory of exponential dichotomies of linear system and Schauder fixed point theorem, we obtain the existence and uniqueness theorem of peri-odic solutions for this kind of functional differential equations.
In Chapter 3, we study a class of neutral Duffing equation, which have periodicity solutions ax"(t)+cx'(t-τ)+bx(t)+g(x(t-τ)))= p(t). By using the theory of coincidence degree, we obtain a sufficient theorem is obtained for the existence of a periodic solution of this kind of equations.
In the last Chapter, we study the existence of multiple positive periodic solutions for functional differential equations with impulsesχ(t)= A(t,x(t))x(t)-λf(t-τ(t)),t≠Tk,k∈N x(τk+)=χ(Tk)+Ek(x(Tk)), t= Tk By using Krasnoselskii fixed point theorem, we obtain the existence of multiple positive periodic solutions for this kind of functional differential equations with impulses. Our results improve some known results or are new.
引文
[1]黄启昌,具有无限时滞的泛函微分方程的周期解的存在性[J].中国科学,1984(14A)：882-889.
[2]王克、黄启昌,Ch空间与具有无限时滞的泛函微分方程的有界性及周期解[J].中国科学,1987(17A)：242-252.
[3]王全义,具有无限时滞的积分微分方程的存在性、唯一性及稳定性[J].应用数学学报,1998(21)：312-318.
[4]石磊,具有无限时滞的泛函微分方程的有界性及周期解[J]科学通报.1990(35)：4.9-411.
[5]范猛,王克,具有无限时滞线性中立型泛函微分方程周期解[J].数学学报.2000(43)：695-701.
[6]Massera J L, The existence of periodic soluitons of systems of differential equa-tions[J].Duke Math J,1950(17):457-475.
[7]Chow S N,Remarks on one dimensional delay-differential equations[J].J Math Appl,1973(41):426-429.
[8]Makay M., periodic solutions of linear differential equations.[J],Diff Int Eqs,1995(8):2177-2187.
[9]Chen Fengde,Sun Dexian., periodic solution of scalar neutral volterra integro-differential equations with infinite delay[J]. Ann Diff Eqs,2003(19):250-255.
[10]彭世国,朱思铭,具有无限时滞泛函微分方程的周期解[J].数学年刊,2002(23)：371-380.
[11]江娇,许建华,韩茂安,具有无穷时滞中立型积分微分方程的周期解[J].数学物理学报,2008(28A)：897-905.
[12]杨喜陶,具有无限时滞的中立型泛函微分方程周期解的存在性、唯一性及稳定性[J].应用数学学报,1998(21)：312-318.
[13]陈凤德,具有无限时滞的非线性积分微分方程的周期解[J].应用数学学报,2003(26)：141-148.
[14]郑祖庥,泛函微分方程理论[M].安徽：安徽教育出版社,1994：1-100.
[15]郭大钧,孙经先,刘兆理,非线性常微分方程泛函方法[M].济南：山东科学技术出版社,2006：190-290.
[16]秦元勋,刘永清,王联,带有时滞的动力系统的运动稳定性[M].北京：科学出版社。1989：14-21.
[17]张正球,庾建设,一类时滞Duffing型方程的周期解[J],高校应用数学学报,1998(13)：389-3925.
[18]唐美兰,刘心歌,刘心笔,一类时滞Duffing型方程的周期解的存在性[J],上饶师范学院学报,2004(24)：12-15.
[19]刘锡平,贾梅,时滞Duffing型方程周期解的存在唯一性[J],数学物理学报,2007(27A)：37-44.
[20]Chen.F.D., On the the periodic solutions of periodic multi-species Kolmogorov type competitive system with delays and feedback control[J].Appl.Math.Com-put,2006(180):366-373.
[21]Fan.M,K.Wang,P.J.,Wong.Y.R.P.Agarwal.Periodicity and stability in periodic n-species Lotka-Volterra competition system with feedback controls and devi-ating arguments [J]. ActaMath.Sin,2003(19):801-822.
[22]Huo.H.F,Li W.T., positive periodic solutions of a class of delay differential system with feedback control[J]. Appl.Math. Comput,2004(148):35-46.
[23]Li.Y.K, Zhu.L.F., Positive periodic solutions for a class of higher-dimensional state-dependent delay functional differential equations with feedback control(J]. Appl. Math. Comput,2004 (159):783-795.
[24]Li. X.Y,Lin X.N.,Jiang.D.Q,Zhang.X.Y.Existence and multiplicity of posi-tive periodic solutions to functional differential equations with impulse ef-fects[J].Nonlinear Anal,2005 (62):683-01.
[25]Wan.A, Jiang D.,Xu X., A new existence theory for positive periodic solutions to functional differential equations[J]. Comput. Math..Appl,2004 (47):1257-262.
[26]Ye.D, Fan M,Wang.H.Y., Periodic solutions for scalar functional differential equations[J].Nonlinear Anal,2005 (62) 1157-181.
[27]Zeng.Z.J. Existence of positive periodic solutions of a class of nonautonomous difference equations[J].Electron. J. Differential Equations,2006 (3):1-8.
[28]Zeng.Z.J,Bi L.,Fan M., Existence of multiple positive periodic solutions for functional differential equations[J]. J.Math.Anal.Appl,2007(325):1378-389.
[29]Zhang.N, Dai B.X.,Qian X.Z. Periodic solutions for a class of higher-dimension functional differential equations with impulses[J].Nonlinear Anal,2008 (68): 629-638.
[30]Nieto.J.J., Impulsive resonance periodic problems of first order[J].Appl. Math. Lett,2002 (15):489-493.
[31]Wang.H.Y., Postive periodic solutions of functional differential equations[J]. J. Differential Equations,2004(202):354-366
[32]Jiang. D.Q,O'Regan D.,Agarwal R.P.,Xu X.J.,On the number of postive periodic solutions of funcitonal differential equations and population mod-els[J].Math.Models Methods Appl.Sci.,2005(15):555-573.
[33]Zeng.Z.J,Li B.,Meng F., Existence of multiple positive periodic solutions for functional differential equations[J].J Math Anal Appl,2007(325):1378-1389.
[34]Zeng Z.J., Existence and multiplicity of positive periodic solutions for a class of higher-dimension functional differential equations with impulses[J].Comp Math Appl,2009(58):1911-1920.
[35]]Guo D., Positive solutions of nonlinear operator equations and its applications to nonlinear integral equations[J]..Adu Math,1984(13):294-310[in Chinese].
[36]Xing Y.,Han M.,Zheng G., Initial value problem for rst-orderintegro-dierential equation of Volterratype on time scales[J].Nonlinear.Anal,2005(60):429-442.
[37]Luo J., Yu J., Global asymptotic stability of nonautonomous mathematical ecological equations with distributed deviating arguments[J] Acta Mathematica Sinica,1998(41):1273-1282.
[38]Weng. P,Liang I. V., The existence and behavior of periodic solution of Hematopoiesis model[J], Mathematiea Applicate,1995 (4):434-439.
[39]Wan A.,Jiang D., Existence of positive periodic solutions for functional differ-ential equations[J], Journal of Mathematics,2002 (56):193-202.
[40]Jiang D., Wei J.J., Existence of positive periodic solutions for Volterra integro-differential equations[J],Aeta Mathematica Scientia,2002(21B):553-560.
[41]Zou Y.,Hu Q.,Zhang R.,On numerical studies o fmulti-point boundary value problem and its fold bifurcation[J],Appl.Math.Comput,2007(185):527-537.
[42]Gupta.C.P.,Tromchuk S.I.,A sharper condition for the solvability of a three point second order boundary value problem[J], J.Math. Anal.Appl,1997(205):586-597.
[43]Xian Xu,Positive solutions for singularm-point boundary value problems with positive parameter[J], J.Math.Anal.Appl,2004(291):352-367.
[44]Mei qiang Feng,Wei gao Ge, Positive solutions for a class of m-point singular boundar yvalue problems[J],Math.Comput.Modelling,2007(46):375-383.
[45]Zhao YuLin, Existence of multiple positive solutions for m-point boundary value problems in Banach spaces[J],J.Comput.Appl.Math,2008(215):79-90.
[46]Zhong li Wei,Chang ci Pang, Positive solutions of some singular m-point bound-ary value problems at non-resonance[J],Appl.Math.Comput,2005(171):433-449.
[47]Deimling K., Nonlinear Functional Analysis[M],Springer-Verlag,NewYork,1985.
[48]Jiang.D.Q,D.O Regan,R.P.Agarwal,X.J.Xu, On the number of positive pe-riodic solutions of functional differential equations and population mod-els[J],Math.Models Methods Appl.Sci,2005(15):555-573.
[49]丁同仁,常微分方程定性方法的应用[M]北京,高等教育出版社,2004：180-185.
[50]Lan K.,Jeffry K.,J.R.L.Webb,Positivesolutions of semilinear differential equa-tions with singula rities,J.Differential uations,1998(148):407-421.
[51]Sawano K. Exponential asymptotie stability for funetional differential equations with infinite retardations.Tohoku Math J,1979(31):363-382.
[2]王克、黄启昌,Ch空间与具有无限时滞的泛函微分方程的有界性及周期解[J].中国科学,1987(17A)：242-252.
[3]王全义,具有无限时滞的积分微分方程的存在性、唯一性及稳定性[J].应用数学学报,1998(21)：312-318.
[4]石磊,具有无限时滞的泛函微分方程的有界性及周期解[J]科学通报.1990(35)：4.9-411.
[5]范猛,王克,具有无限时滞线性中立型泛函微分方程周期解[J].数学学报.2000(43)：695-701.
[6]Massera J L, The existence of periodic soluitons of systems of differential equa-tions[J].Duke Math J,1950(17):457-475.
[7]Chow S N,Remarks on one dimensional delay-differential equations[J].J Math Appl,1973(41):426-429.
[8]Makay M., periodic solutions of linear differential equations.[J],Diff Int Eqs,1995(8):2177-2187.
[9]Chen Fengde,Sun Dexian., periodic solution of scalar neutral volterra integro-differential equations with infinite delay[J]. Ann Diff Eqs,2003(19):250-255.
[10]彭世国,朱思铭,具有无限时滞泛函微分方程的周期解[J].数学年刊,2002(23)：371-380.
[11]江娇,许建华,韩茂安,具有无穷时滞中立型积分微分方程的周期解[J].数学物理学报,2008(28A)：897-905.
[12]杨喜陶,具有无限时滞的中立型泛函微分方程周期解的存在性、唯一性及稳定性[J].应用数学学报,1998(21)：312-318.
[13]陈凤德,具有无限时滞的非线性积分微分方程的周期解[J].应用数学学报,2003(26)：141-148.
[14]郑祖庥,泛函微分方程理论[M].安徽：安徽教育出版社,1994：1-100.
[15]郭大钧,孙经先,刘兆理,非线性常微分方程泛函方法[M].济南：山东科学技术出版社,2006：190-290.
[16]秦元勋,刘永清,王联,带有时滞的动力系统的运动稳定性[M].北京：科学出版社。1989：14-21.
[17]张正球,庾建设,一类时滞Duffing型方程的周期解[J],高校应用数学学报,1998(13)：389-3925.
[18]唐美兰,刘心歌,刘心笔,一类时滞Duffing型方程的周期解的存在性[J],上饶师范学院学报,2004(24)：12-15.
[19]刘锡平,贾梅,时滞Duffing型方程周期解的存在唯一性[J],数学物理学报,2007(27A)：37-44.
[20]Chen.F.D., On the the periodic solutions of periodic multi-species Kolmogorov type competitive system with delays and feedback control[J].Appl.Math.Com-put,2006(180):366-373.
[21]Fan.M,K.Wang,P.J.,Wong.Y.R.P.Agarwal.Periodicity and stability in periodic n-species Lotka-Volterra competition system with feedback controls and devi-ating arguments [J]. ActaMath.Sin,2003(19):801-822.
[22]Huo.H.F,Li W.T., positive periodic solutions of a class of delay differential system with feedback control[J]. Appl.Math. Comput,2004(148):35-46.
[23]Li.Y.K, Zhu.L.F., Positive periodic solutions for a class of higher-dimensional state-dependent delay functional differential equations with feedback control(J]. Appl. Math. Comput,2004 (159):783-795.
[24]Li. X.Y,Lin X.N.,Jiang.D.Q,Zhang.X.Y.Existence and multiplicity of posi-tive periodic solutions to functional differential equations with impulse ef-fects[J].Nonlinear Anal,2005 (62):683-01.
[25]Wan.A, Jiang D.,Xu X., A new existence theory for positive periodic solutions to functional differential equations[J]. Comput. Math..Appl,2004 (47):1257-262.
[26]Ye.D, Fan M,Wang.H.Y., Periodic solutions for scalar functional differential equations[J].Nonlinear Anal,2005 (62) 1157-181.
[27]Zeng.Z.J. Existence of positive periodic solutions of a class of nonautonomous difference equations[J].Electron. J. Differential Equations,2006 (3):1-8.
[28]Zeng.Z.J,Bi L.,Fan M., Existence of multiple positive periodic solutions for functional differential equations[J]. J.Math.Anal.Appl,2007(325):1378-389.
[29]Zhang.N, Dai B.X.,Qian X.Z. Periodic solutions for a class of higher-dimension functional differential equations with impulses[J].Nonlinear Anal,2008 (68): 629-638.
[30]Nieto.J.J., Impulsive resonance periodic problems of first order[J].Appl. Math. Lett,2002 (15):489-493.
[31]Wang.H.Y., Postive periodic solutions of functional differential equations[J]. J. Differential Equations,2004(202):354-366
[32]Jiang. D.Q,O'Regan D.,Agarwal R.P.,Xu X.J.,On the number of postive periodic solutions of funcitonal differential equations and population mod-els[J].Math.Models Methods Appl.Sci.,2005(15):555-573.
[33]Zeng.Z.J,Li B.,Meng F., Existence of multiple positive periodic solutions for functional differential equations[J].J Math Anal Appl,2007(325):1378-1389.
[34]Zeng Z.J., Existence and multiplicity of positive periodic solutions for a class of higher-dimension functional differential equations with impulses[J].Comp Math Appl,2009(58):1911-1920.
[35]]Guo D., Positive solutions of nonlinear operator equations and its applications to nonlinear integral equations[J]..Adu Math,1984(13):294-310[in Chinese].
[36]Xing Y.,Han M.,Zheng G., Initial value problem for rst-orderintegro-dierential equation of Volterratype on time scales[J].Nonlinear.Anal,2005(60):429-442.
[37]Luo J., Yu J., Global asymptotic stability of nonautonomous mathematical ecological equations with distributed deviating arguments[J] Acta Mathematica Sinica,1998(41):1273-1282.
[38]Weng. P,Liang I. V., The existence and behavior of periodic solution of Hematopoiesis model[J], Mathematiea Applicate,1995 (4):434-439.
[39]Wan A.,Jiang D., Existence of positive periodic solutions for functional differ-ential equations[J], Journal of Mathematics,2002 (56):193-202.
[40]Jiang D., Wei J.J., Existence of positive periodic solutions for Volterra integro-differential equations[J],Aeta Mathematica Scientia,2002(21B):553-560.
[41]Zou Y.,Hu Q.,Zhang R.,On numerical studies o fmulti-point boundary value problem and its fold bifurcation[J],Appl.Math.Comput,2007(185):527-537.
[42]Gupta.C.P.,Tromchuk S.I.,A sharper condition for the solvability of a three point second order boundary value problem[J], J.Math. Anal.Appl,1997(205):586-597.
[43]Xian Xu,Positive solutions for singularm-point boundary value problems with positive parameter[J], J.Math.Anal.Appl,2004(291):352-367.
[44]Mei qiang Feng,Wei gao Ge, Positive solutions for a class of m-point singular boundar yvalue problems[J],Math.Comput.Modelling,2007(46):375-383.
[45]Zhao YuLin, Existence of multiple positive solutions for m-point boundary value problems in Banach spaces[J],J.Comput.Appl.Math,2008(215):79-90.
[46]Zhong li Wei,Chang ci Pang, Positive solutions of some singular m-point bound-ary value problems at non-resonance[J],Appl.Math.Comput,2005(171):433-449.
[47]Deimling K., Nonlinear Functional Analysis[M],Springer-Verlag,NewYork,1985.
[48]Jiang.D.Q,D.O Regan,R.P.Agarwal,X.J.Xu, On the number of positive pe-riodic solutions of functional differential equations and population mod-els[J],Math.Models Methods Appl.Sci,2005(15):555-573.
[49]丁同仁,常微分方程定性方法的应用[M]北京,高等教育出版社,2004：180-185.
[50]Lan K.,Jeffry K.,J.R.L.Webb,Positivesolutions of semilinear differential equa-tions with singula rities,J.Differential uations,1998(148):407-421.
[51]Sawano K. Exponential asymptotie stability for funetional differential equations with infinite retardations.Tohoku Math J,1979(31):363-382.