| The fixed point theory of nonlinear operators has a wide range of applica-tions in many areas of mathematics, especially in a variety of nonlinear differential equations and nonlinear integral equations. Because many higher order differen-tial equations in applied mathematics can be transformed to integral equations defined by nonlinear operators through appropriate variable replacement. How-ever, a large number of nonlinear problems arising in the theory and applications are the lack of compactness or continuity. Hence the research on fixed points of nonlinear operators and applications to the differential and integral equations in abstract spaces has a certain theoretical significance and application value.The first chapter introduces the research background of the nonlinear oper-ator equation theory and the basic knowledge of nonlinear functional analysis, including some of the definitions and lemmas used in the proof of the following chapters. More detailed knowledge of nonlinear functional analysis can be found in the references [1-10].The second chapter studies the initial value problem of the semi-linear evo-lution equations in Banach spaces E where A:D(A)→E is a dense and closed linear operator,—A is the infinitesi-mal generator of a Co—semigroup T(t)(t≥0) in E, J=[0, a], x0∈E and (Tu)(t)=∫0tk(t,s)u(s)ds,(Su)(t)=∫0a(t,s)u(s)ds, t∈J, wherek∈C(D,R),D={(t,s)∈J×J:t≥s},and h∈C(J×J,R),R is real number set.M=sup{||T(t)||:t∈[0,a]},k0=max{|k(t,s)|:(t,s)∈D}. By using a fixed point theorem with respect to convex-power condensing oper-ator and the special quality of function e-λt(where λ>0is a constant),the existence of mild solutions and minimal and maximal mild solutions to the initial value problem for a class of semilinear evolution equations of mixed type with noncompact semigroup in Banach spaces is obtained.The third chapter is concerned with the existence of solutions for a class of initial value problems of first-order implicit impulsive integro-differential equa-tions in Banach spaces where f∈C[J×E×E×E,E],J=[t0,t0+a](a>0),t0<t1<…<tm<t0+a<+∞,A1,A2≥0are two contants,Ik∈C[E,E],△u|t=tk=u(tk+)-u(tk-), u(tk+),u(tk-)represent the right and left limits ofu(t)att=tk respectively,(Tu)(t)=∫0t(t,s)u(s)ds,k(t,s)∈C[D1,R+],D1={(t,s)|t,s∈J,t≥s},k0max{t(t,s)|(t,s)∈D1}.The main method is based on Nonch fixed point theo-rem,Gronwall inequity and piecewise estimating. The fourth chapter is concerned with the existence of solutions for a class of nonlinear integral equations of Volterra type in Banach spaces where h0∈PC(J, E),H∈C(D1×E×E×E,E),D1={(t,s)∈J×J10≤s≤t≤a},(Tx)(t)=∫0tk(t,s)x(s)ds,(Sx)(t)=∫0a(t,s)x(s)ds,k∈C(D1,R), h∈C(J×J, R),Ik∈C(E,E),ak∈C(Jk*,R),J*=[tk,a],k=(1,2,…,m). By using Monch fixed point theorem and piecewise estimating,the existence of solutions for the nonlinear impulsive integral equations(4.1.1)is investigated under more widely conditions. |
