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 ∈ [t0,t0 + T], with initial condition ut0 = φ∈ C([-q,0];X), is the tangency condition liminfh↓0 h-1d(S(h)v(0)+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.
DONG Qi-xiang LI Gang School of Math. Sci., Yangzhou Univ., Yangzhou 225002, China
τ是X的一个线性H ausdorff拓扑,在一致τ-O p ia l或τ-UKK条件下,给出了渐近非扩张型映照的不动点定理.由于L1(μ)并不具备通常的O p ia l条件,但是在L1(μ)赋予抽象测度拓扑τ下,(X,τ)满足一致τ-O p ia条l件,从而给出L1(μ)中渐近非扩张型映照的不动点定理.
