CC BY
16
Chelyabinsk Physical and Mathematical Journal. 2020. Vol. 5, iss. 3. P. 363-385.
DOI: 10.47475/2500-0101-2020-15310
COMPLEX POWERS OF MULTIVALUED LINEAR OPERATORS WITH POLYNOMIALLY BOUNDED C-RESOLVENT
M. Kostic
University of Novi Sad, Novi Sad, Serbia marco.s @verat.net
We construct complex powers of multivalued linear operators with polynomially bounded c-resolvent existing on an appropriate region of the complex plane containing the interval (-to, 0]. In our approach, the operator c is not necessarily injective. We clarify the basic properties of introduced powers and analyze the abstract incomplete fractional differential inclusions associated with the use of modified Liuoville right-sided derivatives. We also consider abstract incomplete differential inclusions of second order, working in the general setting of sequentially complete locally convex spaces.
Keywords: complex power of a multivalued linear operator, C-resolvent set, abstract incomplete fractional differential inclusion, abstract incomplete differential inclusion of second order, locally convex space.
1. Introduction and preliminaries
Chronologically, the first results about fractional powers of non-negative multivalued linear operators was given by El H. Alaarabiou [1; 2] in 1991. In these papers, he extended the well known Hirsch functional calculus to the class M of non-negative multivalued linear operators in a complex Banach space. Unfortunately, the method proposed in [1; 2] had not allowed one to consider the product formula and the spectral mapping theorem for powers. Nine years later, in 2000, C. Martinez, M. Sanz and J. Pastor [3] improved a functional calculus established in [1; 2], providing a new definition of fractional powers. A very stable and consistent theory of fractional powers of the operators belonging to the class M has been constructed, including within itself the above-mentioned product formula, spectral mapping theorem, as well as almost all other fundamental properties of fractional powers of non-negative single-valued linear operators. Some later contributions have been given by J. Pastor [4], who considered relations between the multiplicativity and uniqueness of fractional powers of non-negative multivalued linear operators.
The first applications of results from the theory of multivalued linear operators to abstract degenerate differential equations were given by A. Yagi [5] in 1991. In his well-known joint monograph with A. Favini [6], the class of multivalued linear operators A, acting on a complex Banach space (X, || ■ ||), for which (-to, 0] C p(A) and there exist finite numbers M\ > 1,0 E (0,1] such that
||R(A : A)||< Mi(l + |A|)-^, A < 0, (1)
The author is partially supported by grant 174024 of Ministry of Science and Technological Development, Republic of Serbia.
has been thoroughly analyzed (see also [7; 8]). Assuming that (1) is true, the usual von Neumann's expansion argument shows that there exist positive real constants c > 0 and M > 0 such that the resolvent set of A contains an open region
where we have the estimate ||R(A : A)|| = O((1 + |A|)-^), A e Hc,m. Let F be the upwards oriented curve (£ ± i(2M)-1(c — : —to < £ < c}. In [6], A. Favini and A. Yagi define the fractional power A-0, for ReB > 1 — P, by
A0 := (A-0)-1 (ReB > 1 — P); then A-0 e L(E) for ReB > 1 — P, and the semigroup properties A-01 A-02 = A-(01+02), A01 A02 = A01+02 of powers hold for ReB^ ReB2 > 1 — p. The case P e (0,1) occurs in many applications and then we cannot define satisfactorily the power A0 for |ReB| < 1 — p. As explained in the introductory part of paper [9] by A. Favaron and A. Favini, the method of closed extensions used in the pioneering works [10] by A. Balakrishnan and [11] by H. Komatsu cannot be used here for construction of power A0 (ReB e (0,1 — P)). In this place, we would like to observe that the method proposed by F. Periago, B. Straub [12] and C. Martinez, M. Sanz, A. Redondo [13] (cf. also [14]) cannot be of any help for construction of power A0 (ReB e (0,1 — P)), as well. In [9, Section 9], the fractional power A0 has been constructed for |ReB| < 1 — P, provided the validity of condition [9, (H3)]. In general case P e (0,1), the condition (H3) does not hold.
In order to motivate our research, assume that a > — 1 and a closed multivalued linear operator A satisfies:
(0) (0, to) C p(A) and (00) supA>o(1 + |A|)-a||R(A : A)|| < to.
Given P > — 1, e e (0,1], d e (0,1], C e (0,1) and B e (0,n], put Bd := (z e C : |z| < d}, £0 := (z e C : z = 0, arg(z) e (—B, B)} and P^,£,c/ := (£ + in : C > e, n e R, |n| < c'(1 + C)-^}. Then the usual series argument yields that the hypotheses (0)-(00) imply the existence of numbers d e (0,1], c e (0,1), e e (0,1] and M > 0 such that:
(§) Pa,e,c U Bd C p(A), (e, c(1 + e)-a) e and (§§) ||R(A : A)|| < M(1 + |A|)a, A e Pa,e,c U Bd.
Suppose now that X is a Hausdorff sequentially complete locally convex space over the field of complex numbers, SCLCS for short. If Y is also an SCLCS, then we denote by L(X, Y) the space consisting of all continuous linear mappings from X into Y; L(X) := L(X, X). By ©X (©, if there is no risk for confusion), we denote the fundamental system of seminorms which defines the topology of X.
Keeping in mind the above analysis (the notion will be explained a little bit later), it seems reasonable to introduce the following condition:
(H)0 : Let C e L(X) be not necessarily injective, let A be closed, and let CA C AC. There exist real numbers d e (0,1], c e (0,1), e e (0,1] and a > — 1 such that P«,£,cUBd C pc(A), the operator family ((1 + |A|)-a(A — A)-1C : A e P«,£,cUBd} C L(X) is equicontinuous, the mapping A M- (A — A)-1C is strongly analytic on int(Pa,£,c U Bd) and strongly continuous on 5(Pa,e,c U Bd).
3 := (A e C : |ImA| < (2M)-1(c — ReA)^, ReA < c},
The first aim of this paper is construction of complex power (—A)b, b E C of a multivalued linear operator A satisfying the condition (H)0. Although very elegant and elementary, our construction has some serious disadvantages because the introduced powers behave very badly (for example, we cannot expect the additivity property of powers clarified in [15, Remark 2.11]) in the case that the regularizing operator Ci, defined below, is not injective (since the resolvents and C-resolvents of a really multivalued linear operator are not injective, this is the main case in our considerations). The method proposed for construction of power (—A)b is different from that already employed in single-valued linear case [16]; in this paper, we first apply regularization with the operator Ci and follow after that the approach from our joint research paper with C. Chen, M. Li and M. Zigic [15]. In particular, we define any complex power of a multivalued linear operator satisfying (1) and not (H3).
The following sectorial analogon of (H) is most important in applications:
(HS)0 : Let C E L(X) be not necessarily injective, let A be closed, and let CA C AC. There exist real numbers d E (0,1], $ E (0,n/2) and a > —1 such that S U Bd C pc(A), the operator family {(l + |A|)-a(A — A)-1C : A E S U Bdj C L(X) is equicontinuous, the mapping A M (A — A)-1 C is strongly analytic on int(£# U Bd) and strongly continuous on d(£# U Bd).
The construction of power (—A)b, b E C of a multivalued linear operator A for which 0 E int(pc(A)) is without the scope of this paper. For the sake of brevity and better exposition, we will not compare the construction presented here with those appearing in [1-3; 9-12; 14; 16-18], if it makes any sense for doing so.
The second aim of this paper is to show that a great number of resolvent equations and generalized resolvent equations holds for C-resolvents of multivalued linear operators, where C is non-injective, in general. The third and, simultaneously, the main aim of this paper is to continue our recent research study [19] of abstract incomplete fractional degenerate differential equations with modified Liouville right-sided fractional derivatives [20] and abstract incomplete degenerate differential equations of second order. We consider fractionally integrated ^-regularized semigroups generated by the negatives of introduced powers, and provide a few relevant applications of our theoretical results to abstract incomplete degenerate PDEs. Before going any further, the author would like to express his sincere gratitude to Professor Vladimir E. Fedorov for his permanent support and encouragement.
The organization of material is briefly described as follows. In the second section, we collect the basic definitions and results from the theory of multivalued linear operators that are necessary for our further work; in two separate subsections, we consider C-resolvents of multivalued linear operators and generation of fractionally integrated C-semigroups by multivalued linear operators. In the third section, we define the complex powers of operators satisfying (H)0 or (HS)0 and clarify their most intriguing properties. The fourth section of paper is completely devoted to the study of abstract incomplete differential inclusions.
We use the standard notation throughout the paper. Unless specifed otherwise, we assume that X is a Hausdorff sequentially complete locally convex space over the field of complex numbers, SCLCS for short. Let B be the family consisting of all bounded subsets of X, and let pB(T) := supxeBp(Tx), p E ©X, B E B, T E L(X). Then pB(-) is a seminorm on L(X) and the system (pB)(p,B)e®Xxs induces the Hausdorff locally convex topology on L(X). If X is a Banach space, then we denote by ||x|| the norm of an element x E X. The Hausdorff locally convex topology on X*, the dual space of X, defines the system (| ■ |B)BeB of seminorms on X*, where and in the sequel |x*|B := supxeB |(x*,x)|,
x* e X*, B e B. Let us recall that the spaces L(X) and X* are sequentially complete provided that X is barreled [21]. Set gz(t) := tZ-1/r(Z), |_sj := sup(n e Z : n < s}, [s] := inf(n e Z : s < n} (Z > 0, s e R) and C+ := (z e C : Rez > 0}; here, T(-) denotes the Gamma function. Let denote the Kronecker delta.
If V is a general topological vector space, then a function f : Q M V, where Q is an open subset of C, is said to be analytic if it is locally expressible in a neighborhood of any point z e Q by a uniformly convergent power series with coefficients in V. We refer the reader to [20, Section 1.1] and references cited there for the basic information about vector-valued analytic functions. In our approach the space X is sequentially complete, so that the analyticity of a mapping f : Q M X (0 = Q C C) is equivalent with its weak analyticity.
The integration of functions with values in SCLCSs is still an active field of research. In this paper, we follow the approach used in the monograph [22] by C. Martinez and M. Sanz, see pp. 99-102. Concerning the Laplace transform of functions with values in SCLCSs, we refer the reader to [23] and [24]; cf. [25] for the Banach space case. Fairly complete information on abstract degenerate differential equations can be obtained by consulting the monographs [6; 24; 26-29].
Considerable interest in fractional calculus and fractional differential equations has been stimulated due to their numerous applications in engineering, physics, chemistry, biology and other sciences (see e.g. [30-35]). In this paper, we use the modified Liouville right-sided fractional derivatives. Suppose that P > 0 and P e N. Then the Liouville right-sided fractional derivative of order P (see [36, (2.3.4)] for the scalar-valued case) is defined for those continuous functions u : (0, to) m X for which limT^™ JT gf^i-^(t — s)u(t) dt = J™ (t — s)u(t) dt exists and defines a [P]-times continuously differentiable function on (0, to), by
oo
d^l f
D-u(s):=(—1)rel dsM J gfei-e (t — s)u(t) dt, s > 0.
s
We define the modified Liouville right-sided fractional derivative of order P, D-u(s) shortly, for those continuously differentiable functions u : (0, to) m X for which limT^^ J^(t — s)u'(t) dt = f™ (t — s)u'(t) dt exists and defines a [P — 1]-
times continuously differentiable function on (0, to), by
™
dfe-1l f
D-u(s) := (—1)[elds^ J gr^l-e(t — s)u'(t) dt, s > 0;
s
if P = n e N, then D^u and D^u are defined for all n-times continuously differentiable functions u(-) on (0, to), by D-u := D-u := (—1)nd/dn, where d/dn denotes the usual derivative operator of order n (cf. also [36, (2.3.5)]).
2. Multivalued linear operators
In this section, we will present some necessary definitions from the theory of multivalued linear operators. For more details about this topic, we refer the reader to the monographs by R. Cross [37] and A. Favini, A. Yagi [6].
Let X and Y be two sequentially complete locally convex spaces over the field of complex numbers. A multivalued map A : X M P(Y) is said to be a multivalued linear operator (MLO) iff the following holds:
(i) D(A) := (x e X : Ax = 0} is a linear subspace of X;
(ii) Ax + Ay с A(x + y), x,y e D(A) and A Ax с A(Ax), A e C, x e D(A).
If X = Y, then we say that A is an MLO in X. As an almost immediate consequence of definition, we have that the equality AAx + nAy = A(Ax + ny) holds for every x,y e D(A) and for every A,n e C with |A| + |n| = 0. If A is an MLO, then A0 is a linear manifold in Y and Ax = f + A0 for any x e D(A) and f e Ax. Define R(A) := {Ax : x e D(A)}. Then the set N (A) := A-10 = {x e D(A) : 0 e Ax} is called the kernel of A. The inverse A-1 of an MLO is defined by D(A-1) := R(A) and A-1y := {x e D(A) : y e Ax}. It can be easily seen that A-1 is an MLO in X, as well as that N (A-1) = A0 and (A-1)-1 = A. If N (A) = {0}, i.e., if A-1 is single-valued, then A is said to be injective. If A, B : X ^ P(Y) are two MLOs, then we define its sum A + B by D(A + B) := D(A) П D(B) and (A + B)x := Ax + Bx, x e D(A + B). It is clear that A + B is likewise an MLO. We write A С B iff D(A) С D(B) and Ax С Bx for all x e D(A).
Let A : X ^ P(Y) and B : Y ^ P(Z) be two MLOs, where Z is an SCLCS. The product of A and B is defined by D(BA) := {x e D(A) : D(B) П Ax = 0} and BAx := B(D(B) П Ax). Then BA : X ^ P(Z) is an MLO and (bA)-1 = A-1B-1. The scalar multiplication of an MLO A : X ^ P(Y) with the number z e C, zA for short, is defined by D(zA) := D(A) and (zA)(x) := zAx, x e D(A). It is clear that zA : X ^ P(Y) is an MLO and (wz)A = w(zA) = z(wA), z,w e C.
The integer powers of an MLO A : X ^ P(X) is defined recursively as follows: A0 =: I; if An-1 is defined, set D(An) := {x e D(An-1) : D(A) П An-1 x = 0}, and
Anx := (AAn-1)x = U Ay, x e D(An).
y€D(A)nAn-1x
We can prove inductively that (An)-1 = (An-1)-1A-1 = (A-1)n =: A-n, n e N and D((A — A)n) = D(An), n e N0, A e C. Moreover, if A is single-valued, then the above definitions are consistent with the usual definitions of powers of A.
We say that an MLO A : X ^ P(Y) is closed if for any nets (xT) in D(A) and (yT) in Y such that yT e AxT for all т e I we have that the suppositions limT^^ xT = x and yT = y imply x e D(A) and y e Ax (cf. [21] for the notion). As it is well-known any MLO A is closable and its closure A is a closed MLO.
Suppose that A : X ^ P(Y) is an MLO. Then we define the adjoint A* : Y* ^ P(X*) of A by its graph
A* := {(y*,x*) e Y * x X * : (y*,y) = (x*,x) for all pairs (x,y) e A J.
It is simpy verified that A* is a closed MLO, and that (y*,y) = 0 whenever y* e D(A*) and y e A0. It can be easily checked that the equations [6, (1.2)-(1.6)] continue to hold for adjoints of MLOs acting on locally convex spaces. We need the following auxiliary lemma from [24].
Lemma 1. Let П be a locally compact, separable metric space, and let ^ be a locally finite Borel measure defined on П. Suppose that A : X ^ P(Y) is a closed MLO. Let f : П ^ X and g : П ^ Y be ^-integrable, and let g(x) e Af (x), x e П. Then
In e D(A) and Jq g^ eAfn
2.1. C-resolvent sets of multivalued linear operators
In this subsection, we consider the C-resolvent sets of MLOs in locally convex spaces. Our standing assumptions is that A is an MLO in X, as well as that C e L(X) and
C AC AC (this is equivalent to say that, for any (x, y) e X x X, we have the implication (x,y) e A ^ (Cx, Cy) e A; then CAk C AkC for all k e N). Here it is worth noting that we do not require the injectiveness of operator C (cf. [24] for more details concerning this case). Then the C-resolvent set of A, pC(A) for short, is defined as the union of those complex numbers A e C for which
(ii) (A — A) lC is a single-valued linear continuous operator on X.
The operator A M (A — A)-1C is called the C-resolvent of A (A G pC(A)); the resolvent set of A is defined by p(A) := p7(A), R(A : A) = (A — A)-1 (A G p(A)). The basic properties of C-resolvent sets of single-valued linear operators [20; 35] continue to hold in our framework; for instance, if p(A) = 0, then A is closed; it is well known that this statement does not hold if pC(A) = 0 for some (injective, non-injective) C = I (cf. [18, Example 2.2]). Arguing as in the proofs of [6, Theorem 1.7, 1.8, 1.9], we can deduce the following.
Theorem 1. (i) We have
The operator (A — A)-1CA is single-valued on D(A) and (A — A)-1CAx = (A — A)-1Cy, whenever y e Ax and A e pC(A).
(ii) Suppose that A, ^ e pC(A). Then the resolvent equation
(A - A) 1C2x - (ß - A) 1C2x = (ß - A) (A - A) 1C(ß - A) ^x, x G X holds good. In particular, (A - A)-1C(ß - A)-1C = (ß - A)-1C(A - A) 1C.
We can prove the following extension of [20, Proposition 2.1.14] for MLOs in locally convex spaces.
Proposition 1. Let 0 = Q Ç pC(A) be open, and let x G X.
(i) The local boundedness of the mapping A M (A — A)-1Cx, A G Q, resp. the assumption that X is barreled and the local boundedness of the mapping
implies the analyticity of the mapping A M (A — A)-1C3x, A e Q, resp. A M (A — A)-1C3, A e Q. Furthermore, if R(C) is dense in X, resp. if R(C) is dense in X and X is barreled, then the mapping A M (A — A)-1Cx, A e Q is analytic, resp. the mapping A M (A — A)-1C, A e Q is analytic.
(ii) Suppose that R(C) is dense in X. Then the local boundedness of the mapping A M (A — A)-1Cx, A e Q implies its analyticity. Furthermore, if X is barreled, then the local boundedness of the mapping A M (A — A)-1C, A e Q implies its analyticity.
(iii) Suppose that R(C) is dense in X and A is closed. Then we have R(C) C R((A — A)n), n e N and
(i) R(C ) ç R(A -A);
(A -A) 1C A ç A(A -A) 1C - C ç A(A -A) 1C, A g pc (A).
A ^ (A -A)-1 C, A g ft,
dAn-1
dn- 1
-— (A - A)-1Cx = (-1)n-1 (n - 1)^A - A)-nCx, n G N.
(2)
Furthermore, if X is barreled, then R(C) ç Ä((A - A)n), n G N and
dAn-1
dn- 1
(A - A)-1C = (-1)n-1(n - 1)!(A - A)-nC g L(X), n g N.
(3)
Remark 1. Let 0 = Q C pc(A) be open, and let x E X.
(i) Suppose that the operator C is injective. Then the continuity of mapping A M (A — A)-1Cx, A E Q implies its analyticity and
n—1
— (A - A —1Cx = (-1)n—1(n - 1)! (A - A —nCx, n G N
ft' ' -x—1
Furthermore, if X is barreled, then the continuity of mapping A M (A — A) 1C, A E Q implies its analyticity and
dn—1
^ (A -A 1C =(-1)n—1(n - 1)!( A -A " C G L(X), n g N.
(ii) Suppose that A is closed. Then card((A - A) nCx) < 1, A G pC(A), n G N, x G X. This can be proved by induction, observing that (A - A) —10 is a singleton (A G pC(A)) as well as that for each y G (A - A)—(n+1)Cx (A G pC(A), n G N, x G X) we have
(A - A)—(n+1)Cx = y + (A - A)—(n+1)C0 =
= y + (A - A)—1(A - A)—nC0 = y + (A - A)—10 = {y}.
Now we can proceed as in the proof of [18, Corollary 2.8] in order to see that (2) holds, and that (3) holds, provided in addition that X is barreled.
The following generalized resolvent formulae are very important in our work, their validity can be proved inductively.
Theorem 2. (i) Let x G X, k G N0 and A, z G pC(A) with z = A. Then the following holds:
(z -A) (A -A) 1C) k
x
<-l)k (z-a,-c'+.x + £ (-1)'-'«A-A)-'C),C"+I-'x
(z —A) (z — A)k+1
(ii) Let k E N0, x,y E X, y E (A0 — A)kx and A0,z E pc(A) with z = A0. Then the following holds:
(-1)' (z ,r 1C'+1y + v (-1)k—^(Ao -A)—1C*Ck+1—*y k iz -A c y + ^ ( ,fc+1—i .
(z - A)—1ck+1x = ( 1) k (z - A)—1ck+1y +
(z - Ao) i=1 (z - A0j
Now we introduce the following definition.
Definition 1. (Cf. [20, Definition 2.9.4] for single-valued case). Let A be a closed multivalued linear operator on X.
(i) Then we say that A is C-nonnegative iff (-ro, 0) C pC(A) and the family
1
{A(A + A) 1C : A > 0}
is equicontinuous; moreover, a C-nonnegative operator A is called C-positive iff, in addition, 0 G pC(A).
(ii) Let 0 < w < n. Then we say that A is C-sectorial of angle w, in short A G SectC (w), iff C \ C pC (A) and the family j A(A -A 1C : AG } is equicontinuous
for every u < u' < n; if this is the case, then the C-spectral angle of A is defined by wC(A) := inf{u G [0, n) : A G SectC(u)}.
We close this subsection by observing that some properties of C-nonnegative operators established in [15, Proposition 2.4] continue to hold in multivalued linear case. For example, we can prove that:
(i) If 0 G p(C), then A is C-nonnegative iff A is nonnegative.
(ii) If A is C-positive, then the family {(À + C)(A + A)-1C : A > 0} is equicontinuous. Conversely, if the last family is equicontinuous and C is nonnegative, then A is C-nonnegative.
(iii) Let X be barreled. Then the adjoint A* of A is C*-nonnegative in X*. 2.2. Fractionally integrated C-semigroups in locally convex spaces
In this subsection, we collect the basic facts and definitions about (degenerate) fractionally integrated C-semigroups in locally convex spaces. The operator C G L(X) need not be injective.
Definition 2. [38]. Suppose that 0 < a < to and 0 < t < to. Then a strongly continuous operator family (Sa(t))ie[0,r) Ç L(X) is called a (local, if t < to) a-times integrated C-semigroup iff the following holds:
(i) Sa(t)C = CSa(t), t G [0, t), and
(ii) for all x G E and t, s G [0, t) with t + s G [0, t), we have
Sa(t)Sa(s)x
ri+s
- I -
ga(t + s - r)Sa(r)Cxdr.
By a C -regularized semigroup (0-times integrated C -regularized semigroup) we mean any strongly continuous operator family (S0(t) = S(t))ie[0,r) Ç L(X) such that S(t)C = CS (t), t G [0, t ) and S (t + s)C = S (t)S (s) for all t, s G [û,t ) with t + s G [0,t ).
Let 0 < a < to. If t = to, then (Sa(t))t>0 is said to be exponentially equicontinuous (equicontinuous) iff there exists u G R (u = 0) such that the family {e-wiSa(t) : t > 0} is equicontinuous; (Sa(t))ie[0,r) is said to be locally equicontinuous iff for every t' G (0, t) we have that the operator family {Sa(t) : t G [0,t']} is equicontinuous.
The integral generator A of (Sa(t))ie[0,r) is defined by its graph
A := j(x,y) G X x X : Sa(t)x - sa(t)Cx = J Sa(s)yds, t G [0,t) J.
The integral generator A of (Sa(t))ie[0,r) is a closed MLO in X, provided that (Sa(t))ie[0,r) is locally equicontinuous. Furthermore, A Ç C-1AC in the MLO sense, with the equality in the case that the operator C is injective.
By a subgenerator of (Sa(t))ie[0,r) we mean any MLO A in X satisfying the following two conditions:
(A) Sa(t)x — ga+1(t)Cx = /0 Sa(s)yds, whenever t G [0,t) and y G Ax.
(B) For all x G X and t G [0,t), we have /0 Sa(s)xds G D(A) and Sa(t)x — ga+i(t)Cx G A/0i Sa(s)xds.
t
s
0
0
0
If (i))te[0,r) C L(X), resp. (i))te[0,r) C L(X), is strongly continuous and satisfies only (B), resp. (A), with (Sa(t))te[0,r) replaced therein with (S^(i))te[0,r), resp. (S2(t))te[0,r), then we say that (S^(i))te[0,r), resp. (S2(i))te[0,r), is an a-times integrated C-existence family with a subgenerator A, resp., a-times integrated C-uniqueness family with a subgenerator A.
The notion of an exponentially equicontinuous, analytic a-times integrated C-semigroup is introduced in the following definition.
Definition 3. (i) Let v E (0,n], and let (Sa(i))t>0 be an a-times integrated C-semigroup. Then it is said that (Sa(t))t>0 is an analytic a-times integrated C-semigroup of angle v, if there exists a function Sa : M L(X) which satisfies that, for every x E X, the mapping z M Sa(z)x, z E is analytic as well as that:
(a) Sa(t) = Sa(t), t > 0 and
(b) limz^0,zes7 Sa(z)x = i«,0Cx for all 7 E (0, v) and x E X.
(ii) Let (Sa(t))t>0 be an analytic a-times integrated C-semigroup of angle v E (0,n]. Then it is said that (Sa(t))t>0 is an exponentially equicontinuous, analytic a-times integrated C-semigroup of angle v, resp. equicontinuous analytic a-times integrated C-semigroup of angle v, if for every 7 E (0, v), there exists > 0, resp. = 0, such that the family RezSa(z) : z E S7} C L(X) is equicontinuous.
For more details about degenerate fractionally integrated C-semigroups, we refer the reader to [38].
3. Definition and main properties of complex powers of operators satisfying the condition (H)o
Assume that the condition (H)0, resp. (HS)0, holds. Without loss of generality, we may assume that there exists a number A0 E int(pc(A)) \ (Pa,e,c U Bd), resp., A0 E int(pc(A)) \ (£# U Bd). Then we can prove inductively (cf. also Theorem 2(i)) that, for every z E pc(A) \ {A0} :
(z — A)-1C(A0 — A)Cx = (z — A)-1C2x + ± (—1')'(;i(yrC2x. (4)
Strictly speaking, for k =1 this is the usual resolvent equation. Suppose that (4) holds for all natural numbers < k. Then (2) shows that
and we can employ the inductive hypothesis and a simple computation in order to see that (4) holds with k replaced with k + 1 therein. Set
(z - A)—1C(Ao - A)—(k+1)Cx = (z - A)—1C
A=Ao
C1 := C(Ao - A) L"+2JC, if a > -1, and C1 := C, if a = -1.
Then Theorem 1 (i) implies by iteration that C1 commutes with A. Furthermore, the validity of (H)0, resp. (HS)0, implies by Theorem 2 (i) that the following holds:
(H): There exist real numbers d G (0,1], c G (0,1) and e G (0,1] such that Pa,£,cUC pci(A), the operator family {(1 + |A|)-1(A - A)-1Ci : A G Pa,£,c U Bd}'c L(X) is equicontinuous, the mapping A M (A — A)-1 C1 is strongly analytic on the set int(Pa,£,c U Bd) and strongly continuous on d(Pa,£,c U Bd),
(HS): There exist real numbers d G (0,1] and $ G (0,n/2) such that U C pCl(A), the operator family {(1 + |A|)-1(A — A)-1 C1 : A G S^ U Bd} C L(X) is equicontinuous, the mapping A M (A — A)-1 C1 is strongly analytic on int(S# U Bd) and strongly continuous on d(S# U Bd).
So, the condition (H), resp. (HS), holds and C1A C AC1; in particular, —A is C1 -positive, resp. —A, is C1-sectorial of angle n — $ and C pCl(—A).
Put e, c, d) := {£ + in : £ < —e, n = —c(1 + |£|)-a}, ^(a, e, c, d) := {£ + in : £2 + n2 = d2, £ > —e} and r3(a,e,c,d) := {£ + in : £ < — e, n = c(1 + |£|)-a}. The curve r(a,e,c, d) := r1(a,e,c, d) U r2(a,e,c, d) U r3(a,e,c, d) is oriented so that Im(A) increases along r2(a,e,c, d) and that Im(A) decreases along r1(a,e,c, d) and r3(a,e, c, d). Since there is no risk for confusion, we also write r for r(a,e,c, d). We similarly define the curves r1S($,d), r2,S($, d), r3,S($,d) and rS($,d) for $ G (0,n/2) and d G (0,1].
where f (z) is a holomorphic function on an open neighborhood of — (Pa,e,c U
) \ (—to, 0] and the estimate |f (z)| < M|z|-s, z G Ha,e,c,d, holds for some positive number s > 0. Denote by H the class consisting of such functions. Then an application of Cauchy's theorem shows that the definition of fCl (A) does not depend on a particular choice of curve r(a,e, c, d) (with the meaning clear). Furthermore, a standard calculus involving the Cauchy theorem, the Fubini theorem and Theorem 1 (ii) shows that
Given b G C with Re b > 0, set (—A)- := (z-l)Cl (A) and (—A)-0 := C1. By Remark 1 (ii) and the residue theorem, we get (—A)-^ = (—A)-nC1 (n G N); moreover, (—A)-bC1 = C1(—A)-lb (Re b > 0), the mapping b M (—A)-lx, Re b > 0 is analytic for every fixed x G X, and the following holds:
resp.,
Define
x G X,
fa (A)gci (A) = (/g) _ (A)C1, f,g,/g g H.
(6)
x G X, Re b > 0.
Applying the equality (6) once more, we get that
(-A)-1 (-A)-2 = (-A)-f1+b2)C1, Re bi, Reb2 > 0.
It is very simple to prove that
0 < Re b < 1, x G X,
so that the family {(-A)! : 0 < b < 1} is equicontinuous. Define now the powers with negative imaginary part of exponent by
(-A)_ := CT1(-A)-b, Re b > 0.
Then (-A)_b is a closed MLO and (-A)_n = C_1(-A)_nC1 (n G N). We define the powers with positive imaginary part of exponent by
(-A)^ := ((-A)_b)-1 = ((-A)_b)_1Ci, Reb > 0.
Clearly, (-A)n = C_1(-A)nC1 for every n G N, and (-A)b is a closed MLO due to the fact that (-A)_b is a closed MLO (b G C+). Following [22, Definition 7.1.2] and our previous analyses of non-degenerate case [15], we introduce the purely imaginary powers of -A as follows: Let t G R \ {0}. Then the power (-A)ir is defined by
(-A)ir := Cf2(l - A)2(-A)_1(-A)1+iT(1 - A)_2C?,
where (1 - A)2 = C1_1(1 - A)2C1 and (1 - A)_2 = C1_1(1 - A)_2C1. We will later see (cf. (S.4)) that C1(D(A2)) C D((-A)1+ir), so that the closedness of (-A)ir follows from a simple computation involving the closedness of (-A)1+ir.
Further on, the Cauchy integral formula and (2) implies that the operator family {Ak(A - A)_kC1 : A > 0} C L(X) is equicontinuous for all k G N. If y G Akx for some k G N and x G D(Ak), then there exists a sequence (yj)1<j<k in X such that yk = y and (x,y1) G A, (y1,y2) G A, ■ ■ ■, (yk_1 ,yk) G A. Then we can inductively prove with the help of Lemma 1(i) that
Ak (A - A)_kC1X = C1x + ^^ (A - A)_jC1yj, A > 0,
j=1 '
which implies that Ak(A - A) kC1x = C1x, k G N, x G D(Ak); cf.
[15, Lemma 2.7]. The assertion of [15, Lemma 2.5] also holds in our framework.
Now we will reconsider multivalued analogons of some statements established in [15, Theorem 2.8, Theorem 2.10, Lemma 2.14].
(5.1) Suppose Re b = 0. Then it is checked at once that (-A)b C C_1(-A)bC1, with the equality in the case that the operator C1 is injective.
(5.2) Suppose Re b1 < 0 and Re b2 < 0. Then R(C1) C D((-A)z), Rez < 0,
(-A)Cf2x g Cr1(-A)Si+b2C1x = Cr1(-A)Si (-A)C21 x C
C CT1(-A)S1 Cf1(-A)C21 C1x = (-A)^ i (-A)b2C1x, x g X. This, in turn, implies
(-A)S1+i2 C (-A)bi (-A)b2C1, CT1(-A)6c11+b2 C CT1(-A)bi (-A)b2C1
and
(-A)bi +b2 C ^r1 (-A)bi (-A)b2C1. (7)
Let y G (-A)bi (-A)b2x. Thus, C1y G (—A)C11C'r1 (-A)C21 x. This yields the existence of an element u G Cr1(—A)C71 x such that C1z = (-A)C21 x and C1y = (-A)Ci1u. So,
C?y = (-AftC1u = (-Aft (-Aft 6 x, C1y e C^-A)6^2C1x = (-AK +b2C1x and y e C- 1(-A)bl+b2C1 x. Hence,
(-A)bi (-A)b2 C Cr1(-A)bi+b2C1. (8)
(5.3) Suppose now that Re b1 > 0 and Re b2 > 0. Using the equations (7), (8) with b1 and b2 replaced respectively by —b1 and -b2 therein, and taking the inverses after that, it readily follows from (S.2) that (7), (8) holds in this case.
(5.4) Repeating almost literally the arguments from the proof of [15, Theorem 2.8 (ii.2)], we can deduce the following: Suppose that Re b > 0 and k = [Re b], resp. k = [Re b] +1, provided that Re be N, resp. Re b e N. Let x = C1y for some y e D(Ak). Then there exists a sequence (yj)1<j<k in X such that (y,y1) e A, (y1,y2) e A,..., (yk_1,yk) e A. Furthermore, C1(D(Ak)) C D((-A)b) and, for every such a sequence, we have
¿/zb_LRebJ_1(z + A)-1C1yfcdz e (-A)bx. r
(5.5) The assertion of [15, Theorem 2.8 (iii)] is not really interested in multivalued case because (-A)bx is not singleton, in general.
(5.6) Let t e R. Then a straightforward computation involving (S.1) shows that (-A)ir C C_1(-A)iTC1. The equality (-A)ir = C_1(-A)irC1 can be also trivially verified provided that the operator C1 is injective.
(5.7) Let x = C1y for some y e D(A), and let t e R. Keeping in mind (4), (6), (S.4), the residue theorem and Theorem 1, we can prove as in single-valued linear case that:
1
z_1+ir^— (z + A) 1C12xdz e (1 - A)(-A)_1(-A)1+ir(1 -A)_2Clx.
z + 1
Let u e (1 - A)y. Using Lemma 1 and Theorem 1 (i), we get from the above that C_3(1 - A)C1
f (z + A)_1C2xdz
2rn J z + 1 v y r
1 ' — 1+ir z ( _1.
= — I z_1+ir—- (z + A) 6 udz e (-A)irx, 2rn J z + 1 v y
r
so that C1(D(A)) C D((-A)ir). Unfortunately, a great number of important properties of purely imaginary powers established in [15, Theorem 2.10] does not continue to hold in multivalued linear case.
(S.8) Let n e No, let b e C and let Reb e (0,n + 1)\N. Set (1 -b)(2-b)••• (n-b) := 1 for n = 0. Then, for every x e X, we have
Cn( A)_br = (-1)"n! Sinn(n - b) /-,ra_b(, ,)_(n+1)Cn+1rd,
C^-^^ =(1 - b)(2 - b) ... (n - b) n ]t V 61 xdt.
o
This can be shown following the lines of the proof of [39, Theorem 5.27, p. 138].
4. The existence and uniqueness of solutions of abstract incomplete differential inclusions
In this section, we assume that the condition (HS)o holds. Define C1 through (5). Then (HS) holds and we can define the fractional powers of —A as it has been done in the third section of paper.
Following A.V. Balakrishnan [10], define
ft (A) := 1 e-tAY cos sin (tA7 sin n7) = ^ (e-tA7 — e-tAY ), t > 0, A > 0.
1
n v '' 2ni
This function enjoys the following properties:
Q1. |ft(A)| < n-1e-A7et, A > 0, where et := t cos nY > 0.
Q2. | ft (A) | < YtAY e-tAY sin et, A > 0.
Q3. /0°° Anft(A) dA = 0, n G No, t > 0.
Q4. Let m > —1. Then Q1./Q2. together imply that the improper integral fo° Anft(A)(A — A)-1C1 ■ dA is absolutely convergent and defines a bounded linear operator on X (n G N0).
Put now, for 0 < y < 1/2,
S7 (t)x := J ft(A)(A — A)-1CixdA, t> 0, x G X. o
Then (t) G L(X), t > 0 and the following holds: Lemma 2. We have
S7(t)= (e-tzY)C(A), t> 0, 0 < y < 1/2. (9)
Furthermore, (t) can be defined by (9) for all t G £(n/2)-7n, and the mapping t M (t), t G £(n/2)_7n is strongly analytic (0 < y < 1/2).
Proof. Observe that, for every t = t1 + it2 G £(n/2)-Tn and z G C \ {0}, we have
g-tzY < g-|z|Yti cos(7 arg(z))[1-| tan(arg(t))| tan(Yarg(z))]
Keeping this estimate in mind, it is very simple to deform the path of integration rS($, d) into the negative real axis, showing that for each t G £(n/2)-Tn and x G X we have:
— i e-tAY (A + A)-1C1xdz = — i fe-tA7 — e-tAY )(A — A)-1C1xdA. 2^Urs(*,d) 2ni J V /
0
The remaining part of proof is left to the interested reader. □
Set := (n/2) — Y(n — tf), for 0 < y < 1/2.
Theorem 4. Put (0) := C1, SY,z(t)x := /0 $z(t — s)SY(s)xds, x G X, t G £(n/2)-Tn (Z > 0), and SY,0(t) := (t), t G £(n/2)-rn. Then the family {SY(t) : t > 0} is equicontinuous, and there exist strongly analytic operator families (SY(t))tev and
(SY,z(¿))iesV7 such that SY(t) = (t), t > 0 and SY,z(t) = (t), t > 0. Furthermore, the following holds:
(i) S7(ti)S7(t2) = S7(ti + t2)Ci for all ti, t2 G E^7.
(ii) We have limt^0,ies^7_e SY(t)x = C1x, x G D(A), e G (0, ).
(iii) S7(z)(-A)v C '(-A)vS7(z), z G E^7, v G C+.
(iv) If D(A) is dense in X, then (SY (t))t>0 is an equicontinuous analytic C1-regularized semigroup of angle . Moreover, (SY(t))t>0 is a C1-regularized existence family with a subgenerator — (—A)Y and the supposition (x,y) G — (—A)Y implies (C1x,C1y) G A, where A is the integral generator of (SY(t))t>0; otherwise, for every Z > 0, (SY,z(t))t>0 is an exponentially equicontinuous, analytic Z-times integrated C1-regularized semigroup, (SY,z(t))t>0 is a Z-times integrated Coexistence family with a subgenerator — (—A)Y and the supposition (x,y) G — (—A)Y implies (C1x, C1y) G A.
(v) For every x G X, t G E(n/2)_7n and n G N, we have
POO
-1
S7(t)x, — J Ara/t(A)(A — A 1C1xd^ GAn. (10)
(vi) Suppose 0 > 0. Denote by , resp. , the continuity set of (SY(tei0))t>0, resp. (SY(t))tei;. Then, for every x G , the incomplete abstract Cauchy inclusion
{u G Cc((0, w) : X),
D_u(t) G e^(—A)7^u(t), t> 0,
limt^0+ u(t) = C1x,
the set {u(t) : t > 0} is bounded in X,
has a solution u(t) = (tei0)x, t > 0, which can be analytically extended to the sector E^7_|0|. If, additionally, x G , then for every 5 G (0, ) and j G N0, we have that the set {zju(j)(z) : z G E^} is bounded in X.
Proof. The proof of (i) for real parameters t1,t2 > 0 follows almost directly from definition of (■), by applying (6); (v) is an easy consequence of Lemma 1, Theorem 1 (i) and the property Q3. A very simple proof of (iii) is omitted. Set, for |0| < $ and 0 < y < 1/2,
(t)x := / /t,7(A) (A — e%e
A 1C1xdA, x G X, t G E(n/2)_Yn.
Let 01 G (0,$) and 02 G (—$, 0). Define
S7(t)x, t G E( S7(t)x := { Sei,7(te_iY01), if t G e^1 E(^/2)_7n, S,2,7(te_iY02), if t G eiY02E(n/2)_7n.
Then an elementary application of Cauchy formula shows that the operator family (SY(t))tes is well defined; furthermore, (SY(t))tes is strongly analytic and equicontinuous on any proper subsector of E^7 (cf. also the proof of [35, Theorem 2.9.48]). Using Theorem 1 (i), we get that
lim [A(A — A)_1C1x — A(A + 1)_1C1x] = 0
as A ^ (x G D(A)). Taking into account this equality and the proof of [22,
Theorem 5.5.1 (iv), p. 130], we get that limt^0+ SY(t)x = Clx, x G D(A). Now the remaining parts of proofs of (i)-(iii) can be straightforwardly completed.
We will prove (iv) provided that D(A) is dense in E. It is clear that (SY(t))t>0 is an equicontinuous analytic ^-regularized semigroup (SY(t))t>0 of angle . Since, for every t > 0 and x G X,
C1 (—z-Ye-tzY + z(A)x = — (z-Y(A) [(e-iz7)C (A)x - Cix],
we have C1 SY,1(t)x = — (—A)-Y SY(t)x — C1 x , t > 0, x G X. This clearly implies that
(SY,1(t)x, SY,1(t)x — C1x) G — (—A)y, t > 0, x G X, so that (SY,z(t))t>0 is a Z-times integrated Coexistence family with a subgenerator — (—A)Y. The supposition (x,y) G — (—A)y implies C1x = — (—A)-YV and we can similarly prove that (C1x, C1y) G A.
Arguing as in the proof of [15, Theorem 3.5 (i)/(b)'], we get that, for every x G X and t > 0, the following equality holds, with z = tei6> G £(n/2)-Yn,
oo
piöß [■
D-S7(tei0)x = ^ I AYß e"zA7e-in7 - e-zAY(A - A)-iCixdA. (11) i
0
Deforming the path of integration rS($, d) into the negative real axis, as it has been done in the proof of Lemma 2, we get
e-2>7) C (A) = -^ /AYß e-iYßn e-zAY - e-zAY (A -A)-iCixdA. (12) i
oo
0
Since Ci(e-^7)C (A) = (--7ß)C (A)(--7ße-^7)C (A), (11)-(12) immediately implies
that (Vii?/3D-S7(tei0)x,S7(tei0G C-1 (—A)C, t > 0, x G X, i.e.,
i Sy (te x, e D- S7 (tei0) x G ( A)y^, t > 0, x G X.
The proof of (vi) now can be completed through a routine argument. □
Remark 2. (i) If l = Py g N, then the operator (—A)Y/s in the formulation of problem (FP)^ can be replaced with the operator (—A)1 therein; cf. (10).
(ii) Suppose that the operator C1 is injective. Then we can simply prove that (SY,C(t))t>0 is a Z-times integrated C1 -semigroup with a subgenerator — (—A)Y, which implies [38] that the integral generator of (SY,z(t))t>0 is —C-1(—A)YC1 = —(—A)Y. A similar statement holds in the case that y = 1/2, which is further discussed in the following theorem.
Theorem 6. The limit contained in the expression
N
S1/2 (t)x := 1 jim y"sin(tVA)(A — A)-1C1xdA, t> 0, (13)
0
exists in L(X) for every x G X. Put S1/2(0) := C1. Then the family {S1/2(t) : t > 0} is equicontinuous, there exists a strongly analytic operator family (S1/2 (t))tei; such that S1/2(t) = S1/2(t), t > 0 and the following holds:
(i) S1/2(t)S1/2(s) = S1/2(t + s)C1 for all t, s G E^1/2.
(ii) limi^0,ies^1/2_e S1/2(t)x = C1x, x G D(A), e G (0,^1/2).
(iii) S1/2(t)( — .A)v C ( — A)vS1/2(t), t G E^i/2, v G C+.
(iv) If D(A) is dense in X, then (S1/2(t))t>0 is an equicontinuous analytic C1-regularized semigroup of angle . Furthermore, (S1/2(t))t>0 is a C1-regularized existence family with a subgenerator —(—A)1/2 and the supposition (x,y) G —(—A)1/2 implies (C1x,C1y) G a41/2, where a41/2 is the integral generator of (S1/2(t))t>0; otherwise, for every Z > 0, (S1/2,z(t))t>0 is an exponentially equicontinuous, analytic Z-times integrated C1-regularized semigroup, (S1/2,z(t))t>0 is a Z-times integrated Coexistence family with a subgenerator —(—A)1/2 and the supposition (x,y) G —(—A)1/2 implies (C1x,C1y) GA1/2. _
(v) R(S1/2(t)) C Dc(A), t > 0 and, for every x G D(A), the incomplete abstract Cauchy problem
u G Cc((0, w) : X), (P ) : ; u"(t) G —Au(t), t > 0, ( 2) : 1 limt^0+ u(t) = C1x,
the set {u(t) : t > 0} is bounded in X,
has a solution u(t) = S1/2(t)x, t > 0. Moreover, the mapping t M u(t), t > 0 can be analytically extended to the sector E^V2 and, for every 5 G (0, ^1/2) and j G N0, we have that the set {zju(j)(z) : z G E^} is bounded in X.
Proof. First of all, observe that ^1/2 = $/2. Applying the partial integration, (2) and the equicontinuity of family {A2(A — A)_2C1 : A > 0}, we obtain that the limit contained in (13) exists and equals
00
S1/2 (t)x = J / (A, t) (A — A^CxdA, t> 0, x G X, 0
where /(A,t) = 2n_1 t_2[sin(t^A) — tVXcos(tVX)] for A > 0 and t > 0. As in single-valued case, the change of variables x = tv^A shows that the operator family {S1/2(t) : t > 0} is both equicontinuous and strongly continuous. Let (x,y) G A. Then Theorem 1 (i) and an elementary argumentation show that
N
S1/2(t)x — C1x = 1jim /sin(tVA)((A + A)_1C1x — A_1 Cx) dA =
n
0
N • ( r)
= 1 lim i Si^ ^ (A — A)_1C1 ydA. n nJ A v '
0
Keeping in mind the last equality and the equicontinuity of family {S1/2(t) : t > 0}, we get that limt^0 S1/2(t)x = C1x for all x G D(A).
Now we proceed as in the proof of [19, Theorem 2.6 (i)]. Let 0 < 5 < 5 < $/2, 1/2 > y0 > 5/$ and 0 G (—$, (—5)/y0). Then, for every y G (y0, 1/2), we have 0 G (—$, (—5)/y) and y > 5/$. Let e G (0, (n — $)/2) be sufficiently small. Define, for every Y G (y0, 1/2) and x G X,
( ei0Y sin Yn (• » (v_ei9A)-1Cixdv -r ( W t- ( ( /0\ _L
F (A)x I n Jo (Aei®7 +vY cos n7)2+v27 sin2 7n , if arg(A) G ( —'e, (n/2) + 5),
F7(A)x S sin 7n r0 (v_e-ifl A)-1C1xdv .f arg( A ) ^ (_ (n/2) _ 5e)
n J0 (Ae-i»7 +vY cos n7)2+v27 sin2 7n , if arg(A) G ( (n/2) 5,e)
If x G X and arg(A) G (-e, (n/2) + #), resp., arg(A) G (-(n/2) - e), then
f0 Ae^,, , sinYn [o vY(v - eiöA)-iCix
e 7 iSg,7 (t)xdt =-—-^--2-i-— dv, (14)
./0 n Jo (Aeiy7 + vY cos ffYj + v2Y sin Yn
resp.
r e-Ae-(i)xdt = EYE /0 /(V + e-ia-A)"Cix 2 dv. (15)
lo n Jo (Ae-iÖY + vY cos n^) + v2y sm2 Yn
Furthermore,
r>o /»o
ei0Y / e-^7iS0,Y(t)xdt = e"'M e-Ae-i"7'S-^(t)xdt, A G Ee. (16) 00
By (14)-(16), we deduce that the function A M FY(A)x, A G E(n/2)+ is well defined, analytic and bounded by Const^ | A | 1 on sector E(n/2)+£/ (x G X), as well as
1
2ni ir
vr a
S7(z)x = — eAzF7(A)xdA, x G X, z G Ey, y G (yo, 1/2), (17)
where r^ := ^,1 U ^,2, ^,1 := {rei((n/2)+<5/) : r > |z|-1} U {|z|-1e^ : tf G [0, (n/2) + 5']} and r5/,z,2 := {re-i((n/2)+<5/) : r > |z|-1} U {|z|-1e^ : tf G [—(n/2) — 5', 0]} are oriented counterclockwise. The dominated convergence theorem shows that, for every x G X and z G E^/,
v eiö/2 f Az f0 vi/2(v - eiöA iCix , M
lim (z)x =—— e - ^ -n—-dvdA+
2 - 2n2i ,/r Jo A2eiö + v
6/,z,1
1 i1/2!^! _ o-"" a 1 i'
Az
e-iö/2 f A /-0 vi/2(v - e-iöA iCix
+ J^ e L A2e-i^ + v-dvdA := Si/2(z)x.
Define F1/2(A) by replacing the number y with the number 1/2 in definition of FY(A). Then, for every x G X, the function A M F1/2(A)x, A G E(n/2)+ is well defined and analytic on E(n/2)+; furthermore, for each q G © there exists rq G © such that q(F1/2(A)x) < rq(x)Const^1A| 1, A G E(n/2)+^/, x G X [19]. Define (S1/2(z))zeS^2 C L(X) by S1/2(z)x := limY^ 1 - SY(z)x, z G E#/2, x G X; this operator family is equicontinuous on any proper subsector of E^/2 and satisfies additionally that the mapping z M S1/2(z)x, z G E$/2 is analytic for all x G X. Letting y m | — in (17), it is not difficult to prove that
S1 (z)x =—1 eAz Fi (A)xdA, x G X, z G E*,
2 WiV.,
so that the proof of [25, Theorem 2.6.1] implies
00
j e-AtS1 (t)xdt = Fi (A)x, x G X, A > 0. (18)
0
On the other hand, the arguments used in the proof of [22, Theorem 5.5.2, p. 133] show that
00 00
f e-AtS 1 (t)x dt = 1 i (v — A) -1C1x dv = Fi (A)x, x G X, A > 0. (19)
J 2 n J A2 + vK ' 2V/' ' vy
00
Using the uniqueness theorem for the Laplace transform, we obtain from (18), (19) that S1/2(t) = S1/2(t), t > 0. Now the proofs of (i)-(iii) become standard and therefore omitted.
For simplicity, we assume that A is densely defined in (iv). Then the only nontrivial thing that should be proved is that the supposition (x,y) G —(—A)1/2 implies (C1x,C1y) G A^2. So, let (x,y) G —(—A)1/2, i.e., C1x = —(—A)C^/2y. A similar line of reasoning as in the proof of identity [15, (51), p. 489] shows that
C1 / e_AiS7(t)ydt = C1(—A)CYy — A/ e_AtS7(t)ydt, A> 0, y G (0,1/2).
J0 1 J0
Taking the limits of both sides of previous equality when y m 1/2—, we get that
cJ e_AiSV2(t)ydt = C1(—A)C1/2y — A/ e_AiSV2(t)ydt, A> 0. J0 1 J0
Then the uniqueness theorem for the Laplace transform simply implies that
S1/2(t)C1x — C^x = S1/2(s)C1y ds, t > 0, 10
as claimed.
Now we will prove (v) by slightly modifying the arguments used in the corresponding part of proof of [19, Theorem 2.6 (i)]. In order to do that, we will first show that for each x G X we have S1/2(t)x G —AS1/2(t)x, t > 0. Fix temporarily an element x G X. Owing to Theorem 4 (v) and (11), cf. also Remark 2 (i), we have that
D_SY(t)x G—AS7(t)x, t> 0,
i.e.,
d2
g3-1 (s)S'(t + s)x ds G AS7 (t)x, t> 0, y G (Y0,1/2).
dt2
0
Therefore,
g3_ 1 (s)S7'(t + s)x ds G AS7 (t)x, t> 0, y G (Y0,1/2). J0 7
Applying the partial integration, we get
g4_ 1 (s)S(iv)(t + s)xds G —AS7 (t)x, t> 0, y G (Y0, 1/2).
J0 7
The dominated convergence theorem yields by letting y m 1/2— that
pc
/ sSj/v2)(t + s)xds G —AS1/2(t)x, t > 0, 0 1/2
which clearly implies after an application of integration by parts that S1'/2(t)x G —AS1/2(t)x, t > 0, as claimed. By (ii), the function u(t) = S1/2(t)x, t > 0 is a solution of problem (P2) for x G D(A). Furthermore, we obtain by induction that s1/n)(t)x G (—1)nAnS1/2(t)x, t > 0, n G N, x G X, so that R(S1/2(t)) C Dc(A), t > 0. The proof of the theorem is thereby complete. □
30
Example 1. Examples of exponentially bounded integrated semigroups generated by multivalued linear operators can be found in [6, Section 5.3, Section 5.8] (cf. also [40]); multivalued matricial operators on product spaces can also generate exponentially bounded degenerate integrated semigroups (see e.g. [20, Example 3.2.24] for single-valued case). It is also worth noting that multivalued linear operators whose resolvent sets contain certain exponential regions [41] have been considered in [24, Example 3.2.11 (i)] as generators of degenerate local once integrated semigroups. All these examples can serve one to provide possible applications of Theorem 4 and Theorem 6.
Example 2. Assume that (Mp) is a sequence of positive real numbers such that M0 = 1 and that the following conditions are fulfilled:
Mp2 < Mp+1Mp-1, p G N, (M.1)
Mp < AHp sup MiMp-i, p G N, for some A, H > 1, (M.2) 0<i<p
and
sup y Mq-;Mp+1 < (M.3)
q=P+1 PMPM9
For example, we can take the Gevrey sequence (Mp = p!s) with s > 1. Set mp := M^, p G N and u(z) := 0=^ 1 + mir), z G C. Suppose that there exist constants l > 0 and u > 0 satisfying that RHPW = {A G C : ReA > u} C p(A) and the operator family {e-M(1|A|)R(A : A) | A G RHPW} C L(X) is equicontinuous (cf. [20; 35] for a great number of such examples with A being single-valued, and [38, Example 3.25] for purely multivalued linear case, with X being a Frechet space). Let u > u. Then there exists a sufficiently large number n G N such that the expression
S(t) := — i eAt ^Aa dA, t > 0
defines an exponentially equicontinuous C = S(0)-regularized semigroup (S(t))t>0 with a subgenerator A (cf. Lemma 1, Theorem 1 (i) and the proof of [35, Theorem 3.6.4]). It is not difficult to prove that (A — A)-1Cf = /0° e-AtS(t)fdt, ReA > u, f G X, so that Theorem 4 and Theorem 6 can be applied with the operator A replaced with the operator A — u therein. Observe that, even in single-valued linear case, the operator C need not be injective because our assumptions do not imply that A = A generates an ultradistribution semigroup of Beurling class (cf. [24] for the notion).
In this paper, we will not discuss the generation of degenerate fractional regularized resolvent families by the negatives of constructed fractional powers. For more details, cf. [15, Section 3] and [35, Remark 2.9.49].
At the end of paper, we would like to observe that the assertions of [19, Theorem 2.4-Theorem 2.6] can be formulated in the multivalued linear operators setting. For applications, the most important is the following case: X is a Banach space, U C p(A), there exist finite numbers M1 > 1 and v G (0,1] such that (1) holds with the operator A and number P replaced with —A and v therein; see [6, Chapter III, Chapter VI] for a great number of concrete examples. Define the operators SY(■) as before. Then we may conclude the following:
(i) Suppose that > 1 — v. Then (t))ies is an analytic semigroup of
growth order ^ (cf. [19, Definition 2.1] for the notion). Denote by , resp. ^
7>
the continuity set of (S7(tei0))t>0, resp. (S7(t))tei; . Then D(A) C and, for every x G n<9,7, the incomplete abstract Cauchy inclusion (FP/3), with C1 = I, has a solution u(t) = S7(tei0)x, t > 0, which can be analytically extended to the sector E^7_|0|. If, additionally, x G , then for every 5 G (0,<^7) and j G N0, we have that the set {zju(j)(z) : z G E^} is bounded in X.
(ii) Suppose that 1/2 < v < 1. Then the incomplete abstract Cauchy problem (P2), with C1 = I, has a solution u(t), t > 0 for all x G D(A). Moreover, the mapping t M u(t), t > 0 can be analytically extended to the sector E^V2 and, for every 5 G (0,^1/2) and j G N0, we have that the set {zj(1 + |z|2v_2)_1u(j)(z) : z G E5} is bounded in X.
It is very non-trivial to find some necessary and sufficient conditions ensuring the uniqueness of solutions of problems (FP^) and (P2); cf. also [19]. This is an open problem we would like to address to our researchers.
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Article received 30.04.2020
Corrections received 01.06.2020
Челябинский физико-математический журнал. 2020. Т. 5, вып. 3. С. 363-385.
УДК 517.95+517.98 DOI: 10.47475/2500-0101-2020-15310
КОМПЛЕКСНЫЕ СТЕПЕНИ МНОГОЗНАЧНЫХ ЛИНЕЙНЫХ ОПЕРАТОРОВ С ПОЛИНОМИАЛЬНО ОГРАНИЧЕННОЙ C-РЕЗОЛЬВЕНТОЙ
М. Костич
Университет Нови-Сада, Нови-Сад, Сербия marco.s @verat.net
Построены комплексные степени многозначных линейных операторов с полиномиально ограниченной C-резольвентой, существующей на соответствующей области комплексной плоскости, содержащей интервал (-то, 0]. При используемом подходе оператор C не обязательно инъективен. Установлены основные свойства введённых степеней, проанализированы неполные дробные дифференциальные включения с модифицированными правосторонними производными Лиувилля. Рассмотрены также абстрактные неполные дифференциальные включения второго порядка в общей постановке в секвенцильно полных локально выпуклых пространствах.
Ключевые слова: комплексная степень многозначначного линейного оператора, C-'резольвентное множество, абстрактное неполное дробное дифференциальное включение, абстрактное неполное дифференциальное включение второго порядка, локально выпуклое пространство.
Поступила в редакцию 30.04.2020 После переработки 01.06.2020
Сведения об авторе
Костич Марко, профессор, факультет технических наук, Университет Нови-Сада, Нови-Сад, Сербия; e-mail: marco.s@verat.net.
Работа частично поддержана Министерством науки и технологического развития Республики Сербия, грант № 174024.
