The Solution by Iteration of a Composed K Positive Definite Operator Equation in a Banach Space

International Journal of Mathematics and Mathematical Sciences Volume 2010, Article ID 376852,7pages

doi:10.1155/2010/376852

Research Article

The Solution by Iteration of a Composed K-Positive

Definite Operator Equation in a Banach Space

S. J. Aneke

Department of Mathematics, University of Nigeria, Nsukka, Nigeria

Correspondence should be addressed to S. J. Aneke,sylvanus [email protected]

Received 31 May 2010; Accepted 18 August 2010

Academic Editor: S. S. Dragomir

Copyrightq2010 S. J. Aneke. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

The equationLu f, where L AB, with Abeing a K-positive definite operator and B being a linear operator, is solved in a Banach space. Our scheme provides a generalization to the so-called method of moments studied in a Hilbert space by Petryshyn1962, as well as Lax and Milgram1954. Furthermore, an application of the inverse function theorem provides simultaneously a general solution to this equation in some neighborhood of a pointxo, whereLis

Fr´echet differentiable and an iterative scheme which converges strongly to the unique solution of this equation.

1. Introduction

LetHobe a dense subspace of a Hilbert space,H. An operatorT with domainDT⊇Hois said to be continuouslyHo-invertible if the range ofT,RTwithTconsidered as an operator restricted toHo is dense inH andT has a bounded inverse onRT. Let Hbe a complex and separable Hilbert space, and letAbe a linear unbounded operator defined on a dense domainDAinHwith the property that there exist a continuouslyDA-invertible closed linear operatorKwithDA⊆DKand a constantα >0 such that

Au, Ku ≥αKu2_{, u∈DA. 1.1}

The class of K-positive definite operators was first studied by Petryshyn, who proved, interalia, the following theoremsee1.

Theorem 1.1. IfAis a K-pd operator andDA DK, then there exists a constantα > 0 such that, for allu∈DK,

Au ≤αKu. 1.2

Furthermore, the operatorAis closed,RA H, and the equationAu f,f ∈H, has a unique solution.

Chidume and Aneke extended the notion of a K-pd operator to certain Banach spaces see 5. Later, in 2001, we also extended the class of K-pd operators to include the Fr´echet differentiable operators. A new notion—the asymptotically K-pd operators— was also introduced and studied in certain Banach spaces. We proved, among others, the following theorem.

Theorem 1.2see6. Suppose thatX is a real uniformly smooth Banach space. Suppose thatA is an asymptotically K-positive definite operator defined in a neighborhoodUxoof a real uniformly smooth Banach space, X. Define the sequence{xn}by xo ∈ Uxo,xn1 xnrn,n ≥ 0, rn

K−1_y−K−1_{Axn,y∈RA. Then{xn}converges strongly to the unique solution ofAxy∈Uxo.}

In this paper, we consider the composed equation

ABuf, 1.3

where A is K-pd and B is some linear operator in a Banach space E. Our interest is on the existence and uniqueness of solution to the above equation in a Banach space. We also consider an iterative scheme that converges to the unique solution of this equation in an arbitrary Banach space. Our method generalizes the so called method of moments, studied in Hilbert spaces by Petryshyn1and a host of other authors.

2. Preliminaries

LetE be a real normed linear space with dualE∗. We denote by J the normalized duality mapping fromEto 2E∗defined by

Jxf ∈E∗_{:x, fx}2 _f2_{, 2.1}

Lemma 2.1. LetEbe a real Banach space, and letJbe the normalized duality map onE. Then for any givenx, y∈E, the following inequality holds:

xy2

≤ x2_{2y, jxy, ∀jxy∈Jxy. 2.2}

3. Main Result

LetEbe an arbitrary Banach space andAa K-positive definite operator defined in a dense domainDA⊆E. LetBbe a linear unbounded operator such thatDB⊇DA. We prove that the equation

Luf, 3.1

whereL AB, has a unique solution and construct an iterative scheme that converges to the unique solution of this equation. Let

Lu ABuf. 3.2

Multiplying both sides of3.2byA−1_{, we have}

uTug, 3.3

where T A−1_{B, g A}−1_{f. SinceA is continuously invertible, the operatorT A}−1_Bis completely continuous. HenceTis locally lipschitzian and accretive. It follows that3.3has a unique solutionsee7.

IfA B, thenL AB 2A. In this caseLu, Ku 2Au, Ku ≥ 2αKu2

βKu2_{. Thus L is K-positive definite and so the equation Lu f has a unique solution}

see5. Examples of such Aare all positive operators whenK I and are all invertible operators when K A. If A /B, then let E l2, for instance, and define A : l2 → l2 by

Ax ax1, ax2, ax3, . . .forx x1, x2, x3, . . .∈l2anda >0. LetKI, the identity operator, thenAx, x a ∞_i1x2

i ax2 > 1/2ax2. ThusAis K-positive definite. LetBbe any linear operator; in particular, letB : l2 → l2be defined byBx 0, x1, x2, x3, . . .. Then by

3.2and3.3, the equationLuf, whereLAB, has a unique solution.

Next we derive the solution to3.2from the inverse function theorem and construct an iterative scheme which converges to the unique solution of this equation.

Theorem 3.1the inverse function theorem. Suppose thatE,Yare Banach spaces andL:E → Y is such thatL has uniformly continuous Fr´echet derivatives in a neighborhood of some pointuo of

E. Then if Luois a linear homeomorphism of E ontoY, then L is a local homeomorphism of a neighborhoodUuoofuoto a neighborhoodLuo.

Proof. For a sketch of proof of this theorem, see6.

By mimicking the proof of Theorem 3.1 of6, we get that, ifg−Luois sufficiently small,Lughas a unique solutionuuoρ∗, whereρ∗is the limit of the sequenceρo 0,

sufficiently small. It follows that the sequenceun uoρn converges touoρ∗, the unique solution ofLuginUuo. Now

unuoρnuoQρn−1

uoLuo−1g−Luo−Ruo, ρn−1

from Taylors theorem

uoLuo−1gLuoρn−1−L

uoρn−1

uoρn−1

L_u

o−1g−Lun−1

un−1

L_uo−1_g−Lu

n−1

.

3.4

Hence

un1un

L_uo−1_g−Lu

n. 3.5

Special Cases

1IfBI, then3.5becomes

un1un

A_uo−1_g−Au

nun. 3.6

2IfB0, then we have Corollary 3.2 of6.

For the caseB0, we prove the following theorem for an asymptotically K-positive definite operator. Recallsee 6, page 606 the definition of an asymptotically K-pd operator. For simplicity and ease of reference, we repeat the definition.

Definition 3.2. LetEbe a Banach space, and letAbe a linear unbounded operator defined on a dense domainDA⊂E. The operatorAis called asymptotically K-positive definite if there exist a continuouslyDA-invertible closed linear operatorKwithDK⊇DA⊇RAand a constantc >0 such that, forjKu∈JKu,

Kn−1_{Au, jK}n_u≥ck

nKnu2, u∈DA, 3.7

where{kn}is a real sequence such thatkn≥1, limn→ ∞kn1.

We now prove the following theorem for an asymptotically K-positive definite operator equation in an arbitrary Banach space,E.

Theorem 3.3. LetEbe a real Banach space. Suppose thatAis an asymptotically K-positive definite operator defined in a neighborhoodUxoof a real Banach space,E. Define the sequencexn byxo ∈

Proof. By the linearity of K we have

Krn1Krn−Arn. 3.8

UsingLemma 2.1andDefinition 3.2, we obtain

Kn_r

n12Knrn−Kn−1Arn

2

≤ Kn_r n2−2

Kn−1_Ar

n, j

Kn_r

n−Kn−1Arn

≤ Kn_r

n2−2cknKnrn12.

3.9

It follows that

12cknKnrn12 ≤ Knrn2 3.10

or

Kn_r

n12≤12ckn−1Knrn2. 3.11

The last inequality shows that the sequence Krn is monotonically decreasing and hence converges to a real number δ ≥ 0. Hence limn→ ∞Knrn 0. Since K is continuously invertible, thenrn → 0, and since A has a bounded inverse, we have that xn → A−1f, the unique solution ofAxf,f ∈E.

Our next result is a generalization of Theorem 3.6 of Chidume and Aneke 6to an arbitrary real Banach space.

Lemma 3.4Alber-Guerre8. Let{λk}and{γk}be sequences of nonnegative numbers, and let {αk}be a sequence of positive numbers satisfying the condition ∞1 {αk} ∞andγn/αn → 0, as

n → ∞. Let the recursive inequality

λn1≤λn−αnφλn γn, n1,2, . . . 3.12

be given where φλis a continuous and nondecreasing function from R → R such that it is positive onR− {0},φ0 0, limt→ ∞φt ∞. Thenλn → 0, asn → ∞.

Theorem 3.5. Suppose thatEis a real Banach space andAis an asymptotically K-positive definite operator defined in a neighbourhood Ux0 of a real Banach space, E. Suppose that A is Fre´chet differentiable. Define the sequence{xn}byx0∈Ux0,xn1xnrn,n≥0,rnK−1y−K−1Axn,

Proof. By the linearity ofKwe haveKrn1 Krn−Arn. UsingLemma 2.1and the definition of an asymptotically K-positive definite operator, we obtain

Kn_r

n12≤Knrn−Kn−1Arn

2

≤ Kn_r n2−2

Kn−1_Ar

n, jKnrn1

≤ Kn_r n2−2

Kn−1_Ar

n, jKnrn

−2Kn−1Arn, jKnrn1−jKnrn

≤ Kn_r

n2−2cknKnrn2−2

Kn−1_Ar

n, jKnrn1−jKnrn

≤ Kn_r

n2−2cknKnrn22Kn−1ArnjKnrn1−jKnrn.

3.13

Now,

Kn_r

n1−KnrnKnrn1−rn KnK−1Axn1−xn. 3.14

Sincexn1−xn → 0 andjis uniformly continuous, it follows thatjKnrn1−jKnrn → 0 asn → ∞. SinceAis Fr´echet differentiable, thenKn−1_Ar

nis necessarily bounded inUx0, whence

Kn_r

n12≤ Knrn2−2cknKnrn2or. 3.15

We invoke Alber-Guerre lemma,Lemma 3.4, withφt tandλnKnrn2. ThusKnrn → 0 asn → ∞. SinceK has a bounded inverse; thenrn → 0 as n → ∞, that is,Axn → y. Hencexn → A−1y, the unique solution ofAxyinUx0.

Acknowledgment

S. J. Aneke would like to thank the referee for his comments and suggestions, which helped to improve the manuscript.

References

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