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Some surjectivity theorems for compact perturbations of accretive operators
Q
fl36?-545X8-I Sj WJ- .oO 1984 Psrqmon Press Ltd.
SOME SURJECTIVITY THEOREMS FOR COMPACT PERTURBATIONS OF ACCRETIVE OPERATORS NORIMICHI HIRANO Department
of Information
Sciences, Tokyo Institute of Technology, Meguro-ku, Oho-okayama, Japan (Receiued in revised form 1-t September
Key words and phrases:
Tokyo 152,
19S3)
Surjectivity theorems, compact perturbations.
accrerive operators.
1. INTRODUCTION
be a range R(T) sidered the surjectivity
LET E
Banach space, T be a single in E, and C: D(T) + E be solvability of the equation results in [4], [5] and [6]. In
valued m-accretive operator with domain D(T) and an odd, compact mapping. Recently Kartsatos conTX + Cx = f for every f E E and established some the present paper, we consider the equation
Ax+Cx3f,
fEE
(1.1)
where A C E x E is an (multivalued m-accretive operator and C: D(A) + E is a compact mapping. The results in [4] were proved by using Borsuk’s theorem concerning odd operators T and C. The methods employed in this paper also involve degree theory arguments. In Section 3, we establish some surjectivity results by using the theory of Leray-Schauder degree. Operators A and C are not assumed to be odd in Section 3. In Section -t, we consider the case where A and C are odd. 2. PRELIMINARIES
Let E be a Banach space and E* be the dual Banach space of E. In the follouing. denote by + and - strong and weak convergence, respectively. and B(r, 0) the open with center zero and radius r > 0. For a set HC E, the symbols &V, H, EY’H denote boundary of H, the closure of H, and the closed convex hull of H. respectively. Let D subset of E and T a mapping from D into E. Then T is said to be nonexpansive if for X,Y ED,
we ball the be a each
It is known that if E is uniformly convex and T: E + E is nonexpansive, then the mapping I - T is demiclosed, i.e., x, - x and x, - TX, + y imply x - Ti = y (cf. Browder [2]). Let D be a closed subset of E and P be a mapping from E onto D. P is said to be a nonexpansive retraction of E onto D if P is nonexpansive and P2 = P. D is said to be a nonexpansive retract of E if there exists a nonexpansive retraction of E onto D. The duality mapping J of E into E* is defined by J(x) = {x E E*: ( x,x*) Let A be a (multivalued)
operator
=
lkllIl~*Il= Ikll’~.
from E into itself. Then we define D(A) = {x E E: 765
766
N.
H[R-\>o
Ax f $1, R(A) = U{Ax: x E D(A)}, IAxl = inf{i~yl~:J E Ax}and D,(A) = D(A) n B(r, 0) for each r > 0. A is said to be accretive if for each x, y E D(A) and 11E Ax, u E Ay, there exists j E J(x - y) such that (u - u.;) 2 0. An accretive operator A is said to be m-accretive if R(Z + u) = E for all i > 0. Let A be an m-accretive operator in E. Then one can define, for each 1. > 0. a single valued operator JA:E+ D(A) by J;, = (I + IA)-‘. It is known that Jr is nonexpansive for all 1.> 0. Recently Reich [8] prove the following result which is crucial for our argument. THEOREM A. Let E be a reflexive strictly convex Banach space and A be an m-accretive operator with domain D(A) and range R(A) in E. Then there exists a nonexpansive retraction P of E onto ED(A). It is well known that if E is uniformly convex and A C E x E is an m-accretive operator. thenD(A) is convex (cf. [l]). Then we can see that if E is uniformly convex and A C E x E is m-accretive then there exists a nonexpansive retraction P of E onto D(A). Let D be a subset of E. A mapping C is said to be compact if it is continuous and it maps bounded sets of D into relatively compact sets in E. Let D be a subset of E with nonempty interior. T: D+ E be a compact mapping and p E E. Then we denote by d(l - T, D. p) the LeraySchauder degree of I - T at p. 3. MAIN RESULTS THEOREM 1. Let E be a uniformly convex Banach space and A C E X E be an m-accretive operator. Let C: D(A) ---, E be compact and uniformly continuous on bounded subsets of D(A). Suppose that there exist positive constants u. b and r such that ,,${I If (A + C)D,(A)
Ax + Cxl - a&-# 2 r
,s,u,pbIlCli/il~~i~< a.
and
(3.1)
is closed, then B(r, 0) C R(A + C).
Proof. Let f E B(r, 0) and z E D(A). We show that there exists x E E such that Ax + Cx 3 f. Let P be a nonexpansive retraction of E onto D(A). We put a’ = sup{l/Cxj] / jlxll: llxl]2 b} and b’ = sup{llCxll: /Ix/JG b}. Th en, by (3.1), LI’< a. Therefore we can choose positive constants to and k such that to > 1, b(to
-
1) <
abto + r’
k
<
b(to
-
1) -211-d
a’bto + d
and (to - l)b ’ k(b’ + IAz / + llfll) + 2lkli.
(3.2)
(3.3)
where d = 2//.2~/ + 1AZ I + IIf I/ and r’ = r - 11 f (I. We put A = A - f. Then A is also an m-accretive operator. We define a mapping -Jk: E --, D(A) by jkx = (I + kA)-‘. For each x E E and t > 1. we define a mapping S,., by .SX,l(y) = (l/t)j,(x + y) for all y E E. Then since S,., is Lipschitz continuous with Lipschitz constant l/t, S,,, has a unique fixed point LL,,(E E. Therefore we can
Some surjecrivity
theorems
for compact
perturbations
of accretive
operators
767
define a mapping S, by S,(x) = u,,( where ZA,.(is the fixed point of SX.l, i.e., u,,( = (l/f)j,(x + u,.,). It is easy to see that S, is Lipschitz continuous. In fact, for each xi. x2 E E,
Therefore Ilh
- @*II
For each t > 1, we now define an operator 7(t) by T(t)x = S,( -kCP(rx)) for x E E. Then since SI, P are Lipschitz continuous and C is compact, T(t) is a compact mapping on E for each t> 1. Here we prove that d(Z- T(to), B(b,O), 0) = 1. To show this, it is sufficient to show that T(fo)(dB(b, 0)) C B(b, 0). Let x E aB(b, 0) andy = T(fo)x. Then by the definition of T(ra), we have that toy + kA(toy) 3 -kCP(tox) Since A is accretive,
+ y.
(3.4)
(3.4) implies that if u E AZ, jlroy - z/j* S (j, - kCP(rox)
for some j E J(foy - z). Therefore
- ku + y - 2)
we have
(to - l)llYll s wP(~o~)Il
+ Wzl
+ 211~11
= ~(ll~~(~o~~)lllII~(~o~)ll>ll~~~~)ll
If IjP(rox)ll a b, then by the hypothesis,
(3.5) + k/AZ I +2/k/l.
we have that
(fo - l)]]v]t c ka’W(tox)
- ztl + llzll> + klAz I + 21lzll
s ka’(llrox - zll + 11~11) + k1A.z
+ 2~~2~~
s ka’(r,,b + 2]]z]]) + k[Az / + kllfll + 2~~4. Then by (3.2), we obtain that ~IY]] c (k(a’bto + 211~11 + I& If IlP(tox)ll<
b, then IICP(tox)ll G b’. Therefore
+ IIf]]) + 2ltzll>/(~o - 1)
we have that
ll~llc (k(b’ + IAzl) + 211zll)/(to - 1)
T(to) has a fixed point and each t E (1, to], d(Z - T(t), put h(s) = T(sti + (1 - s)fo) transformations on B(b, 0), + x): l]xl]s b, t E (1, ro]}. For
N. HIRANO
768
simplicity, we assume that to - f1 = 1. Then for given E > 0, there exists 6’ > 0 such that if s, t E [0, l] and ]s - tl < 8,
Is - f/ (fl + S)(fl + t>
E/2.
M <
(3.6)
Since P is nonexpansive and C is uniformly continuous on bounded subsets of D(A), CP is uniformly continuous on bounded sets of E. Thus there exists 6 > 0 such that 6 < 6’ and for each X,y E B(rob, 0) such that I/x - y]] < 66, ]lCPx - CPyli < &/2k. Consequently, we obtain that ifs, t E [0, l] and ]s - t] < 6, then
--+(
s
I! (t*
+ X)X) +
-kCP((t,
+:i:,
+
+ &llW
t)
wkw(4
r1 +
s>.r>
x)
+ SIX>+ x) ji CP((t,
-
+ r)x) //
< E/2 + E/2(11 + t) < &.
Therefore h is a homotopy of compact transformations on B(b, 0). To show that is sufficient to show that 0 $ (I - h(t)) d(T(tr), B(b, O), 0) = d(Vf01, B(b, 01, 01, it (dB(b, 0)) for all t E [0, 11. Suppose that x E dB(b, 0) and x = T(r)x for some r E [rt, to]. Then rx = &( -kCP(rx) Since rx E D(A),
+ x).
(3.7)
(3.7) can be written
rx + k/i(rx)
3 -kC(rx)
+x.
(3.8)
Therefore
3 inf{]]u +
C(tx)]]: z4EA(fx)}
2 inf{]]u + C(tx)]]: u EA(tx)} =
IA(tx)
- llfll
+ C(tx) I - i/f/i
2 allfxll + r’.
Thus we have (t - l)b/k
This is a contradiction.
2 arb + r’. While from (3.2), we have that
Hence
b(t - 1) h
b(to
abt + r’
abto+r’
we obtain
-
that
1) <
k
’
d(l-
r(t,),
B(b, 0), 0) = d(Z - T(to),
Some surjectivity
theorems
for compact
perturbations
of accrewe
operators
769
B(b, O),O) = 1. Since tl is arbitrary in (l,ro]. We obtain that d(Z - T(r). B(b, O),O) = d(1 - T(to), B(b, 0,O) = 1 for all t E (1, to]. Thus it follows that for each I E (1. fg], T(r) has a fixed point in B(b, 0). Hence there exists a set {~(:r E (1, to]} C B(b, 0) such that X, = T(t)x, for f E (1, rO]. Then for each r E (1, ro], we have that r.ri E D(A) and r-l --p
+ A(rx,)
+ C(rx,) 3f.
Then we have thatfE (A + C)D,(A). Since (A + C)D,(A) is closed, there exists a solution x E D(A) of the equation Ax + Cx 3 f and this completes the proof. Remark 1. In theorem 1, the condition that E is uniformly convex is needed only to guarantee that D(A) is a nonexpansive retract. Therefore it can be replaced by another condition which guarantees that D(A) is a nonexpansive retract (cf. [7]). Also we note that if C is defined on E, then the assumption on E can be removed. COROLLARY
1. Let E, A and C be as in theorem 1. Suppose that there exist positive constants
such that ,jh_m=(IAx + If (A + C)D,(A)
C.rI - 4-d> = m and li;:p
IIWll~ll < a.
(3.9)
is closed for all r > 0, then R(A + C) = E.
Remark 2. It is easy to see that (3.9) can be replaced by the following condition:
lim IAx + Cxi/jlxll = p jix:/+-X
and
liF’;p
IICxIl/]/x]l < 30.
(3.10)
THEOREM 2. Let E be a Banach space, A C E x E be an m-accretive operator and C: E ---, E be a compact mapping. Suppose that there exist positive constants 6 and r such that
if ]ix]l 2 6, If (A + C)D,(A)
then
inf
(y + Cx,j)/llxII 2’.
(3.11)
yEAx.jE&)
is closed, then B(r, 0) C R(A + C).
Proof. Let f E B(r, 0). We show that there exists x E E such that Ax + Cx 3 f. We fix a positive constant k. For t > 1, let S, be the mapping defined in the proof of theorem 1. For each f > 1, we define an operator r(t) by T(r)x = S,( -kC(tx)) forx E E. Then r(t) is compact for all t > 1. We claim that for each t > 1, T(t) has a fixed point in B(b, 0). To show this, it is sufficient to prove that if x E aB(b, 0), then x # AT(t)x for all J, > 1. Fix f E (1, =) and suppose that x E aB(b, 0) and 1x = T(r)x for some A > 1. Then we have that
k(t - 1)x + ky + kC(rx) = kf, for some y E A(tAx). such that
Let z E A(fx).
Then since y E A(h) (y - z, j) G=0.
(3.12) and A > 1, there exists j E J(x) (3.13)
770
From
N. HIRANO
(3.12)
and (3.13),
we have that kllfll I!xI/ 2 i(f - l)l~xll’ f k(_~ + C(rx), j) 2 rqt - 1)//x/l’ + k(z + C(fX).j) 2 A([ - l)!/xll? + krllf.Kll/f = qt
- 1) II,# + krlixll.
(3.14)
Since llfll G r and //xl/ = 6, (3.14) implies A(t - 1)b 6 0, and this is a contradiction. the argument in the proof COROLLARY
Therefore of theorem
T(r) has a fixed point in B(b. 0) for all t > 1. Then 1, the conclusion follows.
2. Let E, A and C be as in theorem
2. Suppose
lim inf (Y + Cx, X(-.X,VE&/EJ(X) If (A + C)D,(A)
by
that
i>lll4l = =.
(3. IS)
for all r > 0, then R(A + C) = E.
is closed
Some sufficient conditions for the set (A + C)D,(A) to be closed are known. (See theorem A of [4].) In the following theorem we do not need the assumption that the set (A + C)D,(A) be closed. 3. Let E and A be as in theorem 1. Let C: D(A) +E be continuous topology of E to the strong topology of E. Suppose that there exist positive and r which satisfy the condition (3.1). Then B(r, 0) C R(A + C).
THEOREM
from the weak constants c(, b,
Proof. Let fE B(r, 0) and fo, k, A, Jk, T(t) be as in the proof of theorem 1. We note that from the hypothesis, C is uniformly continuous on bounded subsets of D(A). Then by the proof of theorem 1, there exists a subset {x,: t E (1, fa]} of B(b, 0) such that txr E D(A) and xI = r(t)x, for each t E (1, ro]. Then since E is reflexive and {x,: t E (1, to]} is bounded subset of E, we can suppose, without loss of generality, that xi--x0 E B(b, 0), as f+ 1. Since r.u E D(A) for all I > 1, we have thatlim CP(rx,) =lim C(tr,) = Cxo. Now we define a mapping t-l
E + E by .5x = jk(-kCxo x,, we have that
+
S:
x)
r-1
for x E E. Then
= !Lyllxr- &( -kCxo r-1
+
=
/I
!\y
xI - fjk(-kC(trt) i $(
‘I
-kC(tx,)
!iy /I f &(-kC(tr,)
= 0.
From
the definition
of
Sxrll
!\m /IXI -
c hm
S is nonexpansive.
+ x,) /I + xf)
II
+ XJ - &( -kCxo + x,) - &( -kCxo
+ x,) /
+x,)
II (3.16)
theorems forcompact
Some surjectlvlty
Since I - S is demiclosed. completes the proof.
we obtain
perturbations
of accretive
that xg = jk(-kCxo
771
operators
+ xg). i.e. Ax,, + CXO3f and this
COROLLARY 3. Let E be a reflexive C: E -+ E be continuous that there exist positive R(A + C).
Banach space, A be an m-accretive operator and from the weak topology of E to the strong topology of E. Suppose constants b and r which satisfy the condition (3.11). Then B(r, 0) C
Next, we consider the case where the resolvent J;. of A is compact for some A > 0. It is well known that if A C E x E is m-accretive. then the following conditions are equivalent: JA is compact
for some A > 0;
(3.17)
J, is compact
for all 1. > 0;
(3.1s)
for each M > 0, {X E D(A)
: llxil -=cA4 and
l/y/l < M for some y E Ax} is relatively
compact.
(3.19)
THEOREM 4. Let E be a uniformly
convex Banach space and A C E x E be an m-accretive operator which satisfies the condition (3.17). Let C: D(A) +E be uniformly continuous on bounded subsets of D(A). Suppose that there exist positive constants a, b and r which satisfy the condition (3.1). Then B(r, 0) C R(A + C). Proof. Let f E B(r, 0) and let k, ro. A, j,, T(t) be as in the proof of theorem 1. From the condition (3.19), we can see that ji. is compact for all A > 0. Therefore T(t) is compact for r E (1, rO]. Then, by the same argument as in the proof of theorem 1, we have that there exists a set {xI:r E (1, ro3} such that fxf E D(A)x, E B(b, 0) and x, = T(t)x, for all t E (1, fo]. Here we define a mapping T by T.Y = jk(-kCx+ x) for x E E. Then T is a compact mapping. While from the definition of {xr}, we have that !\m l]xr - Tx,li = liim
]]xl - &( -kCPx,
s !\y
XI - +(-kCP(txr) /I
+
j
j,(
-kCP(tx,)
$(-kCP(rx,) Since C and P are uniformly
continuous
+ x,) jl
+ x,)
/I
+-x,) - &( -kCPx,
+x,) on bounded
-&(-kCPx, subsets
+ x,)
Ii
+ x,) 11. of D(A),
(3.20) implies
(3.20) that (3.21)
N. HIRASO
772
Then
since
T is compact,
x4-+x0 E E. Relation x0 E D(A), we obtain proof.
there
exists
a sequence
{x,} C{x,}
such
that
(3.21) implies now that xo = TXO= jk(--KPXO that x0 = Jk(-kCxo + x0), that is A& i CXO3fand
lim I, =1
and
+ XO). Then since this completes the
COROLLARY 4. Let E be a Banach space, A C E x E be a bounded, m-accretive operator which satisfies the condition (3.17) and let C be as in theorem 4. Suppose that there exist positive constants a, b and r which satisfy the condition (3.1). Then B(r, 0) C R(A + C). Proof. Since A is bounded and satisfies (3.17), we have that D(A) is boundedly compact. Then by proposition of 2.5 of [8], D(A) is a nonexpansive retact, that is, there exists a nonexpansive retraction P of E onto D(A). Then by the same argument as in the proof of theorem 4, the conclusion follows. COROLLARY 5. Let E be a Banach space, A C E x E be an m-accretive operator which satisfies the condition (3.17), and C:E - E be uniformly continuous on bounded subsets of E. Suppose that there exist positive constants 6 and r which satisfy the condition (3.11). Then B(r, 0) C R(A + C). 4. RESULTS In this section, we consider A and C are odd. LEMMA. Let E be a Banach D is a nonexpansive retract,
the solvability
Then
MAPPINGS
of the equation
(1.1) in the case that operators
space and D be a closed, convex and symmetric then there exists an odd nonexpansive retraction
Proof. Let P be a nonexpansive by
FOR ODD
retraction
of E onto
Qx = (Px - P(-x))/2,
D. We define
subset of E. If of E onto D.
a mapping
Q : E -+ D
for.r EE.
it is easy to see that Q is an odd, nonexpansive
retraction
of E onto D.
THEOREM 5. Let E be a uniformly convex Banach space, A C E x E be an odd. m-accretive operator and C:D(A) +E be an odd, compact mapping. Suppose that there exist positive constants a, b and r such that lAx + Cxl - aljxlj 3 r, If (A + C),(A)
is closed,
IIxIl2 b.
(4.1)
then B(r, 0) C R(A + C).
Proof. Let f E B(r, 0). We show that there exists x E E such that Ax + Cx 3f. Let P be an odd, nonexpansive retraction of E onto D(A) and choose positive constants to and k such that to > 1 and
b(fo- 1) <
k
abto + r’ where
r’ = r - l/fll. F or each t E (1, to], we define
a mapping
(4.2) T(t) as in the proof of theorem
Some surjectivitr theorems for compact perturbations
of accretive operators
773
1. For each s E [0, 11, we put A(s) = A - sf and J(s) = (I + kA(s))-‘. Here we fix r E (1, to] and for each s E [0, 11, define a mapping S(s) on E by S(s)x = u where u E E is the point such that u = (l/t)J( s ).r i u). Then we can see that for each s E [0, l], S(s) is Lipschitz continuous. Since A(0) is odd, we have that S(0) is also odd. For each s E [0,11. we now define a mapping H(s) by H(s) = S(s)(-kCP(rx)) for x E E. Then H(1) = T(r) and H(O) is odd because S(O), C and P are odd. Since S(s), P are Lipschitz continuous and C is compact, H(s) is compact for all s E [O, 11. We show that H is a homotopy of compact transformations. Let E> 0. Lets, s’ E [0, l] such that /s - ~‘1 6 e(t - l)/kr. Let x E E and put u1 = H(s)x and LL?= H(s’)x. Then there exists o1 E A(tul) and uz E A (Tut) such that tu1+ k(o1-
sf) = -kC(tx)
+ U1
tu2 + k(uz - s’f) = -kC(tx)
+ U?.
and
Then since A is accretive, we have that
Then we obtained that H is a homotopy of compact transformations. Next we show that or all s E [0, 11. Suppose that x E dB (b, 0) and x = H(s)x for some M(Z--H(s))@B@,O))f s E [0, 11. Then TX= J(s)(-kCP(fX) Since rx E D(A),
f x).
(4.4)
(4.4) can be written (t - 1)x + k(A(rx)
+ C(rx))
3 ksf.
Then by (4.1), we have ((t - l)/k)b
> arb + r - s j/f/l2 arb + r’
and this contradicts (4.2). Therefore we obtain that d(H(O), B(b, 0), 0) = d(H(l), B(b, 0), 0). Since H(0) is odd, by Borsuk’s theorem, d(H(O), B(b, 0), 0) is not zero. Then we obtain that H(1) = T(t) has a fixed point in B(b, 0). Therefore it follows that there exists a subset {q:t E (1, to]} of B(b, 0) such that xI = T(t)x, for all t E (1, to]. Then by the same argument in the proof of theorem 1, the conclusion follows. COROLLARY 6.
Let E, A and C be as in theorem 5. Assume that there exists a positive constant
a such that ,,z’;~=(IAx + Cx] - +]l) Then if (A + C)D,(A)
= =.
(4.5)
is closed for all r > 0, R(A + C) = E.
6. Let E be a uniformly convex Banach space, A C E x E be an odd, m-accretive operator which satisfies the condition (3.17), and C: D(A) *E be an odd, continuous mapping. Assume that there exist positive constants a, b and r which satisfy the condition (4.1). Then
THEOREM
B(r, 0) C R(A + C).
N. HIRA~O
774
Proof. The conclusion 1 and 5.
of theorem
6 follows
from the arguments
in the proof
of theorems
COROLLARY 7.Let E be a Banach space, A C E X E be an odd. bounded m-accretive operator which satisfies the condition (3.12), and C: D(A) +E be an odd, continuous mapping. Suppose that there exist positive constants u, b and r which satisfy the condition (4.1). Then B(r. 0) C R(A + C). Acknowledgemen+The suggestions and advice
author wishes to express his hearty thanks in the course of preparing the present paper.
to Professor
W. Takahashi
ior many
kind
REFERENCES 1. BARBU V., Sonlinear semigroups and differential equations in Banach spaces, Edirura Academiei R.S.R.. Bucuresti (1976). 2. BROWDER F. E.. LNonlinear operators and nonlinear equations of evolution in Banach spaces. Proc. Symp. pL[re Marh. 18 (1976). 3. FUCIK S., NECAS J., SOUCEK J. & SOUCEK V., Spectral Analysis of Nonlinear Operators. Lecture Notes in Mathematics. Springer-Verlag. Sew York (1973). 4. K.~RTSATOS A. G., Surjectivitv results for compact perturbations of m-accretive operators. J. marh. Analysis Appk. 78. l-16 (1980). 5. K.~RTSATOS X. G., Mapping theorems involving compact perturbations and compact resolvents of nonlinear operators in Banach spaces, /. .Ctarh. Analysis Appl. 80, 130-146 (1981). 6. KARTSMOS X. G., Some mapping theorems for accretive operators in Banach spaces, /. marh. .-lnalysis Appiic. 82, 169-183 (1981). 7. LLOYD iv. G., Degree Theory, Cambridgy Univ. Press, I\;ew York (1978). 8. REICH S., hsymptotic behavior of sermgroups of nonlinear contractions in Banach spaces, i. marh. Analysis Applic. 53. 277-290 (1976). 9. REICH S., Extension problems for accretive sets in Banach spaces, I. funcr. Analysis 26, 378-395 t 1977).
fl36?-545X8-I Sj WJ- .oO 1984 Psrqmon Press Ltd.
SOME SURJECTIVITY THEOREMS FOR COMPACT PERTURBATIONS OF ACCRETIVE OPERATORS NORIMICHI HIRANO Department
of Information
Sciences, Tokyo Institute of Technology, Meguro-ku, Oho-okayama, Japan (Receiued in revised form 1-t September
Key words and phrases:
Tokyo 152,
19S3)
Surjectivity theorems, compact perturbations.
accrerive operators.
1. INTRODUCTION
be a range R(T) sidered the surjectivity
LET E
Banach space, T be a single in E, and C: D(T) + E be solvability of the equation results in [4], [5] and [6]. In
valued m-accretive operator with domain D(T) and an odd, compact mapping. Recently Kartsatos conTX + Cx = f for every f E E and established some the present paper, we consider the equation
Ax+Cx3f,
fEE
(1.1)
where A C E x E is an (multivalued m-accretive operator and C: D(A) + E is a compact mapping. The results in [4] were proved by using Borsuk’s theorem concerning odd operators T and C. The methods employed in this paper also involve degree theory arguments. In Section 3, we establish some surjectivity results by using the theory of Leray-Schauder degree. Operators A and C are not assumed to be odd in Section 3. In Section -t, we consider the case where A and C are odd. 2. PRELIMINARIES
Let E be a Banach space and E* be the dual Banach space of E. In the follouing. denote by + and - strong and weak convergence, respectively. and B(r, 0) the open with center zero and radius r > 0. For a set HC E, the symbols &V, H, EY’H denote boundary of H, the closure of H, and the closed convex hull of H. respectively. Let D subset of E and T a mapping from D into E. Then T is said to be nonexpansive if for X,Y ED,
we ball the be a each
It is known that if E is uniformly convex and T: E + E is nonexpansive, then the mapping I - T is demiclosed, i.e., x, - x and x, - TX, + y imply x - Ti = y (cf. Browder [2]). Let D be a closed subset of E and P be a mapping from E onto D. P is said to be a nonexpansive retraction of E onto D if P is nonexpansive and P2 = P. D is said to be a nonexpansive retract of E if there exists a nonexpansive retraction of E onto D. The duality mapping J of E into E* is defined by J(x) = {x E E*: ( x,x*) Let A be a (multivalued)
operator
=
lkllIl~*Il= Ikll’~.
from E into itself. Then we define D(A) = {x E E: 765
766
N.
H[R-\>o
Ax f $1, R(A) = U{Ax: x E D(A)}, IAxl = inf{i~yl~:J E Ax}and D,(A) = D(A) n B(r, 0) for each r > 0. A is said to be accretive if for each x, y E D(A) and 11E Ax, u E Ay, there exists j E J(x - y) such that (u - u.;) 2 0. An accretive operator A is said to be m-accretive if R(Z + u) = E for all i > 0. Let A be an m-accretive operator in E. Then one can define, for each 1. > 0. a single valued operator JA:E+ D(A) by J;, = (I + IA)-‘. It is known that Jr is nonexpansive for all 1.> 0. Recently Reich [8] prove the following result which is crucial for our argument. THEOREM A. Let E be a reflexive strictly convex Banach space and A be an m-accretive operator with domain D(A) and range R(A) in E. Then there exists a nonexpansive retraction P of E onto ED(A). It is well known that if E is uniformly convex and A C E x E is an m-accretive operator. thenD(A) is convex (cf. [l]). Then we can see that if E is uniformly convex and A C E x E is m-accretive then there exists a nonexpansive retraction P of E onto D(A). Let D be a subset of E. A mapping C is said to be compact if it is continuous and it maps bounded sets of D into relatively compact sets in E. Let D be a subset of E with nonempty interior. T: D+ E be a compact mapping and p E E. Then we denote by d(l - T, D. p) the LeraySchauder degree of I - T at p. 3. MAIN RESULTS THEOREM 1. Let E be a uniformly convex Banach space and A C E X E be an m-accretive operator. Let C: D(A) ---, E be compact and uniformly continuous on bounded subsets of D(A). Suppose that there exist positive constants u. b and r such that ,,${I If (A + C)D,(A)
Ax + Cxl - a&-# 2 r
,s,u,pbIlCli/il~~i~< a.
and
(3.1)
is closed, then B(r, 0) C R(A + C).
Proof. Let f E B(r, 0) and z E D(A). We show that there exists x E E such that Ax + Cx 3 f. Let P be a nonexpansive retraction of E onto D(A). We put a’ = sup{l/Cxj] / jlxll: llxl]2 b} and b’ = sup{llCxll: /Ix/JG b}. Th en, by (3.1), LI’< a. Therefore we can choose positive constants to and k such that to > 1, b(to
-
1) <
abto + r’
k
<
b(to
-
1) -211-d
a’bto + d
and (to - l)b ’ k(b’ + IAz / + llfll) + 2lkli.
(3.2)
(3.3)
where d = 2//.2~/ + 1AZ I + IIf I/ and r’ = r - 11 f (I. We put A = A - f. Then A is also an m-accretive operator. We define a mapping -Jk: E --, D(A) by jkx = (I + kA)-‘. For each x E E and t > 1. we define a mapping S,., by .SX,l(y) = (l/t)j,(x + y) for all y E E. Then since S,., is Lipschitz continuous with Lipschitz constant l/t, S,,, has a unique fixed point LL,,(E E. Therefore we can
Some surjecrivity
theorems
for compact
perturbations
of accretive
operators
767
define a mapping S, by S,(x) = u,,( where ZA,.(is the fixed point of SX.l, i.e., u,,( = (l/f)j,(x + u,.,). It is easy to see that S, is Lipschitz continuous. In fact, for each xi. x2 E E,
Therefore Ilh
- @*II
For each t > 1, we now define an operator 7(t) by T(t)x = S,( -kCP(rx)) for x E E. Then since SI, P are Lipschitz continuous and C is compact, T(t) is a compact mapping on E for each t> 1. Here we prove that d(Z- T(to), B(b,O), 0) = 1. To show this, it is sufficient to show that T(fo)(dB(b, 0)) C B(b, 0). Let x E aB(b, 0) andy = T(fo)x. Then by the definition of T(ra), we have that toy + kA(toy) 3 -kCP(tox) Since A is accretive,
+ y.
(3.4)
(3.4) implies that if u E AZ, jlroy - z/j* S (j, - kCP(rox)
for some j E J(foy - z). Therefore
- ku + y - 2)
we have
(to - l)llYll s wP(~o~)Il
+ Wzl
+ 211~11
= ~(ll~~(~o~~)lllII~(~o~)ll>ll~~~~)ll
If IjP(rox)ll a b, then by the hypothesis,
(3.5) + k/AZ I +2/k/l.
we have that
(fo - l)]]v]t c ka’W(tox)
- ztl + llzll> + klAz I + 21lzll
s ka’(llrox - zll + 11~11) + k1A.z
+ 2~~2~~
s ka’(r,,b + 2]]z]]) + k[Az / + kllfll + 2~~4. Then by (3.2), we obtain that ~IY]] c (k(a’bto + 211~11 + I& If IlP(tox)ll<
b, then IICP(tox)ll G b’. Therefore
+ IIf]]) + 2ltzll>/(~o - 1)
we have that
ll~llc (k(b’ + IAzl) + 211zll)/(to - 1)
T(to) has a fixed point and each t E (1, to], d(Z - T(t), put h(s) = T(sti + (1 - s)fo) transformations on B(b, 0), + x): l]xl]s b, t E (1, ro]}. For
N. HIRANO
768
simplicity, we assume that to - f1 = 1. Then for given E > 0, there exists 6’ > 0 such that if s, t E [0, l] and ]s - tl < 8,
Is - f/ (fl + S)(fl + t>
E/2.
M <
(3.6)
Since P is nonexpansive and C is uniformly continuous on bounded subsets of D(A), CP is uniformly continuous on bounded sets of E. Thus there exists 6 > 0 such that 6 < 6’ and for each X,y E B(rob, 0) such that I/x - y]] < 66, ]lCPx - CPyli < &/2k. Consequently, we obtain that ifs, t E [0, l] and ]s - t] < 6, then
--+(
s
I! (t*
+ X)X) +
-kCP((t,
+:i:,
+
+ &llW
t)
wkw(4
r1 +
s>.r>
x)
+ SIX>+ x) ji CP((t,
-
+ r)x) //
< E/2 + E/2(11 + t) < &.
Therefore h is a homotopy of compact transformations on B(b, 0). To show that is sufficient to show that 0 $ (I - h(t)) d(T(tr), B(b, O), 0) = d(Vf01, B(b, 01, 01, it (dB(b, 0)) for all t E [0, 11. Suppose that x E dB(b, 0) and x = T(r)x for some r E [rt, to]. Then rx = &( -kCP(rx) Since rx E D(A),
+ x).
(3.7)
(3.7) can be written
rx + k/i(rx)
3 -kC(rx)
+x.
(3.8)
Therefore
3 inf{]]u +
C(tx)]]: z4EA(fx)}
2 inf{]]u + C(tx)]]: u EA(tx)} =
IA(tx)
- llfll
+ C(tx) I - i/f/i
2 allfxll + r’.
Thus we have (t - l)b/k
This is a contradiction.
2 arb + r’. While from (3.2), we have that
Hence
b(t - 1) h
b(to
abt + r’
abto+r’
we obtain
-
that
1) <
k
’
d(l-
r(t,),
B(b, 0), 0) = d(Z - T(to),
Some surjectivity
theorems
for compact
perturbations
of accrewe
operators
769
B(b, O),O) = 1. Since tl is arbitrary in (l,ro]. We obtain that d(Z - T(r). B(b, O),O) = d(1 - T(to), B(b, 0,O) = 1 for all t E (1, to]. Thus it follows that for each I E (1. fg], T(r) has a fixed point in B(b, 0). Hence there exists a set {~(:r E (1, to]} C B(b, 0) such that X, = T(t)x, for f E (1, rO]. Then for each r E (1, ro], we have that r.ri E D(A) and r-l --p
+ A(rx,)
+ C(rx,) 3f.
Then we have thatfE (A + C)D,(A). Since (A + C)D,(A) is closed, there exists a solution x E D(A) of the equation Ax + Cx 3 f and this completes the proof. Remark 1. In theorem 1, the condition that E is uniformly convex is needed only to guarantee that D(A) is a nonexpansive retract. Therefore it can be replaced by another condition which guarantees that D(A) is a nonexpansive retract (cf. [7]). Also we note that if C is defined on E, then the assumption on E can be removed. COROLLARY
1. Let E, A and C be as in theorem 1. Suppose that there exist positive constants
such that ,jh_m=(IAx + If (A + C)D,(A)
C.rI - 4-d> = m and li;:p
IIWll~ll < a.
(3.9)
is closed for all r > 0, then R(A + C) = E.
Remark 2. It is easy to see that (3.9) can be replaced by the following condition:
lim IAx + Cxi/jlxll = p jix:/+-X
and
liF’;p
IICxIl/]/x]l < 30.
(3.10)
THEOREM 2. Let E be a Banach space, A C E x E be an m-accretive operator and C: E ---, E be a compact mapping. Suppose that there exist positive constants 6 and r such that
if ]ix]l 2 6, If (A + C)D,(A)
then
inf
(y + Cx,j)/llxII 2’.
(3.11)
yEAx.jE&)
is closed, then B(r, 0) C R(A + C).
Proof. Let f E B(r, 0). We show that there exists x E E such that Ax + Cx 3 f. We fix a positive constant k. For t > 1, let S, be the mapping defined in the proof of theorem 1. For each f > 1, we define an operator r(t) by T(r)x = S,( -kC(tx)) forx E E. Then r(t) is compact for all t > 1. We claim that for each t > 1, T(t) has a fixed point in B(b, 0). To show this, it is sufficient to prove that if x E aB(b, 0), then x # AT(t)x for all J, > 1. Fix f E (1, =) and suppose that x E aB(b, 0) and 1x = T(r)x for some A > 1. Then we have that
k(t - 1)x + ky + kC(rx) = kf, for some y E A(tAx). such that
Let z E A(fx).
Then since y E A(h) (y - z, j) G=0.
(3.12) and A > 1, there exists j E J(x) (3.13)
770
From
N. HIRANO
(3.12)
and (3.13),
we have that kllfll I!xI/ 2 i(f - l)l~xll’ f k(_~ + C(rx), j) 2 rqt - 1)//x/l’ + k(z + C(fX).j) 2 A([ - l)!/xll? + krllf.Kll/f = qt
- 1) II,# + krlixll.
(3.14)
Since llfll G r and //xl/ = 6, (3.14) implies A(t - 1)b 6 0, and this is a contradiction. the argument in the proof COROLLARY
Therefore of theorem
T(r) has a fixed point in B(b. 0) for all t > 1. Then 1, the conclusion follows.
2. Let E, A and C be as in theorem
2. Suppose
lim inf (Y + Cx, X(-.X,VE&/EJ(X) If (A + C)D,(A)
by
that
i>lll4l = =.
(3. IS)
for all r > 0, then R(A + C) = E.
is closed
Some sufficient conditions for the set (A + C)D,(A) to be closed are known. (See theorem A of [4].) In the following theorem we do not need the assumption that the set (A + C)D,(A) be closed. 3. Let E and A be as in theorem 1. Let C: D(A) +E be continuous topology of E to the strong topology of E. Suppose that there exist positive and r which satisfy the condition (3.1). Then B(r, 0) C R(A + C).
THEOREM
from the weak constants c(, b,
Proof. Let fE B(r, 0) and fo, k, A, Jk, T(t) be as in the proof of theorem 1. We note that from the hypothesis, C is uniformly continuous on bounded subsets of D(A). Then by the proof of theorem 1, there exists a subset {x,: t E (1, fa]} of B(b, 0) such that txr E D(A) and xI = r(t)x, for each t E (1, ro]. Then since E is reflexive and {x,: t E (1, to]} is bounded subset of E, we can suppose, without loss of generality, that xi--x0 E B(b, 0), as f+ 1. Since r.u E D(A) for all I > 1, we have thatlim CP(rx,) =lim C(tr,) = Cxo. Now we define a mapping t-l
E + E by .5x = jk(-kCxo x,, we have that
+
S:
x)
r-1
for x E E. Then
= !Lyllxr- &( -kCxo r-1
+
=
/I
!\y
xI - fjk(-kC(trt) i $(
‘I
-kC(tx,)
!iy /I f &(-kC(tr,)
= 0.
From
the definition
of
Sxrll
!\m /IXI -
c hm
S is nonexpansive.
+ x,) /I + xf)
II
+ XJ - &( -kCxo + x,) - &( -kCxo
+ x,) /
+x,)
II (3.16)
theorems forcompact
Some surjectlvlty
Since I - S is demiclosed. completes the proof.
we obtain
perturbations
of accretive
that xg = jk(-kCxo
771
operators
+ xg). i.e. Ax,, + CXO3f and this
COROLLARY 3. Let E be a reflexive C: E -+ E be continuous that there exist positive R(A + C).
Banach space, A be an m-accretive operator and from the weak topology of E to the strong topology of E. Suppose constants b and r which satisfy the condition (3.11). Then B(r, 0) C
Next, we consider the case where the resolvent J;. of A is compact for some A > 0. It is well known that if A C E x E is m-accretive. then the following conditions are equivalent: JA is compact
for some A > 0;
(3.17)
J, is compact
for all 1. > 0;
(3.1s)
for each M > 0, {X E D(A)
: llxil -=cA4 and
l/y/l < M for some y E Ax} is relatively
compact.
(3.19)
THEOREM 4. Let E be a uniformly
convex Banach space and A C E x E be an m-accretive operator which satisfies the condition (3.17). Let C: D(A) +E be uniformly continuous on bounded subsets of D(A). Suppose that there exist positive constants a, b and r which satisfy the condition (3.1). Then B(r, 0) C R(A + C). Proof. Let f E B(r, 0) and let k, ro. A, j,, T(t) be as in the proof of theorem 1. From the condition (3.19), we can see that ji. is compact for all A > 0. Therefore T(t) is compact for r E (1, rO]. Then, by the same argument as in the proof of theorem 1, we have that there exists a set {xI:r E (1, ro3} such that fxf E D(A)x, E B(b, 0) and x, = T(t)x, for all t E (1, fo]. Here we define a mapping T by T.Y = jk(-kCx+ x) for x E E. Then T is a compact mapping. While from the definition of {xr}, we have that !\m l]xr - Tx,li = liim
]]xl - &( -kCPx,
s !\y
XI - +(-kCP(txr) /I
+
j
j,(
-kCP(tx,)
$(-kCP(rx,) Since C and P are uniformly
continuous
+ x,) jl
+ x,)
/I
+-x,) - &( -kCPx,
+x,) on bounded
-&(-kCPx, subsets
+ x,)
Ii
+ x,) 11. of D(A),
(3.20) implies
(3.20) that (3.21)
N. HIRASO
772
Then
since
T is compact,
x4-+x0 E E. Relation x0 E D(A), we obtain proof.
there
exists
a sequence
{x,} C{x,}
such
that
(3.21) implies now that xo = TXO= jk(--KPXO that x0 = Jk(-kCxo + x0), that is A& i CXO3fand
lim I, =1
and
+ XO). Then since this completes the
COROLLARY 4. Let E be a Banach space, A C E x E be a bounded, m-accretive operator which satisfies the condition (3.17) and let C be as in theorem 4. Suppose that there exist positive constants a, b and r which satisfy the condition (3.1). Then B(r, 0) C R(A + C). Proof. Since A is bounded and satisfies (3.17), we have that D(A) is boundedly compact. Then by proposition of 2.5 of [8], D(A) is a nonexpansive retact, that is, there exists a nonexpansive retraction P of E onto D(A). Then by the same argument as in the proof of theorem 4, the conclusion follows. COROLLARY 5. Let E be a Banach space, A C E x E be an m-accretive operator which satisfies the condition (3.17), and C:E - E be uniformly continuous on bounded subsets of E. Suppose that there exist positive constants 6 and r which satisfy the condition (3.11). Then B(r, 0) C R(A + C). 4. RESULTS In this section, we consider A and C are odd. LEMMA. Let E be a Banach D is a nonexpansive retract,
the solvability
Then
MAPPINGS
of the equation
(1.1) in the case that operators
space and D be a closed, convex and symmetric then there exists an odd nonexpansive retraction
Proof. Let P be a nonexpansive by
FOR ODD
retraction
of E onto
Qx = (Px - P(-x))/2,
D. We define
subset of E. If of E onto D.
a mapping
Q : E -+ D
for.r EE.
it is easy to see that Q is an odd, nonexpansive
retraction
of E onto D.
THEOREM 5. Let E be a uniformly convex Banach space, A C E x E be an odd. m-accretive operator and C:D(A) +E be an odd, compact mapping. Suppose that there exist positive constants a, b and r such that lAx + Cxl - aljxlj 3 r, If (A + C),(A)
is closed,
IIxIl2 b.
(4.1)
then B(r, 0) C R(A + C).
Proof. Let f E B(r, 0). We show that there exists x E E such that Ax + Cx 3f. Let P be an odd, nonexpansive retraction of E onto D(A) and choose positive constants to and k such that to > 1 and
b(fo- 1) <
k
abto + r’ where
r’ = r - l/fll. F or each t E (1, to], we define
a mapping
(4.2) T(t) as in the proof of theorem
Some surjectivitr theorems for compact perturbations
of accretive operators
773
1. For each s E [0, 11, we put A(s) = A - sf and J(s) = (I + kA(s))-‘. Here we fix r E (1, to] and for each s E [0, 11, define a mapping S(s) on E by S(s)x = u where u E E is the point such that u = (l/t)J( s ).r i u). Then we can see that for each s E [0, l], S(s) is Lipschitz continuous. Since A(0) is odd, we have that S(0) is also odd. For each s E [0,11. we now define a mapping H(s) by H(s) = S(s)(-kCP(rx)) for x E E. Then H(1) = T(r) and H(O) is odd because S(O), C and P are odd. Since S(s), P are Lipschitz continuous and C is compact, H(s) is compact for all s E [O, 11. We show that H is a homotopy of compact transformations. Let E> 0. Lets, s’ E [0, l] such that /s - ~‘1 6 e(t - l)/kr. Let x E E and put u1 = H(s)x and LL?= H(s’)x. Then there exists o1 E A(tul) and uz E A (Tut) such that tu1+ k(o1-
sf) = -kC(tx)
+ U1
tu2 + k(uz - s’f) = -kC(tx)
+ U?.
and
Then since A is accretive, we have that
Then we obtained that H is a homotopy of compact transformations. Next we show that or all s E [0, 11. Suppose that x E dB (b, 0) and x = H(s)x for some M(Z--H(s))@B@,O))f s E [0, 11. Then TX= J(s)(-kCP(fX) Since rx E D(A),
f x).
(4.4)
(4.4) can be written (t - 1)x + k(A(rx)
+ C(rx))
3 ksf.
Then by (4.1), we have ((t - l)/k)b
> arb + r - s j/f/l2 arb + r’
and this contradicts (4.2). Therefore we obtain that d(H(O), B(b, 0), 0) = d(H(l), B(b, 0), 0). Since H(0) is odd, by Borsuk’s theorem, d(H(O), B(b, 0), 0) is not zero. Then we obtain that H(1) = T(t) has a fixed point in B(b, 0). Therefore it follows that there exists a subset {q:t E (1, to]} of B(b, 0) such that xI = T(t)x, for all t E (1, to]. Then by the same argument in the proof of theorem 1, the conclusion follows. COROLLARY 6.
Let E, A and C be as in theorem 5. Assume that there exists a positive constant
a such that ,,z’;~=(IAx + Cx] - +]l) Then if (A + C)D,(A)
= =.
(4.5)
is closed for all r > 0, R(A + C) = E.
6. Let E be a uniformly convex Banach space, A C E x E be an odd, m-accretive operator which satisfies the condition (3.17), and C: D(A) *E be an odd, continuous mapping. Assume that there exist positive constants a, b and r which satisfy the condition (4.1). Then
THEOREM
B(r, 0) C R(A + C).
N. HIRA~O
774
Proof. The conclusion 1 and 5.
of theorem
6 follows
from the arguments
in the proof
of theorems
COROLLARY 7.Let E be a Banach space, A C E X E be an odd. bounded m-accretive operator which satisfies the condition (3.12), and C: D(A) +E be an odd, continuous mapping. Suppose that there exist positive constants u, b and r which satisfy the condition (4.1). Then B(r. 0) C R(A + C). Acknowledgemen+The suggestions and advice
author wishes to express his hearty thanks in the course of preparing the present paper.
to Professor
W. Takahashi
ior many
kind
REFERENCES 1. BARBU V., Sonlinear semigroups and differential equations in Banach spaces, Edirura Academiei R.S.R.. Bucuresti (1976). 2. BROWDER F. E.. LNonlinear operators and nonlinear equations of evolution in Banach spaces. Proc. Symp. pL[re Marh. 18 (1976). 3. FUCIK S., NECAS J., SOUCEK J. & SOUCEK V., Spectral Analysis of Nonlinear Operators. Lecture Notes in Mathematics. Springer-Verlag. Sew York (1973). 4. K.~RTSATOS A. G., Surjectivitv results for compact perturbations of m-accretive operators. J. marh. Analysis Appk. 78. l-16 (1980). 5. K.~RTSATOS X. G., Mapping theorems involving compact perturbations and compact resolvents of nonlinear operators in Banach spaces, /. .Ctarh. Analysis Appl. 80, 130-146 (1981). 6. KARTSMOS X. G., Some mapping theorems for accretive operators in Banach spaces, /. marh. .-lnalysis Appiic. 82, 169-183 (1981). 7. LLOYD iv. G., Degree Theory, Cambridgy Univ. Press, I\;ew York (1978). 8. REICH S., hsymptotic behavior of sermgroups of nonlinear contractions in Banach spaces, i. marh. Analysis Applic. 53. 277-290 (1976). 9. REICH S., Extension problems for accretive sets in Banach spaces, I. funcr. Analysis 26, 378-395 t 1977).