A Class Of Second Order Nonlinear Impulsive Evolution Equations On Banach Spaces

In order to study these phenomena that can not be modeled by traditional initial value problems such as the dynamics of populations subjected to abrupt change, such as harvesting, diseases, etc. They are described by partial impulsive differential equations or abstract impulsive evolution equations. To our knowledge there is no work on second order impulsive differential equations on infinite dimensional spaces in literature. It is well-known the second order differential equations is different from the first order equations, we must discuss the unbounded matrix of operators, It is very difficult to discuss.In this thesis, we consider the following nonlinear second order impulsive evolution equations:where D = {t₁ t₂, ..., t_n} (?) (0,T), 0 < t₁ < t₂ < ... < T, (i = 1,2, ... ,n),Δx(t_i) = x(t_i+) - x{t_i-) = x(t_i+) - x(t_i), and Δ(x|·)(t_i) = x|·(U+) - x|·(t_i-) = x|·(t_i+) -x|·(t_i), G_i,H_i represent the jumps in the state x at time t_i.For the different A and B, the equation of (1) is also different, so we discuss the second order nonlinear impulsive differential equation for the following three types.Type I: B is the infinitesimal generator of a strongly continuous semigroup {T(t),t ≥ 0} on Banach space X, and A is a closed linear operator on X with D(B) (?) D(A). We use the results of C₀-semigroup theory and introduce the matrix of operators to discuss it.Type II: B = 0, A = E², E is a closed linear operator on X, and E generates a Co-group {h(t), t ∈ R} on X, 0 ∈ ρ(E). We use the results of C₀-group theory and introduce interpolation operators to discuss it.Type III: B = 0, A is the infinitesimal generator of a cosine family {C(t), t ∈ R} and 0 ∈ρ(A). We use the results of the Cosine family of operators and introduce interpolation space to discuss it.For every type, the existence and uniqueness of the mild solution are given. The continuous dependence with respect to the initial state is also given, In addition, for type I, the existence of optimal control is given. At last, we give an example to illustrate our abstract results.