Results of semigroup of linear equations generating Lipschitz perturbations of linear evolution equations

In this paper, results of -order preserving partial contraction mapping generating Lipschitz perturbations of linear evolution equation was presented.A certain semilinear value problem was studied where A is the infinitesimal generator of a C 0 -semigroup {T (t), t 0} on a Banach space X and f : [t 0 , T ] X X is continuous in t and satisfies a Lipschitz condition in u.We assume A to be independent of t and was extended to the case where A depends on t in a way that insure the existence of an evolution system U (t, s), 0 s t T , for the family {A(t)} t[0,T ] and shows that the initial value problem have a mild solution.