几类微分包含周期解的存在性及可控性研究
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摘要
微分包含是非线性分析理论的重要分支,它与微分方程、最优控制及最优化等其它数学分支有着紧密的联系。微分包含周期解的存在性和可控性是微分包含理论的基本内容。本文主要研究了周期问题,给出了几类微分包含周期解的存在性定理,并对几类微分包含的可控性进行了研究。
1.在有限维空间讨论了半线性发展包含的周期问题。当满足单边Lipschtz条件时,借助于集值分析理论和不动点定理,获得了凸和非凸两种情况下周期解的存在性定理。对于非凸情形,使用单值的Leray-Schauder替换定理获得周期解的存在的充分条件。对于凸情形,利用集值的Leray-Schauder替换定理获得所要的结论。利用Tolstonogov端点连续选择定理,证明了端点周期解的存在性,并证明了端点周期解的稠密性(强松驰定理)。F (t , x)
2.在可分的Banach空间讨论了一类积分微分包含的周期问题。借助于解的积分表示和不动点定理,分别获得了凸和非凸两种情形下周期解存在的充分条件。对于凸情形,利用Kakutani不动点定理,对于非凸情形,利用连续选择的方法和Tychonoff不动点定理。
3.讨论了一类非自治微分包含的周期问题。当向量场满足单边Lipschitz条件时,以Leray-Schauder替换定理(单值形式和集值形式)为工具,获得了凸和非凸两种情形周期解存在的充分条件。利用端点连续选择定理获得端点周期解的存在性,并证明了强松驰定理。F (t , x)
4.讨论了一类发展包含的可控性。处理方法是将所讨论的问题转化为集值积分算子的不动点问题,利用凝聚映射的不动点定理,获得了可控性的充分条件。
5.利用不动点定理讨论了一类积分微分包含的可控性问题。在凸和非凸两种情形下建立了可控性的充分条件。对于凸情形,处理方法是将所讨论的问题转化为集值积分算子的不动点问题,利用Kakutani不动点定理获得可控性。对于非凸情形,是将所讨论的问题转化为单值的积分算子的不动点问题,利用Schauder不动点定理获得所要的结论。
6.利用Galerkin逼近方法,将有限维结果推广到无穷维,在发展三元组框架下,证明了在无穷维空间半线性发展包含周期解的存在性定理。并把得到的结果应用于偏微分包含,给出了一类偏微分包含的周期解存在的充分条件。
Differential inclusion is an important branch of nonlinear analysis, which has close relationships with other branches of mathematics such as differential equation, optimal control and optimization. The existence of periodic solutions and controllability are basic contents of differential inclusions. The main content of this paper is the periodic problems. Particularly, we give the existence theorems of the periodic solutions for several types of differential inclusions. Moreover we study the controllability several classes of differential inclusions.
1. In finite dimensional spaces, we discuss the periodic problems for semi-linear evolution inclusions. WhenF (t , x) satisfies one-side Lipschitz condition, using techniques from multivalued analysis and fixed point theory, we establish the existence theorems for convex and nonconvex cases. In the nonconvex case, we obtain the sufficient conditions for the existence of periodic solutions by using single-valued Leray-Schauder alternative theorem. In the convex case, the desirable resut has been obtained by using set-valued Leray-Schauder alternative theorem. On the basis of Tolstonogov extremal continuous selection theorem, we prove the existence of extermal periodic solutions and the density of extermal periodic solutions (the strong relaxation theorem).
2. In separable Banach Space, we study the periodic problems for a class of integrodifferential inclusions. By applying integral expression of solutions and fixed point theorems, we obtain the sufficient conditions for the existence of periodic solutions both convex and nonconvex case. For convex case, we use Kakutani fixed point theorem. For nonconvex case, the method of continuous selections and Tychonoff fixed point theorem are used.
3. Periodic problems for a class of non-autonomous differential inclusions are debated. Utilizing Leray-Schauder alternative theorem we give the sufficient conditions for the existence of periodic solusions for both convex and nonconvex problems when orientor field F (t , x) satisfy one-side Lipschitz continuous. By the theorem of the continuous selection on extremal points, we give the sufficient conditions for existence of extremal solusions and prove the strong relaxation theorem.
4. The controllability of a class of evolution inclusions has been debated. Our method is based on the transition from the problems discussed to the fixed point problems of set-valued integral operators. By applying a fixed point theorem for condesing maps, the sufficient conditions of controllability are obtained.
5. The controllability of a class of integrodifferential inclusions is studied by fixed point theorem. For convex and nonconvex cases, we establish sufficient conditions for controllability separately. For convex case, our method is based on the transition from the problems discussed to the fixed point problems of set-valued integral operators. By using Kakutani fixed point theorem, we give the sufficient conditions of controllability. For nonconvex case, we convert the problem into fixed point problem of single-valued integral operators and obtain the desirable result by Schauder fixed point theorem.
6. Using Galerkin approximation, we extend the finite dimensional results to infinite dimensional space. Under the framework of evolution triple of spaces, we proved the existence theorem of semi-linear evolution inclusion. Moreover we apply the results obtained to a class of partial differential inclusion, the sufficient conditions of existence for periodic solusions are given.
1.在有限维空间讨论了半线性发展包含的周期问题。当满足单边Lipschtz条件时,借助于集值分析理论和不动点定理,获得了凸和非凸两种情况下周期解的存在性定理。对于非凸情形,使用单值的Leray-Schauder替换定理获得周期解的存在的充分条件。对于凸情形,利用集值的Leray-Schauder替换定理获得所要的结论。利用Tolstonogov端点连续选择定理,证明了端点周期解的存在性,并证明了端点周期解的稠密性(强松驰定理)。F (t , x)
2.在可分的Banach空间讨论了一类积分微分包含的周期问题。借助于解的积分表示和不动点定理,分别获得了凸和非凸两种情形下周期解存在的充分条件。对于凸情形,利用Kakutani不动点定理,对于非凸情形,利用连续选择的方法和Tychonoff不动点定理。
3.讨论了一类非自治微分包含的周期问题。当向量场满足单边Lipschitz条件时,以Leray-Schauder替换定理(单值形式和集值形式)为工具,获得了凸和非凸两种情形周期解存在的充分条件。利用端点连续选择定理获得端点周期解的存在性,并证明了强松驰定理。F (t , x)
4.讨论了一类发展包含的可控性。处理方法是将所讨论的问题转化为集值积分算子的不动点问题,利用凝聚映射的不动点定理,获得了可控性的充分条件。
5.利用不动点定理讨论了一类积分微分包含的可控性问题。在凸和非凸两种情形下建立了可控性的充分条件。对于凸情形,处理方法是将所讨论的问题转化为集值积分算子的不动点问题,利用Kakutani不动点定理获得可控性。对于非凸情形,是将所讨论的问题转化为单值的积分算子的不动点问题,利用Schauder不动点定理获得所要的结论。
6.利用Galerkin逼近方法,将有限维结果推广到无穷维,在发展三元组框架下,证明了在无穷维空间半线性发展包含周期解的存在性定理。并把得到的结果应用于偏微分包含,给出了一类偏微分包含的周期解存在的充分条件。
Differential inclusion is an important branch of nonlinear analysis, which has close relationships with other branches of mathematics such as differential equation, optimal control and optimization. The existence of periodic solutions and controllability are basic contents of differential inclusions. The main content of this paper is the periodic problems. Particularly, we give the existence theorems of the periodic solutions for several types of differential inclusions. Moreover we study the controllability several classes of differential inclusions.
1. In finite dimensional spaces, we discuss the periodic problems for semi-linear evolution inclusions. WhenF (t , x) satisfies one-side Lipschitz condition, using techniques from multivalued analysis and fixed point theory, we establish the existence theorems for convex and nonconvex cases. In the nonconvex case, we obtain the sufficient conditions for the existence of periodic solutions by using single-valued Leray-Schauder alternative theorem. In the convex case, the desirable resut has been obtained by using set-valued Leray-Schauder alternative theorem. On the basis of Tolstonogov extremal continuous selection theorem, we prove the existence of extermal periodic solutions and the density of extermal periodic solutions (the strong relaxation theorem).
2. In separable Banach Space, we study the periodic problems for a class of integrodifferential inclusions. By applying integral expression of solutions and fixed point theorems, we obtain the sufficient conditions for the existence of periodic solutions both convex and nonconvex case. For convex case, we use Kakutani fixed point theorem. For nonconvex case, the method of continuous selections and Tychonoff fixed point theorem are used.
3. Periodic problems for a class of non-autonomous differential inclusions are debated. Utilizing Leray-Schauder alternative theorem we give the sufficient conditions for the existence of periodic solusions for both convex and nonconvex problems when orientor field F (t , x) satisfy one-side Lipschitz continuous. By the theorem of the continuous selection on extremal points, we give the sufficient conditions for existence of extremal solusions and prove the strong relaxation theorem.
4. The controllability of a class of evolution inclusions has been debated. Our method is based on the transition from the problems discussed to the fixed point problems of set-valued integral operators. By applying a fixed point theorem for condesing maps, the sufficient conditions of controllability are obtained.
5. The controllability of a class of integrodifferential inclusions is studied by fixed point theorem. For convex and nonconvex cases, we establish sufficient conditions for controllability separately. For convex case, our method is based on the transition from the problems discussed to the fixed point problems of set-valued integral operators. By using Kakutani fixed point theorem, we give the sufficient conditions of controllability. For nonconvex case, we convert the problem into fixed point problem of single-valued integral operators and obtain the desirable result by Schauder fixed point theorem.
6. Using Galerkin approximation, we extend the finite dimensional results to infinite dimensional space. Under the framework of evolution triple of spaces, we proved the existence theorem of semi-linear evolution inclusion. Moreover we apply the results obtained to a class of partial differential inclusion, the sufficient conditions of existence for periodic solusions are given.
引文
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112 E. Hewitt, K. R. Stromberg. Real and Abstract Analysis— A Modern Treatment of the Theory of Functions of a Real Variable. Springer-Verlag, Berlin, 1978
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116 M. Martelli. A Rothe Type Theorem for Noncompact Acyclic-Valued Map. Boll. Un. Mat. Ital. 1975, 4:70-76
117 F. Hiai, H. Umegaki. Integrals, Conditional Expectations and Martingales of Multivalued Functions J. Multivariate Anal. 1977, 7:149-182
118 N. S. Papageorgiou. Convergence Theorems for Banach Space Valued Integrable Multifunctions. Internat. J. Math. Sci. 1987, 10:433-442
119 M. Forti, P. Nistri. Global Convergence of Neural Networks with Dis-continuous Neuron Activations. IEEE Trans. Circuits Syst. 2003, 150:1421-1435
120 M. Forti, P. Nistri, D. Papini. Global Exponential Stability and Goble Convergence in Finite Time of Delayed Neural Networks with Infinite Gain. IEEE Trans. Neural Networks. 2005, 16:1449-1463
121 Z. Denkowski, S. Mig ó rski, N. S. Papageorgiou. An Introduction to Nonlinear Analysis: Theory, Kluwer/Plenum, New York, 2003
122 Z. Donchev. Qualitative Properties of a Class of Differential Inclusion. Glas. Mat. 1996, 31:269-276
123 N. S. Papageorgiou. Random Differential Inclusion in Banach Spaces. J. Differential Equations. 1986, 65:287-303
124 A. Lasota, Z. Optial. An Application of the Kakutani-Ky Fan Theorem in the Theory of Ordinary Differential Equations. Bull. Acad. Polon. Sci. 1965, 13:781-786
125 B. C. Dhage, A. Boucherif, S. K. Ntouyas. On Periodic Boundary Value Problems of First-Order Perturbed Impulsive Differential Inclusion. Rep. Math. Phys. 2004, 54:221-228