On Perturbation of Non-Linear Equations in Banach Spaces

Condition (1) states that given /eX0 and A>0 there is a ueD(A) satisfying the equation u — lAuBf. It is also known that under condition (1) (with X0 — D(A)) A generates a contraction semigroup on D(A) (cf. [5, Theorem I]). Our first purpose is to discuss sufficient conditions for (1). In general, the direct verification of (1) is not easy. We shall give some conditions on A which implies condition (1). Our conditions seem to be weaker than (1) and hence would be easy to check. We note, however, that our conditions are, in fact, equivalent to (1). Next, given dissipative operators A and J3, we consider the perturbation problem of Kato type; in which one wants to show