SMOOTH BANACH SPACES

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ITERATIVE SOLUTIONS OF K-POSITIVE DEFINITE OPERATOR EQUATIONS IN REAL UNIFORMLY

SMOOTH BANACH SPACES

ZEQING LIU, SHIN MIN KANG, and JEONG SHEOK UME (Received 2 October 2000)

Abstract.Let Xbe a real uniformly smooth Banach space and let T :D(T )⊆X→X be aK-positive deﬁnite operator. Under suitable conditions we establish that the iterative method by Bai (1999) converges strongly to the unique solution of the equationT x=f,f∈ X. The results presented in this paper generalize the corresponding results of Bai (1999), Chidume and Aneke (1993), and Chidume and Osilike (1997).

2000 Mathematics Subject Classiﬁcation. 47H06, 47H07, 47H14.

1. Introduction and preliminaries. LetXbe a real Banach space with a dual space X^∗. The normalized duality mappingJ:X→2^X^∗ is deﬁned by

J(x)=

f∈X^∗:x, f = x²= f²

, x∈X. (1.1)

It is known thatXis uniformly smooth (equivalently,X^∗is uniformly convex) if and only ifJis single-valued and uniformly continuous on any bounded subset ofX.

Chidume and Aneke [3] introduced the concept ofK-positive deﬁnite operators and established the existence of the unique solution of the equationT x=ffor that operator in real separable Banach spaces. Meanwhile they constructed, inLp(orlp) spaces withp≥2, an iteration method which converges strongly to the unique solution, pro- vided thatT andKcommute. Chidume and Osilike [5] gave a new iteration scheme, in separableq-uniformly smooth Banach spaces, which converges strongly to the unique solution of the equationT x=f,f∈X.

Recently, Bai [1] constructed a more general iteration procedure and improved the results of [3,5] to separable uniformly smooth real Banach spaces.

Very recently, Zhou et al. [7] established the following excellent result, which is a generalization of the main result of Chidume and Aneke [3].

Lemma1.1(see [7]). LetXbe a real Banach space and letT be aK-positive deﬁnite operator withD(T )=D(K). Then there exists a constantα >0such that

T x ≤αKx, x∈D(T ). (1.2) Moreover, the operatorT is closed,R(T )=X, and the equationT x=ffor eachf∈X, has a unique solution.

The purpose of this paper is to study the convergence problem of the iteration procedure introduced in [1] forK-positive deﬁnite operators in real uniformly smooth real

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Banach spaces. Our results extend the corresponding results due to Bai [1], Chidume and Aneke [3], and Chidume and Osilike [5].

In what follows, we will also need the following concepts and results.

Deﬁnition1.2(see [3, 7]). LetX be a real Banach space andX1 a subspace of X. An operatorT with domain D(T )⊇X1 is calledcontinuously X1-invertibleif T, as an operator restricted toX1, has a bounded inverse onR(T ). A linear unbounded operatorT with domainD(T )inXand rangeR(T )inXis calledK-positive deﬁniteif there exist a continuouslyD(T )-invertible closed linear operatorKwithD(A)⊆D(K) and a constantc >0 such that

T u, j(Ku)

≥cKu², u∈D(T ), j(Ku)∈J(Ku). (1.3) Let X be a real Banach space. Recall that the modulus of smoothness of X is deﬁned by

ρX(t)=sup 1

2

x+y+x−y

−1 :x, y∈X, x =1,y ≤t

, t≥0. (1.4) Xis said to beuniformly smoothif lim_t→0ρX(t)/t=0. Letp >1 be a real number.X is calledp-uniformly smoothif there exists a constantr >0 such that

ρX(t)≤r t^p, t >0. (1.5) Hilbert spaces,Lp(orlp) spaces, 1< p <∞, and the Sobolev spacesW_p^m, 1< p <∞, are allp-uniformly smooth. It is well known that the class ofp-uniformly smooth real Banach spaces is a proper subclass of that of uniformly smooth real ones.

Lemma1.3(see [4,6]). LetXbe a real uniformly smooth Banach space. Then (i) there exist some positive constantsAandBsuch that

x+y²≤ x²+2

y, J(x)

+Amax

x+y, B ρX

y

, x, y∈X. (1.6) (ii) there exists a continuous nondecreasing functionb:[0,∞)→[0,∞)such that

b(0)=0, b(ct)≤cb(t), c≥1;

x+y²≤ x²+2

y, J(x) +max

x,1

yb(y), x, y∈X. (1.7) Lemma1.4(see [2]). Suppose that{αn}^∞n=0,{βn}^∞n=0, and{ωn}^∞n=0are nonnegative sequences such that

α_n+1≤ 1−ωn

αn+βnωn, n≥0, (1.8)

with{ωn}^∞_n=0⊂[0,1], ^∞_n=0ωn= ∞andlimn→∞βn=0. Thenlimn→0αn=0.

Lemma1.5(see [6]). LetXbe a real Banach space. Then (i) ρX(0)=0,ρX(t)≤t,t >0;

(ii) ρX(t)is convex, continuous, and nondecreasing on[0,∞);

(iii) ρX(t)/tis nondecreasing on(0,∞).

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2. Main results

Theorem2.1. LetXbe a real uniformly smooth Banach space and letT :D(T )⊆ X→Xbe aK-positive deﬁnite operator withD(T )=D(K). Deﬁne a sequence{xn}^∞n=0

iteratively from anyf∈Xandx0∈D(T )by

yn=xn+bnvn, xn+1=yn+anun, n≥0; (2.1) vn=K⁻¹f−K⁻¹T xn, un=K⁻¹f−K⁻¹T yn, n≥0, (2.2) where{an}^∞_n=0and{bn}^∞_n=0are arbitrary nonnegative sequences such that

∞ n=0

an+bn

= ∞; (2.3)

n→∞liman=lim

n→∞bn=0; (2.4)

max an, bn

≤ 1

2c, n≥0; (2.5)

αAmax

1+αanKv0,

1+αbnKv0, B

≤2cKv0, n≥0, (2.6) wherec, α, AandBare the constants appearing in (1.2), (1.3), and (1.6), respectively.

Then the sequence{xn}^∞n=0converges strongly to the unique solution of the equation T x=f.

Proof. It follows fromLemma 1.1that the equationT x=fhas a unique solution inX. Note thatT andKare linear. From (2.1) and (2.2) we have

Kvn+1=f−T xn+1=Kun−anT un, n≥0; (2.7) Kun=f−T yn=Kvn−bnT vn, n≥0. (2.8) In view of (2.8) and (1.2), (1.3), and (1.6), we conclude that

Kun²=Kvn−bnT vn²

≤Kvn²−2bn

T vn, J Kvn

+AmaxKvn+bnT vn, B

ρX

bnT vn

≤

1−2cbnKvn² +Amax

1+αbnKvn, B ρX

αbnKvn

(2.9)

for alln≥0. Using (2.7) and (1.2), (1.3), and (1.6), we have Kv_n+1²=Kun−anT un²

≤Kun²−2an

T un, J Kun

+AmaxKun+anT un, B

ρX

anT un

≤

1−2canKun² +Amax

1+αanKun, B ρX

αanKun

(2.10)

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for alln≥0. SetM= Kv0. We claim that

maxKvn,Kun≤M, n≥0. (2.11)

By virtue of (2.6), (2.9), andLemma 1.5, we get that Ku0²≤

1−2cb0Kv0² +Amax

1+αb0Kv0, B ρX

αb0Kv0

≤

1−2cb0

M²+Amax 1+αb0

M, B αb0M

≤M².

(2.12)

That is, (2.11) is true forn=0. Suppose that (2.11) holds for somen≥0. Using (2.10), (2.6), andLemma 1.5, we infer that

Kvn+1²≤

1−2canKun² +Amax

1+αanKun, B ρX

αanKun

≤

1−2can

M²+Amax 1+αan

M, B αanM

≤M².

(2.13)

From (2.6), (2.9), (2.13), andLemma 1.5, we have Ku_n+1²≤

1−2cb_n+1Kv_n+1² +Amax

1+αbn+1Kvn+1, B ρX

αbn+1Kvn+1

≤

1−2cbn+1

M²+Amax

1+αbn+1 M, B

αbn+1M

≤M².

(2.14)

Therefore (2.11) holds for all n ≥ 0. Since X is uniformly smooth, by (2.4) and Lemma 1.5we conclude that there exist nonnegative sequences{sn}^∞n=0and{tn}^∞n=0

such thatρX(αMan)=snan,ρX(αMbn)=tnbnfor alln≥0 and

n→∞limsn=lim

n→∞tn=0. (2.15)

It follows from (2.5), (2.9), (2.10), and (2.11) that Kvn+1²≤

1−2can

1−2cbnKvn² +

1−2can

Amax

1+αbnKvn, B ρX

αbnKvn +Amax

1+αanKun, B ρX

αanKun

≤ 1−2c

an+bn

+4c²anbnKvn² +Amax{(1+α)M, B}

ρX αMan

+ρX

αMbn

≤ 1−c

an+bnKvn²+L

ansn+bntn

(2.16)

for alln≥0, whereL=Amax{(1+α)M, B}.Let αn=Kvn², ωn=c

an+bn

, βn=L

crn, n≥0, (2.17)

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where

rn=







0, an+bn=0,

an

an+bn

sn+ bn

an+bn

tn, an+bn=0. (2.18) It follows from (2.15) that limn→∞rn=0. That is, limn→∞βn=0.Thus (2.15) can be rewritten in the form

α_n+1≤ 1−ωn

αn+ωnβn, n≥0. (2.19)

Note that (2.3) and (2.5) mean that ^∞_n=0ωn=∞,ωn∈[0,1]. Consequently,Lemma 1.4 ensures thatαn→0 asn→ ∞. That is,

Kvn →0 asn → ∞. (2.20)

It follows from (2.2) and (2.20) that

T xn−f=Kvn →0 asn → ∞. (2.21) Note thatT has a bounded inverse. Thus (2.21) means that xn→T⁻¹f, the unique solution ofT x=f. This completes the proof.

Theorem 2.2. LetX, T, K, f, {xn}^∞n=0, {yn}^∞n=0, {vn}^∞n=0 and {un}^∞n=0 be as in Theorem 2.1. Suppose that {an} and {bn}^∞n=0 are any nonnegative sequences such that (2.3), (2.4), and (2.5) and

max b

αan

, b αbn

≤ 2c max

1,Kv0, n≥0, (2.22) whereb(t)is as in (1.7),αandc are the constants appearing in (1.3) and (1.2), re- spectively. Then the sequence{xn}^∞_n=0converges strongly to the unique solution of the equationT x=f.

Proof. SetM=max{1,Kv0}.As in the proof of Theorem 3 in [1] we have Kv_n+1²≤

1−c

an+bnKvn²+M³α anb

αan

+bnb αbn

, n≥0. (2.23)

Let

αn=Kvn², ωn=c an+bn

, βn=α

cM³rn, n≥0, (2.24) where

rn=









0, an+bn=0,

an

an+bn

b αan

+ bn

an+bn

b αbn

, an+bn=0.

(2.25)

It is easily seen that lim_n→∞βn = 0. The rest of the argument now follows as in the proof of Theorem 2.1 to yield that xn →T⁻¹f as n→ ∞. This completes the proof.

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Remark2.3. Theorems2.1and 2.2extend Theorem 3.3 of Bai [1], Theorem 2 of Chidume and Aneke [3] and Theorem of Chidume and Osilike [5], respectively, in the following ways:

(a) Condition (2.3) is much weaker than ^∞_n=0an= ∞of [1].

(b)Lp(orlp) spaces,p≥2,in [3] andq-uniformly smooth Banach space,q >1, in [5] are replaced by the more general uniformly smooth Banach spaces.

(c) The commutativity condition ofT andKin [3] is dropped.

(d) The iteration methods in [3,5] are special cases of our iteration method.

Acknowledgements. The ﬁrst author is grateful to the Research Institute of Nat- ural Sciences, Gyeongsang National University, in Korea for giving him the opportunity to visit the institute, the second author was supported by Korea Research Foundation Grant (KRF-99-005-D00003), and the third author was supported by KOSEF research project No. (2001-1-10100-005-2).

References

[1] C. Bai,Approximation of a solution for aK-positive deﬁnite operator equation in uniformly smooth separable Banach spaces, J. Math. Anal. Appl.236(1999), no. 2, 236–242.

MR 2000f:47021. Zbl 946.47007.

[2] S. S. Chang, Y. J. Cho, B. S. Lee, J. S. Jung, and S. M. Kang,Iterative approximations of ﬁxed points and solutions for strongly accretive and strongly pseudo-contractive mappings in Banach spaces, J. Math. Anal. Appl.224(1998), no. 1, 149–165.MR 99g:47146.

Zbl 933.47040.

[3] C. E. Chidume and S. J. Aneke,Existence, uniqueness and approximation of a solution for aK-positive deﬁnite operator equation, Appl. Anal. 50 (1993), no. 3-4, 285–294.

MR 95e:47009. Zbl 788.47051.

[4] C. E. Chidume and C. Moore, The solution by iteration of nonlinear equations in uni- formly smooth Banach spaces, J. Math. Anal. Appl.215 (1997), no. 1, 132–146.

MR 98m:47107. Zbl 906.47050.

[5] C. E. Chidume and M. O. Osilike, Approximation of a solution for a K-positive deﬁ- nite operator equation, J. Math. Anal. Appl.210(1997), no. 1, 1–7.MR 98c:47010.

Zbl 901.47002.

[6] Z. B. Xu and G. F. Roach, Characteristic inequalities of uniformly convex and uni- formly smooth Banach spaces, J. Math. Anal. Appl.157 (1991), no. 1, 189–210.

MR 92i:46023. Zbl 757.46034.

[7] H. Y. Zhou, S. M. Kang, and Y. J. Cho,Constructive solvability ofk-positive deﬁnite operator equations in uniformly smooth Banach spaces, submitted to Appl. Math. Lett.

Zeqing Liu: Department of Mathematics, Liaoning Normal University, Dalian, Liaon- ing116029, China

E-mail address:[email protected]

Shin Min Kang: Department of Mathematics, Gyeongsang National University, Chinju660-701, Korea

Jeong Sheok Ume: Department of Applied Mathematics, Ghangwon National Univer- sity, Changwon641-773, Korea

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