Abstract and Applied Analysis
VolumeΒ 2009Β (2009), Article IDΒ 858242, 27 pages
doi:10.1155/2009/858242
Research Article

( π‘Ž , π‘˜ ) -Regularized 𝐢 -Resolvent Families: Regularity and Local Properties

Marko Kostić

Faculty of Technical Sciences, University of Novi Sad, Trg D. Obradovića 6, 21125 Novi Sad, Serbia

Received 29 April 2009; Accepted 25 June 2009

Academic Editor: ViorelΒ Barbu

Copyright Β© 2009 Marko Kostić. 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) image/svg+xml(π‘Ž,π‘˜) -regularized image/svg+xml𝐢 -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 image/svg+xml(π‘Ž,π‘˜) -regularized image/svg+xml𝐢 -resolvent families.

1. Introduction and Preliminaries

In this review, we will report how a large number of known results concerning image/svg+xml(π‘Ž,π‘˜) -regularized resolvents [1–6], -regularized resolvents [7], and (local) convoluted -semigroups and cosine functions [8, 9] can be formulated in the case of general ( π‘Ž , π‘˜ ) -regularized 𝐢 -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 ( π‘Ž , π‘˜ ) -regularized 𝐢 -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 ( π‘Ž , π‘˜ ) -regularized 𝐢 -resolvent families and a fairly general form of the abstract Weierstrass formula.

It is noteworthy that the complete spectral characterization of subgenerators of ( π‘Ž , π‘˜ ) -regularized 𝐢 -resolvent families exists only in the exponential case and that it is not clear, with exception of various types of local convoluted 𝐢 -semigroups and cosine functions [9, 11], in what way one can prove a satisfactory Hille-Yosida theorem for local ( π‘Ž , π‘˜ ) -regularized 𝐢 -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 βˆ— denotes the dual space of 𝐸 , and 𝐴 denotes a closed linear operator acting on 𝐸 . The range and the resolvent set of image/svg+xml𝐴 aredenoted by image/svg+xmlRang(𝐴) and image/svg+xml𝜌(𝐴), respectively; image/svg+xml[𝐷(𝐴)] denotes the Banach space image/svg+xml𝐷(𝐴) equipped with the graph norm. From now on, we assume that image/svg+xml𝐿(𝐸)βˆ‹πΆ is an injective operator which satisfies image/svg+xmlπΆπ΄βŠ†π΄πΆ and employ the convolution like mapping image/svg+xmlβˆ— which is given by image/svg+xmlπ‘“βˆ—π‘”(𝑑)∢=∫ 𝑑0 𝑓(π‘‘βˆ’π‘ )𝑔(𝑠)𝑑𝑠. Recall, the image/svg+xml𝐢 -resolvent set of image/svg+xml𝐴, denoted by image/svg+xml𝜌 𝐢 (𝐴), is defined to be the set of all complex numbers image/svg+xmlπœ† satisfying that the operator image/svg+xmlπœ†βˆ’π΄ is injective and that image/svg+xmlRang(𝐢)βŠ†Rang(πœ†βˆ’π΄). Let us recall that a linear subspace image/svg+xmlπ‘ŒβŠ†π·(𝐴) is called a core for image/svg+xml𝐴 if image/svg+xmlπ‘Œ is dense in image/svg+xml𝐷(𝐴) with respect to the graph norm. Henceforth we identify a closed linear operator image/svg+xml𝐴 with its graph image/svg+xml𝐺(𝐴); given two closed linear operators image/svg+xml𝐴 and image/svg+xml𝐡 on image/svg+xml𝐸, the inclusion image/svg+xmlπ΄βŠ†π΅ means image/svg+xml𝐺(𝐴)βŠ†πΊ(𝐡). If image/svg+xml𝑋 is a closed subspace of image/svg+xml𝐸, then image/svg+xml𝐴 𝑋 denotes the part of image/svg+xml𝐴 in image/svg+xml𝑋, that is, image/svg+xml𝐴 𝑋 ∢={(π‘₯,𝑦)∈𝐴∢π‘₯βˆˆπ‘‹,π‘¦βˆˆπ‘‹}.

We mainly use the following conditions.

(H1):A is densely defined.(H2):image/svg+xml𝜌(𝐴)β‰ βˆ…. (H3):image/svg+xml𝜌 𝐢 (𝐴)β‰ βˆ… and image/svg+xmlRang(𝐢)=𝐸. (H4):image/svg+xml𝐴 isdenselydefinedor 𝜌 𝐢 (𝐴)β‰ βˆ…. (H5):(H1) image/svg+xml∨ (H2) image/svg+xml∨ (H3).(P1):image/svg+xmlπ‘˜(𝑑) is Laplace transformable, that is, it is locally integrable on image/svg+xml[0,∞) and there exists image/svg+xmlπ›½βˆˆβ„ so that image/svg+xmlΜƒπ‘˜(πœ†)=β„’(π‘˜)(πœ†)∢=lim π‘β†’βˆž ∫ 𝑏0 𝑒 βˆ’πœ†π‘‘ π‘˜(𝑑)π‘‘π‘‘βˆΆ=∫ ∞0 𝑒 βˆ’πœ†π‘‘ π‘˜(𝑑)𝑑𝑑 exists for all image/svg+xmlπœ†βˆˆβ„‚ with Put absinf

Let us remind that a function is called a kernel, if for every the supposition implies due to the famous Titchmarsh's theorem [12], the condition supp  implies that is a kernel. Set

2. -Regularized -Resolvent Families

We start with the following definition.

Definition 2.1. Let and let A strongly continuousoperator family is called a (local, if ) -regularized -resolventfamily having as a subgeneratorif and only ifthe following holds: (i) and (ii)(iii) ∫ 𝑅 ( 𝑑 ) π‘₯ = π‘˜ ( 𝑑 ) 𝐢 π‘₯ + 𝑑 0 π‘Ž ( 𝑑 βˆ’ 𝑠 ) 𝐴 𝑅 ( 𝑠 ) π‘₯ 𝑑 𝑠 , 𝑑 ∈ [ 0 , 𝜏 ) , π‘₯ ∈ 𝐷 ( 𝐴 ) . In the case 𝜏 = ∞ , ( 𝑅 ( 𝑑 ) ) 𝑑 β‰₯ 0 is said to be exponentially bounded if, additionally, there exist 𝑀 > 0 and πœ” β‰₯ 0 such that β€– 𝑅 ( 𝑑 ) β€– ≀ 𝑀 𝑒 πœ” 𝑑 , 𝑑 β‰₯ 0 ; ( 𝑅 ( 𝑑 ) ) 𝑑 ∈ [ 0 , 𝜏 ) is said to be nondegenerate if the condition 𝑅 ( 𝑑 ) π‘₯ = 0 , 𝑑 ∈ [ 0 , 𝜏 ) implies π‘₯ = 0 .
From now on, we consider only nondegenerate ( π‘Ž , π‘˜ ) -regularized 𝐢 -resolvent families. Notice that ( 𝑅 ( 𝑑 ) ) 𝑑 ∈ [ 0 , 𝜏 ) is nondegenerate provided that π‘˜ ( 0 ) β‰  0 or that (H5) holds for a subgenerator 𝐴 of ( 𝑅 ( 𝑑 ) ) 𝑑 ∈ [ 0 , 𝜏 ) .
In the case π‘˜ ( 𝑑 ) = 𝑑 𝛼 / Ξ“ ( 𝛼 + 1 ) , where 𝛼 > 0 , and Ξ“ ( β‹… ) denotes the Gamma function, it is also said that ( 𝑅 ( 𝑑 ) ) 𝑑 ∈ [ 0 , 𝜏 ) is an 𝛼 -times integrated ( π‘Ž , 𝐢 ) -resolvent family; in such a way, we unify the notion of (local) 𝛼 -times integrated 𝐢 -semigroups ( π‘Ž ( 𝑑 ) ≑ 1 ) and cosine functions ( π‘Ž ( 𝑑 ) ≑ 𝑑 ) [1, 13, 14]. Furthermore, in the case ∫ π‘˜ ( 𝑑 ) ∢ = 𝑑 0 𝐾 ( 𝑠 ) 𝑑 𝑠 , 𝑑 ∈ [ 0 , 𝜏 ) , where 𝐾 ∈ 𝐿 1 l o c ( [ 0 , 𝜏 ) ) and 𝐾 β‰  0 , we obtain the unification concept for (local) 𝐾 -convoluted 𝐢 -semigroups and cosine functions [15]. In the case π‘˜ ( 𝑑 ) ≑ 1 , ( 𝑅 ( 𝑑 ) ) 𝑑 ∈ [ 0 , 𝜏 ) is said to be a (local) ( π‘Ž , 𝐢 ) -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 β„˜ ( 𝑅 ) the set which consists of all subgenerators of ( 𝑅 ( 𝑑 ) ) 𝑑 ∈ [ 0 , 𝜏 ) .
Then the following holds.
(i) 𝐴 ∈ β„˜ ( 𝑅 ) implies 𝐢 βˆ’ 1 𝐴 𝐢 ∈ β„˜ ( 𝑅 ) . (ii)If 𝐴 ∈ β„˜ ( 𝑅 ) and πœ† ∈ 𝜌 𝐢 ( 𝐴 ) , then (iii)Assume, additionally, that π‘Ž ( 𝑑 ) is a kernel. Then one can define the integral generator  𝐴 of ( 𝑅 ( 𝑑 ) ) 𝑑 ∈ [ 0 , 𝜏 ) by setting The integral generator  𝐴 of ( 𝑅 ( 𝑑 ) ) 𝑑 ∈ [ 0 , 𝜏 ) is a closed linear operator which satisfies 𝐢 βˆ’ 1   𝐴 𝐴 𝐢 = and extends an arbitrary subgenerator of ( 𝑅 ( 𝑑 ) ) 𝑑 ∈ [ 0 , 𝜏 ) . Furthermore,  𝐴 ∈ β„˜ ( 𝑅 ) , if 𝑅 ( 𝑑 ) 𝑅 ( 𝑠 ) = 𝑅 ( 𝑠 ) 𝑅 ( 𝑑 ) , 0 ≀ 𝑑 , 𝑠 < 𝜏 .
Recall that in the case of convoluted 𝐢 -semigroups and cosine functions, the set β„˜ ( 𝑅 ) becomes a complete lattice under suitable algebraic operations and that induced partial ordering coincides with the usual set inclusion. In general, β„˜ ( 𝑅 ) needs not to be finite [9].
Henceforth we assume that the scalar-valued kernels π‘˜ , π‘˜ 1 , π‘˜ 2 , … are continuous on [ 0 , 𝜏 ) , and that π‘Ž β‰  0 in 𝐿 1 l o c ( [ 0 , 𝜏 ) ) .
Assume temporarily πœ† ∈ 𝜌 𝐢 ( 𝐴 ) , π‘₯ ∈ R a n g ( 𝐢 ) , 𝑑 ∈ [ 0 , 𝜏 ) , and put 𝑧 = ( π‘Ž βˆ— 𝑅 ) ( 𝑑 ) π‘₯ .
Following the proof of [1, Lemma  2.2], we have 𝑧 = πœ† ( π‘Ž βˆ— 𝑅 ) ( 𝑑 ) ( πœ† βˆ’ 𝐴 ) βˆ’ 1 π‘₯ βˆ’ ( π‘Ž βˆ— 𝑅 ) ( 𝑑 ) 𝐴 ( πœ† βˆ’ 𝐴 ) βˆ’ 1 π‘₯ = πœ† ( π‘Ž βˆ— 𝑅 ) ( 𝑑 ) ( πœ† βˆ’ 𝐴 ) βˆ’ 1 π‘₯ βˆ’ ( 𝑅 ( 𝑑 ) ( πœ† βˆ’ 𝐴 ) βˆ’ 1 π‘₯ βˆ’ π‘˜ ( 𝑑 ) 𝐢 ( πœ† βˆ’ 𝐴 ) βˆ’ 1 π‘₯ ) = πœ† ( πœ† βˆ’ 𝐴 ) βˆ’ 1 𝐢 ( π‘Ž βˆ— 𝑅 ) ( 𝑑 ) 𝐢 βˆ’ 1 π‘₯ βˆ’ ( ( πœ† βˆ’ 𝐴 ) βˆ’ 1 𝑅 ( 𝑑 ) π‘₯ βˆ’ π‘˜ ( 𝑑 ) ( πœ† βˆ’ 𝐴 ) βˆ’ 1 𝐢 π‘₯ ) , where the last two equalities follow on account of 𝐢 𝐴 βŠ† 𝐴 𝐢 , 𝑅 ( 𝑠 ) 𝐴 βŠ† 𝐴 𝑅 ( 𝑠 ) and 𝑅 ( 𝑠 ) ( πœ† βˆ’ 𝐴 ) βˆ’ 1 𝐢 = ( πœ† βˆ’ 𝐴 ) βˆ’ 1 𝐢 𝑅 ( 𝑠 ) , 𝑠 ∈ [ 0 , 𝜏 ) . Hence, ( πœ† βˆ’ 𝐴 ) 𝑧 = πœ† 𝑧 βˆ’ ( 𝑅 ( 𝑑 ) π‘₯ βˆ’ 𝐢 π‘₯ ) ,

The closedness of 𝐴 implies that (2.3) holds for every 𝑑 ∈ [ 0 , 𝜏 ) and π‘₯ ∈ R a n g ( 𝐢 ) .

Theorem 2.2 (see [1]). (i) Let 𝐴 be a subgenerator of an ( π‘Ž , π‘˜ ) -regularized C -resolvent family ( 𝑅 ( 𝑑 ) ) 𝑑 ∈ [ 0 , 𝜏 ) , and let (H5) hold. Then (2.3) holds for every 𝑑 ∈ [ 0 , 𝜏 ) and π‘₯ ∈ 𝐸 . If 𝜌 𝐢 ( 𝐴 ) β‰  βˆ… , then (2.3) holds for every t ∈ [ 0 , 𝜏 ) and π‘₯ ∈ R a n g ( 𝐢 ) .
(ii) Let 𝐴 be a subgenerator of an ( π‘Ž , π‘˜ 𝑖 ) -regularized 𝐢 -resolvent family ( 𝑅 𝑖 ( 𝑑 ) ) 𝑑 ∈ [ 0 , 𝜏 ) , 𝑖 = 1 , 2 . Then ( π‘˜ 2 βˆ— 𝑅 1 ) ( 𝑑 ) = ( π‘˜ 1 βˆ— 𝑅 2 ) ( 𝑑 ) , 𝑑 ∈ [ 0 , 𝜏 ) , whenever (H4) holds.
(iii) Let ( 𝑅 1 ( 𝑑 ) ) 𝑑 ∈ [ 0 , 𝜏 ) and ( 𝑅 2 ( 𝑑 ) ) 𝑑 ∈ [ 0 , 𝜏 ) be two ( π‘Ž , π‘˜ ) -regularized 𝐢 -resolvent families having 𝐴 as a subgenerator. Then 𝑅 1 ( 𝑑 ) π‘₯ = 𝑅 2 ( 𝑑 ) π‘₯ , 𝑑 ∈ [ 0 , 𝜏 ) , π‘₯ ∈ 𝐷 ( 𝐴 ) , and 𝑅 1 ( 𝑑 ) = 𝑅 2 ( 𝑑 ) , 𝑑 ∈ [ 0 , 𝜏 ) , if (H4) holds.
(iv) Let 𝐴 be a subgenerator of an ( π‘Ž , π‘˜ ) -regularized 𝐢 -resolvent family ( 𝑅 ( 𝑑 ) ) 𝑑 ∈ [ 0 , 𝜏 ) . If π‘˜ ( 𝑑 ) is absolutely continuous and π‘˜ ( 0 ) β‰  0 , then 𝐴 is a subgenerator of an ( π‘Ž , 𝐢 ) -regularized resolvent family on [ 0 , 𝜏 ) .

Remark 2.3. (i) Let ( 𝑅 𝑖 ( 𝑑 ) ) 𝑑 ∈ [ 0 , 𝜏 ) be an ( π‘Ž , π‘˜ 𝑖 ) -regularized 𝐢 -resolvent family with a subgenerator 𝐴 ,    𝑖 = 1 , 2 , and let 𝐷 ( 𝐴 ) β‰  { 0 } . Then π‘˜ 1 = π‘˜ 2 .
(ii) Let ( 𝑅 𝑖 ( 𝑑 ) ) 𝑑 ∈ [ 0 , 𝜏 ) be an ( π‘Ž , π‘˜ 𝑖 ) -regularized 𝐢 -resolvent family with a subgenerator 𝐴 , 𝑖 = 1 , 2 . Then, for every 𝛼 ∈ β„‚ and 𝛽 ∈ β„‚ , ( 𝛼 𝑅 1 ( 𝑑 ) + 𝛽 𝑅 2 ( 𝑑 ) ) 𝑑 ∈ [ 0 , 𝜏 ) is an ( π‘Ž , 𝛼 π‘˜ 1 + 𝛽 π‘˜ 2 ) -regularized 𝐢 -resolvent family with a subgenerator 𝐴 .
(iii) Let ( 𝑅 ( 𝑑 ) ) 𝑑 ∈ [ 0 , 𝜏 ) be an ( π‘Ž , π‘˜ ) -regularized 𝐢 -resolvent family with a subgenerator 𝐴 , and let 𝐿 1 l o c ( [ 0 , 𝜏 ) ) βˆ‹ 𝑏 be a kernel. Then 𝐴 is a subgenerator of an ( π‘Ž , π‘˜ βˆ— 𝑏 ) -regularized 𝐢 -resolvent family ( ( 𝑏 βˆ— 𝑅 ) ( 𝑑 ) ) 𝑑 ∈ [ 0 , 𝜏 ) .
(iv) Let ( 𝑅 ( 𝑑 ) ) 𝑑 ∈ [ 0 , 𝜏 ) be an ( π‘Ž , 𝐢 ) -regularized resolvent family having 𝐴 as a subgenerator. Then ( ( π‘˜ βˆ— 𝑅 ) ( 𝑑 ) ) 𝑑 ∈ [ 0 , 𝜏 ) is an ( π‘Ž , Θ ) -regularized 𝐢 -resolvent family with a subgenerator 𝐴 .
(v) Suppose ( 𝑅 ( 𝑑 ) ) 𝑑 ∈ [ 0 , 𝜏 ) is an ( π‘Ž , π‘˜ ) -regularized 𝐢 -resolvent family with a subgenerator 𝐴 , (H1) or (H3) holds, and π‘Ž ( 𝑑 ) is a kernel. Then the integral generator  𝐴 of ( 𝑅 ( 𝑑 ) ) 𝑑 ∈ [ 0 , 𝜏 ) satisfies  𝐴 = 𝐢 βˆ’ 1 𝐴 𝐢 . Toward this end, let  ( π‘₯ , 𝑦 ) ∈ 𝐴 . Then ∫ 𝑑 0 ∫ π‘Ž ( 𝑑 βˆ’ 𝑠 ) [ π‘˜ ( 𝑠 ) 𝐢 π‘₯ + 𝑠 0 ∫ π‘Ž ( 𝑠 βˆ’ π‘Ÿ ) 𝑅 ( π‘Ÿ ) 𝑦 𝑑 π‘Ÿ ] 𝑑 𝑠 = 𝑑 0 π‘Ž ( 𝑑 βˆ’ 𝑠 ) 𝑅 ( 𝑠 ) π‘₯ 𝑑 𝑠 ∈ 𝐷 ( 𝐴 ) , 𝑑 ∈ [ 0 , 𝜏 ) , and 𝐴 ∫ 𝑑 0 ∫ π‘Ž ( 𝑑 βˆ’ 𝑠 ) [ π‘˜ ( 𝑠 ) 𝐢 π‘₯ + 𝑠 0 ∫ π‘Ž ( 𝑠 βˆ’ π‘Ÿ ) 𝑅 ( π‘Ÿ ) 𝑦 𝑑 π‘Ÿ ] 𝑑 𝑠 = 𝐴 𝑑 0 ∫ π‘Ž ( 𝑑 βˆ’ 𝑠 ) 𝑅 ( 𝑠 ) π‘₯ 𝑑 𝑠 = 𝑅 ( 𝑑 ) π‘₯ βˆ’ π‘˜ ( 𝑑 ) 𝐢 π‘₯ = 𝑑 0 π‘Ž ( 𝑑 βˆ’ 𝑠 ) 𝑅 ( 𝑠 ) 𝑦 𝑑 𝑠 , 𝑑 ∈ [ 0 , 𝜏 ) . Since ( π‘Ž βˆ— 𝑅 ) ( 𝑑 ) 𝑦 ∈ 𝐷 ( 𝐴 ) , ( π‘Ž βˆ— π‘Ž βˆ— 𝑅 ) ( 𝑑 ) 𝑦 ∈ 𝐷 ( 𝐴 ) , 𝐴 ( π‘Ž βˆ— π‘Ž βˆ— 𝑅 ) ( 𝑑 ) 𝑦 = ( π‘Ž βˆ— ( 𝑅 βˆ’ π‘˜ 𝐢 ) ) ( 𝑑 ) 𝑦 , 𝑑 ∈ [ 0 , 𝜏 ) , and π‘Ž βˆ— π‘˜ β‰  0 in 𝐢 ( [ 0 , 𝜏 ) ) , it follows that 𝐢 π‘₯ ∈ 𝐷 ( 𝐴 ) , 𝐴 𝐢 π‘₯ = 𝐢 𝑦 , π‘₯ ∈ 𝐷 ( 𝐢 βˆ’ 1 𝐴 𝐢 ) , and 𝐢 βˆ’ 1  𝐴 𝐢 π‘₯ = 𝐴 π‘₯ = 𝑦 . On the other hand, 𝐢 βˆ’ 1 𝐴 𝐢 is a subgenerator of ( 𝑅 ( 𝑑 ) ) 𝑑 ∈ [ 0 , 𝜏 ) whenever 𝐴 is; this implies 𝐢 βˆ’ 1  𝐴 𝐴 𝐢 βŠ† and proves the claim. If (H2) holds, then  𝐴 = 𝐢 βˆ’ 1 𝐴 𝐢 = 𝐴 . In what follows, we also assume that 𝐡 ∈ β„˜ ( 𝑅 ) and that (H5) holds for 𝐡 and 𝐢 . Proceeding as in the proof of [9, Proposition  2.1.1.6], one gets what follows.
(v.1) 𝐢 βˆ’ 1 𝐴 𝐢 = 𝐢 βˆ’ 1 𝐡 𝐢 and 𝐢 ( 𝐷 ( 𝐴 ) ) βŠ† 𝐷 ( 𝐡 ) . (v.2) 𝐴 and 𝐡 have the same eigenvalues.(v.3)The assumption 𝐴 βŠ† 𝐡 implies 𝜌 𝐢 ( 𝐴 ) βŠ† 𝜌 𝐢 ( 𝐡 ) . (v.4)card ( β„˜ ( 𝑅 ) ) = 1 , if  𝐢 ( 𝐷 ( 𝐴 ) ) is a core for  𝐷 ( 𝐴 ) . (v.5) 𝐴 βŠ† 𝐡 ⇔ 𝐷 ( 𝐴 ) βŠ† 𝐷 ( 𝐡 ) and 𝐴 π‘₯ = 𝐡 π‘₯ , π‘₯ ∈ 𝐷 ( 𝐴 ) ∩ 𝐷 ( 𝐡 ) ; furthermore, the property (v.5) holds whenever { 𝐴 , 𝐡 } βŠ† β„˜ ( 𝑅 ) and π‘Ž ( 𝑑 ) is a kernel.
We refer the reader to [1, page 283] for the definition of (weak) solutions of the problem
where 𝑓 ∈ 𝐢 ( [ 0 , 𝜏 ) ∢ 𝐸 ) , and to [1, page 285] for the notion of spaces 𝐢 𝑛 , π‘˜ ( [ 0 , 𝜏 ) ∢ 𝐸 ) , 𝑛 ∈ β„• , π‘˜ ∈ β„• 0 and 𝐢 𝑛 0 ( [ 0 , 𝜏 ) ∢ 𝐸 ) , a n d 𝑛 ∈ β„• .
Define a subset 𝐴 βˆ— of 𝐸 βˆ— Γ— 𝐸 βˆ— (the use of symbol βˆ— is clear from the context) by image/svg+xml𝐴 βˆ— ∢={(π‘₯ βˆ— ,𝑦 βˆ— )∈𝐸 βˆ— ×𝐸 βˆ— ∢π‘₯ βˆ— (𝐴π‘₯)=𝑦 βˆ— (π‘₯) forall π‘₯∈𝐷(𝐴)}. In the case when image/svg+xml𝐴 is densely defined, image/svg+xml𝐴 βˆ— is a linear mapping from image/svg+xml𝐸 βˆ— into image/svg+xml𝐸 βˆ— .

Lemma 2.4 (see [17]). Let image/svg+xml𝐴 be a closed linear operator. Assume image/svg+xmlπ‘₯ 0 ∈𝐸, image/svg+xml𝑦 0 ∈𝐸, and image/svg+xmlπ‘₯ βˆ— (𝑦 0 )=𝑦 βˆ— (π‘₯ 0 ) for all image/svg+xml(π‘₯ βˆ— ,𝑦 βˆ— )∈𝐴 βˆ— . Then image/svg+xmlπ‘₯ 0 ∈𝐷(𝐴), and image/svg+xml𝐴π‘₯ 0 =𝑦 0 .

Define the mapping image/svg+xml𝐾 𝐢 ∢𝐢([0,𝜏)∢𝐸)→𝐢([0,𝜏)∢𝐸) by image/svg+xml𝐾 𝐢 π‘’βˆΆ=π‘˜βˆ—πΆπ‘’,π‘’βˆˆπΆ([0,𝜏)∢𝐸). Then image/svg+xml𝐾 𝐢 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 image/svg+xmlπ‘“βˆˆπΆ([0,𝜏)∢𝐸), image/svg+xml𝐴 is a subgenerator of a (local) image/svg+xml(π‘Ž,π‘˜) -regularized image/svg+xml𝐢 -resolvent family image/svg+xml(𝑅(𝑑)) π‘‘βˆˆ[0,𝜏) , and (H5) holds. Then (2.4) has a unique solution if and only if image/svg+xmlπ‘…βˆ—π‘“βˆˆRang(𝐾 𝐢 ).
(ii) (cf. also [18]) Assume image/svg+xmlπ‘›βˆˆβ„•, image/svg+xmlπ‘“βˆˆπΆ([0,𝜏)∢𝐸), image/svg+xml𝐴 is a subgenerator of a (local) image/svg+xml𝑛 -times integrated image/svg+xml(π‘Ž,𝐢) -resolvent family image/svg+xml(𝑅(𝑑)) π‘‘βˆˆ[0,𝜏) , and (H5) holds. Then (2.4) has a unique solution if and only if image/svg+xml𝐢 βˆ’1 (π‘…βˆ—π‘“)∈𝐢 𝑛+10 ([0,𝜏)∢𝐸).
(iii) Let the assumptions of the item (i) of this theorem hold, and let image/svg+xmlπ‘˜βˆˆπΆ 𝑛0 ([0,𝜏)∢𝐸). Then image/svg+xml𝐢 βˆ’1 (π‘…βˆ—π‘“)∈𝐢 (𝑛+1) ([0,𝜏)∢𝐸) if and only if image/svg+xml𝐢 βˆ’1 (π‘…βˆ—π‘“)∈𝐢 𝑛+10 ([0,𝜏)∢𝐸).
(iv) Let (H5) hold. Assume that image/svg+xmlπ‘›βˆˆβ„•, image/svg+xml𝐴 is a subgenerator of an image/svg+xml𝑛 -times integrated image/svg+xml(π‘Ž,𝐢) -regularized resolvent, and image/svg+xmlπ‘Žβˆˆπ΅π‘‰ loc ([0,𝜏)∢𝐸), respectively image/svg+xml𝐴 is a subgenerator of an image/svg+xml(π‘Ž,𝐢) -regularized resolvent family. Assume, further, that image/svg+xml𝐢 βˆ’1 π‘“βˆˆπΆ (𝑛+1) ([0,𝜏)∢𝐸), image/svg+xml𝑓 (π‘˜βˆ’1) (0)∈𝐷(𝐴 𝑛+1βˆ’π‘˜ ) and image/svg+xml𝐴 𝑛+1βˆ’π‘˜ 𝑓 (π‘˜βˆ’1) (0)∈Rang(𝐢), image/svg+xml1β‰€π‘˜β‰€π‘›+1, respectively image/svg+xml𝐢 βˆ’1 π‘“βˆˆπ΄πΆ loc ([0,𝜏)∢𝐸). Then (2.4) has a unique solution.
(v) Assume that (H5) holds, image/svg+xml𝐴 is a subgenerator of an image/svg+xml(π‘Ž,π‘˜) -regularized image/svg+xml𝐢 -resolvent family, image/svg+xmlπ‘˜(𝑑) is absolutely continuous, and image/svg+xmlπ‘˜(0)β‰ 0. If image/svg+xml𝐢 βˆ’1 π‘“βˆˆπΆ 1 ([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 image/svg+xmlπ‘˜(𝑑) and image/svg+xmlπ‘Ž(𝑑) satisfy (P1), and let image/svg+xml(𝑅(𝑑)) 𝑑β‰₯0 be a strongly continuous operator family satisfying image/svg+xml‖𝑅(𝑑)‖≀𝑀𝑒 πœ”π‘‘ ,𝑑β‰₯0, for some image/svg+xml𝑀>0 and image/svg+xmlπœ”β‰₯0. Put image/svg+xmlπœ” 0 ∢=max(πœ”, abs (π‘Ž), abs (π‘˜)). (i)Suppose image/svg+xml𝐴 is a subgenerator of the exponentially bounded image/svg+xml( a , k ) -regularized image/svg+xml𝐢 -resolvent family image/svg+xml(𝑅(𝑑)) 𝑑β‰₯0 , and (H5) holds. Then, for every image/svg+xmlπœ†βˆˆβ„‚ with image/svg+xmlReπœ†>πœ” 0 and image/svg+xmlΜƒπ‘˜(πœ†)β‰ 0, the operator image/svg+xmlπΌβˆ’Μƒπ‘Ž(πœ†)𝐴 is injective, image/svg+xmlRang(𝐢)βŠ†Rang(πΌβˆ’Μƒπ‘Ž(πœ†)𝐴), and image/svg+xml𝐴 (ii)Assume that (2.5)-(2.6) hold. Then image/svg+xml(π‘Ž,π‘˜) is a subgenerator of the exponentially bounded image/svg+xmlC -regularized image/svg+xml(𝑅(𝑑)) 𝑑β‰₯0 . -resolvent family image/svg+xml(π‘Ž,π‘˜)

The preceding theorem enables one to establish the real and complex characterization of subgenerators of (locally Lipschitz continuous) exponentially bounded image/svg+xml𝐢 -regularized image/svg+xmlπ‘˜(𝑑) -resolvent families [1, 9, 12]:

Theorem 2.7. (i) Let image/svg+xmlπ‘Ž(𝑑) and image/svg+xmlπœ” 0 β‰₯max(0, abs (π‘Ž), abs (π‘˜)). satisfy (P1), and let image/svg+xmlπœ†βˆˆβ„‚ Assume that, for every image/svg+xmlReπœ†>πœ” 0 with image/svg+xmlΜƒπ‘˜(πœ†)β‰ 0, and image/svg+xmlπΌβˆ’Μƒπ‘Ž(πœ†)𝐴 the operator image/svg+xmlRang(𝐢)βŠ†Rang(πΌβˆ’Μƒπ‘Ž(πœ†)𝐴). is injective and that image/svg+xmlΞ₯∢{πœ†βˆˆβ„‚βˆΆReπœ†>πœ” 0 }→𝐿(𝐸) If there exists an analytic function image/svg+xmlΞ₯(πœ†)=Μƒπ‘˜(πœ†)(πΌβˆ’Μƒπ‘Ž(πœ†)𝐴) βˆ’1 𝐢,πœ†βˆˆβ„‚,Reπœ†>πœ” 0 with:
(i.1)image/svg+xmlβ€–Ξ₯(πœ†)‖≀𝑀|πœ†| π‘Ÿ ,πœ†βˆˆβ„‚,Reπœ†>πœ” 0 , , (i.2)image/svg+xml𝑀>0 for some image/svg+xmlπ‘Ÿβ‰₯βˆ’1, and image/svg+xml𝛼>1, then, for every image/svg+xml𝐴 image/svg+xml(π‘Ž,π‘˜βˆ—π‘‘ 𝛼+π‘Ÿβˆ’1 /Ξ“(𝛼+π‘Ÿ)) is a subgenerator of a norm continuous, exponentially bounded image/svg+xml𝐢 -regularized image/svg+xmlπ‘˜(𝑑) -resolvent family.
(ii) Suppose image/svg+xmlπ‘Ž(𝑑) and image/svg+xml𝐴 satisfy (P1) and (H2) or (H3) holds, and image/svg+xml(π‘Ž,Θ) is a subgenerator of an exponentially bounded image/svg+xml𝐢 -regularized image/svg+xml(𝑅(𝑑)) 𝑑β‰₯0 -resolvent family image/svg+xml‖𝑅(𝑑+β„Ž)βˆ’π‘…(𝑑)β€–β‰€π‘€β„Žπ‘’ πœ”(𝑑+β„Ž) ,𝑑β‰₯0,β„Žβ‰₯0, forsome 𝑀>0,πœ”β‰₯0. (2.7) which satisfies the next condition:
Then there exists image/svg+xmlξ‚»1 Μƒπ‘Ž(πœ†)βˆΆπœ†>π‘Ž,Μƒπ‘˜(πœ†)Μƒπ‘Ž(πœ†)β‰ 0ξ‚ΌβŠ†πœŒ 𝐢 (𝐴), (2.8)themapping πœ†β†¦π»(πœ†)∢=Μƒπ‘˜(πœ†)(πΌβˆ’Μƒπ‘Ž(πœ†)𝐴) βˆ’1 𝐢,πœ†>π‘Ž,Μƒπ‘˜(πœ†)Μƒπ‘Ž(πœ†)β‰ 0 (2.9) such that is infinitely differentiable and
(iii) Suppose image/svg+xml𝐴 and image/svg+xml(π‘Ž,Θ) satisfy (P1) and (2.8)–(2.10) holds. Then image/svg+xml𝐢 is a subgenerator of an exponentially bounded image/svg+xml(𝑅(𝑑)) 𝑑β‰₯0 -regularized image/svg+xml𝑀>0 -resolvent family image/svg+xmlπœ”β‰₯0, which satisfies (2.7).
(iv) Suppose image/svg+xmlπ‘˜(𝑑) , image/svg+xmlπ‘Ž(𝑑) image/svg+xml𝐴 and image/svg+xml𝐴 satisfy (P1), and image/svg+xml(π‘Ž,π‘˜) is densely defined. Then image/svg+xml𝐢 is a subgenerator of an exponentially bounded image/svg+xml(𝑅(𝑑)) 𝑑β‰₯0 -regularized image/svg+xml‖𝑅(𝑑)‖≀𝑀𝑒 πœ”π‘‘ ,𝑑β‰₯0 -resolvent family image/svg+xmlπ‘Žβ‰₯max(0, abs (π‘Ž), abs (π‘˜)) which satisfies image/svg+xmlπ‘Ž βˆ—π‘› if and only if there exists image/svg+xml𝑛 such that (2.8)–(2.10) hold.

Denote by image/svg+xmlπ‘Ž(𝑑), the image/svg+xmlπ‘›βˆˆβ„•, th convolution power of the kernel image/svg+xmlπ‘Ž(𝑑),𝑏(𝑑), image/svg+xml𝑐(𝑑) 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 image/svg+xml∫ ∞0 𝑒 βˆ’π›½π‘‘ |𝑏(𝑑)|𝑑𝑑<∞ and image/svg+xml𝛽β‰₯0. satisfy (P1), and let image/svg+xml𝛼=̃𝑐 βˆ’1 ξ‚΅1 𝛽 if ξ€œ ∞0 𝑐(𝑑)𝑑𝑑>1 𝛽,𝛼=0 otherwise , (2.11) for some image/svg+xmlΜƒπ‘Ž(πœ†)=̃𝑏(1/̃𝑐(πœ†)), Let and let image/svg+xml𝐴 image/svg+xml(𝑏,π‘˜) Let image/svg+xml𝐢 be a subgenerator of a image/svg+xml(𝑅 𝑏 (𝑑)) 𝑑β‰₯0 -regularized image/svg+xml‖𝑅 𝑏 (𝑑)‖≀𝑀𝑒 πœ” 𝑏 𝑑 ,𝑑β‰₯0, -resolvent family image/svg+xml𝑀>0 satisfying image/svg+xmlπœ” 𝑏 β‰₯0, for some image/svg+xml𝑐(𝑑) and image/svg+xmlπ‘˜ 1 (𝑑) and let (H2) or (H3) hold. Assume, further, that image/svg+xmlΜƒπ‘˜ 1 (πœ†)=1 πœ†Μƒπ‘(πœ†)Μƒπ‘˜ξ‚΅1