Affine Periodic Solutions For Dissipative-repulsive Systems And Differential Inclusions

Affine periodicity describes a more complex motion than periodicity. It is widely used in acoustic, electromagnetic and other physical phenomena. As an important research direction in the theory of nonlinear analysis, differen-tial inclusion has a wide range of applications in many fields such as physics, mechanics and economics. Research on the existence of periodic solutions of differential inclusions is a fundamental issue. In the present thesis, we study the existence of affine periodic solutions of affine dissipative-repulsive systems and the existence of affine periodic solutions of differential inclusions.In Chapter 2, we firstly consider the following differential equationWe give the definitions of affine dissipative-repulsive (functional) differ-ential equations. According to the special structure of affine periodic systems, we construct a Poincare mapping and prove the existence of fixed points of the Poincare mapping by applying the topological degree theory. Then we prove that an affine periodic and affine dissipative-repulsive system admits affine periodic solutions. For functional differential equations, we get a similar result. In this way, the classic result on the existence of periodic solutions of dissipative-repulsive systems is extended to the case of affine periodic solution-s. In the last part of Chapter 2, some examples are given to understand the affine periodic systems and a method of Lyapunov function is given to verify the affine dissipative-repulsive condition.In Chapter 3, we consider the following differential inclusion where f(t, x):R × Rn??(Rn) is affine periodic and upper semi-continuous function, ?(Rn) is the set of all non-empty compact convex subsets in Rn. Firstly, we use continuous functions to approximate the upper semi-continuous function on the right side of differential inclusion by Zaremba Theorem ([68]). Then we convert the original problem into the initial value problem of ordi-nary differential equation by Filippov Thoerem ([23]). Finally, we prove the existence of fixed points of Poincare mapping by applying Horn's fixed point theorem. In this way, we obtain the existence of affine periodic solutions of differential inclusion. We prove that if the solutions of differential inclusion are uniformly ultimately bounded, then there exist affine periodic solutions of differential inclusion.We consider the following functional differential inclusion with delay where xt(?)= x(t+?),??[-?,0],?>0,F:R × C([-?,0],Rn)?Comp(Rn) is affine periodic and lower semi-continuous function, Comp(Rn) is the set of all non-empty compact subsets in Rn.When we consider the functional differential inclusions with nonconvex right-hand side, Continuous Selection Theorem for non-convex and non-empty closed values lower semi-continuous set-valued functions given by A. Bressan and G. Colombo ([5]) can be used to convert the original problem into the problem of functional differential equations, and then we use a method of finite-dimensional approach to treat the differential inclusion under consideration as a finite-dimensional differential equation. Thus we overcome the problem of Poincare mapping without compactness, which is due to the infinite dimen-sional phase space of the system. By using Horn's fixed point theorem, we prove that if the solutions of differential inclusion are locally affine dissipative and satisfy a boundedness condition, then there exist affine periodic solutions of differential inclusion.At last, we consider the following functional differential inclusion with infinite delay is affine periodic and upper semi-continuous with respect to (t,?). F maps bounded sets in C([-?,0],Rn) into bounded sets in Rn. We use the method of finite-dimensional approach mentioned above and prove that if the solu-tions of differential inclusion are uniformly bounded and uniformly ultimately bounded, then there exist affine periodic solutions of differential inclusion.