On the range of sums of accretive and continuous operators in Banach spaces

Nonlrneor

Anal.vsrs,

Theory,

Muhods

& A~ph,lrcor,ons.

Vol.

19,

No.

I, PP.

I-Y.

0362-%6X/92

1992.

$5.00

: 1992PergamonPm\

Printed m GreatBritain.

+ .OO Ltd

ON THE RANGE OF SUMS OF ACCRETIVE AND CONTINUOUS OPERATORS IN BANACH SPACES CLAUDIO Department

of Mathematics, (Received

Key words and phrases:

University

1 February

m-Accretive

H.

MORALES

of Alabama

in Huntsville,

1991; received for publication and demicontinuous

mappings,

Huntsville, 4 October uniformly

AL 35899, U.S.A. 1991) convex

Banach

spaces.

LET X BE A real Banach space and let X* be its dual space. An operator T: D(T) C X + 2x is said to be accretive if for each x, y E D(T) and for u E T(x) and u E T(y) there exists j E J(u - v) such that (U - u, j> L 0

where J:X + 2 .Y*is the normalized

duality

mapping

which is defined

JO4 = L/ E X*: = 1142,

Iljll =

by

114~

duality pairing. If in addition, the range of (see [4, 91). Here (*, -) denotes the generalized I + AT is precisely X for all 2 > 0, then T is said to be m-accretive. For this type of operators, the resolvent Jx = (I + AT)-’ is a single-valued nonexpansive mapping whose domain is all X. Also the set R(T) = ( y: y E T(x), x E D(T)] denotes the range of T. The purpose of this paper is to study the solvability of nonlinear functional equations of the type z E S(x) + T(x) (*) where T is generally a multivalued m-accretive operator (without any continuity assumptions) and S, an operator which satisfies some kind of sign condition. Equation (*) has been extensively studied for the last fifteen years. Perhaps, one of the earliest works of this nature, for Hilbert spaces, can be found in [l-3]. Nevertheless, the exploration for resolving the aforementioned problem has continued its development to more general types of operators in Banach spaces. Some of these results have been obtained by Browder [5], Gupta and Hess [7], Calvert and Gupta [6] and Reich [12]. Our main objective here is to establish some new results along with extensions of some others already known. It seems to be important to mention that these result can be applied to nonlinear elliptic boundary value problems and Hammerstein integral equations (see [6, lo]). Throughout this paper, we use A and int(A) to denote, respectively, the closure and the interior of a subset A of X and we also use B(x; r) to denote the open ball centered at x E X with radius r > 0. Occasionally, we use the notation IA / = inf (llxll : x E A) for an arbitrary subset A of X. Before we establish our results, we need some basic definitions. Let J: X + 2x* be the normalized duality mapping. We say that J satisfies condition (Z) (see [6]) if there exists a function @: X + [0, 00) such that for U, v E X, sup(lIj

- j*ll : j E J(u), j* E J(v)) 5 @(u - u).

2

We should mapping

C.

mention

that

H. MORALES

if X = LP(sZ) with

Q a bounded

subset

of IRN, then

the duality

(where l/p + l/q = 1 and 2 5 p < 00) defined by J(U) = IuI’-‘sgn ~jlull~-~, satisfies condition (I) (see [6]). A mapping T: D(T) C X -+ X is said to be bounded if it maps bounded subsets of D(T) into bounded subsets of X and compact if it is continuous and maps bounded subsets of D(T) into relatively compact subsets of X. The operator T is said to be demicontinuous if it is continuous from the strong topology of X to the weak topology of X and completely continuous if u, , u E D(T) and U, --t u weakly imply that TM,, + Tu strongly. Finally the operator T is said to be demiclosed if u, E D(T) with u, + u E X and Tu, + u E X weakly imply that u E D(T) and Tu = u. In the sequel, we need a result for arbitrary Banach spaces which is a particular case of [5, lemma 21. LEMMA 1. Let X be a Banach space and let J: X + 2x* be the normalized duality mapping. Let (u,) be a sequence in X and (an) a sequence of positive numbers converging to zero as n + co. Let r > 0 and suppose that for every w E X with IIwII I r there exists a constant C, such that

for all n E tN and somej, E J(u,). Then the sequence (u,) is bounded. We should mention that theorem 1 below extends related results of Calvert [6, theorem 2.11, Kartsatos [8, theorems 1 and 31, and Reich [12, proposition apparently wider family of Banach spaces.

and Gupta 1.131 to an

THEOREM 1. Let X be a Banach space whose duality mapping satisfies condition (I) and let T: D(T) c X + 2x be m-accretive with (I + T)-’ compact. Suppose g: D(T) -+ X is a continuous and bounded mapping for which there exists a function y: IR+ + IR+ with y(r) + 0 as r -+ COsatisfying the following: for every w E D(T) there exists a constant C(w) > 0 such that (g(u),j> for all u E D(T)

2 -C(w)

(1)

- Y(ll4l>ll~ll

c R(T + g) and int(R(T))

and j E J(u - w). Then R(T)

C R(T + g).

= z - g(T + (l/n)l)-‘(x). Proof. Let z E X. Then define the operator A ,,: X + X by/l,,(x) We first show that A, has a fixed point. To see this, we prove that the set E(n) = lx E X: A,(x) To this end, let x E E(n).

= Ax for some A > 1) is bounded.

Then Ax + g(T + (l/n)l)-‘(x)

for some A > 1. By letting

u = (T + (l/n)Z)-‘(x), (AU + (Un)u

for some j E J(u - w). Then (l/n)llu

- 4*

= z

we may choose

+ g(u),j>

u E T(u) such that

= (z,j>

for y E T(w), we have

5 IIIYII + Wn)llwll

+ llzlllIl~ - 4

+ C(w) + Y(ll4)ll4~

Accretive

which implies so that

that

E(n)

is bounded.

and continuous

Consequently,

z E T(u,)

+ g(u,)

First of all, let z E T(W) for some w E D(T). so that (u, - z,j> + (g(u,),j> which yields to (l/n)llu,

- wl12 5 wdwll

3

operators

for each n E N there

exists

u,, E D(T)

+ (l/n)u,.

(2)

Then we choose + (l/n)(u,,j>

u, E T(u,) and j E .Z(u, - w) = 0,

II% - WII + C(w) + Y(ll~nIl)ll~nII.

Therefore (l/n)u, -+ 0 as n + co. This implies that z E R(T + g). Secondly, let z E int(R(T)). Then there exists r > 0 such that z + h E int(R(7’)) for every h E B(0; r). Let z + h E T(w,) while z satisfies (2). Then by choosing u,, E T(u,), we may write z = v, + g(u,) This combined

+ (l/n&.

with (1) implies (v, - (z + h),j,)

+

+ (l/n)(u,,j,)

= -(kj,)

and (kj,) for some j, E J(u,

5 (1/4n)llwhl12 + C(w,) + v(ll~,ll)ll%Ill

- w,). Now by choosing (h,j,*)

= M,j,)

+ (h,j,*

5 w4)llw,l12

j,* E .Z(U,) and using condition

(I) we obtain

-j,>

+ C(w,) + IPll@(%)

+ r(ll%J)ll~nII.

If we let o(, = ~(llu,ll) and C, = (1/4)11~,1/~ + C(w,) + llhll@(w,,), we may use lemma 1 to derive that (u,) is bounded. Thus, the boundedness of g, the compactness of (I + T)-’ and the fact that u, = (I + T - z)-‘[(l - l/n)u, - g(u,)], allows us to select a convergent subsequence (u,,) of (u,), say, u,+ -+ u E D(T). (I + T - z)-’ are both continuous, we obtain that u E D(T) and

Since g and

z E T(u) + g(u), which completes

the proof.

Our next theorem involves the same type of conclusions as in theorem 1. Here, we relax the continuity assumption on g by imposing an extra condition on the space X. THEOREM 2. Let X be a reflexive Banach space whose duality mapping satisfies condition (I) and let T: D(T) c X + 2x be m-accretive with (I + T)-’ completely continuous. Suppose g is a demicontinuous and bounded mapping from X into X which satisfies (1). Then Z?(T) c R( T + g) and int(R( T)) C R( T + g). Proof. Let w0 E D(T). Then by replacing T(x) by T(x + wo), g(x) by g(x + wO) and D(T) that 0 E D(T). Consequently, by D(T) - wet we may assume without loss of generality, assumption (1) may be written as follows: for each w E D(T) there exists a constant C(w) > 0 such that k(u),j) 2 -C(w) - ytllu + woll)ll~ + wall (3)

C. H. MORALES

4

for all u E D(T) and j E .Z(U - w). Now, let z E X and let A,: X -+ X be defined A,(x) = (T + (l/n)Z)-‘(z - g(x)). Since A, is a compact operator, it would be sufficient show that the set E(n) = (x E X:A,(x) = Zx for some L > 1)

by to

is bounded. To see this, let (T + (l/n)Z)-‘(z - g(x)) = Ax for some A > 1. This implies that z = u + g(x) + @/n)x for some u E r(nx), and by choosing j E J&Y) and w = 0 in (3), we get (U + g(x) + (A/n)x,j)

= GLj>

and

which implies

that

Therefore E(n) must be bounded say U, so that

for each n E M. This means the operator

A, has a fixed point,

2 = U, + g@,) + (l/n&

(4)

for some u, E T(u,). Then the first part of the conclusion follows now as in the proof of theorem 1. Secondly, if z E int(R(T)), for every 12E B(O; r) and some r > 0, we may suppose z f h E T(+,) for some wh E D(T), while z satisfies (4). Then, for j, E J(u, - w,) we have (v, - (2 + h),j,)

+ (g(u,),j,,)

+ (l/n)(u,,j,)

= -(h,j,)

and (h,j,) If we choose j,* E J(u,),

5 (1~4~)11%12 + C(w,) condition

+ VdlKI + wclll)ll~,+

wall.

(I) implies

5 U/4)IIwhl12+ CC%) + llfm(wh) + Y(Il% + wdl)Il% + wall.

(h,j,*)

If ]Iu~II + a3 as n -+ 00, we select a constant ch

z

(1/4)11

Wh112

+

C(w,)

+

II~~ll~(W/J+ Y(lh, + woll)llw,ll

and a!, = ~(llu, + wOJI) such that (h, j,*> 5 cyIIu,,II + C, for all n E N. Therefore implies that (u,) is bounded. Also from (4) U, = (T + I)-‘[(1

- l/n)!.& + z - g(u,l)],

lemma

1

(5)

and since ((1 - l/n)u, + z - g(u,)) is bounded, we may select a subsequence (u,,] of lu,f such that (1 - l/nk)unk + z - g(u,,) -+ u weakly as k -+ M. Then, since (T + I)-’ is completely continuous, unk -+ (T + Z)-‘(u) strongly as k -+ 00. Therefore, by letting u = (T + Z)-‘(u), equation (5) implies that z E T(U) + g(u). We should mention that theorem 2 improves theorem 4 of Kartsatos [8], however, we should also mention that this proof seems to be using the fact that if u E D(S) and A E (0, l), then ,?ZJE D(S), where D(S) denotes the domain of the operator S. This is not necessarily true under the assumptions of Kartsatos. Before we state our next result we need the following definition. We say that an operator T: D(T) c X -+ 2x satisfies condition (I) if there exists a function y: iR+ + Rt with y(r) + 0 as r + COsuch that: for every t E R(T) and every a E D(T) there exists a constant C(n, z) > 0

Accretive and continuous operators

so that (u - z,j> for u E D(T),

2 -C(U)

-

rcll4l)ll4

u E T(U) and j E J(u - a).

The following theorem involves the notion of “almost” equal sets, in the sense that they have the same closures and interiors. This means, for two subsets A, and A, of X we write A 1 = A2 of cl(A,) = cl(A,) and int(A,) = int(A,). We begin by stating a lemma of Reich [12]. LEMMA 2. (cf. [12, lemma

closed

1 .l].) Let X be a Banach space for which each nonempty bounded subset has the fixed point property for nonexpensive self-mapping and let c X -+ 2x be m-accretive. Suppose v, E Tu, where (u,) is bounded and v, -+ v. Then

convex

T: D(T)

u E R(T). THEOREM 3. Let X be a Banach

space whose duality mapping satisfies condition (I) and each nonempty bounded closed convex subset has the fixed point property for nonexpansive selfmappings. Suppose the operators S: D(S) c X -+ 2x and T: D(T) C X + 2x are accretive such that S + T is m-accretive. If either S or T satisfies condition (I-) with D(T) C D(S) or D(S) c D(T), respectively, then R(S + T) = R(S) + R(T). Proof. (i) Suppose T satisfies condition (I) and let z E R(S) + Z?(T). Since S + T is m-accretive, for each n E N there exists u, E D(S) fl D(T) such that z E SU, + (l/n)u,. Therefore, we may choose u0 E R(S), v, E Su,, w. E R(T) and w, E Tu,, for n E N, so that

ug + wg = u, + w, + (l/n&. We now select u0 E D(S) so that u0 E Su,. Then in view of our assumptions, positive constant C(u,, z) and a function y for which (-l/n)
= (v, - ug + w, - w,,j,> 2 (v, - vo,j,> 2 -C(uo,

where j, E J(u,

there exists a

z) -

+ (w, - wo,j,>

Y(ll~,ll)ll4ll

- uo). This yields

w411kI - uol12 5 a%l,4 + (l/n)lluollII&I - uoll+ Y(ll&711)II~,II which implies that (l/n)u, + 0 as n -+ 03. Consequently, (ii) Let z E int(R(S) + R(T)). Then there exists r > 0 every h E &O; r). Since S + T is m-accretive, there z E SU, + Tu, + (l/n)u, . This means, we may choose u, z = u, + w, + (l/n)u,

z E R(S + T). such that z + h E R(S) + R(T) for exists u, E D(S) fl D(T) so that E Su, and w, E Tu, such that

for n E N.

On the other hand, for each h E &O; r) we may choose v,(h) E R(S), w,(h) E R(T) for which z + h = v,(h) + w,(h). Now, in view of the assumptions on S and T, we select u,(h) E D(S) with u,(h) E S,(h) for which there exists a positive constant C(h) and a function y satisfying (u, + w, - (uo(h) + wo(h)),j,)

2 -C(h)

-

~(ll~,ll>ll4l

C. H. MORALES

6 where j, E J(u,

- u,(h)).

Therefore, ((-l/n)u,

- h,j,)

2 -C(h)

-

Y(ll%J)ll~,ll

and thus 5 U~4~)ll~,(~)l12 + C(h) + Y(ll%zll)llKJl.

(k.iJ

Now the proof lemma

of theorem 1 carries 2 completes the proof.

over.

This

means

(u,) is bounded,

and consequently

Theorem 3 remains to be true if the assumption of S satisfying condition (I) with D(T) c D(S) is replaced by assuming that both, S and T, satisfy condition (I) without restriction on their domains. In view of this, theorem 3 extends [5, theorem 3; 12, theorems 1.4 and 1.7; 6, theorem 1.11. PROPOSITION 1. Let X be a Banach space, let T: D(T) c X ---* X be a bounded, demicontinuous and m-accretive operator and let S :D(S) c D(T) + X be demiclosed so that (I + S)-’ exists and is compact with the property: there exists p: IR+ -+ R+ with p(r) -+ 0 as r + co such that for every v E D(S) there exists a constant C(v) > 0 satisfying

Gu

- Su,j)

for u E D(S) and j E J(u - u). Then R(T Proof.

- P(ll~ll)ll~ll

2 -C(u)

(6)

+ S + ~1) = X for every E > 0.

Let z E X and let n E N. We first solve the approximating

problem

TJ,,u f Su + EU,, = z where J, = (I + (l/n)T)-‘. strongly accretive operator

To this end, let U,, = TJ,,+ &I. Since defined on X, we define A,: X -+ X by A,(x)

for a fixed n E N. As before,

= (S + I)-‘(z

it remains

To see this, let x E E(n).

U,, is a continuous

and

+ x - U,(x))

to show that the set

E(n) = lx E X:4,(x) is bounded.

(7)

= Ax for some /I > 1)

Then

(A - 1) + S(Ax) + U,(x) = z for some A > 1. Choose

u E D(S) and let u = Ax. Then

(1 - 1-%2&j> In view of the assumptions

+ (SU,j)

for j E J(u - u) we have,

+

= (z,j>.

on S and U,, we derive

(1 + A-Ye - l))llu - ul125 [llzll + lldl + llsull + II~,(~-‘~)ll1ll~ - UII

+ C(u) + P(ll~ll)ll~ll. Since the set (U,(K’v): A > 1) is bounded, so is E(n). This means, A, satisfies the LeraySchauder condition on the boundary of some closed ball and thus has a fixed point in X (e.g. see [l 11). Therefore equation (7) has a solution for each n E IN and E > 0. That is TJ,,u,

+ Su, •t EU, = z.

1

Accretive and continuous operators

An argument similar to the one used for the boundedness of E(n), shows that the sequence itself is bounded. Since ( Un(un)) is also bounded, it follows from the equation U, = (S + I)-‘[z

(u,)

+ U, - Un(Un)lr

and the compactness of (S + I)-’ that (u,) may be assumed to be convergent, say, to u0 E D(S). The fact that llJn~n - uOll % Iju, - ~~11+ lIZ,u, - uOll also implies that J,,u,, + u,, as n + co. This combined with the assumptions on S and T completes the proof. A restriction in the space X with a relaxation us to obtain the following proposition.

of the demicontinuity

of the operator

T, allows

PROPOSITION 2. Let X* be a uniformly convex space and let S be as in proposition T: D(T) c X + X is bounded and m-accretive, then R(T + S + EZ) = X.

1. If

Proof. We should just mention that if X* is uniformly convex and U, -+ u0 with (TJ,(u,)) bounded, then u0 E D(T) and TJn(un) -+ TuO weakly. This is sufficient to carry over the proof of proposition 1.

convex, let T: D(T) C X + X be bounded and m-accretive. Suppose the operator S satisfies the assumptions of proposition 1. Further assume that there exist constants C, , C, > 0 and k E (0, 1) such that THEOREM 4. Let X* be uniformly

(Su, J(Tu))

2 -kllTull*

-

C,IITull -

C,

(8)

for u E D(S + T). Then Z?(S + T) = Z?(S) + R(T). Proof.

We know from proposition

2 that the equation

Su, + Tu, + (l/n)u,

= z

(9)

has a solution for each n E Kl. We first assume z E Z?(S) + Z?(T). Then there exist u,, E D(S) and u0 E D(T) so that SU, + Tu, + (l/n)u, = Su, + TV,. Now the accretiveness of T and condition (6) on S imply that

(l/Nil% - uol12 5 11% + (l/n)u, - Q/i llu, - uoll+ C(u,)+ P(llu,ll)llu,ll, which concludes that the sequence equation (9) we obtain

((l/n)u,]

(1 - k)llTu,? This means (Tu,] is actually on S and T we may derive

bounded,

is bounded.

5 [llz - (l/n)u,ll

On the other hand,

+

C,lllTu,I/ +

and so is the sequence

if we apply (8) to

C2.

(Su,]. Now using the assumptions

(l/~)llunII5 ~kJ~ll~,II + Ptll~,ll) + llSu, - %ll IIJ(u,/liu,li)- J(dd

- udtu,II)II

+ II%, - Td IIJCu,/llu,ll> - J(u,/IidI - Q,/IIu,I/)II. This implies

that (l/n)u,

+ 0 as n -+ 00 and thus z E R(S + T).

C. H. MORALES

8

For the second part, we first refer to a property ~:I?+ --+ R+ by flu(f) = suPMx

of the duality

+ X0) - 4.4

:

mapping

J. Define

a function

llxll 5 1, llxoll5 f1.

The function P(I) is nondecreasing in t and ~(0) = 0. Due to the uniform continuity of J on bounded sets, p(t) is continuous at 0. Furthermore, p(t) is well-defined for all t > 0 in virtue of the positive homogeneity of J. Therefore

ILO + x0) - J(x)II 5 k4lhA for llxll I 1. We now assume that z E int(R(S) + R(T)). Then there exists r > 0 so that z + h E R(S) + R(T) for every h E &O; r). From proposition 2, we may select u, E D(S) such that for n E N. z = SU, + Tu, + (l/n)u, Also, for each h E L?(O;r), there exist u,(h) E D(S) and v,(h) E D(T) Su,(h) + Tv,(h). Then in view of the assumptions on S and T, we have (h, 44)

5 -
- G,(h),

5 C@,(h)) + (Tu,

J(u,)>

+ P(lln,ll)ilu,li - To,(h), J(u,

- (Tu,

- Tv,(h), J(u,))

+ (Su,

- Su,(h),

J(u,

such

- n,(h))

that

z + h =

- J(u,)>

- v,(h)) - J(u,)),

yielding (h, J(u,)>

5 C@,(h))

+ [P(llu,il)

+ /ITn, - TGh)ll This implies

+ IISu, - Sn,(h)II IIJ(un/lln,II IIJ(un/llu,II

- tU)/llu,II)

- u,(h)/llu,II)

- J(n,~/llu,I/)II

J(u,/llu,ll)lllll~nli.

-

that

(h> J(u,)>

5 C(n,(h))

+ [P(lln,ll)

where Dh = sup IISu, - Su,(h)II

+ Dh~(I/uo(h)ll/IIunII)

+

and Eh = sup IlTu, - Tv,(h)ll.

Suppose tha; (u,) is not a bounded sequen:e. Then without that IIu,\I + co. For each k E IN, we define the set Sk = Vr E &O; 4: l(h,

Eh~c1(llvo(h)ll/IIu,ll)lIIu,ll

loss of generality,

we may assume

~(u,))l I k + [~(llu,ll)+ 2k~(k/j(u,ll)lllu,II, for all ~1.

Then each S, is a closed subset of&O; r) @O; r). Therefore, by the Baire category This means, we may choose h, E X and Let w be an arbitrary element in &O;

and the union of all of them is precisely the closed ball principle, there exists k, E N such that int(S,,,) # 0. 6 > 0 so that B(h,; 6) is contained in Sk,. 6). Then it can be easily seen that

(~3 J(u,))

2 2k, + ~(llu,ll)lln,ll

where y(llu,,ll) = 2[/3(llu,II) + 2k,p(k,/llu,,11)] for all n E N. Since y(llu,,II) + 0 as h + to, lemma 1 implies that the sequence (u,) is bounded. This means, the equation u, = (S + I)-‘(2

+ (1 - l/n)u,

- Tu,)

and the compactness of (S + I)-’ allow us to assume that u, + u0 E D(S). Also Su, + Tu, + z as n -+ 00 and S + T is a closed operator. Then u,, E D(S) and z E R(S + T).

Accretive

and continuous

operators

Note added inproof. Professor Kartsatos has mentioned to the author in the proof) if (1) holds with u E X instead of u E D(S).

that theorem

9 4 in [8] is true (without

any changes

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