Note on the representation of semi-groups of non-linear operators
Shinnosuke Ôharu · Proceedings of the Japan Academy Series A Mathematical Sciences · 1966
Comm. by Kinjir Kuu, ..., Dee. 12, 1966) 1o Let X be a Banach space and let {T()}_0 be a family of non-linear operators from X into itself satisfying the following conditions:1 II T()x-T()y II 0, x, y e X,(3) There exists a dense subset D in X such that for each x e D, the right derivative D T()x-lira h-( T( + h)x-T()x)h--*O/ exists and it is continuous for >__0.Then we shall call this family {T()}0 a non-linear contraction semi-group.Definition.We define the infinitesimal generator A of a non-linear contraction semi-group {T()}_0 by Ax lim Ax whenever the limit exists, where Aah-( T(h)-I). We denote the domain of A by D(A).Lately J. W. Neuberger _1 gave the following result: If {T()}_0 is a non-linear contraction semi-group, *) then for each x e X and each 0 lim limsup [[ (I-(/n)A)-x T()x I[-0.It is well known that if (T()}0 is a linear contraetion semi- group of class (Co), then for each x X and each >__0 lim(I-(/n)A)-'x= T()x (see [2).In this paper we shall give the representation of this type for non-linear contraction semi-groups.The main results are the follwing Theorem.Let {T()}_ be a non-linear contraction semi-group and let A be the infinitesimal generator such that (I-oA)=X for some o O. Then for each 0 there exists an inverse operator (I-A)and its unique extension L() onto X, which is a contrac- tion operator, and T() is represented by *) In his paper the following condition is assumed:(3) There is a dense subset D of X such that if x is in D, then the derivative Tr()x is continuous with domain [0, ).