Zeros of operators in Banach spaces

Appl. Math. Lett. Vol. 5, No. 1, pp. 55-57, 1992 Printed in Great Britain. All rights reserved

ZEROS

0893-9659192 $5.00 + 0.00 Copyright@ 1992 Pergamon Press plc

OF OPERATORS

IN BANACH

JONG-SHENQ

SPACES

Guo

Institute of Applied Mathematics National Tsing Hua University, Hsinchu, Taiwan 30043, R.O.C. JEN-CHIH

YAO

Department of Applied Mathematics National Sun Yat-sen University, Taiwan 80424, R.O.C.

(Received July 1991) Abstract-In this paper, we derive certain conditions under which an operator from a Banach space to its topological conjugate space possesses a zero. We also obtain some surjectivity results for such operators.

Let B be a real reflexive Banach space, B* its topological conjugate space endowed with the weak-star topology and (u, V) the paring between u E B and w E B*. Let T be an operator from B into B*. As has been noted by K. Deimling in [l], knowledge about such operators gives more flexibility, for example, in the study of weak solutions of differential equations and integral equations, etc. The operator T is said to be continuotis on finite-dimensional subspaces if it is continuous on every finite-dimensional subspace U of B. For any subspace M of B, let PM denote the injection of M into B and P& be the dual of PM. We use B,(z) to denote the ball with radius r and center at x. For any subset D of B, D denotes the closure of D. We now state and prove the first result of this paper. THEOREM

1.

Suppose

that

the following

conditions

are satisfied:

(i) T is continuous on finite-dimensional subspaces, (ii) for each {x~} weakly convergent to 2, liim_izf(y,Tx,) (iii) (x - y, TX - Ty) # 0 for all x # y, (iv) there exists 10 E B such that kzy>i (v)

Then

for any finite-dimensionalsubspace X E M.

[(Q,Tx)I

5 (y,Tx)

for all y E B,

> 0,

M of B containing

x0, jIT;cl~jl

-+ 00 as ~~x~~ ---t 00 and

there exists x E B such that TX = 0.

PROOF.

Let A be the family of all ordered by inclusion. For each M dimM < co, we may without loss of identify M* with M. For any x, y E

subspaces of B containing xc partially : M + M* is continuous. Since generality assume that M is a Euclidean space and we can M with x # y, we have finite-dimensional

E A, TM =

1(x - Y, TM~ Hence TM is one-toone, and therefore condition (V). Consequently, TM is onto Let BM = {XV : M C V E A} and let of sets {WC~BM : M E A} has the finite M E A be such that U U V C M. Then

TnnY)I

=

P&TPM

1(x -

Y, Tx - TY)I > 0.

is open by [l, Theorem 4.31. But TM is also closed by M* and we have a unique 2~ E M such that TMXM = 0. WCIBM denote the weak closure of BM. Then the family intersection property. Indeed, for U, V E A we can let 0 # WC~BM C wclBu n wclBv.

This work was partially supported by the National Science Council grants NSC 80-0208-M-007-09 0415-E155-07.

and NSC SO-

Typeset by d,+#-Q$

55

YAO

J.-S. Guo, J.-C.

56

For each M E A, since TMXM = 0 we have O=

I(xo,TMzM)I

=

l(~o,T~.w)I.

Therefore, by condition (iv), there exists T > 0 independent to M E A such that_ ]]xM]] 5 r for all M E A. Consequently, WC~BM c Bp(0) f or all M E A. Since B is reflexive, B,(O) is weakly compact. It follows that nME,,WCIBM # 0. Let x E nMEAwclBM. For any u E B, fix M E A such that x, u E M. Since x E WC~BM, by Alaoglu’s Theorem, there is a sequence {xCn} E BM such that x, converges to x weakly. Let M, E A be such that x, E M,,. Since TM,x, = 0 for all n, we have, by condition (ii), 0 = llnm_iEf(u - x, TM, xn ) = lirrizf(u

- x, TX,)

5 (u - x, Tx).

Consequently,

(u - x,Tx)

If the condition result.

2 0 for all u E B. Hence TX = 0 and the result

(iv) of Theorem

COROLLARY 2. Suppose

1 is strengthened,

that the following

then

conditions

I surjectivity

are satisfied:

(i) T is continuous on fInitedimensional subspaces, (ii) for each (2”) weakly convergent to x, linm_izf(y,Tz,) (iii) (iv)

follows.

we have the following

5 (y,Tx)

for all y E B,

(x - y, TX - Ty) # 0 for all x # y, there exist xc E B and ,8 > 0 such that ~~in&nr ~(x~,Tt)~/~~~~~~ > 0.

Then T is onto B’. Other results related to Corollary 2 are the following: [2, Chap. 1, Theorem 4.31 which that if T is maximal monotone and coercive, then T is onto B’; [l, Theorem 12.11 which that if T is hemicontinuous, monotone and coercive, then T is onto B’; [l, Corollary 12.11 states that if T is hemicontinuous, monotone and IITx/l-t 03 as ~~x~~ ---+co, then T is onto Next, we have: COROLLARY 3. Suppose (i) for each sequence

that

the following

conditions

{xn} weakly convergent lin&f(y,Tx,)

states states which B’.

are satisfied:

to x, then 5 (y,Tx),

Y E B,

(ii) the function x I+ (x, TX) is sequentially weakly lower semicontinuous (iii) (x - y, TX - Ty) # 0 for all x # y, (iv) there exist xc E B and /3 > 0 such that ;~II>: ~(x~,Tx)~/~~x~~~ > 0.

on B,

Then T is onto B. PROOF. It follows from [3, Theorem 21 that any operator T satisfying (i) and (ii) must be The result then follows from Corolnecessarily continuous on finite-dimensional subspaces. lary 2. I We note that the operator to employ the open mapping not be locally one-to-one and variational inequality theory THEOREM 4. Suppose

that

T above is assumed to be one-to-one or locally one-to-one in order theorem to obtain the above results. In many situations, T need the above argument is not applicable. For these cases, we use the to derive the following results. the following

conditions

are satisfied:

(i) T is continuous on finite-dimensional subspaces, (ii) for each {xn} weakly convergent to x, liiiizf(y, (iii)

there exist x0 E B such that ,,zlii~

Then T is onto B’.

TX,) 5 (y, TX) for all y E B,

(x - x0, TX - Txo)/ljx - x01) = co.

Zerosin Banach PROOF. Let A inclusion. without For any

57

spaces

Again, it suffices to show that 0 E TB. be the family of all finite-dimensional subspaces of B containing to partially ordered by For each M E A, TM = P&TPM : M -+ M* is continuous. Since dimM < 00, we may loss of generality assume that M is a Euclidean space and we can identify M’ with M. x E M, we have

(x--o,T~x--TMXO)

= (a:-xo,Tx-Txo)

as II41- 00.

112 - x011

11x - xoll Therefore, for all x E Let BM the family For each

loo

by [4, Chap. I, Corollary 4.31, there exists XM E M such that (x - ZM,TMXM) 2 0 M. Consequently, TMXM = 0. = {XV : M C V E A} and let WC~BM denote the weak closure of BM. Then clearly of sets {wc~BM : M E A} has the finite intersection property. M E A, since TMXM = 0 we have (xM

-

xo, TM~M

-

T~xo)

(xM

=

IIXM- xoll

-

xor Txo)

1jxM -

I

x011

IPoll~

Therefore, by condition (iii), there exists T > 0 independent to M E A such that IIxMII 5 T B,(O) is for all M E A. Consequently, WC~BM C BP(O) f or all M E A. Since B is reflexive, weakly compact. It follows that nMEA WC~BM # 0. Hence the result follows as in the proof of Theorem 1. I The following

corollary

is a direct consequence

COROLLARY 5. Suppose

that

the following

of Theorem

conditions

4.

are satisfied:

(i) for each {+} convergent weakly to z, {TX,} converges to TX in the weak-star (ii) there exist x0 E B such that ,,$f~oo(x - xo,Tx - Txo)/llx - ~011= 00.

topology,

Then T is onto B*. REFERENCES 1.

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