Product integrals of continuous resolvents: Existence and nonexistence

Let {I?λA(t)]^?1:0≦λ≦Λ, 0≦t≦T} be a family of resolvents of bounded linear m-dissipative operatorsA(t) on a Banach spaceX. Suppose that the map(λ,t,x←I?λA(t)]^?1 x is jointly continuous. Then we show it is not necessarily true that for eachx∈X: (1) the product integral lim _{n → ∞} Π _i=1 ⁿ I - (t/n)A(it/n)]^?1 x exists, (2) the initial value problemy′(t)=A(t)y(t), y(0)=x has a strong solution.