几类具有偏差变元的微分方程解的振动性
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摘要
随着科学技术的进步与发展,在物理学、自动控制、生物学、医学和经济学等许多自然学科和边缘学科领域中提出了大量的由微分方程描述的具体数学模型.微分方程是用来描述自然现象变化规律的一种有力工具,由于寻求其通解十分困难,故从理论上探讨解的性态一直是近年来研究的热点问题.
本文将利用各种已知技术研究具有偏差变元的微分方程的振动性和非振动性理论.我们的工作主要集中在两方面：一方面是具有偏差变元常微分方程的振动性理论；另一方面是具有偏差变元偏微分方程的振动性理论.本文由四章组成,主要内容如下：
第一章概述常(偏)微分方程的应用背景和国内、外研究现状以及本人的主要工作
第二章通过使用广义Riccati变换和积分算子,我们获得了一类具有连续偏差变元和阻尼项二阶中立型微分方程的振动准则,其中一些结果不同于许多已有结果,并且可应用于不能被现有结果所包含的一些情况中.
第三章我们考虑一类具有连续偏差变元中立型双曲微分方程,得到了两种情况下一些边值问题的任一解在圆柱域上振动的充分条件.
第四章研究了具有依赖于未知函数偏差变元的高阶微分方程的非振动性.
With the increasing development of natural science, in the field of natural science including physics, theory of control, biology, medicine, economies and edging field, many mathematical models which are described by differential equations are proposed. Differential equations are powerful tools that describe the law of nature, but it is difficult to find their general solutions. Therefore, there has been an increasing interest in the study of the nature of solutions of differ-ential equation in theory.
This paper will use a variety of known techniques research the oscillation and nonoscilla-tion of differential equations with deviating arguments. This dissertation focuses on two sides: one is the oscillation of ordinary differential equations with deviating arguments, the other is the partial differential equations with deviating arguments. The paper is made up of four chapters. Main contents are as follows:
In chapter one, we give a survey to the development and current state of oscillation of or-dinary(partial) differential equations with deviating arguments, as well as the main work status.
In chapter two, using some integral operators and generalized Riccati technique, we estab-lish some new oscillation criteria for a second order neutral differential equation with deviating arguments. The obtained results are different from most known ones and can be applied to many cases which are not covered by existing.
In chapter three, we consider certain neutral hyperbolic equations with continuous deviat-ing arguments, and sufficient conditions presented for every solution of some boundary value problems under two different cases to be oscillatory in a cylindrical domain.
In chapter four, nonoscillation of higher order differential equation with deviating argu-ments depending on the unknown function is consider.
本文将利用各种已知技术研究具有偏差变元的微分方程的振动性和非振动性理论.我们的工作主要集中在两方面：一方面是具有偏差变元常微分方程的振动性理论；另一方面是具有偏差变元偏微分方程的振动性理论.本文由四章组成,主要内容如下：
第一章概述常(偏)微分方程的应用背景和国内、外研究现状以及本人的主要工作
第二章通过使用广义Riccati变换和积分算子,我们获得了一类具有连续偏差变元和阻尼项二阶中立型微分方程的振动准则,其中一些结果不同于许多已有结果,并且可应用于不能被现有结果所包含的一些情况中.
第三章我们考虑一类具有连续偏差变元中立型双曲微分方程,得到了两种情况下一些边值问题的任一解在圆柱域上振动的充分条件.
第四章研究了具有依赖于未知函数偏差变元的高阶微分方程的非振动性.
With the increasing development of natural science, in the field of natural science including physics, theory of control, biology, medicine, economies and edging field, many mathematical models which are described by differential equations are proposed. Differential equations are powerful tools that describe the law of nature, but it is difficult to find their general solutions. Therefore, there has been an increasing interest in the study of the nature of solutions of differ-ential equation in theory.
This paper will use a variety of known techniques research the oscillation and nonoscilla-tion of differential equations with deviating arguments. This dissertation focuses on two sides: one is the oscillation of ordinary differential equations with deviating arguments, the other is the partial differential equations with deviating arguments. The paper is made up of four chapters. Main contents are as follows:
In chapter one, we give a survey to the development and current state of oscillation of or-dinary(partial) differential equations with deviating arguments, as well as the main work status.
In chapter two, using some integral operators and generalized Riccati technique, we estab-lish some new oscillation criteria for a second order neutral differential equation with deviating arguments. The obtained results are different from most known ones and can be applied to many cases which are not covered by existing.
In chapter three, we consider certain neutral hyperbolic equations with continuous deviat-ing arguments, and sufficient conditions presented for every solution of some boundary value problems under two different cases to be oscillatory in a cylindrical domain.
In chapter four, nonoscillation of higher order differential equation with deviating argu-ments depending on the unknown function is consider.
引文
[1]郑祖庥.泛函微分方程理论[M].合肥：安徽教育出版社,1994.
[2]J. K. Hale. Theory of functional differential equations[M]. New York:Springer,1977.
[3]R. P. Agarwal, S. R. Grace and D. O'Regan. Oscillation theory for second order dynamic equa-tions[M]. London and New York:Talyor & Francis,2003.
[4]燕居让.常微分方程振动理论[M].太原：山西教育出版社,1992.
[5]V. N. Shevelo. Oscillation of Solutions of Differential Equations with Deviating Arguments[M]. Kiev: Naukova Dumka,1978.
[6]G. S. Ladde, V. Lakshmikantham and B. G. Zhang. Oscillation theory of differential equations with deviating arguments[M]. New York:Marcel Dekker,1987.
[7]A. I. Zahariev and D. D. Bainov. Oscillating properties of the solutions of a class of neural type functional differential equations[J]. Bull. Austral. Math. Soc.,1980(22):365-372.
[8]Z. Xu and P. Weng. Oscillation of second order neutral equations with distributed deviating argu-ment[J]. J. Comput. Appl. Math.,2007(202):460-477.
[9]Ch. G. Philos. Oscillation theorems for linear differential equations of second order[J]. Arch. Math.(Basel), 1989(53):482-492.
[10]J. S. W. Wong. On Kamenev-type oscillation theorems for second order differential equations with damping[J]. J. Math. Anal. Appl.,2001(258):244-257.
[11]F. Lu and F. Meng. Oscillation theorems for superlinear second-order damped differential equa-tions[J]. Appl. Math. Comput.,2007(189):796-804.
[12]P. Wang. Oscillation criteria for second order neutral equations with distributed deviating argu-ments[J]. J. Comput. Appl. Math.,2004(47):1935-1946.
[13]J. H. Wu. Theory and applications of partial functional differential equations[M]. New York:Springer, 1977.
[14]D. P. Mishev and D. D. Bainov. Oscillation properties of the solutions of hyperbolic equations of neutral type[C]. Colloq. Math. Soc. Janos Bolyai(Proceedings of the Colloquium on Qualitative Theory of Differential Equations, Szeged,1984). Amsterdam,47(2),771-780.
[15]D. D. Bainov and D. P. Mishev. Oscillation Theory for neutral differential equations with delay[M]. Bristol:Adam Hilger,1991.
[16]Y. Shoukaku. Forced oscillation of certain neutral hyperbolic equations with continuous distributed deviating arguments[J]. Math. Comput. Modelling,2009(49):1211-1220.
[17]P. G. Wang. Oscillation of certain neutral hyperbolic equations[J]. Indian J. Pure Appl. Math., 2000(31):949-956.
[18]L. H. Deng, W. G. Ge and P. G. Wang. Oscillation of hyperbolic equations with continuous deviating argument under the Robin boundary condition[J]. Soochow J. Math.,2003(29):1-6.
[19]P. G. Wang and Y. H. Yu. Oscillation criteria for a nonlinear hyperbolic equation boundary value problem[J]. Appl. Math. Lett.,1999(12):91-98.
[20]X. Z. Liu and X. L. Fu. Oscillation criteria for nonlinear inhomogeneous hyperbolic equations with distributed deviating arguments[J]. J. Appl. Math. Stoch. Anal.,1996(9):21-31.
[21]N. T. Markova and P. S. Simeonov. Existence of nonoscillatory solutions tending to zero at ∞ for differential equations with retarded arguments depending on the unknown function[J]. Comm. Appl. Anal,2007,11(1):113-120.
[22]Z. Xu. Oscillation and nonoscillation of second order differential equations with delay depending on the unknown function[J]. J. Comput. Appl. Math.,2008(214):371-380.
[23]D. D. Bainov, N. T. Markova and P. S. Simeonov. Asymptotic behavior of the nonoscillation solutions of second order differential equations with delay depending on the unknown function[J]. J. Comput. Appl. Math.,1998(91):87-96.
[24]X. Fan, W. T. Li and C. Zhong. A classification scheme for positive solutions of second order nonlinear iterative differential equations[J]. Electron. J. Differ. Equ.,2000,2000(25):1-14.
[25]N. T. Markova and P. S. Simeonov. Asymptotic behavior of oscillatory solutions of n-th order dif-ferential equations with deviating arguments depending on the unknown function[J]. Comm. Appl. Anal.,2007,11(2):235-246.
[26]Z. Xu. Nonlinear oscillation of second order differential equations with state-dependent delays[J]. Appl. Math. Comput.,2008(206):198-204.
[27]J. Luo. Asymptotic behavior of solutions of second order quasilinear differential equations with delay depending on the unknown function[J]. J. Comput. Appl. Math.,2006(185):133-143.
[28]F. Hartung, T. Krisztin, H.-O. Walther, et al. Functional differential equations with state-dependent delays:Theory and Applications[M]. North Holland:Elsevier B. V.,2006.
[29]M. Bohner and S. H. Saker. Oscillation of second order nonlinear delay differential equations of Emden-Fowler type[J]. Advances in Dynamical Systems and Applications,2006(1):163-182.
[30]W. Li and R. P. Agarwal. Interval oscillation criteria for second order nonlinear differential equations with damping[J]. J. Comput. Appl. Math.,2000(20):217-230.
[31]K. Deimling. Nonlinear functional analysis[M]. Berlin:Springer,1985.
[2]J. K. Hale. Theory of functional differential equations[M]. New York:Springer,1977.
[3]R. P. Agarwal, S. R. Grace and D. O'Regan. Oscillation theory for second order dynamic equa-tions[M]. London and New York:Talyor & Francis,2003.
[4]燕居让.常微分方程振动理论[M].太原：山西教育出版社,1992.
[5]V. N. Shevelo. Oscillation of Solutions of Differential Equations with Deviating Arguments[M]. Kiev: Naukova Dumka,1978.
[6]G. S. Ladde, V. Lakshmikantham and B. G. Zhang. Oscillation theory of differential equations with deviating arguments[M]. New York:Marcel Dekker,1987.
[7]A. I. Zahariev and D. D. Bainov. Oscillating properties of the solutions of a class of neural type functional differential equations[J]. Bull. Austral. Math. Soc.,1980(22):365-372.
[8]Z. Xu and P. Weng. Oscillation of second order neutral equations with distributed deviating argu-ment[J]. J. Comput. Appl. Math.,2007(202):460-477.
[9]Ch. G. Philos. Oscillation theorems for linear differential equations of second order[J]. Arch. Math.(Basel), 1989(53):482-492.
[10]J. S. W. Wong. On Kamenev-type oscillation theorems for second order differential equations with damping[J]. J. Math. Anal. Appl.,2001(258):244-257.
[11]F. Lu and F. Meng. Oscillation theorems for superlinear second-order damped differential equa-tions[J]. Appl. Math. Comput.,2007(189):796-804.
[12]P. Wang. Oscillation criteria for second order neutral equations with distributed deviating argu-ments[J]. J. Comput. Appl. Math.,2004(47):1935-1946.
[13]J. H. Wu. Theory and applications of partial functional differential equations[M]. New York:Springer, 1977.
[14]D. P. Mishev and D. D. Bainov. Oscillation properties of the solutions of hyperbolic equations of neutral type[C]. Colloq. Math. Soc. Janos Bolyai(Proceedings of the Colloquium on Qualitative Theory of Differential Equations, Szeged,1984). Amsterdam,47(2),771-780.
[15]D. D. Bainov and D. P. Mishev. Oscillation Theory for neutral differential equations with delay[M]. Bristol:Adam Hilger,1991.
[16]Y. Shoukaku. Forced oscillation of certain neutral hyperbolic equations with continuous distributed deviating arguments[J]. Math. Comput. Modelling,2009(49):1211-1220.
[17]P. G. Wang. Oscillation of certain neutral hyperbolic equations[J]. Indian J. Pure Appl. Math., 2000(31):949-956.
[18]L. H. Deng, W. G. Ge and P. G. Wang. Oscillation of hyperbolic equations with continuous deviating argument under the Robin boundary condition[J]. Soochow J. Math.,2003(29):1-6.
[19]P. G. Wang and Y. H. Yu. Oscillation criteria for a nonlinear hyperbolic equation boundary value problem[J]. Appl. Math. Lett.,1999(12):91-98.
[20]X. Z. Liu and X. L. Fu. Oscillation criteria for nonlinear inhomogeneous hyperbolic equations with distributed deviating arguments[J]. J. Appl. Math. Stoch. Anal.,1996(9):21-31.
[21]N. T. Markova and P. S. Simeonov. Existence of nonoscillatory solutions tending to zero at ∞ for differential equations with retarded arguments depending on the unknown function[J]. Comm. Appl. Anal,2007,11(1):113-120.
[22]Z. Xu. Oscillation and nonoscillation of second order differential equations with delay depending on the unknown function[J]. J. Comput. Appl. Math.,2008(214):371-380.
[23]D. D. Bainov, N. T. Markova and P. S. Simeonov. Asymptotic behavior of the nonoscillation solutions of second order differential equations with delay depending on the unknown function[J]. J. Comput. Appl. Math.,1998(91):87-96.
[24]X. Fan, W. T. Li and C. Zhong. A classification scheme for positive solutions of second order nonlinear iterative differential equations[J]. Electron. J. Differ. Equ.,2000,2000(25):1-14.
[25]N. T. Markova and P. S. Simeonov. Asymptotic behavior of oscillatory solutions of n-th order dif-ferential equations with deviating arguments depending on the unknown function[J]. Comm. Appl. Anal.,2007,11(2):235-246.
[26]Z. Xu. Nonlinear oscillation of second order differential equations with state-dependent delays[J]. Appl. Math. Comput.,2008(206):198-204.
[27]J. Luo. Asymptotic behavior of solutions of second order quasilinear differential equations with delay depending on the unknown function[J]. J. Comput. Appl. Math.,2006(185):133-143.
[28]F. Hartung, T. Krisztin, H.-O. Walther, et al. Functional differential equations with state-dependent delays:Theory and Applications[M]. North Holland:Elsevier B. V.,2006.
[29]M. Bohner and S. H. Saker. Oscillation of second order nonlinear delay differential equations of Emden-Fowler type[J]. Advances in Dynamical Systems and Applications,2006(1):163-182.
[30]W. Li and R. P. Agarwal. Interval oscillation criteria for second order nonlinear differential equations with damping[J]. J. Comput. Appl. Math.,2000(20):217-230.
[31]K. Deimling. Nonlinear functional analysis[M]. Berlin:Springer,1985.