Let X be a Banach space,A:D(A)X→X the generator of a compact C0-semigroup S(t):X→X,t≥0,D a locally closed subset in X,and f:(a,b)×X→X a function of Caratheodory type.The main result of this paper is that a ne...Let X be a Banach space,A:D(A)X→X the generator of a compact C0-semigroup S(t):X→X,t≥0,D a locally closed subset in X,and f:(a,b)×X→X a function of Caratheodory type.The main result of this paper is that a necessary and sufficient condition in order to make D a viable domain of the semilinear differential equation of retarded type u'(t)=Au(t)+f(t,u(t-q)),t∈[to,to+T],with initial condition uto=φ∈C([-q,0];X),is the tangency condition lim infh10 h^-1d(S(h)v(O)+hf(t,v(-q));D)=0 for almost every t∈(a,b)and every v∈C([-q,0];X)with v(0),v(-q)∈D.展开更多
基金Supported by the National Natural Science Foundation of China(10571150)the Natural Science Foundation of Jiangsu Education Committee(07KJB110131)
文摘Let X be a Banach space,A:D(A)X→X the generator of a compact C0-semigroup S(t):X→X,t≥0,D a locally closed subset in X,and f:(a,b)×X→X a function of Caratheodory type.The main result of this paper is that a necessary and sufficient condition in order to make D a viable domain of the semilinear differential equation of retarded type u'(t)=Au(t)+f(t,u(t-q)),t∈[to,to+T],with initial condition uto=φ∈C([-q,0];X),is the tangency condition lim infh10 h^-1d(S(h)v(O)+hf(t,v(-q));D)=0 for almost every t∈(a,b)and every v∈C([-q,0];X)with v(0),v(-q)∈D.
