Existence Of Mild Solutions Of Impulsive Differential Equations Banach Space,

The impulsive differential equations can be used to model processes which are subjected to abrupt changes at some moments in the continuous evolution. During recent years, the theory of the impulsive differential equations has been an object of extensive attention and investigation because of its wide practical applications in many fields such as theoretical physics, population dynamics and control theory. At the beginning of 90's, the nonlocal condition, which is a generalization of the classical initial condition, was motivated by physical problems. The pioneering work on nonlocal condition is due to Byszewski, J.H.Liu, etc.In consideration of the existence and controllability of impulsive differential equations, many results depend on the compactness or equicontinuity of the operator semigroup and also have compact request to impulsive functions. Since it is quite difficult to determine whether a semigroup is compact, we try to reduce the request to the compactness of semigroup. At the same time, we make lower restrictions to impulsive functions.In this work we deal with the impulsive equations with nonlocal conditions,without the compactness or equicontinuity of the operator semigroup. Our main tools are theory of operator semigroup, measures of noncompactness, mutivalued analysis and the fixed point theory. There are two chapers in this work, which discuss the impulsive equations of single valued functions and mutifunctions, respectively.In Chapter One we discuss the impulsive equations as following, in a Banach space X . Here A is the infinitesimal generator of a strongly continuous semigroup of bounded linear operators T (t ), t≥0 in a Banach space, I k, k = 1,2, ,m are impulsive functions.In section 1.2 we first recall some definitions and facts about the measures of noncompactness and interval-continuous functions in PC ( [ 0, b ];X ). In this section an important property of the measures of noncompactness in PC ( [ 0, b ];X ) is proved(see Lemma 2.4).In section 1.3 we get the existence of impulsive differential equations by Monch fixed point theorem in PC ( [ 0, b ];X ). We remark that in our proof the operator semigroup is not necessarily compact and we also get rid of the compactness of f and I k, k = 1,2, , m.This improves and extends some knowing results.In Chapter Two we study the existence of mild solutions to the following semilinear evolution differential inclusion, where { A(t)} _{t∈[ 0,b]} is a family of linear operators generating an evolution system {U (t , s ):(t , s )∈Δ} and F is a multifunction.In section 2.2 we give some facts about multivalued mapping. In section 2.3 we make full use of the properties of semicompact sets in ( [ ])L1 0, b ;X and obtain the sufficient conditions to the existence of mild solutions when g is completely continuous, without the compactness and equicontinuity of evolution system {U (t , s ):(t , s )∈Δ} (see Theorem 3.1). In section 2.4 the existence for mild solutions of impulsive differential inclusions is obtained by use of the condensing mapping theorem for mutifunctions when g is Lipschitz continuous(see Theorem 4.1).