几类中立型差分方程的振动性与非振动性
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摘要
本文首先讨论具变系数中立型差分方程
Δ(x_n-c_nx_n-r)+p_nx_(n-k)-q_nx_(n-l)=0。 (Ⅰ)通过建立一些引理，获得了方程(Ⅰ)所有解振动的几个新的充分条件。
接下来，通过建立一些Riccati型差分不等式，研究了二阶中立型差分方程
Δ(r_nΔ(x_n+p_nx_(n-k)))+q_nx_(n-Υ_((n)))=0 (Ⅱ)的振动性。
最后，我们讨论奇数阶中立型差分方程
Δ~m(x_n-p_nx_(n-Υ))+q_nx_(n-σ)=0 (m是奇数)。 (Ⅲ)得到了一些新的振动条件。
我们获得的定理改进和推广了相应文献中的结果，而且方程(Ⅱ)和方程(Ⅲ)的正解存在性的定理是第一次给出。文中的例子说明了我们结果的有效性。
In this paper ,firstly,we discuss the neutral difference equation with variable coefficients
By establishing some new lemmas, we obtain several new sufficient conditions for the oscillation of all solutions of equation (I).
Secondly,by building some Riccati type difference inequalities,we investigate the oscillation of second-order neutral difference equation
Finally, we discuss the odd-order neutral difference equation
and obtain some new sufficient conditions for the oscillation of all solutions of equation (III).
Our results improve and extend the all known results in related literature. Moreover the theorems for the existence of a positive solution of equtions(II)and (III) are firstly obtained. Some examples given in the paper illustrate the effect of our results.
Δ(x_n-c_nx_n-r)+p_nx_(n-k)-q_nx_(n-l)=0。 (Ⅰ)通过建立一些引理，获得了方程(Ⅰ)所有解振动的几个新的充分条件。
接下来，通过建立一些Riccati型差分不等式，研究了二阶中立型差分方程
Δ(r_nΔ(x_n+p_nx_(n-k)))+q_nx_(n-Υ_((n)))=0 (Ⅱ)的振动性。
最后，我们讨论奇数阶中立型差分方程
Δ~m(x_n-p_nx_(n-Υ))+q_nx_(n-σ)=0 (m是奇数)。 (Ⅲ)得到了一些新的振动条件。
我们获得的定理改进和推广了相应文献中的结果，而且方程(Ⅱ)和方程(Ⅲ)的正解存在性的定理是第一次给出。文中的例子说明了我们结果的有效性。
In this paper ,firstly,we discuss the neutral difference equation with variable coefficients
By establishing some new lemmas, we obtain several new sufficient conditions for the oscillation of all solutions of equation (I).
Secondly,by building some Riccati type difference inequalities,we investigate the oscillation of second-order neutral difference equation
Finally, we discuss the odd-order neutral difference equation
and obtain some new sufficient conditions for the oscillation of all solutions of equation (III).
Our results improve and extend the all known results in related literature. Moreover the theorems for the existence of a positive solution of equtions(II)and (III) are firstly obtained. Some examples given in the paper illustrate the effect of our results.
引文
1．马知恩，种群生态学的数学建模与研究[M]，安徽教育出版社，1996．
2．裴新澍，生物控制进化论[M]，科学出版社，1998．
3．徐克学，生物数学[M]，科学出版社，1999．
4．张金城，郑美特(译)，电力系统稳定性的控制[M]，电力工业出版社，1982．
5．卢强，王仲鸿，韩英泽，输电系统最优控制[M]，科学出版社，1982．
6．仝茂达，线性系统理论和设计[M]，中国科学技术大学出版社，1998．
7．郑大钟，线性系统理论[M]，清华大学出版社，2000．
8．胡健伟，汤怀民，微分方程数值方法[M]，科学出版社，2001．
9. Ladas G., Oscillation of difference equations with postive and negative coefficients, Rocrymountain J. of Math.20(4),(1990),1051-1061.
10. Chuanxi G. and Ladas G., Oscillation of difference equation with positive and negative coefficients,Matematiche (catania). 44(1989),293-309.
11. Chen M. P. and Zhang B. G., Oscillation and comparison theorems of difference equations with positive and negative coefficients, Bulletin of the Institute of Math. cademia sinia.22(40,(1994),295-30.
12. Zhang B. G. and Wang H., The existence of oscillatory and nonoscillatory sollutions of neutral difference equations. Chinese J.Math. 24(4), (1996), 377-393.
13. Chen M. P., Lalli B. S. and Yu J. S., Oscillation in neutral delay difference equations with variable coefficients, Computers Math. Applic. 29(3), (1995), 5-11.
14. Yu J. S. and Wang Z.C.,Asymptotic behavior and oscillation in delay difference equations. Func.Ekvac.37(1992),241-248.
15. Agarwal R. P., Difference Equations and Inequalities: Theory, Methods, and Applications, Marcel Dekker, New York.(1994)
16. Gyri I. and Ladas G., Oscillation Theory of Delay Differential Equations with Applications, Claredon Press, Oxford. (1991).
17. Tang X.H.,Yu J.S. and Peng D.H.,Oscillation and Nonoscillation of Neutral difference equations with positive and negative coefficients, Computers Math. Applic., 39(2000), 169-181.
18. Lalli B. S., Zhang B. C. and Li J. Zh., On the oscillation of solutions and existence of positive solutions of neutral difference equations, J. Math. Anal. Appl., 158, 213-233(1991).
19. Lalli B. S., Zhang B. G, On existence of positive solutions and bounded oscillation for neutral difference equations, J. Math. Anal. Appl., 166, 272-287(1992).
20. Yori I. G. and Ladas G., Oscillation theory of delay differential equations with application[M], Clarendon Press, Oxford, 1991.
21. Erbe L.H.,Kong Q.K.and zhang B.G.,Oscillation theory for functional differential equation[M], Marcel Dekker, Inc. New Yok,1995.
22．张广，高英，差分方程的振动理论[M]，高等教育出版社，2001．
23. Zhang Z.and Li Q.,Oscillation theorems for second-order advanced functional difference equation, Computers Math. Applic.,36(1998),11-18.
24. Zhang Z.and Zhang J.,Oscillation criteria for second-order functional difference equations with "summation small" coefficient, Computers Math. Applic., 38 (1999),25-31.
25. Zhang B.G. and Cheng S.S.,Oscillation criteria and comparision theorems for delay difference equations, Fasciculi Math., 25,13-32(1995).
26. Zhang B.G. and Zhou Y.,Oscillation and nonoscillstion for second-order linear difference equations, Computers Math. Applic.,39(2000)1-7.
27. Zhou Y., Oscillations of higher-order linear difference equations[J],Computers Math. Applic., 42(2001)323-331.
28．张炳根，杨博，非线性高阶差分方程的振动性[J]，数学年刊，20A：1(1999)，71-80．
29．刘玉记，奇数阶中立型差分方程的线性化振动性[J]，高校应用数学学报A辑，2002，17(1)：37-42．
30. Tang X.H.,Yu J.S.,A further result on the oscillation of delay difference equations,Computers Math. Applic., 38(1999)229-237.
31. Tang X.H.,Cheng S.S.,An osxillation criteria for linear difference equations with oscillating coefficients, J. Comput. Appl. Math., 132 (2001) 319-329.
32. Tang X.H. Zhang R.Y.,New oscillation criteria for delay difference equations,Computrs Math.Applic.,42(2001) 1319-1330.
33. Shen J.H.,Luo Z.G.,Some oscillation criteria for difference equations, Computers Math. Applic., 40(2000)713-719.
34. Luo Z.G.,Shen J.H., New results for oscillation of delay difference equations, Computers Math. Applic., 41(2001) 553-561.
35. Tang X. H.,Oscillation of nonlinear delay difference equations,J.Math.Anal.Appl.,249,476-490(2000).
36. Tang X.H.,Yu J.S.,Oscillation of delay difference equations in a critical state, Applied math. letters, 13(2000)9-15.
37. Tang X.H.,Yu J.S.,Oscillation of delay difference equations,J.Hokkaido Math., Vol. 29(2000)213-228.
38. Tang X.H.,Yu J.S.,Oscillation of delay difference equations,Computers Math.Applic.,37(1999)11-20.
39．唐先华，具有变系数差分方程解的振动性，中南工业大学学报，Vol．29，No．3，June 1998．
40. Yu J.S.,Tang X.H.,Sufficient conditions for the oscillation of linear delay difference equations with oscillating coeffients, J. Math. Anal. Appl.,250,735-742(2000).
41. Tang X.H.,Yu J.S.,New oscillation criteria for delay difference equations,Dynamics of Continuous,Discrete and Impulse,7(2000)525-531.
42．林晓艳，申建华，一类二阶中立型时滞差分方程的有界振动，湖南师范大学自然科学学报，Vol．24，No．2，Jun 2001．
43. Shen J.H.,Tang X.H.,New oscillation criteria for linear delay differential equations, Computers Math. Applic., Vol. 36, No. 6, 53-61(1998).
44. Luo Z.G., Shen J.H.,New oscillation criteria for delay difference equations, J.Math.Anal. Appl.,264,85-95(2001).
45. Yu J.S.,Zhang B.G. and Qian X. Z.,Oscillation of delay difference equations with oscillating coefficients,it J.Math.Anal.Appl.,177,432-444(1993).
46．贺新光，罗治国，无界时滞差分方程解的振动性，湖南师大学报自然科学版，Vol．24 No．3 Sep．，2001．
47．王其如，二阶非线性微分方程的振动准则，数学学报，Vol．44，No．2，March，2001．
48．张晓声，王泽，一类具有正负系数的中立型方程的振动性，北京大学学报(自然科学版)，第38卷，第1期，2002年1月．
49．周勇，俞元洪，一阶中立型时滞微分方程的振动定理，数学研究与评论，Vol．21，No．1，Feb．2001．
50. Huang C.,Oscillation and nonoscillation for second-order linear differential equations, J.Math.Anal.Appl.,210,712-723(1997).
51. Luo Z.G., Shen J.H.,New oscillation criteria for differential equations with piecewise constant argument, Tam Kang Journal Math.,Vol.32,Num.4,Winter 2001.
52. Luo Z.G., Shen J.H.,New razumikhim type theorems for impulsive functional differential equations,Appli.math. Comput.,125(2002)375-386.
53. Shen J.H.,Stavroulakis I.P.,Oscillatory and nonoscillatory delay equations with piecewise constannt argument, J.Math.Anal.Appl.,248,385-401(2000).
54．张玉珠，燕居让，具有连续变量的差分方程振动性的判据，数学学报，Vol．38，No．3，May，1995．
55．郑洁钢，关于分断常变元之时滞微分方程解的振动性，湖南师范大学自然科学学报，V01．24，No．4，Dec．2001．
56．申建华，具有连续变量差分方程振动性的比较定理及应用，科学通报，1996年8月．
57．唐先华，庾建设，线性脉冲差分方程解的振动性与稳定性，应用数学，2001，14(1)：28-32．
58．谢海明，廖六生，脉冲时滞差分方程的振动性，数学理论与应用，Vol．21，No．1，March，2001．
2．裴新澍，生物控制进化论[M]，科学出版社，1998．
3．徐克学，生物数学[M]，科学出版社，1999．
4．张金城，郑美特(译)，电力系统稳定性的控制[M]，电力工业出版社，1982．
5．卢强，王仲鸿，韩英泽，输电系统最优控制[M]，科学出版社，1982．
6．仝茂达，线性系统理论和设计[M]，中国科学技术大学出版社，1998．
7．郑大钟，线性系统理论[M]，清华大学出版社，2000．
8．胡健伟，汤怀民，微分方程数值方法[M]，科学出版社，2001．
9. Ladas G., Oscillation of difference equations with postive and negative coefficients, Rocrymountain J. of Math.20(4),(1990),1051-1061.
10. Chuanxi G. and Ladas G., Oscillation of difference equation with positive and negative coefficients,Matematiche (catania). 44(1989),293-309.
11. Chen M. P. and Zhang B. G., Oscillation and comparison theorems of difference equations with positive and negative coefficients, Bulletin of the Institute of Math. cademia sinia.22(40,(1994),295-30.
12. Zhang B. G. and Wang H., The existence of oscillatory and nonoscillatory sollutions of neutral difference equations. Chinese J.Math. 24(4), (1996), 377-393.
13. Chen M. P., Lalli B. S. and Yu J. S., Oscillation in neutral delay difference equations with variable coefficients, Computers Math. Applic. 29(3), (1995), 5-11.
14. Yu J. S. and Wang Z.C.,Asymptotic behavior and oscillation in delay difference equations. Func.Ekvac.37(1992),241-248.
15. Agarwal R. P., Difference Equations and Inequalities: Theory, Methods, and Applications, Marcel Dekker, New York.(1994)
16. Gyri I. and Ladas G., Oscillation Theory of Delay Differential Equations with Applications, Claredon Press, Oxford. (1991).
17. Tang X.H.,Yu J.S. and Peng D.H.,Oscillation and Nonoscillation of Neutral difference equations with positive and negative coefficients, Computers Math. Applic., 39(2000), 169-181.
18. Lalli B. S., Zhang B. C. and Li J. Zh., On the oscillation of solutions and existence of positive solutions of neutral difference equations, J. Math. Anal. Appl., 158, 213-233(1991).
19. Lalli B. S., Zhang B. G, On existence of positive solutions and bounded oscillation for neutral difference equations, J. Math. Anal. Appl., 166, 272-287(1992).
20. Yori I. G. and Ladas G., Oscillation theory of delay differential equations with application[M], Clarendon Press, Oxford, 1991.
21. Erbe L.H.,Kong Q.K.and zhang B.G.,Oscillation theory for functional differential equation[M], Marcel Dekker, Inc. New Yok,1995.
22．张广，高英，差分方程的振动理论[M]，高等教育出版社，2001．
23. Zhang Z.and Li Q.,Oscillation theorems for second-order advanced functional difference equation, Computers Math. Applic.,36(1998),11-18.
24. Zhang Z.and Zhang J.,Oscillation criteria for second-order functional difference equations with "summation small" coefficient, Computers Math. Applic., 38 (1999),25-31.
25. Zhang B.G. and Cheng S.S.,Oscillation criteria and comparision theorems for delay difference equations, Fasciculi Math., 25,13-32(1995).
26. Zhang B.G. and Zhou Y.,Oscillation and nonoscillstion for second-order linear difference equations, Computers Math. Applic.,39(2000)1-7.
27. Zhou Y., Oscillations of higher-order linear difference equations[J],Computers Math. Applic., 42(2001)323-331.
28．张炳根，杨博，非线性高阶差分方程的振动性[J]，数学年刊，20A：1(1999)，71-80．
29．刘玉记，奇数阶中立型差分方程的线性化振动性[J]，高校应用数学学报A辑，2002，17(1)：37-42．
30. Tang X.H.,Yu J.S.,A further result on the oscillation of delay difference equations,Computers Math. Applic., 38(1999)229-237.
31. Tang X.H.,Cheng S.S.,An osxillation criteria for linear difference equations with oscillating coefficients, J. Comput. Appl. Math., 132 (2001) 319-329.
32. Tang X.H. Zhang R.Y.,New oscillation criteria for delay difference equations,Computrs Math.Applic.,42(2001) 1319-1330.
33. Shen J.H.,Luo Z.G.,Some oscillation criteria for difference equations, Computers Math. Applic., 40(2000)713-719.
34. Luo Z.G.,Shen J.H., New results for oscillation of delay difference equations, Computers Math. Applic., 41(2001) 553-561.
35. Tang X. H.,Oscillation of nonlinear delay difference equations,J.Math.Anal.Appl.,249,476-490(2000).
36. Tang X.H.,Yu J.S.,Oscillation of delay difference equations in a critical state, Applied math. letters, 13(2000)9-15.
37. Tang X.H.,Yu J.S.,Oscillation of delay difference equations,J.Hokkaido Math., Vol. 29(2000)213-228.
38. Tang X.H.,Yu J.S.,Oscillation of delay difference equations,Computers Math.Applic.,37(1999)11-20.
39．唐先华，具有变系数差分方程解的振动性，中南工业大学学报，Vol．29，No．3，June 1998．
40. Yu J.S.,Tang X.H.,Sufficient conditions for the oscillation of linear delay difference equations with oscillating coeffients, J. Math. Anal. Appl.,250,735-742(2000).
41. Tang X.H.,Yu J.S.,New oscillation criteria for delay difference equations,Dynamics of Continuous,Discrete and Impulse,7(2000)525-531.
42．林晓艳，申建华，一类二阶中立型时滞差分方程的有界振动，湖南师范大学自然科学学报，Vol．24，No．2，Jun 2001．
43. Shen J.H.,Tang X.H.,New oscillation criteria for linear delay differential equations, Computers Math. Applic., Vol. 36, No. 6, 53-61(1998).
44. Luo Z.G., Shen J.H.,New oscillation criteria for delay difference equations, J.Math.Anal. Appl.,264,85-95(2001).
45. Yu J.S.,Zhang B.G. and Qian X. Z.,Oscillation of delay difference equations with oscillating coefficients,it J.Math.Anal.Appl.,177,432-444(1993).
46．贺新光，罗治国，无界时滞差分方程解的振动性，湖南师大学报自然科学版，Vol．24 No．3 Sep．，2001．
47．王其如，二阶非线性微分方程的振动准则，数学学报，Vol．44，No．2，March，2001．
48．张晓声，王泽，一类具有正负系数的中立型方程的振动性，北京大学学报(自然科学版)，第38卷，第1期，2002年1月．
49．周勇，俞元洪，一阶中立型时滞微分方程的振动定理，数学研究与评论，Vol．21，No．1，Feb．2001．
50. Huang C.,Oscillation and nonoscillation for second-order linear differential equations, J.Math.Anal.Appl.,210,712-723(1997).
51. Luo Z.G., Shen J.H.,New oscillation criteria for differential equations with piecewise constant argument, Tam Kang Journal Math.,Vol.32,Num.4,Winter 2001.
52. Luo Z.G., Shen J.H.,New razumikhim type theorems for impulsive functional differential equations,Appli.math. Comput.,125(2002)375-386.
53. Shen J.H.,Stavroulakis I.P.,Oscillatory and nonoscillatory delay equations with piecewise constannt argument, J.Math.Anal.Appl.,248,385-401(2000).
54．张玉珠，燕居让，具有连续变量的差分方程振动性的判据，数学学报，Vol．38，No．3，May，1995．
55．郑洁钢，关于分断常变元之时滞微分方程解的振动性，湖南师范大学自然科学学报，V01．24，No．4，Dec．2001．
56．申建华，具有连续变量差分方程振动性的比较定理及应用，科学通报，1996年8月．
57．唐先华，庾建设，线性脉冲差分方程解的振动性与稳定性，应用数学，2001，14(1)：28-32．
58．谢海明，廖六生，脉冲时滞差分方程的振动性，数学理论与应用，Vol．21，No．1，March，2001．