On Regularized Quasi-Semigroups and Evolution Equations

and Applied Analysis 3 Here are some useful examples of regularized quasi-semigroups. Example 2.2. Let {Tt}t≥0 be an exponentially bounded strongly continuous C-semigroup on Banach space X, with the generator A. Then K s, t : Tt, s, t ≥ 0, 2.5 defines a C-quasi-semigroup with the generator A s A, s ≥ 0, and so D D A . Example 2.3. Let X BUC R , the space of all bounded uniformly continuous functions on R with the supremum-norm. Define C,K s, t ∈ B X , by Cf x e−x 2 f x , K s, t f x e−x 2 f ( t2 2st x ) , s, t ≥ 0. 2.6 One can see that {K s, t }s,t≥0 is a regularized C-quasi-semigroup of operators on X, with the infinitesimal generator A s f 2sḟ on D, where D {f ∈ X : ḟ ∈ X}. Example 2.4. Let {Tt}t≥0 be a strongly continuous semigroup of operators on Banach space X, with the generator A. If C ∈ B X is injective and commutes with Tt, t ≥ 0, then K s, t : Ces t−Ts , s, t ≥ 0, 2.7 is a C-quasi-semigroup with the generator A s ATs. Thus D D A . In fact, for x ∈ D, boundedness of C implies that CA s x lim t→ 0 Ces t−Tsx − Cx t C lim t→ 0 es t−Tsx − x t C d ds |t 0 Ts t − Ts x CATsx. 2.8 Now injectivity of C implies that A s x ATsx, and so D D A . Example 2.5. Let {Tt}t≥0 be a strongly continuous exponentially bounded C-semigroup of operators on Banach space X, with the generator A. For s, t ≥ 0, define K s, t T ( g s t − g s ), s, t ≥ 0, 2.9 where g t ∫ t 0 a s ds, and a ∈ C 0,∞ , with a t > 0. We have K s, 0 T 0 C and the C-semigroup properties of {T t }t≥0 imply that CK r, s t CT ( g r t s − g r ) CT ( g r t s − g t r g t r − g r ) T ( g r t s − g t r )T(g t r − g r ) K r t, s K r, t . 2.10 4 Abstract and Applied Analysis So {K s, t }s,t≥0 is a C-quasi-semigroup the other properties can be also verified easily . Also D D A and for x ∈ D, A s x a s Ax. Some elementary properties of regularized quasi-semigroups can be seen in the following theorem. Theorem 2.6. Suppose {K s, t }s,t≥0 is aC-quasi-semigroup with the generator {A s }s≥0 on Banach space X. Then i for any x ∈ D and s0, t0 ≥ 0, K s0, t0 x ∈ D and K s0, t0 A s x A s K s0, t0 x; 2.11