(ProQuest: ... denotes non-US-ASCII text omitted.)
Marko Kostic 1
Recommended by Viorel Barbu
Faculty of Technical Sciences, University of Novi Sad, Trg D. Obradovica 6, 21125 Novi Sad, Serbia
Received 29 April 2009; Accepted 25 June 2009
1. Introduction and Preliminaries
In this review, we will report how a large number of known results concerning (a,k) -regularized resolvents [1-6], C -regularized resolvents [7], and (local) convoluted C -semigroups and cosine functions [8, 9] can be formulated in the case of general (a,k) -regularized C -resolvent families.
The paper is organized as follows. In Theorem 2.2, Remark 2.3, and Theorems 2.5, 2.6, and 2.7, we analyze the properties of subgenerators of (a,k) -regularized C -resolvent families and slightly improve results from [1]. With a view to further study the problem describing heat conduction in materials with memory and the Rayleigh problem of viscoelasticity in L∞ type spaces, we prove in Theorem 2.8 several different forms of subordination principles [10]. The main objective in Theorems 2.9-2.12, 2.26, 2.28, and 2.32 is to continue the researches raised in [3] and [5, 6]. Our main contributions are Theorems 2.16-2.17, 2.20-2.25, 2.27, and 2.30 clarifying the basic regularity properties of (a,k) -regularized C -resolvent families and a fairly general form of the abstract Weierstrass formula.
It is noteworthy that the complete spectral characterization of subgenerators of (a,k) -regularized C -resolvent families exists only in the exponential case and that it is not clear, with exception of various types of local convoluted C -semigroups and cosine functions [9, 11], in what way one can prove a satisfactory Hille-Yosida theorem for local (a,k) -regularized C -resolvent families.
Throughout this paper E denotes a nontrivial complex Banach space, L(E) denotes the space of boundedlinear operators from E into E,E* denotes the dual space of E, and A denotes a closed linear operator acting on E. The range and the resolvent set of A aredenoted by Rang (A) and ρ(A), respectively; [D(A)] denotes the Banach space D(A) equipped with the graph norm. From now on, we assume that L(E)∋C is an injective operator which satisfies CA⊆AC and employ the convolution like mapping * which is given by f*g(t):=∫0t f(t-s)g(s)ds. Recall, the C -resolvent set of A, denoted by ρC (A), is defined to be the set of all complex numbers λ satisfying that the operator λ-A is injective and that Rang (C)⊆ Rang (λ-A). Let us recall that a linear subspace Y⊆D(A) is called a core for A if Y is dense in D(A) with respect to the graph norm. Henceforth we identify a closed linear operator A with its graph G(A); given two closed linear operators A and B on E, the inclusion A⊆B means G(A)⊆G(B). If X is a closed subspace of E, then AX denotes the part of A in X, that is, AX :={(x,y)∈A:x∈X, y∈X}.
We mainly use the following conditions.
(H1):: A is densely defined.
(H2):: ρ(A)≠∅.
(H3):: ρC (A)≠∅ and Rang (C)¯=E.
(H4):: A is densely defined or ρC (A)≠∅.
(H5):: (H1) ⋁ (H2) ⋁ (H3).
(P1):: k(t) is Laplace transformable, that is, it is locally integrable on [0,∞) and there exists β∈... so that k...(λ)=[Lagrangian (script capital L)](k)(λ):=lim b[arrow right]∞∫0be-λt k(t)dt:=∫0∞e-λt k(t)dt exists for all λ∈... with Re λ>β. Put abs(k):= inf{Re λ:k...(λ) exists}.
Let us remind that a function k∈Lloc 1 ([0,τ)) is called a kernel, if for every [varphi]∈C([0,τ)), the supposition ∫0t k(t-s)[varphi](s)ds=0, t∈[0,τ), implies [varphi]≡0; due to the famous Titchmarsh's theorem [12], the condition 0∈ supp k implies that k(t) is a kernel. Set Θ(t):=∫0t k(s)ds, t∈[0,τ).
2. (a,k) -Regularized C -Resolvent Families
We start with the following definition.
Definition 2.1.
Let 0<τ≤∞, k∈C([0,τ)), k≠0, and let a∈Lloc 1 ([0,τ)), a≠0. A strongly continuousoperator family (R(t))t∈[0,τ) is called a (local, if τ<∞ ) (a,k) -regularized C -resolventfamily having A as a subgeneratorif and only ifthe following holds:
(i) R(t)A⊆AR(t), t∈[0,τ), R(0)=k(0)C, and CA⊆AC,
(ii) R(t)C=CR(t), t∈[0,τ),
(iii): R(t)x=k(t)Cx+∫0t a(t-s)AR(s)x ds, t∈[0,τ), x∈D(A).
In the case τ=∞,(R(t))t≥0 is said to be exponentially bounded if, additionally, there exist M>0 and ω≥0 such that ||R(t)||≤Meωt , t≥0;(R(t))t∈[0,τ) is said to be nondegenerate if the condition R(t)x=0, t∈[0,τ) implies x=0.
From now on, we consider only nondegenerate (a,k) -regularized C -resolvent families. Notice that (R(t))t∈[0,τ) is nondegenerate provided that k(0)≠0 or that (H5) holds for a subgenerator A of (R(t))t∈[0,τ) .
In the case k(t)=tα /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):=∫0t K(s)ds,t∈[0,τ), where K∈Lloc 1 ([0,τ)) and K≠0, we obtain the unification concept for (local) K -convoluted C -semigroups 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 [Weierstrass p](R) the set which consists of all subgenerators of (R(t))t∈[0,τ) .
Then the following holds.
(i) A∈[Weierstrass p](R) implies C-1 AC∈[Weierstrass p](R).
(ii) If A∈[Weierstrass p](R) and λ∈ρC (A), then [figure omitted; refer to PDF]
(iii): Assume, additionally, that a(t) is a kernel. Then one can define the integral generator A... of (R(t))t∈[0,τ) by setting [figure omitted; refer to PDF] The integral generator A... of (R(t))t∈[0,τ) is a closed linear operator which satisfies C-1 A...C=A... and extends an arbitrary subgenerator of (R(t))t∈[0,τ) . Furthermore, A...∈[Weierstrass p](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 [Weierstrass p](R) becomes a complete lattice under suitable algebraic operations and that induced partial ordering coincides with the usual set inclusion. In general, [Weierstrass p](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 1 ([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)-1 x- (a*R)(t)A(λ-A)-1 x =λ(a*R)(t)(λ-A)-1 x- (R(t)(λ-A)-1 x-k(t)C(λ-A)-1 x) =λ(λ-A)-1 C(a*R)(t)C-1 x- ((λ-A)-1 R(t)x-k(t)(λ-A)-1 Cx), where the last two equalities follow on account of CA⊆AC, R(s)A⊆AR(s) and R(s)(λ-A)-1 C=(λ-A)-1 CR(s), s∈[0,τ). Hence, (λ-A)z=λz-(R(t)x-Cx),
[figure omitted; refer to PDF]
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)¯.
(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) Let A 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, then A 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∈... and β∈...,(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 1 ([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) -regularized C -resolvent family with a subgenerator A, (H1) or (H3) holds, and a(t) is a kernel. Then the integral generator A... of (R(t))t∈[0,τ) satisfies A...=C-1 AC. Toward this end, let (x,y)∈A.... Then ∫0t a(t-s)[k(s)Cx+∫0s a(s-r)R(r)y dr]ds=∫0t a(t-s)R(s)x ds∈D(A), t∈[0,τ) , and A∫0t a(t-s)[k(s)Cx+∫0s a(s-r)R(r)y dr]ds=A∫0t a(t-s)R(s)x ds=R(t)x-k(t)Cx=∫0t a(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-1 AC) , and C-1 ACx=A...x=y. On the other hand, C-1 AC is a subgenerator of (R(t))t∈[0,τ) whenever A is; this implies C-1 AC⊆A... and proves the claim. If (H2) holds, then A...=C-1 AC=A. In what follows, we also assume that B∈[Weierstrass p](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-1 AC=C-1 BC 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 ([Weierstrass p](R))=1, if C(D(A...)) is a core for D(A...).
(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}⊆[Weierstrass p](R) and a(t) is a kernel.
We refer the reader to [1, page 283] for the definition of (weak) solutions of the problem
[figure omitted; refer to PDF] where f∈C([0,τ):E) , and to [1, page 285] for the notion of spaces Cn,k ([0,τ):E), n∈..., k∈...0 and C0n ([0,τ):E), and n∈....
Define a subset A* of E* ×E* (the use of symbol * is clear from the context) by A* :={(x* ,y* )∈E* ×E* :x* (Ax)=y* (x) for all x∈D(A)}. In the case when A is densely defined, A* is a linear mapping from E* into E* .
Lemma 2.4 (see [17]).
Let A be a closed linear operator. Assume x0 ∈E,y0 ∈E, and x* (y0 )=y* (x0 ) for all (x* ,y* )∈A* . Then x0 ∈D(A), and Ax0 =y0 .
Define the mapping KC :C([0,τ):E)[arrow right]C([0,τ):E) by KC u:=k*Cu, u∈C([0,τ):E). Then KC is linear, bounded, and injective.
Keeping in mind Lemma 2.4 and the proofs of [1, Theorem 2.7, Corollary 2.9, Remark 2.10, Corollary 2.11, and Corollary 2.13], we have the following.
Theorem 2.5.
(i) Suppose f∈C([0,τ):E), A is a subgenerator of a (local) (a,k) -regularized C -resolvent family (R(t))t∈[0,τ) , and (H5) holds. Then (2.4) has a unique solution if and only if R*f∈ Rang (KC ).
(ii) (cf. also [18]) Assume n∈..., f∈C([0,τ):E), A is a subgenerator of a (local) n -times integrated (a,C) -resolvent family (R(t))t∈[0,τ) , and (H5) holds. Then (2.4) has a unique solution if and only if C-1 (R*f)∈C0n+1 ([0,τ):E).
(iii) Let the assumptions of the item (i) of this theorem hold, and let k∈C0n ([0,τ):E). Then C-1 (R*f)∈C(n+1) ([0,τ):E) if and only if C-1 (R*f)∈C0n+1 ([0,τ):E).
(iv) Let (H5) hold. Assume that n∈..., A is a subgenerator of an n -times integrated (a,C) -regularized resolvent, and a∈BVloc ([0,τ):E), respectively A is a subgenerator of an (a,C) -regularized resolvent family. Assume, further, that C-1 f∈C(n+1) ([0,τ):E),f(k-1) (0)∈D(An+1-k ) and An+1-kf(k-1) (0)∈Rang (C), 1≤k≤n+1, respectively C-1 f∈ACloc ([0,τ):E). Then (2.4) has a unique solution.
(v) Assume that (H5) holds, A is a subgenerator of an (a,k) -regularized C -resolvent family, k(t) is absolutely continuous, and k(0)≠0. If C-1 f∈C1 ([0,τ):E), then there exists a unique solution of (2.4).
The proof of following theorem follows from a standard application of Laplace transform techniques.
Theorem 2.6.
Let k(t) and a(t) satisfy (P1), and let (R(t))t≥0 be a strongly continuous operator family satisfying ||R(t)||≤Meωt , t≥0, for some M>0 and ω≥0. Put ω0 :=max (ω,abs(a),abs(k)).
(i) Suppose A is a subgenerator of the exponentially bounded (a,k) -regularized C -resolvent family (R(t))t≥0 , and (H5) holds. Then, for every λ∈... with Re λ>ω0 and k...(λ)≠0, the operator I-a...(λ)A is injective, Rang (C)⊆Rang (I-a...(λ)A), [figure omitted; refer to PDF] [figure omitted; refer to PDF] and R(s)R(t)=R(t)R(s), t,s≥0.
(ii) Assume that (2.5)-(2.6) hold. Then A is a subgenerator of the exponentially bounded (a,k) -regularized C -resolvent family (R(t))t≥0 .
The preceding theorem enables one to establish the real and complex characterization of subgenerators of (locally Lipschitz continuous) exponentially bounded (a,k) -regularized C -resolvent families [1, 9, 12]:
Theorem 2.7.
(i) Let k(t) and a(t) satisfy (P1), and let ω0 ≥max (0,abs(a),abs(k)). Assume that, for every λ∈... with Re λ>ω0 and k...(λ)≠0, the operator I-a...(λ)A is injective and that Rang (C)⊆Rang (I-a...(λ)A). If there exists an analytic function Υ:{λ∈...: Re λ>ω0 }[arrow right]L(E) with:
(i.1): Υ(λ)=k...(λ)(I-a...(λ)A)-1 C, λ∈...,Re λ>ω0 ,
(i.2): ||Υ(λ)||≤M|λ|r , λ∈...,Re λ>ω0 , for some M>0 and r≥-1, then, for every α>1, A is a subgenerator of a norm continuous, exponentially bounded (a,k*tα+r-1 /Γ(α+r)) -regularized C -resolvent family.
(ii) Suppose k(t) and a(t) satisfy (P1) and (H2) or (H3) holds, and A is a subgenerator of an exponentially bounded (a,Θ) -regularized C -resolvent family (R(t))t≥0 which satisfies the next condition:
[figure omitted; refer to PDF] Then there exists a≥ω0 such that [figure omitted; refer to PDF] [figure omitted; refer to PDF] is infinitely differentiable and [figure omitted; refer to PDF]
(iii) Suppose k(t) and a(t) satisfy (P1) and (2.8)-(2.10) holds. Then A is a subgenerator of an exponentially bounded (a,Θ) -regularized C -resolvent family (R(t))t≥0 which satisfies (2.7).
(iv) Suppose M>0 , ω≥0, k(t) and a(t) satisfy (P1), and A is densely defined. Then A is a subgenerator of an exponentially bounded (a,k) -regularized C -resolvent family (R(t))t≥0 which satisfies ||R(t)||≤Meωt , t≥0 if and only if there exists a≥max (0,abs(a),abs(k)) such that (2.8)-(2.10) hold.
Denote by a*n the n th convolution power of the kernel a(t), n∈..., and see [10] for the definition of completely positive functions and the notion used in the subsequent theorem and examples. An insignificant technical modification of the proofs of [1, Theorem 3.7] and [10, Theorems 4.1, 4.3, 4.5] (cf. also [7, Lemma 4.2]) implies the next subordination principles.
Theorem 2.8.
(i) Let a(t), b(t), and c(t) satisfy (P1), and let ∫0∞e-βt |b(t)|dt<∞ for some β≥0. Let [figure omitted; refer to PDF] and let a...(λ)=b...(1/c...(λ)), λ≥α. Let A be a subgenerator of a (b,k) -regularized C -resolvent family (Rb (t))t≥0 satisfying ||Rb (t)||≤Meωb t , t≥0, for some M>0 and ωb ≥0, and let (H2) or (H3) hold. Assume, further, that c(t) is completely positive and that there exists a function k1 (t) satisfying (P1) and [figure omitted; refer to PDF]
Let [figure omitted; refer to PDF] Then A is a subgenerator of an exponentially bounded, locally Lipschitz continuous (a,1*k1 ) -regularized C -resolvent (Ra (t))t≥0 , and there exists Ma ≥1 such that [figure omitted; refer to PDF] respectively, for every [straight epsilon]>0, there exists M[straight epsilon] ≥1 such that [figure omitted; refer to PDF] Furthermore, if A is densely defined, then A is a subgenerator of an exponentially bounded (a,k1 ) -regularized C -resolvent (Ra (t))t≥0 which fulfills (2.14), respectively, (2.15).
(ii) Suppose α≥0, A is a subgenerator of an exponentially bounded α -times integrated C -semigroup, a(t) is completely positive and satisfies (P1), and k(t) satisfies (P1) and k...(λ)=a...(λ)α , λ sufficiently large. Then A is a subgenerator of a locally Lipschitz continuous, exponentially bounded (a,t*k) -regularized C -resolvent family ((a,t*a*n ) -regularized C -resolvent family if α=n∈..., respectively, (a,t) -regularized C -resolvent family if α=0 ). If, additionally, A is densely defined, then A is a subgenerator of an exponentially bounded (a,1*k) -regularized C -resolvent family ((a,1*a*n ) -regularized C -resolvent family if α=n∈..., respectively, (a,C) -regularized resolvent family if α=0 ).
(iii) Suppose α≥0 and A is a subgenerator of an exponentially bounded α -times integrated C -cosine function. Let Lloc 1 ([0,∞))∋c be completely positive, and let a(t)=(c*c)(t), t≥0. (Given Lloc 1 ([0,∞))∋a in advance, such a function c(t) always exists provided a(t) is completely positive or a(t)≠0 is a creep function and a1 (t) is log-convex.) Assume that k(t) satisfies (P1), and k...(λ)=c...(λ)α /λ,λ sufficiently large. Then A is a subgenerator of a locally Lipschitz continuous, exponentially bounded (a,t*k) -regularized C -resolvent family ((a,t*c*n ) -regularized C -resolvent family if α=n∈..., resp. (a,t) -regularized C -resolvent family if α=0 ). If, additionally, A is densely defined, then A is a subgenerator of an exponentially bounded (a,1*k) -regularized C -resolvent family ((a,1*c*n ) -regularized C -resolvent family if α=n∈..., resp. (a,C) -regularized resolvent family if α=0 ).
Denote by Ap the realization of the Laplacian with Dirichlet or Neumann boundary conditions on Lp ([0,π]n ), 1≤p<∞. By [19, Theorem 4.2], Ap generates an exponentially bounded α -times integrated cosine function for every α≥(n-1)|(1/2)-(1/p)|. Assume further that c∈BVloc ([0,∞)) and that m(t) is a bounded creep function with m0 =m(0+)>0. Thanks to [10, Proposition 4.4, page 94], we have that there exists a completely positive function b(t) such that dm*b=1. After the usual procedure, the problem [10, (5.34)] describing heat conduction in materials with memory is equivalent to
[figure omitted; refer to PDF] where a(t)=(b*dc)(t), t≥0, and f(t) contains r*b as well as the temperature history. In what follows, we assume that
(i) p≠2,
(ii) Γb =∅ or Γf =∅,
(iii): there exists a completely positive function c1 (t) such that a(t)=(c1 *c1 )(t), t≥0.
We refer the reader to [10, pages 140-141] for the analysis of the problem (2.16) in the case: p=2 and m, c∈[Bernoulli].... Applying Theorem 2.8(iii), one gets that Ap is the integral generator of an exponentially bounded (a,1*[Lagrangian (script capital L)]-1 (c...1 (λ)(n-1)|(1/2)-(1/p)| /λ)(t)) -regularized resolvent family, where [Lagrangian (script capital L)]-1 denotes the inverse Laplace transform. Notice also that [10, Lemma 4.3, page 105] implies that, for every β∈[0,1], the function λ...(c...1(λ)β /λ) is the Laplace transform of a Bernstein function and that the function k(t) appearing in the formulations of Theorem 2.8(ii)-(iii) always exists. On the other hand, an application of [9, Proposition 2.1.3.12] gives that there exists ω>0 such that Ap is the integral generator of an exponentially bounded (ω-Ap )-[left ceiling](1/2)(n-1)|(1/2)-(1/p)|[right ceiling] -regularized cosine function; herein [left ceiling]s[right ceiling]=inf {k∈...:s≤k}, s∈.... Using Theorem 2.8(iii) again, we have that Ap is the integral generator of an exponentially bounded (a,(ω-Ap )-[left ceiling](1/2)(n-1)|(1/2)-(1/p)|[right ceiling] ) -regularized resolvent family, and Theorem 2.5(iv) can be applied. In both approaches, regrettably, we must restrict ourselves to the study of pure Dirichlet or Neumann problem. It is also worthwhile to note that Theorem 2.8(iii) can be applied in the analysis of the Rayleigh problem of viscoelasticity in L∞ type spaces; as a matter of fact, the operator A defined on [10, page 136] generates an exponentially bounded α -times integrated cosine function in L∞ ((0,∞)) for all α>0.
Approximation type theorem for exponentially bounded (a,k) -regularized C -resolvent families follows from Theorem 2.6 and [12, Theorem 1.7.5, page 42], and the representation formulae for exponentially bounded (a,k) -regularized C -resolvent families are consequences of the Post-Widder inversion (the Phragamén-Doetsch inversion). For further information, see [2, 12, 20, 21].
Using the argumentation given in [3, 5], one can prove the following assertions.
Theorem 2.9.
(i) Suppose that the next conditions hold.
(i.1): The mapping t...|k(t)|, t∈[0,τ), is nondecreasing.
(i.2): There exist [straight epsilon]a,k >0 and ta,k ∈[0,τ) such that [figure omitted; refer to PDF]
(i.3): A is a subgenerator of an (a,k) -regularized C -resolvent family(R(t))t∈[0,τ) , and (H5) holds.
(i.4): lim sup t[arrow right]0+ ||R(t)||/|k(t)|<∞.
Then [figure omitted; refer to PDF] [figure omitted; refer to PDF]
(ii) Suppose A is a subgenerator of an (a,k) -regularized C -resolvent family (R(t))t∈[0,τ) satisfying ||R(t)||=O(k(t)), t[arrow right]0+, min (a(t),k(t))>0, t∈(0,τ), and (H5) holds. Then (2.18)-(2.19) hold.
Theorem 2.10.
(i) Suppose A is a subgenerator of an (a,k) -regularized C -resolvent family (R(t))t∈[0,τ) satisfying ||R(t)||=O(k(t)), t[arrow right]0+ and min (a(t),k(t))>0, t∈(0,τ). Then lim t[arrow right]0+ (a*R)(t)x/(a*k)(t)=Cx,x∈D(A)¯.
(ii) Suppose A is a subgenerator of an (a,k) -regularized C -resolvent family (R(t))t∈[0,τ) satisfying ||R(t)||=O(k(t)), t[arrow right]0+, min (a(t),k(t)) >0, t∈(0,τ) and (H5) holds. If x∈D(A)¯,y∈E and lim t[arrow right]0+ (R(t)x-k(t)Cx)/(a*k)(t)=y, then x∈D(A) and y=Ax.
(iii) Suppose E is reflexive, A is a subgenerator of an (a,k) -regularized C -resolvent family (R(t))t∈[0,τ) satisfying ||R(t)||=O(k(t)), t[arrow right]0+, R(s)R(t)=R(t)R(s), 0≤t, s<τ, min (a(t),k(t))>0, t∈(0,τ), and (H5) holds. If x∈D(A)¯ and lim t[arrow right]0+ ||(R(t)x-k(t)Cx)/(a*k)(t)||<∞, then x∈D(A).
Theorem 2.11 (cf. also [22]).
Suppose α>0 and A is a subgenerator of an α -times integrated C -semigroup (Sα (t))t∈[0,τ) , respectively, an α -times integrated C -cosine function (Cα (t))t∈[0,τ) , which satisfies lim sup t[arrow right]0+ ||Sα (t)||/tα <∞, respectively, lim sup t[arrow right]0+ ||Cα (t)||/tα <∞. Then, for every x∈D(A) such that Ax∈D(A)¯: [figure omitted; refer to PDF] [figure omitted; refer to PDF]
Theorem 2.12.
Suppose M>0, ω≥0, A is a densely defined subgenerator of an (a,k) -regularized C -resolvent family (R(t))t≥0 which satisfies ||R(t)||≤Meωt , t≥0; B∈L(E), Rang (B)⊆Rang (C), and BCx=CBx, x∈D(A). Suppose, further, that there exist a function b(t) satisfying (P1) and a number ω0 ≥ω such that b...(λ)=a...(λ)/k...(λ), λ>ω0 , k...(λ)≠0. Then the operator A+B is a subgenerator of an(a,k) -regularized C -resolvent family (RB (t))t≥0 which satisfies ||RB (t)||≤M/1-γeμt , t≥0 , [figure omitted; refer to PDF]
Remark 2.13.
In order to prove Theorem 2.9(i) and (2.20)-(2.21) in the case of nondensely subgenerators, it is enough to notice that [3, (2.1), page 219] holds for every z∈D(A)¯ and that [3, (2.2), page 219] holds for every x∈D(A) such that Ax∈D(A)¯. On the other hand, if (2.20), resp. (2.21), holds for some x∈D(A), then it is obvious that Ax∈D(A)¯. This implies that the representation formulae (2.20) and (2.21) are best possible in some sense.
Given α∈(0,π], set Σα :={λ∈...:λ≠0, |argλ|<α}.
Definition 2.14 (cf. also [23, Definition 5.1]).
Let 0<α≤π/2, and let (R(t))t≥0 be an (a,k) -regularized C -resolvent family. Then it is said that (R(t))t≥0 is an analytic (a,k) -regularized C -resolvent family of angle α, if there exists an analytic function R:Σα [arrow right]L(E) which satisfies
(i) R(t)=R(t), t>0 ,
(ii) lim z[arrow right]0,z∈Σγ R(z)x=k(0)Cx for all γ∈(0,α) and x∈E.
It is said that (R(t))t≥0 is an exponentially bounded, analytic (a,k) -regularized C -resolvent family, respectively, bounded analytic (a,k) -regularized C -resolvent family, of angle α, if for every γ∈(0,α), there exist Mγ >0 and ωγ ≥0, resp. ωγ =0, such that ||R(z)||≤Mγeωγ Rez , z∈Σγ .
Since no confusion seems likely, we also write R(·) for R(·). The next proposition can be proved by means of the arguments given in [7, Section 3] and [10, Chapter 2].
Proposition 2.15.
Suppose k(t) and a(t) satisfy (P1), lim λ[arrow right]+∞ λk...(λ)=k(0)≠0, A is densely bounded, A∉L(E) , and there exists ω0 ≥max (0,abs(k), abs(a)) such that ∫0∞e-ωt |a(t)|dt<∞. Assume that A is a subgenerator of an exponentially bounded, analytic (a,k) -regularized C -resolvent family (R(t))t≥0 of angle α∈(0,π/2] and that there exists ω≥ω0 such that [figure omitted; refer to PDF] Then the function a...(λ) can be extended to a meromorphic function defined on the sector ω+Σπ/2+α .
It is worthwhile to mention that it is not clear, all assumptions of Proposition 2.15 being satisfied, whether A must be a subgenerator of an (a,C) -regularized resolvent family on [0,τ) (cf. Theorem 2.2(iv). Further on, let us notice that the assertions (i) and (ii) of [10, Theorem 2.2, page 57] still hold in the case of exponentially bounded, analytic (a,C) -regularized resolvent families.
The subsequent theorem clarifies the basic analytical properties of (a,k) -regularized C -resolvent families. Notice only that the assertion which naturally corresponds to [7, Lemma 3.7] (cf. also [10, Corollary 2.2, page 53]) does not seem attainable in the case of a general (a,k) -regularized C -resolvent family.
Theorem 2.16.
Suppose α∈(0,π/2], k(t) and a(t) satisfy (P1), (H5) holds, and k...(λ) can be analytically continued to a function g:ω+Σπ/2+α [arrow right]..., where ω≥max (0, abs(k),abs(a)). Suppose, further, that A is a subgenerator of an analytic (a,k) -regularized C -resolvent family (R(t))t≥0 of angle α and that (2.23) holds. Set [figure omitted; refer to PDF] Then N is an open connected subset of .... Assume that there exists an analytic function a...:N[arrow right]... such that a...(λ)=a...(λ), λ∈..., Re λ>ω. Then the operatorI-a...(λ)A is injective for every λ∈N, Rang (C)⊆Rang (I-a...(λ)C-1 AC) for every λ∈N1 :={λ∈N:a...(λ)≠0}, [figure omitted; refer to PDF] [figure omitted; refer to PDF] [figure omitted; refer to PDF]
Proof.
By Theorem 2.6(i), it follows that, for every λ∈... with Re λ>ω, the operator I-a...(λ)A is injective and that Rang (C)⊆Rang (I-a...(λ)A). Since (2.23) holds, one yields that the function q:{λ∈...: Re λ>ω}[arrow right]L(E) given by q(λ)=∫0∞e-λt R(t)dt, λ∈...,Re λ>ω has an analytic extension q...(λ):ω+Σπ/2+α [arrow right]L(E) such that sup λ∈ω+Σπ/2+γ ||(λ-ω)q...(λ)||<∞ for all γ∈(0,α) [12]. The set N is open and connected ([9], Subsection 2.1.4), and clearly, the mapping F(λ):=q...(λ)/g(λ),λ∈N, is analytic. Denote by V the set which consists of all complex numbers λ∈N such that I-a...(λ)A is injective, Rang (C)⊆ Rang (I-a...(λ)A), and F(λ)=(I-a...(λ)A)-1 C. Let ρC (A)∋μ satisfy a...(μ)≠0. Then [figure omitted; refer to PDF] [figure omitted; refer to PDF] [figure omitted; refer to PDF] and the uniqueness theorem for analytic functions implies that (2.28)-(2.29) hold for every λ∈N and that (2.30) holds for every λ∈N such that a...(λ)≠0. Let (I-a...(λ)A)x=0 for some λ∈N and x∈D(A). Thanks to (2.28), Cx=0, x=0, and I-a...(λ)A is injective. Assume, for the time being, λ∈N and a...(λ)≠0. Then (2.29)-(2.30) hold, and one gets (I-a...(λ)A)CF(λ)y= (I-a...(λ)A)F(λ)Cy =C2 y-(1/a...(λ)-1/a...(μ))(1/a...(μ)-A)-1 CF(λ)y+ a...(λ)(1/a...(λ)-1/a...(μ))[-CF(λ)y+1/a...(μ)(1/a...(μ)-A)-1 CF(λ)y] -1/a...(μ)(1/a...(μ)-A)-1C2 y. Then (2.30) implies (I-a...(λ)A)CF(λ)y =C2 y-1/a...(μ)(1/a...(μ)-A)-1C2 y +1/a...(μ)(1/a...(μ)-A)-1 CF(λ)y -a...(λ)/a...(μ)[-CF(λ)y+1/a...(μ)(1/a...(μ)-A)-1 CF(λ)y] :=C2 y+Rλ,μ . Clearly, Rλ,μ =0 if and only if C2 y-CF(λ)y-a...(λ)(1/a...(μ)-A)CF(λ)y+(a...(λ)/a...(μ))CF(λ)y=0. In order to see that the last equality is true, one can again apply (2.30). Thereby, (I-a...(λ)A)CF(λ)y=C2 y, λ∈N,a...(λ)≠0, and as an outcome, we obtain that the operator I-a...(λ)A is injective for all λ∈N and that Rang (C)⊆Rang (I-a...(λ)C-1 AC) for all λ∈N with a...(λ)≠0, as required. The estimates (2.25)-(2.27) follow by using the argumentation given in the proof of [9, Theorem 2.1.4.4].
Vice versa, we have the following theorem which can be proved as in the case of convoluted C -semigroups [9].
Theorem 2.17.
Assume k(t) and a(t) satisfy (P1), ω≥max (0,abs(k), abs(a)) and α∈(0,π/2]. Assume, further, that A is a closed linear operator and that, for every λ∈... with Re λ>ω and k...(λ)≠0, we have that the operator I-a...(λ)A is injective and that Rang (C)⊆Rang (I-a...(λ)A). If there exists an analytic function q:ω+Σπ/2+α [arrow right]L(E) such that [figure omitted; refer to PDF] [figure omitted; refer to PDF] [figure omitted; refer to PDF] then A is a subgenerator of an exponentially bounded, analytic (a,k) -regularized C -resolvent family of angle α.
Example 2.18 (cf. also [9, Theorem 2.1.4.7]).
Let β∈(0,2), α>0, k(t)=tα /Γ(α+1), and let a(t)=tβ-1 /Γ(β). Let A be densely defined. Then A is a subgenerator of an exponentially bounded, analytic (a,k) -regularized C -resolvent family of angle γ if and only if for every δ∈(0,γ), there exist Mδ >0 and ωδ ≥0 such that [figure omitted; refer to PDF] the mapping λ...(λβ -A)-1 C,λ∈(ωδ +Σπ/2+δ )1/β is analytic (continuous).
Let (Mp ) be a sequence of positive real numbers such that M0 =1 and that
(M.1): Mp2 ≤Mp+1Mp-1 , p∈...,
(M.2): Mn ≤AHnmin p, q∈..., p+q=nMpMq , n∈..., for some A>1,and H>1,
(M.3) : ∑p=1∞ (Mp-1 /Mp )<∞.
The Gevrey sequences (p!s ), (pps ), and (Γ(1+ps)) satisfy the above conditions, where s>1. Put mp :=(Mp /Mp-1 ), p∈...; by (M.1), (mp ) is increasing, and (M.3)[variant prime] implies ∑p=1∞(1/mp )<∞. The associated function of (Mp ) is defined by M(λ):=sup p∈...0 ln (|λ|p /Mp ),λ∈...\{0}, M(0):=0. As is known, the function t...M(t), t≥0, is increasing, absolutely continuous, lim t[arrow right]∞ M(t)=+∞ and lim t[arrow right]∞ (M(t)/t)=0. For consistency of terminology with [24], we also employ the sequence (Lp :=Mp1/p ) and set ωL (t):=∑p=0∞ (tp /Lpp ), t≥0.
We need the following family of kernels. Define, for every l>0, the next entire function of exponential type zero ωl (λ):=∏p=1∞ 1+lλ/mp , λ∈.... Then
[figure omitted; refer to PDF] and this implies that |ωl (λ)|≥eM(l|λ|) , λ∈..., Re λ≥0. It is noteworthy that, for every α∈(0,π/2), p∈...0 and λ∈Σπ/2+α , |1+lλ/mp |≥l|Im λ|/mp ≥l(1+tanα)-1 |λ|/mp .
This yields
[figure omitted; refer to PDF] Put now [figure omitted; refer to PDF] Then, for every l>0, 0∈ supp kl and kl is infinitely differentiable in t≥0.
Definition 2.19.
Let (R(t))t∈[0,τ) be a (local) (a,k) -regularized C -resolvent family having A as a subgenerator, and let the mapping t...R(t), t∈(0,τ), be infinitely differentiable (in the uniform operator topology). Then it is said that (R(t))t∈[0,τ) is of class CL , resp. of class CL , if and only if for every compact set K⊆(0,τ) there exists hK >0, resp. for every compact set K⊆(0,τ) and for every h>0 : [figure omitted; refer to PDF] (R(t))t∈[0,τ) is said to be ρ-hypoanalytic, 1≤ρ<∞, if (R(t))t∈[0,τ) is of class CL with Lp =p!ρ/p .
By the proof of the scalar-valued version of the Pringsheim theorem, it follows that the mapping t...R(t), t∈(0,τ) is real analytic if and only if (R(t))t∈[0,τ) is ρ -hypoanalytic with ρ=1.
The main objective in Theorems 2.20-2.24 is to enquire into the basic differential properties of (a,k) -regularized C -resolvent families.
Theorem 2.20 ([25]).
SupposeA is a closed linear operator, k(t) and a(t) satisfy (P1), r≥-1, and there exists ω≥max (0,abs(k),abs(a)) such that, for every z∈{λ∈...:Re λ>ω, k...(λ)≠0}, we have that the operator I-a...(z)A is injective and that Rang (C)⊆Rang (I-a...(z)A). If, additionally, for every σ>0, there exist Cσ >0,Mσ >0 and an open neighborhood Ωσ,ω of the region [figure omitted; refer to PDF] and an analytic mapping hσ :Ωσ,ω [arrow right]L(E) such that hσ (λ)=k...(λ)(I-a...(λ)A)-1 C,Re λ>ω, k...(λ)≠0 and that ||hσ (λ)||≤Mσ|λ|r λ∈Λσ,ω , then, for every ζ>1, A is a subgenerator of a norm continuous, exponentially bounded (a,k*tζ+r-1 /Γ(ζ+r)) -regularized C -resolvent family (R(t))t≥0 satisfying that the mapping t...R(t), t>0 is infinitely differentiable.
Theorem 2.21.
Suppose k(t) and a(t) satisfy (P1), (H5) hold and A is a subgenerator of an (a,k) -regularized C -resolvent family (R(t))t≥0 satisfying ||R(t)||≤Meω[variant prime] t , t≥0 for appropriate constants ω[variant prime] ≥max (0,abs(k),abs(a)), and M>0. If there exists ω>ω[variant prime] such that, for every σ>0, there exist Cσ >0 and Mσ >0 so that
(i) there exist an open neighborhood Ωσ,ω of the region Λσ,ω , and the analytic mappings fσ :Ωσ,ω [arrow right]...,gσ :Ωσ,ω [arrow right]..., and hσ :Ωσ,ω [arrow right]L(E) such that fσ (λ)=k...(λ), λ∈..., Re λ≥ω and gσ (λ)=a...(λ), λ∈..., Re λ≥ω,
(ii) for every λ∈Λσ,ω with Re λ≤ω, the operator I-a...(λ)A is injective and Rang (C)⊆Rang (I-a...(λ)A),
(iii): hσ (λ)=fσ (λ)(I-gσ (λ)A)-1 C, λ∈Λσ,ω ,
(iv) ||hσ (λ)||≤Mσ |Im λ|, λ∈Λσ,ω , Re λ≤ω, and max (|fσ (λ)|,|gσ (λ)|)≤Mσ , λ∈Λσ,ω ,
then the mapping t...R(t)x, t>0 is infinitely differentiable for every fixed x∈D(A2 ). Furthermore, if D(A2 ) is dense in E, then the mapping t...R(t), t>0 , is infinitely differentiable.
Proof.
Assume σ>0, [varsigma]>0,ω0 >ω, and put Γσ1 :={λ∈...:Re λ=2Cσ -σln (- Im λ), -∞< Im λ≤-e(2Cσ /σ) },Γσ2 :={λ∈...:Re λ=ω0 , -e(2Cσ /σ) ≤Im λ≤e(2Cσ /σ) },Γσ3 :={λ∈...:Re λ=2Cσ -σln ( Im λ),e(2Cσ /σ) ≤Im λ<+∞},Γσ :=Γσ1 ∪ Γσ2 ∪ Γσ3 , and Γk,σ :={λ∈Γσ :|λ|≤k}, k∈.... The curves Γσ and Γk,σ are oriented so that Im λ increases along Γσ and Γk,σ , k∈.... Set, for a sufficiently large k0 ∈...,Sσk (t):=(1/2πi)∫Γk,σeλt (hσ (λ)/λ2 )dλ, t≥0, k≥k0 . One can simply prove that (dj /dtj )Sσk (t)=(1/2πi)∫Γk,σeλtλj-2hσ (λ)dλ, t≥0, k≥k0 , j∈.... Let k0 <k<l. Then (iv) implies [figure omitted; refer to PDF] for all j∈...0 . Since |Im λ|1-σt|λ|j-2 ~|Im λ|j-1-σt , |λ|[arrow right]∞, λ∈Γσ , one gets that, for every j∈...0 and t>j/σ, the sequence ((dj /dtj )Sσk (t))k is convergent in L(E) and that the convergence is uniform on every compact subset of [j/σ+[varsigma],∞). Put Sj,σ (t):=lim k[arrow right]∞ (dj /dtj )Sσk (t), j∈...0 , t>j/σ. Then it is obvious that (d/dt)Sj,σ (t)=S(j+1),σ (t), j∈...0 , t>(j+1)/σ+[varsigma]. This implies that the mapping t...S0,σ (t), t>(j+1/σ)+[varsigma] is j -times differentiable and that (dj /dtj )S0,σ (t)=Sj,σ (t),t>(j+1)/σ)+[varsigma]. On the other hand, it is clear that, for every σ>0, x∈D(A2 ) , and λ∈{z∈Ωσ,ω :gσ (z)≠0}, [figure omitted; refer to PDF] By (2.41), we get that, for every x∈D(A2 ) and t>0, [figure omitted; refer to PDF] With (iv) and the residue theorem in view, it follows that, for every t>0 and x∈D(A2 ), [figure omitted; refer to PDF] Put R2 (t):=∫0t (t-s)R(s)x ds, x∈E, t≥0. By Theorem 2.6(i), we get that [figure omitted; refer to PDF] This implies that the function λ...hσ (λ)/λ2 is bounded on some right half plane. Taking into account (2.41), we have that, for every t≥0 and x∈D(A2 ), [figure omitted; refer to PDF] By (2.43)-(2.45), S0,σ (t)=R2 (t), t>0. The arbitrariness of σ implies that the mapping t...R2 (t), t>0 is infinitely differentiable, finishing the proof.
Using the argumentation given in [25], one can prove the following theorems.
Theorem 2.22.
Suppose k(t) and a(t) satisfy (P1), A is a subgenerator of a (local) (a,k) -regularized C -resolvent family (R(t))t∈[0,τ) , ω≥max (0, abs(k), abs(a)) , and m∈.... Denote, for every [straight epsilon]∈(0,1) and a corresponding K[straight epsilon] >0, [figure omitted; refer to PDF] Assume that, for every [straight epsilon]∈(0,1), there exist C[straight epsilon] >0,M[straight epsilon] >0, an open neighborhood O[straight epsilon],ω of the region G[straight epsilon],ω :={λ∈...:Re λ≥ω,k...(λ)≠0}∪{λ∈F[straight epsilon],ω :Re λ≤ω}, and analytic mappings f[straight epsilon] :O[straight epsilon],ω [arrow right]...,g[straight epsilon] :O[straight epsilon],ω [arrow right]... and h[straight epsilon] :O[straight epsilon],ω [arrow right]L(E) such that
(i) f[straight epsilon] (λ)=k...(λ), Re λ>ω;g[straight epsilon] (λ)=a...(λ), Re λ>ω,
(ii) for every λ∈F[straight epsilon],ω , the operator I-g[straight epsilon] (λ)A is injective and Rang (C)⊆Rang (I-g[straight epsilon] (λ)A),
(iii): h[straight epsilon] (λ)=f[straight epsilon] (λ)(I-g[straight epsilon] (λ)A)-1 C, λ∈G[straight epsilon],ω ,
(iv) ||h[straight epsilon] (λ)||≤M[straight epsilon](1+|λ|)me[straight epsilon]|Re λ| , λ∈F[straight epsilon],ω ,Re λ≤ω and ||h[straight epsilon] (λ)||≤M[straight epsilon](1+|λ|)m , λ∈..., Re λ≥ω.
Then (R(t))t∈[0,τ) is of class CL .
Theorem 2.23.
Suppose k(t) and a(t) satisfy (P1), A is a subgenerator of a (local) (a,k) -regularized C -resolvent family (R(t))t∈[0,τ) , ω≥max (0, abs(k), abs(a)) , and m∈.... Denote, for every [straight epsilon]∈(0,1), ρ∈[1,∞) and a corresponding K[straight epsilon] >0, [figure omitted; refer to PDF] Assume that, for every [straight epsilon]∈(0,1), there exist C[straight epsilon] >0,M[straight epsilon] >0, an open neighborhood O[straight epsilon],ω of the region G[straight epsilon],ω,ρ :={λ∈...:Re λ≥ω, k...(λ)≠0}∪{λ∈F[straight epsilon],ω,ρ :Re λ≤ω}, and analytic mappings f[straight epsilon] :O[straight epsilon],ω [arrow right]...,g[straight epsilon] :O[straight epsilon],ω [arrow right]... and h[straight epsilon] :O[straight epsilon],ω [arrow right]L(E) such that the conditions (i)-(iv) of Theorem 2.22 hold with F[straight epsilon],ω , resp. G[straight epsilon],ω , replaced by F[straight epsilon],ω,ρ , respectively, G[straight epsilon],ω,ρ . Then (R(t))t∈[0,τ) is of class CL .
Theorem 2.24.
Suppose α>0, j∈..., and (R(t))t∈[0,τ) is a (local) (a,k) -regularized C -resolvent family with a subgenerator A. Set [figure omitted; refer to PDF]
Then (R(t))t∈[0,τ) is an (a,k*(tα-1 /Γ(α))) -regularized C -resolvent family with a subgenerator A. Furthermore, if the mapping t...R(t), t∈(0,τ) is j -times differentiable, then the mapping t...Rα (t), t∈(0,τ) is likewise j -times differentiable. If this isthe case, then we have, for every t∈[0,τ), b∈(0,t), and x∈E:
[figure omitted; refer to PDF] and we have the following.
(i) If (R(t))t∈[0,τ) is of class CL , resp. of class CL , then (Rα (t))t∈[0,τ) is likewise of class CL , resptivley, of class CL .
(ii) If (R(t))t∈[0,τ) is ρ -hypoanalytic, 1≤ρ<∞, then (Rα (t))t∈[0,τ) is likewise ρ -hypoanalytic.
Before going further, notice that we can slightly reformulate Theorem 2.16 and Theorems 2.21-2.23 in the case when the functions k...(λ) and a...(λ) possess the meromorphic extensions on the corresponding regions defined in formulation of mentioned theorems. Having in mind [9, Theorem 2.1.1.11, Theorem 2.1.1.14], we have the following interesting analogue of [25, Theorem 2.8] which cannot be so easily interpreted in the case of general (a,k) -regularized C -resolvent families.
Theorem 2.25.
(i) Let A be a subgenerator of a local K -convoluted C -cosine function (CK (t))t∈[0,τ) , 0∈ supp K, K∈C∞ ((0,τ)) (K∈Cj ((0,τ)), j∈... ) resp. K is of class CL (CL ), and let K=K1|[0,τ) for an appropriate complex-valued function K1 ∈Lloc1 ([0,2τ)). (Put Θ1 (t)=∫0tK1 (s)ds; since it makes no misunderstanding, we will also write K and Θ, for K1 and Θ1 , respectively, and denote by K*K the restriction of this function to any subinterval of [0,2τ). ) Let the mapping t...CK (t), t∈(0,τ), be infinitely differentiable (j -times differentiable, j∈... ), respectively, and let (CK (t))t∈[0,τ) be of class CL (CL ). Then A is a subgenerator of a local (K*K) -convoluted C2 -cosine function (CK*K (t))t∈[0,2τ) satisfying that the mapping t...CK*K (t), t∈(0,2τ), is infinitely differentiable ((j-1) -times differentiable), resp. (CK*K (t))t∈[0,2τ) is of class CL (CL ). Furthermore, the suppositions j∈... and K∈Cj ((0,τ))∩Cj-1 ([0,τ)) imply the following: if the mapping t...CK (t), t∈(0,τ), is j -times differentiable, then the mapping t...CK*K (t), t∈(0,2τ) is likewise j -times differentiable.
(ii) Suppose α≥0, j∈..., and A is a subgenerator of a local α -times integrated C -cosine function (Cα (t))t∈[0,τ) . Then A is a subgenerator of a local (2α) -times integrated C2 -cosine function (C2α (t))t∈[0,2τ) and the following holds.
(ii.1): If the mapping t...Cα (t), t∈(0,τ) is infinitely differentiable (j -times differentiable, j∈... ), then the mapping t...C2α (t), t∈(0,2τ) is infinitely differentiable ((j-1) -times differentiable; j -times differentiable, provided α≥j ).
(ii.2): If (Cα (t))t∈[0,τ) is of class CL , resp. CL , then (C2α (t))t∈[0,2τ) is likewise of class CL , resp. CL .
(ii.3): Assume α∈...0 , j∈..., and the mapping t...Cα (t), t∈(0,τ) is infinitely differentiable (j -times differentiable). Then the mapping t...C2α (t), t∈(0,2τ), is j -times differentiable.
Proof.
The first part of (i) can be proved by passing to the theory of semigroups (see [9, Theorem 2.1.1.11] and [25, Theorem 2.8]). So, let us assume j∈..., K∈Cj ((0,τ))∩Cj-1 ([0,τ)),τ0 ∈(0,τ), and let the mapping t...CK (t), t∈(0,τ) be j -times differentiable. By [9, Theorem 2.1.1.14], A is a subgenerator of a local (K*K) -convoluted C2 -cosine function (CK*K (t))t∈[0,2τ) , which is given by [figure omitted; refer to PDF] Since the mapping t...CK (t), t∈(0,τ) is j -times differentiable and K∈Cj ((0,∞)), we have that the mapping t...CK*K (t), t∈(0,τ) is also j -times differentiable. Arguing as in [25, Theorem 2.8], one gets that the mappings t...CK (τ0 )CK (t-τ0 ), t∈(τ0 ,2τ0 ), t...(∫0t-τ0 +∫0τ0 )K(t-r)CK (r)Cdr, t∈(τ0 ,2τ0 ) and t...∫0t-τ0 K(r+2τ0 -t)CK (r)Cdr, t∈(τ0 ,2τ0 ) are j -times differentiable.
Let f(t):=∫2τ0 -tτ0 K(r+t-2τ0 )CK (r)Cdr, t∈(τ0 ,2τ0 ). Using the fact that K∈C1 ((0,τ))∩C([0,τ)), we have f[variant prime] (t)=∫2τ0 -tτ0K[variant prime] (r+t-2τ0 )CK (r)Cdr+K(0)CK (2τ0 -t)C, t∈(τ0 ,2τ0 ). Repeating this procedure leads us to the fact that the mapping t...f(t), t∈(τ0 ,2τ0 ) is j -times differentiable, and this completes the proof of (i). The proof of (ii) in the case α∈... follows immediately from (i) with K(t)=(tα-1 /Γ(α)) while the proof of (ii) in the case α=0 is much easier [16].
Suppose that min (a(t),k(t))>0, t∈(0,τ) and that A is a subgenerator of an (a,k) -regularized C -resolvent family (R(t))t∈[0,τ) . We define the Favard class Fa,k by setting [figure omitted; refer to PDF] Equipped with the norm |·|a,k :=||·||+sup t∈(0,τ) (||R(t)·-k(t)C·||/(a*k)(t)),Fa,k becomes a Banach space, and in the case when ||R(t)||=O(k(t)), t∈[0,τ), we have D(A)⊆Fa,k . The proof of [5, Theorem 3.4] immediately implies the following assertion.
Theorem 2.26.
Assume min (a(t),k(t))>0, t∈(0,τ), abs(k)=abs(a)=0, A is a subgenerator of an (a,k) -regularized C -resolvent family (R(t))t≥0 satisfying ||R(t)||=O(1), t≥0 and (H5) holds.
(i) Let x∈Fa,k . Then [figure omitted; refer to PDF]
(ii) Assume, in addition, that the mapping a...:(0,∞)[arrow right](0,∞) is surjective and that sup t>0 (1*a)(t)/(a*k)(t)<∞. Then (2.52) implies Cx∈Fa,k .
The assertion (ii) of the next theorem improves [5, Theorem 4.6].
Theorem 2.27 (cf. [26, Theorem 4.2] and Proposition 2.12.7).
(i) Suppose A is a subgenerator of a (local, global exponentially bounded) (a,k) -regularized C -resolvent family (R(t))t∈[0,τ) , D(A) and Rang (C) are dense in E and α>0. Then A* is a subgenerator of a (local, global exponentially bounded) (a,k*0 (tα-1 /Γ(α))) -regularized C* -resolvent family (Rα* (t))t∈[0,τ) , which is given by [figure omitted; refer to PDF]
(ii) Suppose A is a subgenerator of a (local, global exponentially bounded) (a,k) -regularized C -resolvent family (R(t))t∈[0,τ) , and D(A) and Rang (C) are dense in E. Then the part of A* in D(A* )¯ is a subgenerator of a (local, global exponentially bounded) (a,k) -regularized CD(A* )¯* -resolvent family in E* .
(iii) Suppose E is reflexive, D(A) and Rang (C) are dense in E, k(t) and a(t) satisfy (P1), and A is a subgenerator of a (local, global exponentially bounded) (a,k) -regularized C -resolvent family (R(t))t∈[0,τ) . Then A* is a subgenerator of a (local, global exponentially bounded) (a,k) - regularized C* -resolvent family (of the same exponential type, in the second case).
Suppose, for the time being, that a∈C([0,τ)) and denote, for every λ∈..., by s(t,λ) the unique continuous solution of the equation [figure omitted; refer to PDF] Put r(t,λ):=k(t)+λ∫0t s(t-v,λ)k(v)dv, t∈[0,τ). Arguing as in [5, Section 5], one can simply verify the validity of the next theorem.
Theorem 2.28.
(i) Let A be a subgenerator of an (a,k) - regularized C -resolvent family(R(t))t∈[0,τ) , and let (H5) hold. If the operator r(t,λ)C-R(t) is bijective for some t∈[0,τ) and λ∈..., then λ∈ρ(A).
(ii) Let A be a densely defined subgenerator of an (a,k) -regularized C -resolvent family (R(t))t∈[0,τ) , and let (H5) hold. If Rang (λ-A)¯≠E for some λ∈..., then, for every t∈[0,τ), Rang (r(t,λ)C-R(t))¯≠E.
(iii) Let A be a subgenerator of an (a,k) -regularized C -resolvent family (R(t))t∈[0,τ) . Then the assumption Ax=λx, for some x∈E and λ∈..., implies r(t,λ)Cx=R(t)x, t∈[0,τ).
For further information concerning duality and spectral properties of (a,k) -regularized resolvent families, we refer to [5].
Proposition 2.29 (cf. [9, Proposition 2.1.1.17]).
Suppose ±A are subgenerators of (local, global exponentially bounded) (a,k) - regularized C - resolvent families (R± (t))t∈[0,τ) , and A2 is closed. Then A2 is a subgenerator of a (local, global exponentially bounded) (a*a,k) -regularized C -resolvent family (R(t))t∈[0,τ) , which is given by R(t)x:=1/2(R+ (t)x+R- (t)x), x∈E, t∈[0,τ).
Proof.
Clearly, (R(t))t∈[0,τ) is a strongly continuous operator family, CA2 ⊆A2 C, R(0)=k(0)C, R(t)C=CR(t) and R(t)A2 ⊆A2 R(t), t∈[0,τ). Let x∈D(A2 ). Then we have [figure omitted; refer to PDF] This completes the proof.
The next version of the abstract Weierstrass formula extends [15, Theorem 11].
Theorem 2.30.
(i) Assume that k(t) and a(t) satisfy (P1), and there exist M>0 and ω>0 such that |k(t)|≤Meωt ,t≥0. Assume, further, that there exist a number ω[variant prime] ≥ω and a function a1 (t) satisfying (P1) and a...1 (λ)=a...(λ), λ∈..., Re λ>ω[variant prime] . Let A be a subgenerator of an exponentially bounded (a,k) -regularized C -resolvent family (C(t))t≥0 , and let (H5) hold. Then A is a subgenerator of an exponentially bounded, analytic (a1 ,k1 ) -regularized C -resolvent family (R(t))t≥0 of angle (π/2), where
[figure omitted; refer to PDF] [figure omitted; refer to PDF]
(ii) Assume k(t) satisfy (P1), β>0 , and there exist M>0 and ω>0 such that |k(t)|≤Meωt , t≥0. Let A be a subgenerator of an exponentially bounded (t2β-1 /Γ(2β),k) -regularized C -resolvent family (C(t))t≥0 , and let (H5) hold. Then A is a subgenerator of an exponentially bounded, analytic (tβ-1 /Γ(β),k1 ) -regularized C -resolvent family (R(t))t≥0 of angle π/2, where k1 (t) and R(t) are defined through (2.56) and (2.57).
Proof.
Since k(t) is continuous and exponentially bounded, one can use the substitution r=s/t and the dominated convergence theorem after that to deduce that, for every s≥0, [figure omitted; refer to PDF] This implies k1 ∈C([0,∞)). Moreover, k1 (t) is a kernel since [figure omitted; refer to PDF] Let x∈E be fixed. Then, for every s≥0, [figure omitted; refer to PDF] By (2.60), (R(t))t≥0 is a strongly continuous, exponentially bounded operator family. Furthermore, one can employ Theorem 2.6(i) and [12, Proposition 1.6.8] to obtain that, for every λ∈... with Re λ>β2 and k...1 (λ)≠0, [figure omitted; refer to PDF] By Theorem 2.6(ii), we get that (R(t))t≥0 is an exponentially bounded (a1 ,k1 ) -regularized C -resolvent family with a subgenerator A, and the remnant of the proof of (i) may be carried out by modifying the corresponding part of the proof of [15, Theorem 11]; the assertion (ii) follows from (i) with a(t)=t2β-1 /Γ(2β).
Notice that a1 (t)=∫0∞ s(e-s2 /4t /2πt3/2 )a(s)ds, t>0, whenever the function a(t) is exponentially bounded.
Example 2.31.
(i)(Reference[15]) Let E=Lp (...), 1≤p≤∞, α∈(-1,1), and a(t)=(tα /Γ(α+1)). Consider the next multiplication operator with maximal domain in E :
[figure omitted; refer to PDF] Assume s∈(1,2), δ=1/s,Mp =p!s , and Kδ (t):=[Lagrangian (script capital L)]-1 (e-λδ )(t), t≥0. Then A generates a global (not exponentially bounded) (a,Kδ ) -regularized resolvent family since, for every τ∈(0,∞), A generates a local (a,Kδ ) -regularized resolvent family on [0,τ). In order to show this, designate by M(t) the associated function of the sequence (Mp ) and put Λα,β,γ :={λ∈...:Re λ≥(M(αλ)/γ)+β}, α>0, β>0, γ>0. Clearly, there exists a constant Cs >0 such that M(λ)≤Cs|λ|1/s , λ∈.... Given τ>0, choose α>0 and β>0 such that τ≤cos (δπ/2)/Csα1/s as well as that Λα,β,1 ⊆ρ(A) and that the resolvent of A is bounded on the set {λα+1 :λ∈Λα,β,1 }. Put Γ:=∂(Λα,β,1 ), and assume that the curve Γ is upward oriented. Define, for every f∈E, x∈... and t∈[0,cos (δπ/2)/Csα1/s ), [figure omitted; refer to PDF] Then one can straightforwardly check that (Rδ (t))t∈[0,τ) is a local (a,Kδ ) -regularized resolvent family generated by A. Arguing in the same way, we get that there exists τ0 >0 such that A generates a local (a,K1/2 ) -regularized resolvent family on [0,τ0 ), where K1/2 (t):=[Lagrangian (script capital L)]-1 (e-λ1/2 )(t), t≥0.
(ii) [References [9, 27]] Let A(p!s ) and E(p!s ) be as in [27, Example 1.6] with Mp =p!s (s>1). Let β∈(0,1), and let, for every l>0,kl (t)=[Lagrangian (script capital L)]-1 (1/∏p=1∞ (1+lλ/ps/β ))(t), t≥0, (see (2.36)-(2.37) and a(t)=tβ-1 /Γ(β). Then it is obvious that there exist l[variant prime] >0 and K>0 such that ||λk...l[variant prime] (λ)(I-a...(λ)A)-1 ||≤K, λ∈Σπ/2β . This in combination with Theorem 2.17 implies that, for every l>l[variant prime] , the operator A(p!s ) generates an analytic (a,kl ) -regularized resolvent of angle (π/2)((1/β)-1). In the meantime, A(p!s ) does not generate an exponentially bounded (a,tα /Γ(α+1)) -regularized resolvent (α≥0) since A(p!s ) is not stationary dense.
(iii) [References [9, 28]; cf. also [29, Example 2.20]] Suppose E:=L2 [0,π] and A:=-Δ with the Dirichlet or Neumann boundary conditions, β∈[1/2,1), α>1+β, a(t)=(tβ-1 /Γ(β)), and [figure omitted; refer to PDF]
Define hα,β :Σπ/2β [arrow right]... by setting: hα,β (λ)=hα,β (λ), λ∈Σπ/2β , λ≠n2/β , n∈..., and hα,β (n2/β )=0, n∈.... Then the function hα,β (λ) is analytic, and there exists a constant C>0 such that [figure omitted; refer to PDF] Let k(t)=[Lagrangian (script capital L)]-1 (hα,β (λ))(t),t≥0. By Theorem 2.17, it follows that A generates an exponentially bounded, analytic (a,k) -regularized resolvent (R(t))t≥0 of angle (π/2)(1/β-1). Using the inverse Laplace transform, one can simply prove that ||R(t)||=O(tα-1 +tα+β-1 ), t≥0. Since Δ generates a cosine function, we are in a position to apply Theorem 2.17 to deduce that Δ generates an exponentially bounded, analytic (a,k) -regularized resolvent of angle (π/2)(1/β-1). By Proposition 2.29, we have that the biharmonic operator Δ2 , equipped with the suitable boundary conditions, generates an exponentially bounded, analytic (a*a,k) -regularized resolvent of angle (π/2)(1/β-1). Then the use of Theorem 2.30(ii) enables one to see that there exists a continuous kernel k1 (t) such that Δ2 generates an exponentially bounded, analytic (a,k1 ) -regularized resolvent family of angle π/2. Keeping in mind the fact that -Δ2n generates an analytic C0 -semigroup of angle π/2 (cf. for example [30, page 215]), one can prove that, for every n∈..., there exists an exponentially bounded kernel kn (t) such that the polyharmonic operator Δ2n generates an exponentially bounded, (a,kn ) -regularized resolvent family of angle π/2 [15].
It has recently been proved that, in the case β=1, there exists an exponentially bounded continuous kernel K(t) such that A generates an exponentially bounded, analytic K -convoluted semigroup of angle π/2 [25]. Let us consider now the case β∈(1,2) and a(t)=(tβ-1 /Γ(β)). Choose a number a∈(1/2,1/β) and after that a number s∈(1,1/βa). Put kl (t):=[Lagrangian (script capital L)]-1 (1/∏p=1∞ 1+lλ/ps )(t), t≥0. Arguing as in [25], one yields that the function h(λ)=∏n=0∞(n2 -λ)/(n2 +λ), λ∈...\{±n2 :n∈...0 }; h(n2 )=0, n∈..., is analytic, and that there exists M>0 such that, for every γ∈(0,π(1/β-1/2)), there exist Mγ >0 and cγ >0 such that, for every λ∈Σγ ,
[figure omitted; refer to PDF] Furthermore, there exists an exponentially bounded continuous kernel k(t) such that k...(λ)=k...l (λ)h(λβ ), λ∈...,Re λ>0. By (2.66), it follows that A generates an exponentially bounded, analytic (a,k) -regularized resolvent of angle π(1/β-1/2). Furthermore, an application of Theorem 2.17 gives that Δ generates an exponentially bounded, analytic (a,k) -regularized resolvent of angle π(1/β-1/2). By Proposition 2.29, we have that Δ2 generates an exponentially bounded, analytic (a*a,k) -regularized resolvent of angle π(1/β-1/2). Arguing as in the case β∈[1/2,1), we have that, for every n∈..., there exists an exponentially bounded kernel kn (t) such that the polyharmonic operator Δ2n generates an exponentially bounded, (a,kn ) -regularized resolvent family of angle π/2. In the case β=2, it is known that A cannot be the generator of any exponentially bounded convoluted cosine function [15]; the case β∈(0,1/2) requires an additional analysis. Finally, it is worth noting that we can incorporate the above results in the study of the equation [figure omitted; refer to PDF] where Dtβ denotes the Caputo fractional derivative [29].
The next theorem generalizes [6, Theorem 3.6, Corollary 3.8] (cf. also [31, Theorem 2.1] and [32, Theorem 3]).
Theorem 2.32.
(i) Assume C([0,∞))∋a satisfies (P1), (H5) holds, B∈L(E), Rang (B)⊆Rang (C) that and A is a subgenerator of an exponentially bounded (a,a) -regularized C -resolvent family (R(t))t≥0 . Assume, further, that there exists ω≥0 such that, for every h≥0 and for every function f∈C([0,∞):E),
(Ma) ∫0h R(h-s)C-1 Bf(s)ds∈D(A),
(Mb) ||A∫0h R(h-s)C-1 Bf(s)ds||≤eωtμB (h)||f||[0,h] , t≥0, where ||f||[0,h] :=sup t∈[0,h] ||f(t)||,μB (t):[0,∞)[arrow right][0,∞) is continuous, nondecreasing and satisfies μB (0)=0,
(Mc) there exists an injective operator C1 ∈L(E) such that Rang (C1 )⊆Rang (C) and that C1 A(I+B)⊆A(I+B)C1 .
Then A(I+B) is a subgenerator of an exponentially bounded (a,a) -regularized C1 -resolvent family (S(t))t≥0 which satisfies the following integral equation [figure omitted; refer to PDF]
(ii) Let A be a subgenerator of an exponentially bounded, once integrated C -cosine function and let ω, B, and C1 be as in (i). Then A(I+B) is a subgenerator of an exponentially bounded, once integrated C1 -cosine function.
Remark 2.33.
(i) Assume that A is a subgenerator of an exponentially bounded (a,a) -regularized C -resolvent family (R(t))t≥0 and that a Banach space (Z,|·|Z ) satisfies the conditions (Za), (Zb), and (Zc) given in the formulation of [6, Definition 4.1]. (In particular, these conditions hold for [D(A)]. ) Then (Ma) and (Mb) are fulfilled if C-1 B∈L(X,Z).
(ii) (References [32, 33]) Let B∈L(E), and let BC=CB.
(ii.1): Assume that BA is a subgenerator of a (local) (a,k) -regularized C -resolvent family, and (H5) holds for BA and C. Then AB is a subgenerator of an (a,k) -regularized C -resolvent family.
(ii.2): Assume that AB is a subgenerator of a (local) (a,k) -regularized C -resolvent family and (H5) holds for AB and C. Then BA is a subgenerator of an (a,k) -regularized C -resolvent family, provided ρ(BA)≠∅.
The proof of the next generalization of [15, Proposition 3] is provided for the sake of completeness.
Theorem 2.34.
Assume that τ∈(0,∞],Lloc 1 ([0,τ))∋a1 is a kernel, Lloc 1 ([0,τ))∋k is a kernel, a(t)=(a1 *a1 )(t), t∈[0,τ), and k1 (t)=(k*a1 )(t), t∈[0,τ). Put ...9C;≡(0IA0), ...9E;≡(C00C) , and assume that (H5) holds. Then A is a subgenerator of an (a,k) -regularized C -resolvent family (R(t))t∈[0,τ) if and only if ...9C; is a subgenerator of an (a1 ,k1 ) -regularized ...9E; -resolvent family (S(t))t∈[0,τ) . If this is the case, then we have [figure omitted; refer to PDF] and the integral generators of (R(t))t∈[0,τ) and (S(t))t∈[0,τ) , denoted respectively by B and [Bernoulli], satisfy [Bernoulli]=(0IB0).
Proof.
It is immediately verified that (S(t))t∈[0,τ) is a nondegenerate, strongly continuous operator family in E×E which satisfies S(t)...9C;⊆...9C;S(t) and S(t)...9E;=...9E;S(t), 0≤t<τ. Furthermore, the function k1 (t) is a continuous kernel, S(0)=0=k1 (0)...9E;, and ...9E;...9C;⊆...9C;...9E;. Let x∈D(A), and let y∈E. Then a simple computation involving (H5) shows that, for every t∈[0,τ), [figure omitted; refer to PDF] Assume now that ...9C; is a subgenerator of an (a1 ,k1 ) -regularized ...9E; -resolvent family (S(t))t∈[0,τ) . Put S(t)=(S1 (t)S2 (t)S3 (t)S4 (t))t∈[0,τ) , where Si (t)∈L(E), i∈{1,2,3,4} , and 0≤t<τ . A simple consequence of S(t)...9E;=...9E;S(t), t∈[0,τ) is Si (t)C=CSi (t), t∈[0,τ), i∈{1,2,3,4}. Since S(t)...9C;⊆...9C;S(t), t∈[0,τ), one gets [figure omitted; refer to PDF] Hence, S3 (t)x=S2 (t)Ax, x∈D(A), and S3 (t)y=AS2 (t)y, y∈E, 0≤t<τ. This implies that, for every x∈D(A),S3 (t)Ax=AS2 (t)Ax=AS3 (t)x, t∈[0,τ). Thereby, S3 (t)A⊆AS3 (t), t∈[0,τ), and (R(t)≡S3 (t)+k(t)C)t∈[0,τ) is a strongly continuous operator family in E satisfying R(0)=k(0)C, R(t)C=CR(t) and R(t)A⊆AR(t), 0≤t<τ. Since, for every λ∈..., λ∈ρ...9E; (...9C;) if and only if λ2 ∈ρC (A) [25], we have that (H5) holds for ...9C; and ...9E;. Since, for every x∈E and y∈E, [figure omitted; refer to PDF] one gets (a1 *S3 )(t)x=S1 (t)x-k1 (t)Cx, (a1 *S4 )(t)x=S2 (t)x, A(a1 *S3 )(t)x=S3 (t)x and A(a1 *S2 )(t)x =S4 (t)x-k1 (t)Cx, 0≤t<τ. Hence, A(a*R)(t)x =A(a*(S3 +kC))(t)x =A(a1 *a1 *(S3 +kC))(t)x =A(a1 *(S1 -k1 C+(a1 *k)C))(t)x =S3 (t)x =R(t)x-k(t)Cx, t∈[0,τ). This implies that (R(t))t∈[0,τ) is a nondegenerate operator family, and we finally get that (R(t))t∈[0,τ) is an (a,k) -regularized C -resolvent family with a subgenerator A. The remnant of the proof follows from a slight technical modification of the final part of the proof of [15, Proposition 3].
Remark 2.35.
(i) Let τ=∞, and let k(t) and a1 (t) be exponentially bounded. Then (R(t))t∈[0,τ) is exponentially bounded if and only if (S(t))t∈[0,τ) is exponentially bounded.
(ii) Let j∈..., α>0, a1 (t)=tα-1 /Γ(α) and k∈Cj ((0,τ)), respectivley, k∈C∞ ((0,τ)), and let the mapping t...R(t), t∈(0,τ) be j -times differentiable, respectivley infinitely differentiable. Then the mapping t...S(t),t∈(0,τ) is also j -times differentiable, resp. infinitely differentiable. Furthermore, if k(t) is of class CL , resp. CL (ρ -hypoanalytic, 1≤ρ<∞ ) and (R(t))t∈[0,τ) is of class CL , resp. CL (ρ -hypoanalytic), then (S(t))t∈[0,τ) is also of class CL , resp. CL (ρ -hypoanalytic).
(iii) Let a1 (t)=1/πt and k1 (t)=tn-(1/2) /Γ(n-(1/2)), n∈.... Then Theorem 2.34 enables one to discuss the maximal interval of existence of a local (a1 ,k1 ) -regularized C -resolvent family and to construct an example of a local (a1 ,k1 ) -regularized C -resolvent family (R(t))t∈[0,τ) which cannot be extended beyond the interval [0,τ); combining with [25, Examples 1, 3, 5] and [34, Theorem 3.1], it is possible to construct examples of infinitely differentiable, nonanalytic (a1 ,k1 ) -regularized C -resolvent families and examples of (pseudo)differential operators generating (a1 ,k1 ) -regularized C -resolvent families of class CL .
(iv) Assume a1 (t)=1/πt, A is a subgenerator of a (local) K -convoluted C -semigroup (SK (t))t∈[0,τ) , k(t)=∫0t K(s)ds, t∈[0,τ), and (H3) holds (see Theorem 2.2(iii). Let k1 (t) possess the same meaning as in Theorem 2.34. Then, for every x∈D(A) and y∈E, the system of integral equations
[figure omitted; refer to PDF] has a unique solution.
[1] C. Lizama, "Regularized solutions for abstract Volterra equations," Journal of Mathematical Analysis and Applications , vol. 243, no. 2, pp. 278-292, 2000.
[2] C. Lizama, "On approximation and representation of K -regularized resolvent families," Integral Equations and Operator Theory , vol. 41, no. 2, pp. 223-229, 2001.
[3] C. Lizama, J. Sánchez, "On perturbation of K -regularized resolvent families," Taiwanese Journal of Mathematics , vol. 7, no. 2, pp. 217-227, 2003.
[4] C. Lizama, H. Prado, "Rates of approximation and ergodic limits of regularized operator families," Journal of Approximation Theory , vol. 122, no. 1, pp. 42-61, 2003.
[5] C. Lizama, H. Prado, "On duality and spectral properties of (a,k) -regularized resolvents," preprint
[6] C. Lizama, V. Poblete, "On multiplicative perturbation of integral resolvent families," Journal of Mathematical Analysis and Applications , vol. 327, no. 2, pp. 1335-1359, 2007.
[7] M. Li, Q. Zheng, J. Zhang, "Regularized resolvent families," Taiwanese Journal of Mathematics , vol. 11, no. 1, pp. 117-133, 2007.
[8] I. Cioranescu, G. Lumer, "Problèmes d'évolution régularisés par un noyau général K(t) . Formule de Duhamel, prolongements, théorèmes de génération," Comptes Rendus de l'Académie des Sciences. Série I. Mathématique , vol. 319, no. 12, pp. 1273-1278, 1994.
[9] M. Kostic, " Generalized semigroups and cosine functions ," to appear
[10] J. Prüss Evolutionary Integral Equations and Applications , vol. 87, of Monographs in Mathematics, pp. xxvi+366, Birkhäuser, Basel, Switzerland, 1993.
[11] I. Cioranescu, G. Lumer, "Regularization of evolution equations via kernels K(t) , K -evolution operators and convoluted semigroups, generation theorems," in Seminar Notes in Functional Analysis and PDE, pp. 45-52, Louisiana State Univ., Baton Rouge, La, USA, 1994.
[12] W. Arendt, C. J. K. Batty, M. Hieber, F. Neubrander Vector-Valued Laplace Transforms and Cauchy Problems , vol. 96, of Monographs in Mathematics, pp. xii+523, Birkhäuser, Basel, Switzerland, 2001.
[13] M. Kim Abstract Volterra equations , Ph.D. thesis, Louisiana State University, Baton Rouge, La, USA, 1995.
[14] H. Oka, "Linear Volterra equations and integrated solution families," Semigroup Forum , vol. 53, no. 3, pp. 278-297, 1996.
[15] M. Kostic, S. Pilipovic, "Convoluted C -cosine functions and semigroups. Relations with ultradistribution and hyperfunction sines," Journal of Mathematical Analysis and Applications , vol. 338, no. 2, pp. 1224-1242, 2008.
[16] S. W. Wang, M. C. Gao, "Automatic extensions of local regularized semigroups and local regularized cosine functions," Proceedings of the American Mathematical Society , vol. 127, no. 6, pp. 1651-1663, 1999.
[17] C.-C. Kuo, "A note on weak solutions of second order abstract Cauchy problems," to appear in Taiwanese Journal of Mathematics
[18] Y.-C. Li, S.-Y. Shaw, " N -times integrated C -semigroups and the abstract Cauchy problem," Taiwanese Journal of Mathematics , vol. 1, no. 1, pp. 75-102, 1997.
[19] V. Keyantuo, M. Warma, "The wave equation in Lp -spaces," Semigroup Forum , vol. 71, no. 1, pp. 73-92, 2005.
[20] M. Kim, "Trotter-Kato type approximations of convoluted solution operator families," Communications of the Korean Mathematical Society , vol. 19, no. 2, pp. 293-305, 2004.
[21] C. Müller, "Approximation of local convoluted semigroups," Journal of Mathematical Analysis and Applications , vol. 269, no. 2, pp. 401-420, 2002.
[22] J.-C. Chang, S.-Y. Shaw, "Optimal and non-optimal rates of approximation for integrated semigroups and cosine functions," Journal of Approximation Theory , vol. 90, no. 2, pp. 200-223, 1997.
[23] M. Kim, "Remarks on Volterra equations in Banach spaces," Communications of the Korean Mathematical Society , vol. 12, no. 4, pp. 1039-1064, 1997.
[24] V. Barbu, "Differentiable distribution semi-groups," Annali dela Scuola Normale Superiore di Pisa , vol. 23, pp. 413-429, 1969.
[25] M. Kostic, "Differential and analytical properties of semigroups of linear operators," preprint
[26] I. Miyadera, M. Okubo, N. Tanaka, "On integrated semigroups which are not exponentially bounded," Proceedings of the Japan Academy, Series A, Mathematical Sciences , vol. 69, no. 6, pp. 199-204, 1993.
[27] P. C. Kunstmann, "Stationary dense operators and generation of non-dense distribution semigroups," Journal of Operator Theory , vol. 37, no. 1, pp. 111-120, 1997.
[28] B. Bäumer, "Approximate solutions to the abstract Cauchy problem," Evolution Equations and Their Applications in Physical and Life Sciences (Bad Herrenalb, 1998) , vol. 215, of Lecture Notes in Pure and Appl. Math., pp. 33-41, Marcel Dekker, New York, NY, USA, 2001.
[29] E. Bazhlekova Fractional evolution equations in Banach spaces , Ph.D. thesis, Eindhoven University of Technology, Eindhoven, UK, 2001.
[30] R. deLaubenfels Existence Families, Functional Calculi and Evolution Equations , vol. 1570, of Lecture Notes in Mathematics, pp. xvi+234, Springer, Berlin, Germany, 1994.
[31] J. Liang, T.-J. Xiao, F. Li, "Multiplicative perturbations of local C -regularized semigroups," Semigroup Forum , vol. 72, no. 3, pp. 375-386, 2006.
[32] Y. Xin, C. Liang, "Multiplicative perturbations of C -regularized resolvent families," Journal of Zheijang University SCIENCE , vol. 5, no. 5, pp. 528-532, 2004.
[33] A. Rhandi, "Multiplicative perturbations of linear Volterra equations," Proceedings of the American Mathematical Society , vol. 119, no. 2, pp. 493-501, 1993.
[34] Q. Zheng, Y. Li, "Abstract parabolic systems and regularized semigroups," Pacific Journal of Mathematics , vol. 182, no. 1, pp. 183-199, 1998.
You have requested "on-the-fly" machine translation of selected content from our databases. This functionality is provided solely for your convenience and is in no way intended to replace human translation. Show full disclaimer
Neither ProQuest nor its licensors make any representations or warranties with respect to the translations. The translations are automatically generated "AS IS" and "AS AVAILABLE" and are not retained in our systems. PROQUEST AND ITS LICENSORS SPECIFICALLY DISCLAIM ANY AND ALL EXPRESS OR IMPLIED WARRANTIES, INCLUDING WITHOUT LIMITATION, ANY WARRANTIES FOR AVAILABILITY, ACCURACY, TIMELINESS, COMPLETENESS, NON-INFRINGMENT, MERCHANTABILITY OR FITNESS FOR A PARTICULAR PURPOSE. Your use of the translations is subject to all use restrictions contained in your Electronic Products License Agreement and by using the translation functionality you agree to forgo any and all claims against ProQuest or its licensors for your use of the translation functionality and any output derived there from. Hide full disclaimer
Copyright © 2009 Marko Kostic et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
Abstract
We introduce the class of (local) (a,k) -regularized C -resolvent families and discuss its basic structural properties. In particular, our analysis covers subjects like regularity, perturbations, duality, spectral properties and subordination principles. We apply our results in the study of the backwards fractional diffusion-wave equation and provide several illustrative examples of differentiable(a,k) -regularized C -resolvent families.
You have requested "on-the-fly" machine translation of selected content from our databases. This functionality is provided solely for your convenience and is in no way intended to replace human translation. Show full disclaimer
Neither ProQuest nor its licensors make any representations or warranties with respect to the translations. The translations are automatically generated "AS IS" and "AS AVAILABLE" and are not retained in our systems. PROQUEST AND ITS LICENSORS SPECIFICALLY DISCLAIM ANY AND ALL EXPRESS OR IMPLIED WARRANTIES, INCLUDING WITHOUT LIMITATION, ANY WARRANTIES FOR AVAILABILITY, ACCURACY, TIMELINESS, COMPLETENESS, NON-INFRINGMENT, MERCHANTABILITY OR FITNESS FOR A PARTICULAR PURPOSE. Your use of the translations is subject to all use restrictions contained in your Electronic Products License Agreement and by using the translation functionality you agree to forgo any and all claims against ProQuest or its licensors for your use of the translation functionality and any output derived there from. Hide full disclaimer