时标上几类动力方程解的振动性、存在性及边值问题
|
摘要
近年来,时标上动力方程作为数学的一个新兴领域得到了人们的普遍重视。1988年,Stefan Hilger率先提出时标及时标上的微积分理论,并初步建立了时标上动力方程的基本理论。从本世纪初开始,这一理论受到关注,目前,这一理论正得到迅速的发展。一方面,它统一和推广了现有的微分和差分、微分方程和差分方程的理论,另一方面,时标上动力方程的研究对于刻画真实现象和过程的数学模型有着重要应用,例如:时标上的种群动力学、流行病模型、金融消费过程的数学模型等。总之,时标和时标上动力方程理论有着广阔的应用前景。
在现有时标理论的基础上,论文分别就时标上动力方程解的振动性、解的存在性以及边值问题进行了研究。
首先讨论了时标上一类二阶非线性中立型动力方程解的振动性,得到了解振动的充分条件,并且给出了实际应用的例子。
其次给出了时标上一类二阶中立型时滞动力方程解振动的充分条件,同时给出了例子加以说明。
然后运用Banach压缩映射原理研究了时标上一类高阶带强迫项中立型时滞动力方程非振动解的存在性,得到了方程非振动解存在的判别结果。
进一步讨论了时标上一类具有非线性中立项的高阶变系数变时滞动力方程解的强迫振动与非振动解的存在性,所得结果推广了已知的一些结论。
最后,研究了时标上一类二阶非线性边值问题及时标上的半直线上奇异m-点边值问题的正解存在性,得到了方程正解存在的判别结果。
Recently, the study of dynamic equations on time scales is an area of mathematics that has received a lot of attention. Theory of dynamic equations on time scales, founded by Stefan Hilger in 1988, is undergoing a rapid development as it provides a powerful tool not only to unify the existing theories of both differential equations and difference equations but also to generalize the discussion to a wide class of equations on time scales. In addition, the study of dynamic equations on time scales has lead to several important applications related to mathematical modeling of real phenomena. For instance, investigation of population dynamics, financial assumption process, and epidemic models have been involved with the concept of dynamic equations on time scales. Clearly, theory of dynamic equations on time scales has been and are being proved to be of great importance and potential application.
In this thesis, oscillation properties、existence properties of solutions of dynamic equations and boundary value problems on time scales are researched.
Firstly, oscillation properties of solutions for a class of second order nonlinear neutral dynamic equations on time scales are researched, a series of sufficient conditions for oscillation of the solutions are established. These results are illustrated by examples.
Secondly, some sufficient conditions for oscillation of a class of second order neutral delay dynamic equations on time scales are obtained. These results are explained by examples.
The existence of nonoscillatory solutions for a class of high-order forced neutral delay dynamic equations on time scales are studied by using Banach contraction mapping principle. Some sufficient conditions to satisfy existence of nonoscillatory solutions of these equations are derived.
A class of forced high-order nonlinear neutral variable delay dynamic equations with variable coefficient on time scales is studied. The sufficient conditions are given for the forced oscillation and existence of nonoscillatory solutions. These results generalize the corresponding known results.
Finally, the existence of positive solutions for a class of second order nonlinear boundary value problems on time scales and nonlinear singular m-point boundary value problems on the half-line on time scales are considered. Some sufficient conditions to satisfy existence of positive solutions of these equations are derived.
在现有时标理论的基础上,论文分别就时标上动力方程解的振动性、解的存在性以及边值问题进行了研究。
首先讨论了时标上一类二阶非线性中立型动力方程解的振动性,得到了解振动的充分条件,并且给出了实际应用的例子。
其次给出了时标上一类二阶中立型时滞动力方程解振动的充分条件,同时给出了例子加以说明。
然后运用Banach压缩映射原理研究了时标上一类高阶带强迫项中立型时滞动力方程非振动解的存在性,得到了方程非振动解存在的判别结果。
进一步讨论了时标上一类具有非线性中立项的高阶变系数变时滞动力方程解的强迫振动与非振动解的存在性,所得结果推广了已知的一些结论。
最后,研究了时标上一类二阶非线性边值问题及时标上的半直线上奇异m-点边值问题的正解存在性,得到了方程正解存在的判别结果。
Recently, the study of dynamic equations on time scales is an area of mathematics that has received a lot of attention. Theory of dynamic equations on time scales, founded by Stefan Hilger in 1988, is undergoing a rapid development as it provides a powerful tool not only to unify the existing theories of both differential equations and difference equations but also to generalize the discussion to a wide class of equations on time scales. In addition, the study of dynamic equations on time scales has lead to several important applications related to mathematical modeling of real phenomena. For instance, investigation of population dynamics, financial assumption process, and epidemic models have been involved with the concept of dynamic equations on time scales. Clearly, theory of dynamic equations on time scales has been and are being proved to be of great importance and potential application.
In this thesis, oscillation properties、existence properties of solutions of dynamic equations and boundary value problems on time scales are researched.
Firstly, oscillation properties of solutions for a class of second order nonlinear neutral dynamic equations on time scales are researched, a series of sufficient conditions for oscillation of the solutions are established. These results are illustrated by examples.
Secondly, some sufficient conditions for oscillation of a class of second order neutral delay dynamic equations on time scales are obtained. These results are explained by examples.
The existence of nonoscillatory solutions for a class of high-order forced neutral delay dynamic equations on time scales are studied by using Banach contraction mapping principle. Some sufficient conditions to satisfy existence of nonoscillatory solutions of these equations are derived.
A class of forced high-order nonlinear neutral variable delay dynamic equations with variable coefficient on time scales is studied. The sufficient conditions are given for the forced oscillation and existence of nonoscillatory solutions. These results generalize the corresponding known results.
Finally, the existence of positive solutions for a class of second order nonlinear boundary value problems on time scales and nonlinear singular m-point boundary value problems on the half-line on time scales are considered. Some sufficient conditions to satisfy existence of positive solutions of these equations are derived.
引文
1 S. Hilger. Ein Ma?kettenkalkül mit Anwendung auf Zentrumsmannigfaltigkeiten. PhD thesis, Universit?t Würzburg, 1988
2 F. M. Atici, D. C. Biles, A. Lebedinsky. An Application of Time Scales to Economics. Math. Comput. Modelling, 2006,43(7-8):718-726
3 S. Hilger. Analysis on Measure Chains-a Unified Approach to Continuous and Discrete Calculus. Results Math., 1990,18:18-56
4张炳根.测度链上微分方程的进展.中国海洋大学学报,2004,34:907-912
5 M. Bohner, A. Peterson. Dynamic Equations on Time Scales: An Introduction with Applications. Boston: Birkh?user, MA, 2001
6 M. Bohner, A. Peterson. Advances in Dynamic Equations on Time Scales. Boston: Birkh?user, MA, 2003
7 B. Kaymakcalan, V. Lakshmikantham, S. Sivasundaram. Dynamic Systems on Measure chains. Kluwer Academic Publishers, Boston, MA, 1996
8 R. P. Agarwal, M. Boher, D. O'Regan, A. Peterson. Dynamic Equations on Time Scales: a Survey. J. Comput. Appl. Math., 2002,141(1-2):1-26
9 Z. Q. Zhu, Q. R, Wang. Existence of Nonoscillatory Solutions to Neutral Dynamic Equations on Time Scales. J. Math. Anal. Appl., 2007,335(2):751-762
10 R. P. Agarwal, M. Bohner, D. O'Regan. Time Scale Systems on Infinite Intervals. Nonlinear Anal., 2001,47(2):837-848
11 F. M. Aticl, D. C. Biles. First Order Dynamic Inclusions on Time Scales. J. Math. Anal. Appl., 2004,292(1):222-237
12 J. P. Sun, W. T. Li. Positive Solutions for System of Nonlinear First-Order PBVPs on Time Scales. Nonlinear Anal., 2005,62(1):131-139
13 Z. C. Hao, T. J. Xiao, J. Liang. Existence of Positive Solutions for Singular Boundary Value Problem on Time Scales. J. Math. Anal. Appl., 2007,325(1):517-528
14 D. R. Anderson, R. I. Avery. An Even-Order Three-Point Boundary Value Problem onTime Scales. J. Math. Anal. Appl., 2004,291(2):514-525
15 C. J. Chyan. Uniqueness Impiles Existence on Time Scales. J. Math. Anal. Appl., 2001, 258:359-365
16 D. R. Anderson. Taylor Polynomials for Nabla Dynamic Equations on Time Scales. Panamer. Math. J., 2002,12:17-27
17 C. J. Chyan, J. Henderson. Eigenvalue Problems for Nonlinear Diferential Equations on a Measure Chain. J. Math. Anal. Appl., 2000,245:547-559
18 P. W. Eloe, Q. Sheng, J. Henderson. Notes on Crossed Symmetry Solutions of the Two-Point Boundary Value Problems on Time Scales. J. Difference Equations Appl., 2003,9: 29-48
19 B. G. Zhang, X. H. Deng. Oscillation of Delay Differential Equations on Time Scales. Math. Comput. Modelling, 2002,36(11-13):1307-1318
20刘爱莲,朱思铭,吴洪武.二阶时标动力系统的振动准则.中山大学学报(自然科学版), 2004,43(2):9-12
21 S. H. Saker. Oscillation of Nonlinear Dynamic Equations on Time Scales. Appl. Math. Comput., 2004,148(1):81-91
22 Y. Sahiner. Oscillation of Second-Order Delay Differential Equations on Time Scales. Nonlinear Anal., 2005,63(5-7):1073-1080
23 S. H. Saker. Oscillation Criteria of Second-Order Half-Linear Dynamic Equations on Time Scales. J. Comput. Appl. Math., 2005,177(2):375-387
24 S. H. Saker. Oscillation of Second-Order Nonlinear Neutral Delay Dynamic Equations on Time Scales. J. Comput. Appl. Math., 2006,187(2):123-141
25 Z. L. Han, S. R. Sun, B. Shi. Oscillation Criteria for a Class of Second-Order Emden-Fowler Delay Dynamic Equations on Time Scales. J. Math. Anal. Appl., 2007, 334(2):847–858
26 L. Erbe, A. Peterson. Green’s Functions and Comparison Theorems for Differential Equations on Measure Chains. Dynam. Contin. Discrete Impuls. Systems, 1999,6: 121-137
27 L. Erbe, A. Peterson. Positive Solutions for a Nonlinear Differential Equation on aMeasure Chain. Math. Comput. Modelling, 2000,32(5-6):571-585
28 R. P. Agarwal, D. O'Regan. Triple Solutions to Boundary Value Problems on Time Scales. Appl. Math. Lett., 2000,13(4):7-11
29 R. P. Agarwal, D. O'Regan. Nonlinear Boundary Value Problems on Time Scales. Nonlinear. Aanl., 2001,44(4):527-535
30 C. J. Chyan, J. Henderson. Twin Solutions of Boundary Value Problems for Differential Equations on Measure Chains. J. Comput. Appl. Math., 2002,141(1-2): 123-131
31 R. I. Avery, J. Henderson. Two Positive Fixed Points of Nonlinear Operator on Ordered Banach Spaces. Comm. Appl. Nonlinear Anal., 2001,8:27-36
32 F. M. Atici, G. Sh. Guseinov. On Green’s Functions and Positive Solutions for Boundary Value Problems on Time Scales. J. Comput. Appl. Math., 2002(1-2),141: 75-99
33 Z. C. Hao, J. Liang, T. J. Xiao. Existence Results for Time Scale Boundary Value Problem. J. Comput. Appl. Math., 2006,197(1):156-168
34 D. R. Anderson. Solutions to Second-Order Three-Point Problems on time scales. J. Difference Equations Appl., 2002,8:673-688
35 E. R. Kaufmann. Positive Solutions of a Three-Point Boundary Value Problem on a Time Scale. Electronic J. Differential Equations, 2003,82:1-11
36孙红蕊,李万同.测度链上非线性m ?点边值问题的正解.数学学报,2006,49(2): 369-380
37 Z. M. He. Existence of Two Solutions of m ? Point Boundary Value Problem for Second Order Dynamic Equations on Time Scales. J. Math. Anal. Appl., 2004,296(1): 97-109
38 R. I. Avery, D. R. Anderson. Existence of Three Positive Solutions to a Second-Order Boundary Value Problem on a Measure Chain. J. Comput. Appl. Math., 2002,141(1-2): 65-73
39 R. I. Avery, C. J. Chyan, J. Henderson. Twin Solutions of Boundary Value Problems for Ordinary Differential Equations and Finite Difference Equations. Comput. Math.Appl., 2001,42(3-5):695-704
40 H. Amann. Fixed Point Equations and Nonlinear Eigenvalue Problems in Ordered Banach Space. Siam Review, 1976,18:620-709
41 H. R. Sun, W. T. Li. Positive Solutions for Nonlinear Three-Point Boundary Value Problem on Time Scales. J. Math. Anal. Appl., 2004,299(2):508-524
42 Y. K. Chang, W. T. Li. Existence Results for Second-Order Dynamic Inclusion with m?Point Boundary Value Conditions on Time Scales. Appl. Math. Lett., 2007,20(8): 885-891
43 P. Amster, C. Rogers, C. C. Tisdell. Existence of Solutions to Boundary Value Problems for Dynamic Systems on Time Scales. J. Math. Anal. Appl., 2005,308(2): 565-577
44 Z. M. He, L. Li. Multiple Positive Solutions for the One-Dimensional p ? Laplacian Dynamic Equations on Time Scales. Math. Comput. Modelling, 2007,45(1-2):68-79
45 H. R. Sun, W. T. Li. Multiple Positive Solutions for p ? Laplacian m ? Point Boundary Value Problems on Time Scales. Appl. Math. Comput., 2006,182(1):478-491
46 Z. M. He, X. M. Jiang. Triple Positive Solutions of Boundary Value Problems for p ?Laplacian Dynamic Equations on Time Scales. J. Math. Anal. Appl., 2006,32(2): 911-920
47 C. D. Ahlbrandt, C. Morian. Partial Differential Equations on Time Scales. J. Comput. Appl. Math., 2002,141:35-55
48 B. Jackson. Partial Dynamic Equations on Time Scales. J. Comput. Appl. Math., 2006, 186:391-415
49 L. Erbe, A. Peterson. S. H. Saker. Oscillation Criteria for Second-Order Nonlinear Delay Dynamic Equations. J. Math. Anal. Appl., 2007,333(1):505-522
50 Y. Bolat, ?. Akin. Oscillatory Behaviour of a Higher-Order Nonlinear Neutral Type Functional Difference Equation with Oscillating Coefficients. Appl. Math. Lett., 2004, 17(9):1073-1078
51 X. P. Wang. Oscillation for Higher Order Nonlinear Delay Differential Equations. Appl. Math. and Comput., 2004,157(1):287-294
52 S. R. Grace, B. S. Lalli. Oscillation of Higher-Order Nonlinear Difference Equations. Math. Comput. Modelling, 2005,41(4-5):485-491
53 T. Candan, R. S. Dahiya. Oscillation Behavior of Higher Order Neutral Differential Equations. Appl. Math. Comput., 2005,167(2):1267-1280
54杨军,关新平,刘树堂.具强迫项高阶非线性中立型差分方程的振动性与渐进性.纯粹数学与应用数学,1999,15(4):29-35
55 Y. G. Sun, S. H. Saker. Forced Oscillation of Higher Order Nonlinear Difference Equations. Appl. Math. Comput., 2007,187(2):868-872
56 Y. G. Sun, A. B. Mingarelli. Oscillation of Higher-Order Forced Nonlinear Differential Equations. Appl. Math. Comput., 2007,190(1):905-911
57 Y. Zhou, Y. Q. Huang. Existence for Nonoscillatory Solutions of Higher-Order Nonlinear Neutral Difference Equations. J. Math. Anal. Appl., 2003,280(1):63-76
58 Y. Zhou, B. G. Zhang. Existence of Nonoscillatory Solutions of Higher-Order Neutral Delay Difference Equations with Variable Coefficients. Comput. Math. Appl., 2003, 45(1-2):991-1000
59 Y. Zhou. Existence of Nonoscillatory Solutions of Higher-Order Neutral Difference Equations with General Coefficients. Appl. Math. Lett., 2002,15(7):785-791
60 Y. Zhou, B. G. Zhang. Existence of Nonoscillatory Solutions of Higher-Order Neutral Differential Equations with Positive and Negative Coefficients. Appl. Math. Lett., 2002, 15(7):867-874
61 R. P. Agarwal, S. R. Grace. The Oscillation of Higher-Order Differential Equations with Deviating Arguments. Comput. Math. Appl., 1999,38(3-4):185-199
62谭合理.测度链上高阶时滞动力方程解的振动性和渐近性.[湘潭大学硕士论文]. 2005: 17-18
63 D. Guo, V. Lakshmikantham. Nonlinear Problems in Abstract Cones. Academic Press, San Diego, 1988
64 R. W. Leggett, L. R. Williams. Multiple Positive Fixed Points of Nonlinear Operators on Ordered Banach Spaces. Indiana Univ. Math. J., 1979,28:673-688
65 K. Deimling. Nonlinear Functional Analysis. Springer-Verlag, New York, 1985
66 J. L. Ren, W. G. Ge, B. X. Ren. Existence of Positive Solutions for Quasi-Linear Boundary Value Problems. Acta Math. Appl. Sinica, 2005,21(3):353-358
67 S. H. Hong. Triple Positive Solutions of Three-Point Boundary Value Problems for p ?Laplacian Dynamic Equations on Time Scales. J. Comput. Appl. Math., 2007,206: 967-976
68 Y. S. Liu. Existence and Unboundedness of Positive Solutions for Singular Boundary Value Problems on Half-Line. Appl. Math. Comput., 2003,1404:543-556
2 F. M. Atici, D. C. Biles, A. Lebedinsky. An Application of Time Scales to Economics. Math. Comput. Modelling, 2006,43(7-8):718-726
3 S. Hilger. Analysis on Measure Chains-a Unified Approach to Continuous and Discrete Calculus. Results Math., 1990,18:18-56
4张炳根.测度链上微分方程的进展.中国海洋大学学报,2004,34:907-912
5 M. Bohner, A. Peterson. Dynamic Equations on Time Scales: An Introduction with Applications. Boston: Birkh?user, MA, 2001
6 M. Bohner, A. Peterson. Advances in Dynamic Equations on Time Scales. Boston: Birkh?user, MA, 2003
7 B. Kaymakcalan, V. Lakshmikantham, S. Sivasundaram. Dynamic Systems on Measure chains. Kluwer Academic Publishers, Boston, MA, 1996
8 R. P. Agarwal, M. Boher, D. O'Regan, A. Peterson. Dynamic Equations on Time Scales: a Survey. J. Comput. Appl. Math., 2002,141(1-2):1-26
9 Z. Q. Zhu, Q. R, Wang. Existence of Nonoscillatory Solutions to Neutral Dynamic Equations on Time Scales. J. Math. Anal. Appl., 2007,335(2):751-762
10 R. P. Agarwal, M. Bohner, D. O'Regan. Time Scale Systems on Infinite Intervals. Nonlinear Anal., 2001,47(2):837-848
11 F. M. Aticl, D. C. Biles. First Order Dynamic Inclusions on Time Scales. J. Math. Anal. Appl., 2004,292(1):222-237
12 J. P. Sun, W. T. Li. Positive Solutions for System of Nonlinear First-Order PBVPs on Time Scales. Nonlinear Anal., 2005,62(1):131-139
13 Z. C. Hao, T. J. Xiao, J. Liang. Existence of Positive Solutions for Singular Boundary Value Problem on Time Scales. J. Math. Anal. Appl., 2007,325(1):517-528
14 D. R. Anderson, R. I. Avery. An Even-Order Three-Point Boundary Value Problem onTime Scales. J. Math. Anal. Appl., 2004,291(2):514-525
15 C. J. Chyan. Uniqueness Impiles Existence on Time Scales. J. Math. Anal. Appl., 2001, 258:359-365
16 D. R. Anderson. Taylor Polynomials for Nabla Dynamic Equations on Time Scales. Panamer. Math. J., 2002,12:17-27
17 C. J. Chyan, J. Henderson. Eigenvalue Problems for Nonlinear Diferential Equations on a Measure Chain. J. Math. Anal. Appl., 2000,245:547-559
18 P. W. Eloe, Q. Sheng, J. Henderson. Notes on Crossed Symmetry Solutions of the Two-Point Boundary Value Problems on Time Scales. J. Difference Equations Appl., 2003,9: 29-48
19 B. G. Zhang, X. H. Deng. Oscillation of Delay Differential Equations on Time Scales. Math. Comput. Modelling, 2002,36(11-13):1307-1318
20刘爱莲,朱思铭,吴洪武.二阶时标动力系统的振动准则.中山大学学报(自然科学版), 2004,43(2):9-12
21 S. H. Saker. Oscillation of Nonlinear Dynamic Equations on Time Scales. Appl. Math. Comput., 2004,148(1):81-91
22 Y. Sahiner. Oscillation of Second-Order Delay Differential Equations on Time Scales. Nonlinear Anal., 2005,63(5-7):1073-1080
23 S. H. Saker. Oscillation Criteria of Second-Order Half-Linear Dynamic Equations on Time Scales. J. Comput. Appl. Math., 2005,177(2):375-387
24 S. H. Saker. Oscillation of Second-Order Nonlinear Neutral Delay Dynamic Equations on Time Scales. J. Comput. Appl. Math., 2006,187(2):123-141
25 Z. L. Han, S. R. Sun, B. Shi. Oscillation Criteria for a Class of Second-Order Emden-Fowler Delay Dynamic Equations on Time Scales. J. Math. Anal. Appl., 2007, 334(2):847–858
26 L. Erbe, A. Peterson. Green’s Functions and Comparison Theorems for Differential Equations on Measure Chains. Dynam. Contin. Discrete Impuls. Systems, 1999,6: 121-137
27 L. Erbe, A. Peterson. Positive Solutions for a Nonlinear Differential Equation on aMeasure Chain. Math. Comput. Modelling, 2000,32(5-6):571-585
28 R. P. Agarwal, D. O'Regan. Triple Solutions to Boundary Value Problems on Time Scales. Appl. Math. Lett., 2000,13(4):7-11
29 R. P. Agarwal, D. O'Regan. Nonlinear Boundary Value Problems on Time Scales. Nonlinear. Aanl., 2001,44(4):527-535
30 C. J. Chyan, J. Henderson. Twin Solutions of Boundary Value Problems for Differential Equations on Measure Chains. J. Comput. Appl. Math., 2002,141(1-2): 123-131
31 R. I. Avery, J. Henderson. Two Positive Fixed Points of Nonlinear Operator on Ordered Banach Spaces. Comm. Appl. Nonlinear Anal., 2001,8:27-36
32 F. M. Atici, G. Sh. Guseinov. On Green’s Functions and Positive Solutions for Boundary Value Problems on Time Scales. J. Comput. Appl. Math., 2002(1-2),141: 75-99
33 Z. C. Hao, J. Liang, T. J. Xiao. Existence Results for Time Scale Boundary Value Problem. J. Comput. Appl. Math., 2006,197(1):156-168
34 D. R. Anderson. Solutions to Second-Order Three-Point Problems on time scales. J. Difference Equations Appl., 2002,8:673-688
35 E. R. Kaufmann. Positive Solutions of a Three-Point Boundary Value Problem on a Time Scale. Electronic J. Differential Equations, 2003,82:1-11
36孙红蕊,李万同.测度链上非线性m ?点边值问题的正解.数学学报,2006,49(2): 369-380
37 Z. M. He. Existence of Two Solutions of m ? Point Boundary Value Problem for Second Order Dynamic Equations on Time Scales. J. Math. Anal. Appl., 2004,296(1): 97-109
38 R. I. Avery, D. R. Anderson. Existence of Three Positive Solutions to a Second-Order Boundary Value Problem on a Measure Chain. J. Comput. Appl. Math., 2002,141(1-2): 65-73
39 R. I. Avery, C. J. Chyan, J. Henderson. Twin Solutions of Boundary Value Problems for Ordinary Differential Equations and Finite Difference Equations. Comput. Math.Appl., 2001,42(3-5):695-704
40 H. Amann. Fixed Point Equations and Nonlinear Eigenvalue Problems in Ordered Banach Space. Siam Review, 1976,18:620-709
41 H. R. Sun, W. T. Li. Positive Solutions for Nonlinear Three-Point Boundary Value Problem on Time Scales. J. Math. Anal. Appl., 2004,299(2):508-524
42 Y. K. Chang, W. T. Li. Existence Results for Second-Order Dynamic Inclusion with m?Point Boundary Value Conditions on Time Scales. Appl. Math. Lett., 2007,20(8): 885-891
43 P. Amster, C. Rogers, C. C. Tisdell. Existence of Solutions to Boundary Value Problems for Dynamic Systems on Time Scales. J. Math. Anal. Appl., 2005,308(2): 565-577
44 Z. M. He, L. Li. Multiple Positive Solutions for the One-Dimensional p ? Laplacian Dynamic Equations on Time Scales. Math. Comput. Modelling, 2007,45(1-2):68-79
45 H. R. Sun, W. T. Li. Multiple Positive Solutions for p ? Laplacian m ? Point Boundary Value Problems on Time Scales. Appl. Math. Comput., 2006,182(1):478-491
46 Z. M. He, X. M. Jiang. Triple Positive Solutions of Boundary Value Problems for p ?Laplacian Dynamic Equations on Time Scales. J. Math. Anal. Appl., 2006,32(2): 911-920
47 C. D. Ahlbrandt, C. Morian. Partial Differential Equations on Time Scales. J. Comput. Appl. Math., 2002,141:35-55
48 B. Jackson. Partial Dynamic Equations on Time Scales. J. Comput. Appl. Math., 2006, 186:391-415
49 L. Erbe, A. Peterson. S. H. Saker. Oscillation Criteria for Second-Order Nonlinear Delay Dynamic Equations. J. Math. Anal. Appl., 2007,333(1):505-522
50 Y. Bolat, ?. Akin. Oscillatory Behaviour of a Higher-Order Nonlinear Neutral Type Functional Difference Equation with Oscillating Coefficients. Appl. Math. Lett., 2004, 17(9):1073-1078
51 X. P. Wang. Oscillation for Higher Order Nonlinear Delay Differential Equations. Appl. Math. and Comput., 2004,157(1):287-294
52 S. R. Grace, B. S. Lalli. Oscillation of Higher-Order Nonlinear Difference Equations. Math. Comput. Modelling, 2005,41(4-5):485-491
53 T. Candan, R. S. Dahiya. Oscillation Behavior of Higher Order Neutral Differential Equations. Appl. Math. Comput., 2005,167(2):1267-1280
54杨军,关新平,刘树堂.具强迫项高阶非线性中立型差分方程的振动性与渐进性.纯粹数学与应用数学,1999,15(4):29-35
55 Y. G. Sun, S. H. Saker. Forced Oscillation of Higher Order Nonlinear Difference Equations. Appl. Math. Comput., 2007,187(2):868-872
56 Y. G. Sun, A. B. Mingarelli. Oscillation of Higher-Order Forced Nonlinear Differential Equations. Appl. Math. Comput., 2007,190(1):905-911
57 Y. Zhou, Y. Q. Huang. Existence for Nonoscillatory Solutions of Higher-Order Nonlinear Neutral Difference Equations. J. Math. Anal. Appl., 2003,280(1):63-76
58 Y. Zhou, B. G. Zhang. Existence of Nonoscillatory Solutions of Higher-Order Neutral Delay Difference Equations with Variable Coefficients. Comput. Math. Appl., 2003, 45(1-2):991-1000
59 Y. Zhou. Existence of Nonoscillatory Solutions of Higher-Order Neutral Difference Equations with General Coefficients. Appl. Math. Lett., 2002,15(7):785-791
60 Y. Zhou, B. G. Zhang. Existence of Nonoscillatory Solutions of Higher-Order Neutral Differential Equations with Positive and Negative Coefficients. Appl. Math. Lett., 2002, 15(7):867-874
61 R. P. Agarwal, S. R. Grace. The Oscillation of Higher-Order Differential Equations with Deviating Arguments. Comput. Math. Appl., 1999,38(3-4):185-199
62谭合理.测度链上高阶时滞动力方程解的振动性和渐近性.[湘潭大学硕士论文]. 2005: 17-18
63 D. Guo, V. Lakshmikantham. Nonlinear Problems in Abstract Cones. Academic Press, San Diego, 1988
64 R. W. Leggett, L. R. Williams. Multiple Positive Fixed Points of Nonlinear Operators on Ordered Banach Spaces. Indiana Univ. Math. J., 1979,28:673-688
65 K. Deimling. Nonlinear Functional Analysis. Springer-Verlag, New York, 1985
66 J. L. Ren, W. G. Ge, B. X. Ren. Existence of Positive Solutions for Quasi-Linear Boundary Value Problems. Acta Math. Appl. Sinica, 2005,21(3):353-358
67 S. H. Hong. Triple Positive Solutions of Three-Point Boundary Value Problems for p ?Laplacian Dynamic Equations on Time Scales. J. Comput. Appl. Math., 2007,206: 967-976
68 Y. S. Liu. Existence and Unboundedness of Positive Solutions for Singular Boundary Value Problems on Half-Line. Appl. Math. Comput., 2003,1404:543-556
© 2004-2018 中国地质图书馆版权所有 京ICP备05064691号 京公网安备11010802017129号 地址：北京市海淀区学院路29号 邮编：100083 电话：办公室：(+86 10)66554848；文献借阅、咨询服务、科技查新：66554700 |