Rotating Periodic Solutions For Impulsive Differential Inclusions

In the study of the real world, people found that there are some natural phenomena which may take a sudden change under some certain conditions. And by normal differential equations, people can hardly describe this kind of systems. Hence a new type of differential equations is needed, and this type of differential equations is called the impulsive differential equations.The theory of impulsive differential equations was found in 1960s by Mil’man and Myshkis in their paper on the stability of motion in the pres-ence of impulses. The most important highlight of this type of equations is that it can show us the difference of the system when it takes a sudden change. Hence this kind of equations plays a very important part in the study of Demography, Physics, Biology and Control Theory. Besides differential e-quations, differential inclusion is also a powerful tool in the study of natural phenomena, especially when the status of system is uncertain or multi-valued.In this paper, we will consider differential equations and differential inclu-sions with impulsive conditions, and prove the existence of periodic solutions and rotating-periodic solutions when the system is dissipative.This thesis is organized as follows.In Chapter 1, we make a brief introduction of rotating-periodicity and impulse, and recall some basic definitions and important results.In Chapter 2, we consider dissipative impulsive differential equations and respectively dissipative functional impulsive differential equations. They are andIn Section 2.1, we prove an existence theorem of periodic solutions for dissipative impulsive differential equations.Theorem 0.0.1 Let (0.0.8) be a periodic impulsive system, f and{Ii)i∈Z1 satisfy the local Lipschitz condition in x. Assume{Ï„i(x)} and{Ii(x)} satisfy the hypotheses i) andii). Then, if system (0.0.8) is dissipative, system (0.0.8) has an T-periodic solution.Rotating-periodicity is a more generalized type of periodicity. In Section 2.2, we prove an existence theorem of rotating-periodic solutions on dissipative impulsive differential equations.Theorem 0.0.2 Let (0.0.8) be a (Q,T)-rotating-periodic impulsive sys-tem,and assume f and{Ii}i∈z1 satisfy the local Lipschitz condition in x,{Ti{x)} and{Ii(x)} satisfy the hypotheses i) and ii). If system (0.0.8) is Q-dissipative, then it admits a (Q,T)-rotating-periodic solution.At the end of Chapter 2, we consider dissipative functional impulsive differential equations, and prove the existence theorem of rotating-periodic solutions for this kind of systems.Theorem 0.0.3 If system (0.0.9) is (Q,T)-dissipative, then it admits a (Q,T)-rotating-periodic solution.Just as we talked before, for some problems, which the status may be un-certain or multi-valued, they can be described by differential inclusions. Hence in Chapter 3, we consider differential inclusions and functional differential in-clusion systems with impulsive of the forms. and We prove that if they are dissipative, then they have rotating periodic solu-tions.In Section 3.2, we prove an existence theorem of rotating-periodic solu-tions for dissipative impulsive differential inclusion systems.Theorem 0.0.4 Let f(t,x):R×Rn â†'K(Rn) be an upper semi-continuous and (Q, T)-rotating-periodic function. It means that for all (t, x)∈R×Rn where Q∈GL(n), T> 0 is a constant. And assume solutions of the impulsive differential inclusion is uniformly ultimate bounded. Then, differential inclusion (0.0.14) admits a (Q, T)-rotating-periodic solution.Then, similar to Chapter 2, we consider functional differential inclusion systems under same conditions, and prove the existence of rotating-periodic solutions.Theorem 0.0.5 For F:R×CT([-r,0])â†'Comp(Rn); if the following hypotheses hold:(H1) F is lower semi-continuous;(H2) For any t∈R, φ∈CÏ„([-r,0])with||φ>||≤M, there exists an L(M) such that d(F(t,φ),0)≤L(M);(H3) For any t∈R, φ∈CT([-r,0]), we have F(t+T,φ)= QF(t,Q-1φ), where Q∈GL(Rn), T> 0 is a constant.Let Do(?)D1 is a bounded subset of Rn, Do is closed and D1 is a convex open set. If solutions of functional differential inclusion (0.0.12)-(0.0.13) is D1-bounded and D1-D0 local rotating dissipative, then functional differential inclusion (0.0.12)-(0.0.13) admits a (Q,T)-rotating periodic solution in Do.In the last chapter, we consider the following 1D-nonconvex variational problem where the Lagrangian L depends on x"’, and f:[0.1]×R4â†'R1 is continuous, x∈W2,1(0,1),x(0)=x0, x(1)= x1, x’" exists a.e. on (0,1). The research on this problem has positive meaning for the development of nonconvex varia-tional problem and Mather theory for nonconvex Hamiltonian systems. Under certain appropriate conditions, by using integral-minimax method, we pro-vide a sufficient condition for the existence of the minimizer of the nonconvex variational problem. In fact, our result can be generalized to more general nonconvex variational problems whose Lagrangian depends on derivatives of higher order.