-Regularized C-Resolvent Families: Regularity and Local Properties

and Applied Analysis 3 In the case k t t/Γ α 1 , where α > 0, and Γ · denotes the Gamma function, it is also said that R t t∈ 0,τ is an α-times integrated a,C -resolvent family; in such a way, we unify the notion of local α-times integrated C-semigroups a t ≡ 1 and cosine functions a t ≡ t 1, 13, 14 . Furthermore, in the case k t : ∫ t0K s ds, t ∈ 0, τ , where K ∈ Lloc 0, τ and K/ 0, we obtain the unification concept for local K-convoluted Csemigroups and cosine functions 15 . In the case k t ≡ 1, R t t∈ 0,τ is said to be a local a,C -regularized resolvent family with a subgenerator A cf. also 16 for the definition which does not include the condition ii of Definition 2.1 . Designate by ℘ R the set which consists of all subgenerators of R t t∈ 0,τ . Then the following holds. i A ∈ ℘ R implies C−1AC ∈ ℘ R . ii If A ∈ ℘ R and λ ∈ ρC A , then R t λ −A −1C λ −A −1CR t , t ∈ 0, τ . 2.1 iii Assume, additionally, that a t is a kernel. Then one can define the integral generator  of R t t∈ 0,τ by setting  : { ( x, y ) ∈ E × E : R t x − k t Cx ∫ t 0 a t − s R s yds, t ∈ 0, τ } . 2.2 The integral generator  of R t t∈ 0,τ is a closed linear operator which satisfies C−1ÂC  and extends an arbitrary subgenerator of R t t∈ 0,τ . Furthermore,  ∈ ℘ R , if R t R s R s R t , 0 ≤ t, s < τ. Recall that in the case of convoluted C-semigroups and cosine functions, the set ℘ R becomes a complete lattice under suitable algebraic operations and that induced partial ordering coincides with the usual set inclusion. In general, ℘ R needs not to be finite 9 . Henceforth we assume that the scalar-valued kernels k, k1, k2, . . . are continuous on 0, τ , and that a/ 0 in Lloc 0, τ . Assume temporarily λ ∈ ρC A , x ∈ Rang C , t ∈ 0, τ , and put z a ∗ R t x. Following the proof of 1, Lemma 2.2 , we have z λ a ∗ R t λ −A −1x − a ∗ R t A λ −A −1x λ a ∗ R t λ −A −1x − R t λ −A −1x − k t C λ −A −1x λ λ −A −1C a ∗R t C−1x − λ −A −1R t x− k t λ −A −1Cx ,where the last two equalities follow on account of CA ⊆ AC,R s A ⊆ AR s and R s λ −A −1C λ −A −1CR s , s ∈ 0, τ . Hence, λ −A z λz − R t x − Cx , ∫ t 0 a t − s R s x ds ∈ D A , A ∫ t 0 a t − s R s x ds R t x − k t Cx. 2.3 The closedness of A implies that 2.3 holds for every t ∈ 0, τ and x ∈ Rang C . Theorem 2.2 see 1 . i Let A be a subgenerator of an a, k -regularized C-resolvent family R t t∈ 0,τ , and let (H5) hold. Then 2.3 holds for every t ∈ 0, τ and x ∈ E. If ρC A / ∅, then 2.3 holds for every t ∈ 0, τ and x ∈ Rang C . 4 Abstract and Applied Analysis ii Let A be a subgenerator of an a, ki -regularized C-resolvent family Ri t t∈ 0,τ , i 1, 2. Then k2 ∗ R1 t k1 ∗ R2 t , t ∈ 0, τ , whenever (H4) holds. iii Let R1 t t∈ 0,τ and R2 t t∈ 0,τ be two a, k -regularized C-resolvent families having A as a subgenerator. Then R1 t x R2 t x, t ∈ 0, τ , x ∈ D A , and R1 t R2 t , t ∈ 0, τ , if (H4) holds. iv LetA be a subgenerator of an a, k -regularized C-resolvent family R t t∈ 0,τ . If k t is absolutely continuous and k 0 / 0, thenA is a subgenerator of an a,C -regularized resolvent family on 0, τ . Remark 2.3. i Let Ri t t∈ 0,τ be an a, ki -regularized C-resolvent family with a subgenerator A, i 1, 2, and let D A / {0}. Then k1 k2. ii Let Ri t t∈ 0,τ be an a, ki -regularized C-resolvent family with a subgenerator A, i 1, 2. Then, for every α ∈ C and β ∈ C, αR1 t βR2 t t∈ 0,τ is an a, αk1 βk2 regularized C-resolvent family with a subgenerator A. iii Let R t t∈ 0,τ be an a, k -regularized C-resolvent family with a subgenerator A, and let Lloc 0, τ b be a kernel. Then A is a subgenerator of an a, k ∗ b -regularized C-resolvent family b ∗ R t t∈ 0,τ . iv Let R t t∈ 0,τ be an a,C -regularized resolvent family having A as a subgenerator. Then k ∗ R t t∈ 0,τ is an a,Θ -regularized C-resolvent family with a subgenerator A. v Suppose R t t∈ 0,τ is an a, k -regularizedC-resolvent familywith a subgenerator A, H1 or H3 holds, and a t is a kernel. Then the integral generator  of R t t∈ 0,τ satisfies  C−1AC. Toward this end, let x, y ∈ Â. Then ∫ t0a t − s k s Cx ∫s 0a s − r R r y dr ds ∫ t 0a t − s R s x ds ∈ D A , t ∈ 0, τ , and A ∫ t 0a t − s k s Cx ∫s 0a s − r R r y dr ds A ∫ t 0a t − s R s x ds R t x − k t Cx ∫ t 0a t − s R s y ds, t ∈ 0, τ . Since a ∗ R t y ∈ D A , a ∗ a ∗ R t y ∈ D A , A a ∗ a ∗ R t y a ∗ R − kC t y, t ∈ 0, τ , and a ∗ k / 0 in C 0, τ , it follows that Cx ∈ D A , ACx Cy, x ∈ D C−1AC , and C−1ACx Âx y. On the other hand, C−1AC is a subgenerator of R t t∈ 0,τ whenever A is; this implies C−1AC ⊆  and proves the claim. If H2 holds, then  C−1AC A. In what follows, we also assume that B ∈ ℘ R and that H5 holds for B and C. Proceeding as in the proof of 9, Proposition 2.1.1.6 , one gets what follows. v.1 C−1AC C−1BC and C D A ⊆ D B . v.2 A and B have the same eigenvalues. v.3 The assumption A ⊆ B implies ρC A ⊆ ρC B . v.4 card ℘ R 1, if C D  is a core for D  . v.5 A ⊆ B ⇔ D A ⊆ D B and Ax Bx, x ∈ D A ∩D B ; furthermore, the property v.5 holds whenever {A,B} ⊆ ℘ R and a t is a kernel. We refer the reader to 1, page 283 for the definition of weak solutions of the problem