几类高阶非线性差分方程的定性研究
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摘要
本篇硕士论文主要针对G．Ladas等人提出的一些公开问题与猜想，考察几类高阶时滞差分方程的正解的性质，如有界持久性、全局吸引性、全局渐近稳定性、一致持久存在性等。这些工作或是部分回答了G．Ladas等人提出的公开问题，或是对公开问题的更一般的情形进行探讨，使得对这些公开问题的研究工作向前推进。
全文共分六章。
第一章，我们给出本文所要研究的问题的理论与实际背景及目前国内外在该问题上的研究进展，并给出本文所使用的一些重要概念、记号。
第二章，我们讨论一类高阶非线性时滞差分方程的有界持久性，利用已有文献的相关结论及分析的技巧，给出了这类方程的正解有界持久的新的充分条件，对更一般的情形，改进了已有的相关结果。
第三、四章，我们考虑两类非线性差分模型的正解的全局吸引性，利用极限理论，得到了方程的正解全局吸引的充分条件。作为应用，部分回答了G．Ladas等人提出的一个公开问题。
第五章，考察一类非线性时滞差分方程的正解的全局渐近稳定性，利用极限理论、迭代理论、级数理论和已有文献的相关结果，得出了该方程的正解全局渐近稳定的充分条件。并将其应用于有理递归序列，得出相应的结果，从而对一公开问题的更一般的情形做出了部分回答。
最后，我们考察一类具多时滞的离散Lotka-Volterra互惠模型。利用迭代理论、M—矩阵理论、单调理论，得到了该差分系统一致持久存在的充分条件和其解全局吸引的充分条件。
This thesis directs towards some open problems and conjectures by G.Ladas etal. We consider some properties of positive solutions of several kinds of higher order delay nonlinear difference equations, for example, boundedness and persistence, global attractivity, global asymptotic stability, permanence etc. Some known results are improved, and some conjectures by GXadas are proved partially here. We also investigate the more general situation of the open problems by G.Ladas, and push forward the study of the open problems and conjectures.
The whole thesis is composed of six chapters.
In the first chapter, we introduce the theoretical and practical background of problems and their researching situations. We also list some important notions and signs.
In the second chapter, we consider the boundedness and persistence of a class of higher order delay nonlinear difference equation. Ultimating some known results and skills of analyzing, we obtain a sufficient condition for boundedness and persistence of its positive solutions, and improve some known results.
In the third and fourth chapters, we study the attractivity of two kinds of difference equations. The sufficient conditions for attractivity are obtained by application of limit theory and monotone theory. In the application, we answer an open problem by GXadas.
In the fifth chapter, we investigate a class of higher order delay nonlinear difference equation. By limit theory ,iterate theory, progression theory and some known results, we obtain some sufficient conditions for global attractivity and global asymptotic stability. We also apply our results to the rational recursive sequence, answer partially the more general situation of an open problem by G.Ladas.
In the last chapter, we consider a class of discrete-time Lotka-Volterra facultative mutualism system with different delays. Some sufficient conditions for permanence and global attractivity of its positive solutions are obtained.
全文共分六章。
第一章，我们给出本文所要研究的问题的理论与实际背景及目前国内外在该问题上的研究进展，并给出本文所使用的一些重要概念、记号。
第二章，我们讨论一类高阶非线性时滞差分方程的有界持久性，利用已有文献的相关结论及分析的技巧，给出了这类方程的正解有界持久的新的充分条件，对更一般的情形，改进了已有的相关结果。
第三、四章，我们考虑两类非线性差分模型的正解的全局吸引性，利用极限理论，得到了方程的正解全局吸引的充分条件。作为应用，部分回答了G．Ladas等人提出的一个公开问题。
第五章，考察一类非线性时滞差分方程的正解的全局渐近稳定性，利用极限理论、迭代理论、级数理论和已有文献的相关结果，得出了该方程的正解全局渐近稳定的充分条件。并将其应用于有理递归序列，得出相应的结果，从而对一公开问题的更一般的情形做出了部分回答。
最后，我们考察一类具多时滞的离散Lotka-Volterra互惠模型。利用迭代理论、M—矩阵理论、单调理论，得到了该差分系统一致持久存在的充分条件和其解全局吸引的充分条件。
This thesis directs towards some open problems and conjectures by G.Ladas etal. We consider some properties of positive solutions of several kinds of higher order delay nonlinear difference equations, for example, boundedness and persistence, global attractivity, global asymptotic stability, permanence etc. Some known results are improved, and some conjectures by GXadas are proved partially here. We also investigate the more general situation of the open problems by G.Ladas, and push forward the study of the open problems and conjectures.
The whole thesis is composed of six chapters.
In the first chapter, we introduce the theoretical and practical background of problems and their researching situations. We also list some important notions and signs.
In the second chapter, we consider the boundedness and persistence of a class of higher order delay nonlinear difference equation. Ultimating some known results and skills of analyzing, we obtain a sufficient condition for boundedness and persistence of its positive solutions, and improve some known results.
In the third and fourth chapters, we study the attractivity of two kinds of difference equations. The sufficient conditions for attractivity are obtained by application of limit theory and monotone theory. In the application, we answer an open problem by GXadas.
In the fifth chapter, we investigate a class of higher order delay nonlinear difference equation. By limit theory ,iterate theory, progression theory and some known results, we obtain some sufficient conditions for global attractivity and global asymptotic stability. We also apply our results to the rational recursive sequence, answer partially the more general situation of an open problem by G.Ladas.
In the last chapter, we consider a class of discrete-time Lotka-Volterra facultative mutualism system with different delays. Some sufficient conditions for permanence and global attractivity of its positive solutions are obtained.
引文
[1] Ladas G. Open problems and conjectures. J.Diff.Eq.App., 1995,3:317-321
[2] Kocic V L, Ladas G, Rodrignes I W. On rational recursive sequences. J Math Anal,1993,173: 127-157
[3] Ladas G, Qian C. Oscillation and global stability in a delay Logistic equation. Dynam Stabibity Systems, 1994,9(2): 153-162
[4] Chen M P ,and Yu J S . Oscillation and global attractibity of a delay Logisitic difference equation. J.Difference Equations Appl., 1995,1:227-237
[5] Erbe L H, Zhang B G . Oscillation of discrete analogues of delay equations. Diff.Int.Eqs., 1989,2:300-309
[6] Gyori I , Ladas G . Oscillation Delay Difference Equation. Oxford:Oxford Science Publications, 1991,132-150
[7] Kocic V L, Ladas G. Oscillation and global attractivity in a discrete model of Nicholoson's blowflies. Appl.Anal., 1990,38:21-23
[8] Ladas G. Recent developments in the oscillation of delay difference equations. New York:Marcel Dekker, 1990,321-332
[9] Philos,Chen G . On oscillations of some difference equations. Funkcialaj Ekvacioj, 1991,34:157-172
[10] Yu J S, Zhang B G, Qian X Z. Oscillations of some difference equations with oscillating coefficients. J.Math.Anal. Appl.,1993,177:432-444
[11] Yu J S, Zhang B G, Wang Z C. Oscillations of difference equations.Applicable Analysis, 1994,53:117-124
[12] Gyori I, Ladas G, Vlhos P N. Global attractivity in a delay difference equation. Nonlinear Analysis Theory and Applications, 1991,17(5):473-479
[13] Erbe L H, Xia H ,Yu J S. Global stability of a linear nonautonomous delay diff- erence equation. J.Difference Equations Appl., 1995,1:151-161
[14] Kocic V L, Ladas G. Global attractivity in second-order difference equations. J.Math.Anal.Appl., 1993,180:144-150
[15] Kulenovic M R S ,Ladas G, Sficas Y G. Global attractivity in population dyna- mics. Computers Math.Appl., 1989,18:925-928
[16] Kuruklis S A. The asymptotic stability of x_(n+1)-ax_n+bx_(n-k)=0. J.Math. Anal. Appl., 1984,188:719-731
[17] Sojoseph W H ,Yu J S. On the stability and uniform persistence of a discrete model of Nicholson's blowflies. J.Math.Anal.Appl., 1995,193:233-244
[18] 庾建设．非自治时滞微分方程的渐近稳定性．科学通报，1997，42：1248-1251
[19] Ladas G, Qaian C, Vlahos P N, Yan J. Stability of solution of linear nonauton- omous equations. Appl.Anal., 1991,41:183-191
[20] Yu J S, Wang Z C. Asymptotic behacior and oscillation in delay difference equations.Funkcial Ekvac., 1994,37:117-124
[21] Zhang B G . Stability of infinite delay difference systems. Nonl.Anal.TMA, 1994, 22:1121-1129
[22] Freedman H I , Wu J . Persistence and global asymptotical stability of single species dispersal models with stage structure. Quarterly of Applied Mathemartics and Applications ,1991,49:351-371
[23] Zhou Z, Yu J S , Huang L H. Asymptotic behaviou of delay difference systems. Computers Math.Applic, 2001,42:283-290
[24] Yu J S. Global attractivity of the zero solution of a class of functional diffrtential equations and its application. SCI. china.ser.A,1996,39:225-237
[25] Zhou Z, Yu J S, Wang Z C. Global attractivity of neutral difference equations. Computers Math. Applic, 1998,36(6):1-10
[26] So J W, Yu J S. Global attractivity for a population model with time delay. Proc. Amer.Math.soc, 1995,123:2687-2694
[27] Ladas G. Open Problems and conjectures. Journal of Difference Equations and Applications,1996,14(5): 113-121
[28] Kocic V L, Ladas G. Global Behavior of Nonlinear Difference Equation of Higher Order with Application. Dordrecht:Kluwer Academic Publishets, 1993,232-231
[29] Ladas G. Open problems and Conjectures [A].In:Proceedings of the First International Conference on Difference Equations[C]. Basel:Gorden and Breach Science Publishers, 1995,337-349
[30] 李先义．一类非线性时滞差分分方程解的若干性质．应用数学，2000，13(1)：27-30
[31] Li Xianyi. A conjecture by G.Ladas. Appl.Math.-JCU, 1998,13B: 39-44
[32] Ou Chunhua(欧春华). Boundedness,persistence and asymptotic stability on the delay difference equation x_(n+1)=f(x_n,x_(n-1),…,x_(n-k)). Ann.of Diff.Eqs., 1999, 15(1):38-47
[33] Devault R, Ladas G ,Schultz S W. On the recursive sequence. In:Proceedierence of the second international conference on difference equation. Gordon and Breach Publisher,1995,334-351
[34] Kocic V L, Stutson D .Global Behavior of solutions of a nonlinear second-order difference equation. Journal of Mathematical Amalysis and Applications, 2000, 246:608-624
[35] 李先义，金银来．对G．Ladas的一个开问题的解答．数学杂志，2002,22(1)：50-52
[36] 李先义．一类高阶时滞差分方程的有界持久性与全局渐近稳定性．应用数学和力学，2002，23(11)：1188-1194
[37] Agarwal R P . Difference Equations and Inequalities[M]. New York: Kekker, 1992:546-554
[38] 霍海峰，李万同．一类非线性差分方程的持续生存与全局吸引．纯粹数学与应用数学，2000，16(3)：45-49
[40] Li Longtu. GLOBAL ASYMPTOTIC STABILITY OF x_(n+1)=F(x_n)g(x_(n-1)). Ann.of Diff.Eqs., 1998,14,(3):518-525
[41] 李万同．非线性时滞差分方程的持续生存和渐近性质．应用数学和力学，2003，11：1126-1132
[42] Camouzis E, Ladas G, Rodrigues I W. On the rational recursive Computer Math Appl,1994,28(1):37-43
[43] Zhang Decun,Shi Bao, Gai Mingjiu.On the rational recursive sequence [J].Indian J Pure Appl Math,2001,32(5):657-663
[44] Debasis Mukherjee.Permanence and global attractivity for facultativemutualism system with delay. Math. Meth.Appl.Sci.,2003;26:1-9
[45] Chen L S, Lu Z Y, Wang W D. The effect of delays on the permanence for Lotka-Volterra systems.Appl.Math.Lett., 1995,8(4):71-73
[46] Tokumaru H,Adachi N,Ametmiya T.Macroscopic stability of interconnected systems.In:IFAC 6~(th) World Congress.Boston: Porch,1975,551-559
[2] Kocic V L, Ladas G, Rodrignes I W. On rational recursive sequences. J Math Anal,1993,173: 127-157
[3] Ladas G, Qian C. Oscillation and global stability in a delay Logistic equation. Dynam Stabibity Systems, 1994,9(2): 153-162
[4] Chen M P ,and Yu J S . Oscillation and global attractibity of a delay Logisitic difference equation. J.Difference Equations Appl., 1995,1:227-237
[5] Erbe L H, Zhang B G . Oscillation of discrete analogues of delay equations. Diff.Int.Eqs., 1989,2:300-309
[6] Gyori I , Ladas G . Oscillation Delay Difference Equation. Oxford:Oxford Science Publications, 1991,132-150
[7] Kocic V L, Ladas G. Oscillation and global attractivity in a discrete model of Nicholoson's blowflies. Appl.Anal., 1990,38:21-23
[8] Ladas G. Recent developments in the oscillation of delay difference equations. New York:Marcel Dekker, 1990,321-332
[9] Philos,Chen G . On oscillations of some difference equations. Funkcialaj Ekvacioj, 1991,34:157-172
[10] Yu J S, Zhang B G, Qian X Z. Oscillations of some difference equations with oscillating coefficients. J.Math.Anal. Appl.,1993,177:432-444
[11] Yu J S, Zhang B G, Wang Z C. Oscillations of difference equations.Applicable Analysis, 1994,53:117-124
[12] Gyori I, Ladas G, Vlhos P N. Global attractivity in a delay difference equation. Nonlinear Analysis Theory and Applications, 1991,17(5):473-479
[13] Erbe L H, Xia H ,Yu J S. Global stability of a linear nonautonomous delay diff- erence equation. J.Difference Equations Appl., 1995,1:151-161
[14] Kocic V L, Ladas G. Global attractivity in second-order difference equations. J.Math.Anal.Appl., 1993,180:144-150
[15] Kulenovic M R S ,Ladas G, Sficas Y G. Global attractivity in population dyna- mics. Computers Math.Appl., 1989,18:925-928
[16] Kuruklis S A. The asymptotic stability of x_(n+1)-ax_n+bx_(n-k)=0. J.Math. Anal. Appl., 1984,188:719-731
[17] Sojoseph W H ,Yu J S. On the stability and uniform persistence of a discrete model of Nicholson's blowflies. J.Math.Anal.Appl., 1995,193:233-244
[18] 庾建设．非自治时滞微分方程的渐近稳定性．科学通报，1997，42：1248-1251
[19] Ladas G, Qaian C, Vlahos P N, Yan J. Stability of solution of linear nonauton- omous equations. Appl.Anal., 1991,41:183-191
[20] Yu J S, Wang Z C. Asymptotic behacior and oscillation in delay difference equations.Funkcial Ekvac., 1994,37:117-124
[21] Zhang B G . Stability of infinite delay difference systems. Nonl.Anal.TMA, 1994, 22:1121-1129
[22] Freedman H I , Wu J . Persistence and global asymptotical stability of single species dispersal models with stage structure. Quarterly of Applied Mathemartics and Applications ,1991,49:351-371
[23] Zhou Z, Yu J S , Huang L H. Asymptotic behaviou of delay difference systems. Computers Math.Applic, 2001,42:283-290
[24] Yu J S. Global attractivity of the zero solution of a class of functional diffrtential equations and its application. SCI. china.ser.A,1996,39:225-237
[25] Zhou Z, Yu J S, Wang Z C. Global attractivity of neutral difference equations. Computers Math. Applic, 1998,36(6):1-10
[26] So J W, Yu J S. Global attractivity for a population model with time delay. Proc. Amer.Math.soc, 1995,123:2687-2694
[27] Ladas G. Open Problems and conjectures. Journal of Difference Equations and Applications,1996,14(5): 113-121
[28] Kocic V L, Ladas G. Global Behavior of Nonlinear Difference Equation of Higher Order with Application. Dordrecht:Kluwer Academic Publishets, 1993,232-231
[29] Ladas G. Open problems and Conjectures [A].In:Proceedings of the First International Conference on Difference Equations[C]. Basel:Gorden and Breach Science Publishers, 1995,337-349
[30] 李先义．一类非线性时滞差分分方程解的若干性质．应用数学，2000，13(1)：27-30
[31] Li Xianyi. A conjecture by G.Ladas. Appl.Math.-JCU, 1998,13B: 39-44
[32] Ou Chunhua(欧春华). Boundedness,persistence and asymptotic stability on the delay difference equation x_(n+1)=f(x_n,x_(n-1),…,x_(n-k)). Ann.of Diff.Eqs., 1999, 15(1):38-47
[33] Devault R, Ladas G ,Schultz S W. On the recursive sequence. In:Proceedierence of the second international conference on difference equation. Gordon and Breach Publisher,1995,334-351
[34] Kocic V L, Stutson D .Global Behavior of solutions of a nonlinear second-order difference equation. Journal of Mathematical Amalysis and Applications, 2000, 246:608-624
[35] 李先义，金银来．对G．Ladas的一个开问题的解答．数学杂志，2002,22(1)：50-52
[36] 李先义．一类高阶时滞差分方程的有界持久性与全局渐近稳定性．应用数学和力学，2002，23(11)：1188-1194
[37] Agarwal R P . Difference Equations and Inequalities[M]. New York: Kekker, 1992:546-554
[38] 霍海峰，李万同．一类非线性差分方程的持续生存与全局吸引．纯粹数学与应用数学，2000，16(3)：45-49
[40] Li Longtu. GLOBAL ASYMPTOTIC STABILITY OF x_(n+1)=F(x_n)g(x_(n-1)). Ann.of Diff.Eqs., 1998,14,(3):518-525
[41] 李万同．非线性时滞差分方程的持续生存和渐近性质．应用数学和力学，2003，11：1126-1132
[42] Camouzis E, Ladas G, Rodrigues I W. On the rational recursive Computer Math Appl,1994,28(1):37-43
[43] Zhang Decun,Shi Bao, Gai Mingjiu.On the rational recursive sequence [J].Indian J Pure Appl Math,2001,32(5):657-663
[44] Debasis Mukherjee.Permanence and global attractivity for facultativemutualism system with delay. Math. Meth.Appl.Sci.,2003;26:1-9
[45] Chen L S, Lu Z Y, Wang W D. The effect of delays on the permanence for Lotka-Volterra systems.Appl.Math.Lett., 1995,8(4):71-73
[46] Tokumaru H,Adachi N,Ametmiya T.Macroscopic stability of interconnected systems.In:IFAC 6~(th) World Congress.Boston: Porch,1975,551-559