Existence And Uniqueness Of Unbounded Solution Of Nonlinear System

This paper studies existence and uniqueness of unbounded solution and structural stability for nonlinear system with nonuniform exponential dichotomy, and discusses ex-istence and uniqueness of unbounded solution for perturbed nonautonomous impulsive systems with unbounded nonlinear terms. This paper includes four chapters.In chapter 1, we briefly introduce the research background and some lemmas of this dissertation.In chapter 2, up till now, many scholars studied nonlinear system with nonuniform exponential dichotomy theory, but their essential conditions are based on the assumption that nonlinear term satisfies|f(t,x)|≤μe-ε|t|. Different from standard consideration, in this chapter, we prove existence and uniqueness of unbounded solution for the system with a relatively conservative assumption that |f(t, x)|≤μ. To the best of the authors’ knowledge, there is no relevant paper using nonuniform exponential dichotomy theory to study the unbounded solution problem of nonlinear system.In chapter 3, we prove if linear system x(t)= A(t)x(t) admits nonuniform exponen-tial dichotomy, then nonlinear system is structural stability.In chapter 4, Fenner and Pinto (1999) proved that if linear impulsive system satisfies IS condition with bounded nonlinear term f(t,x,η), then perturbed nonlinear impulsive system has a unique bounded solution. What happens if f(t,x,η) is unbounded? In this chapter, we prove that if linear impulsive system admits exponential dichotomy and nonlinear term satisfies |f(t,x,η)|≤μeβ|t|+M, then perturbed nonlinear impulsive system has a unique unbounded solution. It’s worth mentioning that the method in Fenner and Pinto cannot be applied to our unbounded case.