Bull. Kyushu Inst. Tech.
(M. & N. S.) No. 30, 1983, pp. 11-18
ON RANDOM LINEAR EVOLUTION EQUATIONS
By
Shigeru IToH
(Received Oct. 25, l982)
1. Introduction
The generation theorem of (linear) Co-semigroups for closed linear operators in . Banach spaces is well-known (i.e., the Hille-Yoside theorem) (cf. Hille-Phillips [10], Dunford-Schwartz [5], Yosida [27], Tanabe [26], Masuda [19]). The extension of the Hille-Yosida theorem to time dependent cases given by Kato is also now well-known (cf. Kato [18], Tanabe [26], Masuda [19]).
In this paper we study evolution equations containing random parameters in separable Banach spaces and show the existence of solutions of such equations. These types of equations arise naturally in the theory of partial differential equations whose coeMcients include random parameters.
In section 3 we treat time independent cases, while in section 4 time dependent cases.
2. Preliminaries
Let X be a Banach space and B(X) be the set of bounded linear operators on X. A mapping S: [O, oo)Å~X.X with the following properties is said to be a (Co-)semigroup of type (M, 6) on X:
( i ) For any t l}i O, S( t) e B(X) and ll S( t) Il s{; MeB' ;
( ii ) For any t År- O, s ;}l O, S( t) S( s) = S(t + s) and S(O) =: I (the identity operator of X) ;
(iii) For any sÅrmO, xGX, limS(t)x==S(s)x.
tes
The Hille-Yosida theorem is stated as follows: A linear operator -A in X is an in- finitesimal generator of a semigroup of type (M, 6) on X if and only if A is densely defined, closed and has resolvents satisfying
Il (A + Z)'" ll s M(A - 6)-n
for-every 2.År6 and n= 1, 2,... (cf. Hille-Phillips [10], Dunford-Schwartz [5], Yosida [27],
Tanabe [26], Masuda [19]). IfX=H (Hilbert space), then an operator A in H is accretive
(cÅí Tanabe [26], Masuda [19], Itoh [15]) if and only if for each x, yE D(A) (the domain
ofA), Re(Ax-Ay, x-- y)2O, where (•, •) is the inner product ofH. IfA is linear, densely
12 Shigeru lToH
defined and accretive, and if there is no proper extension of A that is also a linear ac- cretive operator in H, then A is called maximal accretive. In this case A is necessarily closed (cf. Tanabe [26, Theorem 2.1.1]). It is known that A is maximal accretive if and only if for some (all) ZÅrO, the range ofA+ZI is H (cf. Tanabe [26, Proposition 2.1.4]).
Let (9, .s]1) be a measurable space. A mappingf: 9.X is called measurable if for Jany closed subset C of X,f-'(C) = {to e9:f(tu) e C} e.s2e'. It is clear thatfis measurable
if and only if for any Borel subset B of X,f-i(B)E.g?'. If X is separable, other defini- tions of measurability of mappingsf: 9.X are equivalent to the above (cf. Hille-Phillips [10, pp. 72-73], Bharucha-Reid [2, pp. 14-16]). Let D be a subset of X. A mapping A: 9Å~D.X is called a random operator if for every xeD, A(•)x: 9.X is measurable.
A random operator A is called continuous (linear, etc.) if for each tue9, A(to): D.X is continuous (linear, etc.).
In this paper derivatives are strong derivatives. Let [O, T] be a finite closed interval of the real line R. A mapping u: [O, T] Å~9.X is said to satisfy condition (C, 9) (or (Ci, 9)) if for any te [O, T], u(t, •) is measurable and for any co e 9, u(•, to) is (strongly) continuous (or continuously differentiable), For the interval [O, oo) we also adopt similar definitions.
Let Z be another Banach space. We denote by B(Z, X) the set of bounded linear operators of Z into X. A mapping S: [O, T].B(Z, X) is called strongly continuous (or strongly continuously differentiable) if for each z E Z, S( • )z is continuous (or continuously differentiable). S is called norm continuous if for each se [O, T],
lim II S(t) ---- S(s)ilz,x=O, t-s
where ll • llz,x is the operator norm of B(Z, .X).
3. Randomevolutionequations(I)
In the rest of this paper let X be a separable Banach space and Ybe a dense linear subspace of X. The following lemma is easy to prove, but for the sake of completeness we give its proof.
LEMMA 3.1. Let A: Y-ÅrX be a one to one, onto, ciosed linear operator. Then Y becomes a separable Banach space by the new norm ll•llA defined by llyllA==llyR+
ll Ay ll 07 e Y).
PRooF. Since A is a closed linear operator, Yis clearly a Banach space YA by this
norm. Then AeB(YA, X) by the closed graph theorem (cf. 'Hille-Phillips [10, Theorem
2.12.3], Dunford-Schwartz [5, Theorem II.2.4], Yosida [27, Theorem II.6.1]). Since A
is one to one and onto, A-ieB(X, YA) by the open mapping theorem (cf. Hille-Phillips
[10, Theorem2,12.1], Dunford-Schwartz[5, Theoremll.2.1], Yosida[27, Theorem
On Random Linear Evolution Equations 13
II.5]). Thus YA is separable because X is separable.
The result of the following proposition is of fundamental importance throughout
this paper.
PRoposmoN 3.2. Let A: S2 Å~ Y--ÅrX be a closed linear random operator such that for any co E 2, A(co) is one to one and onto. Then the mapping J: S:2 Å~ X---ÅrX defined by J(co)x
=A(co)-'x (co E 9, xeX) is a linear random operator.
PRooF. Fix an element coES[2. Define anorm !l•ilA on Yby lly[IA=llyH+11A(co)yll (ye Y), By Lemma 3.1, Ybecomes a separable Banach space YA by this norm. More- over, for every coe9, A(co)EB(YA, X) and A(co)" EB(X, YA). Then it is known that for each xEX, J(•)x: 9.YA is measurable (cf. Han"s [9], Itoh [15, Lemma 2.1]). Since the injection of YA into X is continuous, J( •)x is measurable as a mapping of 9 into X.
Now we present a random version of the Hille-Yosida theorem.
THEoREM3.3. Let A:2Å~Y.X be a closed linear random operator. Suppose
that there exist functions M: 9.[1, oo) and 6: 2--•R such that sup {6(to): tuG2}Åqoo and for each tuE9, -A(to) generates a semigroup of type (M(co), 6(tu)). Then the
following holds :
(i) There exists a unique mapping S: [O, co)Å~9Å~X.X with S(t, w)YcY(t;}rO, tuE9) such that for each tuG9, S(•, tu) isasemigroup of type (M(tu), 6(tu)) and for each t ;}i O, x E X, S(t, • )x is measurable ;
(ii) For any yE Y, define a mapping u: [O, oo)Å~S2--ÅrX by u(t, co) :S(t, co)y (t;}rO, coE S2). Then u satisfies condjtion (Ci, S;2) and this u is the unique mapping such that for each coE2, u(O, co)==y and
du(t, co)/dt = -A(co)u(t, cD)=: -S(t, co)A(tu)y (t }) O) .
PRooF. Since sup {6(co): coE9}Åq oo, for sufficiently small ZÅrO, the range of I+
AA((o) is X for all coest!. Thus the mapping JA:9xX.X defined by Jz(co)x=:
(I+AA((o))-'x (tue9,xeX) is a random operator by Proposition3.2. Defines S:
[O, oo)Å~9Å~X-X by
S(t, cD)x == lim J,/.(co)nx (t l;l O, co e S2, xE X) .
n- co
Then for any t;) O and x E X, S(t, • )x is measurable (cf. Hille-Phillips [10, Theorem 3.5.4], Himmelberg [11, Theorem 6.5]), and for any tu E 9, S( • , tu) is a semigroup of type (M(to), 6(w)) (cf. Hille-Phillips [10, Theorem 12.3.l], Tanabe [26, Theorem 3.1.4], Masuda [19, Theo rem 2.2. 1]). It is known that for any co e si!, y G Y, S( • , co)y e Y is continuously
differentiable and is uniquely determined by S(O, tu)y=y and ,
dS(t, co)yldt= --A(co)S(t, co)y = -S(t, co)A(co)y
14 Shigeru lToH
for tÅr O.
REMARK 3.4. Let H be a separable Hilbert space and ]Y be a dense linear subspace of H. Let A: 9Å~ Y.H be a closed linear random operator. Suppose there exists a fun- ction 6: S2 -ÅÄR such that sup {6((D) : tu G 9} Åq co and for each (D e 9, A(tu)+6( co)I is maximal accretive. Then it is easily seen that the corresponding S(•, co) in Theorem 3.3 is a semigroup of type (1, 6(tu)).
THEoREM 3.5. Let A: 2Å~ Y.X be as in Theorem 3.3, B: S;}Å~X.X be a bounded linear random operator, andf: [O, T] Å~ 9.X be a mapping satisfying condition (Ci, 2).
Then for any measurable mapping v: 9--ÅrX with v(9)c Y, there exists a unique mapping u: [O, T] Å~9-ÅrX satig. fyjng condition (Ct, S;2) such that for any co eS2, u(O, co)=v(co) and du(t, tu)ldt + A(co)u(t, co) + B( co)u(t, co) =f(t, co) (O s{ t K T) .
PRooF. For every coeS2, -(A(co)+B(co)) generates a semigroup T(•,co) of type (M(to), 6(to)+M(tu)llB(tu)11) (cf. Hille-Phillips [10, Theorem 13.2.1], Dunford-Schwartz [5, Theorem VIII.1.19], Tanabe [26, Theorem 3.4.1]), and for any t2O,
co
T(t, co) = 2 S.(t, e)), n=O
where So(t, tu) :S(t, tu) is the semigroup generated by --A(co) by Theorem 3.3 and Sn(t, CO)` -'S8 S(t-'s, co)B(co)Sn-i(s, co)ds (n=1, 2,.••)
(cf. Hille-Phillips [10, Theorem13.4.1], Dunford-Schwartz [5, Theorem VIII.1.19], Tanabe [26, Theorem 3.4.2]). Thus for every t;}iO, xEX, T(t, •)x is measurable (cf.
(Hille-Phillips [10, Theorem 3.5.4]). Define u: [O, T] Å~9.X by u(t, co) = T(t, co)v(tu)+S8 T(t--s, tu)f(s, co)ds
for tE [O, T], cD e S;2. Then this u is the desired solution (cf. Tanabe [26, Theorems 3.2.1 and 3.2.2]).
By using a similar method as above, we obtain the following theorem (cf. Tanabe [26, Theorems 3.2.1 and 3.2.3]).
THEoREM 3.6. Let A and B be as in Theorem 3.5 andf: [O, T] Å~ S2.X be a mapping
satisfying condition (C, S2) such that for any tG [O, T], (D E 9, f(t, co) G Yand A(co)f( • , (D):
[O, T].X is continuous. Then for any measurable mapping v: 2.X with v(9)cY,
there exists a unique mapping u: [O, T] Å~ su.X satisfying condition (Ci, 2) such that for
each co G S2, u(O, co)=v(co) and
On Random Linear Evolution Equations 15
du(t, co)ldt+A(co)u(t, co)+B(co)u(t, co)==f(t, co) (O :f{ tS T) .
REMARK 3.7. Let A be as in Theorem 3.6. Suppose there exists a norm on Ywhich is stronger than the original one and makes Ya Banach space Yi. Letf: [O, T] Å~ S2--År Yi be a mapping satisfying condition (C, 9). Then for every toeS2, A(tu)eB(Yi, X), hence A(co)f( • , co) : [O, T] --ÅrX is continuous.
4. Random evolution equations (II)
In this section we treat time dependent cases. The corresponding original deter- ministic cases were given by Kato [18] (cf. Tanabe [26], Masuda [19]). For the sake of simplicity we state results for operators generating semigroups of type (1, 6). With some appropriate modifications, a little more general results may be obtained as in Kato [18]
(cf. Tanabe [26], Masuda [19]).
THEoREM4.1. Let A: [O, T]Å~9Å~Y.X be a mapping with the following pro-
pertles :
(i) For any tG [O, T], ye Y, A(t, •)y is measurable;
(ii) There exists a function 6: S;2---ÅrR such that sup {6(co): coES2!}Åqco and for any te [O, T], tu e 9, -A(t, co) generates a semigroup of type (1, 6(co));
(iii) For any co e 9, there exists a norm ll • ll. making Ya Banach space Y. such that the injection of Y. into X is continuous and A(•, (o): [O, T].B(Y., X) is norm con- tinuous. Moreover, there exist a strongly differentiable mapping S.: [O, T]-ÅrB(Y., X) and a strongly continuous mapping B.: [O, T]-ÅrB(X) such that for any te [O, T], S.(t):
Y..X is an isomorphism and S.(t)A(t, tu)S.(t)-' ==A(t, to)+B.(t).
Then there exists a unique fa mily of operators U( t, s, co) e B(X) (O :E{; s s; t sg T, co G S2) satisfying the following conditions:
(a) For each xGX, OSsf!gtf{ T, U(t, s, •)x is measurable;
(b) For each coE2, U(t, s, co) is strongly continuous in s, t, with U(s, s, tu)==I and ll U(t, s, co)ll f{g; efi(ca)(t-s);
(c) For each co E 9, O s;l s s{l r sg t fE{ T, U(t, s, co) == U(t, r, cD) U(r, s, co) ;
(d) For each coG2, yG Y, U(t, s, co)yeYis continuously differentiable in s, t, with OU(t, s, co)ylOs= U(t, s, co)A(s, co)y
and
0U(t, s, co)ylOt :-A(t, co)U(t, s, co)y (Os;;sff{:tsT).
PRooF. For any positive integer n, consider a partition {tY•}r•=o of [O, T] defined by
O=:t6ÅqtrÅq•••Åqt: = T, tr• -t7-i=T/n
16 Shigeru lToH
(i=1, 2,..., n). Let A.: [O, T] Å~2x Y.X be a mapping defined by A.(t, tu)y =
A(tr•-i, co)y (co G S;2, tr•-i f{; tÅq t7• , yG Y, i '-- 1, 2,..., n) and A.( T, co)y =A(t:-,, co) (co e S2, ye Y). Then, by condition (iii), for every coe9,
lim il A.(t, to) - A(t, to) ll y.,x == O
n -- co
