JOURNAL OF CONCRETE AND APPLICABLE MATHEMATICS VOLUME 2- 2004

262 Pages·4.266 MB·English

VOLUME 2,NUMBER 3 JULY 2004

ISSN:1548-5390 PRINT,1559-176X ONLINE

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JOURNAL OF CONCRETE AND APPLICABLE MATHEMATICS,VOL.2,NO.3,191-212, 191
2004,COPYRIGHT 2004 EUDOXUS PRESS,LLC
Strong And Weak Solutions Of
Abstract Cauchy Problems
1
Chung-Cheng Kuo
Department of Mathematics, Fu-Jen University
Hsin-Chuang, Taipei, Taiwan
e-mail: [email protected]
Sen-Yen Shaw
Graduate School of Engineering, Lunghwa University of Science and Technology
Gueishan, Taoyuan, Taiwan
e-mail: [email protected]
Running head: Strong and weak solutions of abstract Cauchy problems
Abstract. Let α be a positive number, C be a bounded linear injection
on a Banach space X, and let A : D(A) ⊂ X → X be a closed linear operator
commuting with C. Under suitable conditions on A (such as, C−1AC = A,
ρ(A)6=∅,D(A)=X),wediscussconnectionsamong: (i)Abeingthegenerator
of an α-times integrated C-semigroup on X; (ii) the existence of unique strong
solution of u0(t) = Au(t) + j (t)Cx, t > 0; u(0) = 0, for all x ∈ D(A);
α−1
(iii) the existence of unique strong solution of v0(t) = Av(t)+j (t)Cx+j ∗
α α
Cg(t), t>0; v(0)=0, for all x∈X; (iv) the existence of unique weak solution
oftheabstractCauchyproblem: w0(t)=Aw(t)+j (t)Cx+j ∗Cg(t), t>
α−1 α−1
0; v(0) = 0, for all x ∈ X. Here j (t) = tα/Γ(α+1) and g is any function in
α
L1 ([0,∞),X). Applications to concrete examples are also demonstrated.
loc
2000 Mathematics Subject Classification: 47D60, 47D62.
Key words and phrases: α-times integrated C-semigroup, generator, abstract
Cauchy problem, strong solution, weak solution.
1 Introduction
Let X be a Banach space with norm k · k, and let B(X) be the set of
all bounded linear operators from X into itself. Consider the abstract Cauchy
problem (ACP):
(cid:26)u0(t)=Au(t)+f(t), t>0;
(ACP(f,x))
u(0)=x,
1Research supported in part by the National Science Council of Taiwan.
192 C.KUO,S.SHAW
where A : D(A) ⊂ X → X is a closed linear operator and f is an X-valued
function on [0,∞).
Let X denote the Banach space D(A) equipped with the graph norm
1
kxk = kxk+kAxk for x ∈ D(A). A function u is called a strong solution
X1
of ACP(f,x) if u ∈ C1((0,∞),X)∩C([0,∞),X ) and satisfies ACP(f,x). In
1
the case where A is densely defined, u is called a weak solution of ACP(f,x) if
u is continuous, and for every x∗ ∈ D(A∗) the function hu(·),x∗i is absolutely
continuous and satisfies
(cid:26) dhu(t),x∗i=hu(t),A∗x∗i+hf(t),x∗i,a.e.t>0;
dt
u(0)=x.
The ACP is closely related to the theory of operator semigroups. Arendt
[14, A-II, Theorem 1.1] proved that ACP(0,x) has a unique strong solution for
every x ∈ D(A) if and only if the part A of A in X (i.e. the restriction of
1 1
A with domain D(A ) := {x ∈ D(A);Ax ∈ X }) generates a C -semigroup on
1 1 0
X . Moreover, these two conditions are also equivalent to that A generates a
1
C -semigrouponX,providedthatAhasnonemptyresolventsetρ(A)[14,A-II,
0
Corollary 1.2]. Ball [2] proved that A has dense domain and ACP(f,x) has a
unique weak solution for every f ∈ L1([0,∞),X) and x ∈ X if and only if A
generates a C -semigroup on X.
0
RecentlyDaviesandPang[3],MiyaderaandTanaka([16],[17]),deLauben-
fels ([4], [5]) have studied C-semigroups (also called regularized semigroups)
and their connections with the ACP. A result of deLaubenfels [4, Theorem 4.1]
states that if A is the generator of a C-semigroup, then A commutes with C
and ACP(0,x) has a unique strong solution for each initial value x in C(D(A)).
TanakaandMiyadera[17,Corollary2.2]thenshowedthatincaseρ(A)6=∅,the
converse of the last statement is also true.
The aim of this paper is to prove generalizations of the aforementioned
resultstoα-times(α>0)integratedC-semigroups. Thisclassisageneralization
oftheclassofα-timesintegratedsemigroups,whichhavebeenstudiedin[1],[6],
[8],and[15]forα∈N,and[7],[12],and[13]forα∈R . Thispaperservesasa
+
continuationof[9],inwhichbasicpropertiesofα-timesintegratedC-semigroups
as well as characterization of their generators have been discussed.
Let C ∈B(X) be an injective operator and α a positive number. A family
S(·)={S(t);t≥0} in B(X) is called an α-times integrated C-semigroup if
(1.1) S(·)x:[0,∞)−→X is continuous for each x∈X;
(1.2) S(0)=0, S(t)C =CS(t), and for all x∈X, t, s≥0
1 (cid:18)Z t+s Z t Z s(cid:19)
S(t)S(s)x= − − (t+s−r)α−1S(r)Cxdr.
Γ(α)
0 0 0
An α-times integrated C-semigroup S(·) is said to be nondegenerate if
(1.3) S(t)x=0 for all t>0 implies x=0.
Note that for a nondegenerate α-times integrated C-semigroup, the identity
S(0) = 0 follows automatically from (1.3) and the functional equation in (1.2)
and so it is superfluous in condition (1.2) in the nondegenerate case.