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criterion for compactness of sets in the spacesLP(g,dt, X),
SIAM J. MATH. ANAL.

(C)

1982 Society for Industrial and Applied Mathematics

0036-1410/82/1305-0007 $01.00/0

Vol. 13, No. 5, September 1982

COMPACT PERTURBATIONS OF m-ACCRETIVE OPERATORS IN GENERAL BANACH SPACES* S. GUTMAN Abstract. The evolution problem au A u(t) Bu(t), u(0) x, where A is a nonlinear m-accretive operator and B is a nonlinear completely continuous operator B: C([0, T]; X)- LP(0, T; X), is studied and various results on a local and global existence of solutions are established. A generalization of the Weyl-Riesz criterion for compactness of sets in the spaces LP(g,dt, X), 0 Jx (I + XA) A x (I-Jx) and lax[- lim x 0 Ax for x X. Set/}(A)- {x X: IAxI < m }./(A) is called the generalized domain of A. Identify X with its natural embedding in X** and define operator A by

-,

Ax-(zX** "V[6,n]A,wF(6-x), (n-z,w)_>0}. The following proposition is a standard result for accretive operators (see, e.g., [2,

pp. 0.7-0.8]): PROPOSITION 2.1. Let A be m-accretive. Then

1) D(A)C6(A)CD(A). 2) IlAxxll O. 3) IAxl-inf( yll "y2x} for x(A). then IAxol 0 for =/= 0. (see, e.g., [9]). THEOREM 6.1. Let A be an m-accretive operator in L(f). Let T>0, t[0, T] and g(x, y, ’, s)" f f [0, T N Nt be continuous of all its variables satisfying [g(x,y, r,s )1-< M(1 + Isle ), s N for some M>0 and fl [0, 1[. Then the initial value problem

-

+Au

g(x,y,’r,u(x,’r))dxd’r,

u(x,O)-uo(x)D(A )

-

has an integral solution defined on a maximal interval of existence [0, T] or [0, Tmax[, Tmax n,

n-- 1,2,3-.., and

gn(x,y,’r,u(x,’r))dxd’r for u into

C([0, T] L(a)). LEMMA 6.1. The operator B, is a completely continuous operator from C([0, T],L(2))

itself. Proof.

Since gn is bounded and uniformly continuous in 2 [0, T] N, the operator B is a bounded operator in C([0, T]; L(2)). Moreover, B can be considered as a bounded operator from C([0, T]; El(f])) into C([0, T] 2). The image of every bounded set of C([0, T]; Ll(2)) is a bounded set of equicontinuous functions in C([0, T] 2). By Ascoli’s theorem we get that B is a compact operator into C([0, T] 2). Since

m-ACCRETIVE OPERATORS

797

C([0, T]) is isometric to C([0, T]; C()) and C([0, T]; C())CC([O,T];L(2)), we have that B is a compact operator from C([0, T]; Ll()) into itself. To show that Bn is a continuous operator note that every function u G C([0, T]; L(2)) can be considered as a function in the space L([0, T] 2). For fixed n and y, g(x,y, u(x, -))" [0, T f Nt is a bounded measurable function. Let qk q, k--, oe, in C([O,T];L()); then pkq in L([0, T]Xf), and therefore, there exists a subsequence {pi}] of {p}] that converges to p almost everywhere in [0, T]2. Therefore, g(x,y,,%(x,)) g,( x, y, q( x, $ )) as i almost everywhere in [0, T] for 2. By Lebesgue dominated convergence theorem, B,% ](y, t) Bnq ](y, t) as it of functions, family is an equicontinuous Since every (y,t)[O, Tlf. {[B,%]}=l --, f therefore, Bnp, and, in uniformly B,% [0, T] that follows B,%] B,p], m, in C([0, T]; L(2)). But this is sufficient for the continuity of B,. Proof of Theorem 6.1. It is enough to prove (see Theorem 1.2) that the operator

,

,

-

,

g(x,y,r,u(x,r))dxdr is a causal and completely continuous from C([0, T]; L(a)) into C([0, T]; Ll(f)). From the estimate [g(x,y,t,s)lN }, then m(a(N,))_N, we have

.

[[Bq](Y,t)-[Bq](Y,t)] 0 such that for every A CR with

,

proved.

LEMMA A.3. Let FC LP(N, dt, X) satisfy the conditions (i)-(iv) of Theorem A. 1. Then for every N and e > O, the set B(t) { f(t) f F } is relatively compact in X. a>0 we can choose by Lemmas A.1 and A.2 a 8>0 such that Given Proof. \ [f’x(N A)-jqp