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 aneke@yahoo.com 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 ThenAis called K-positive definitesee, e.g., 1. If K I the identity operator onH, then1.1reduces toAu, u ≥αu2, and in this caseAis called positive definite. Positive definite operators have been studied by various authorssee, e.g.,1–4. It is clear that the class of K-pd operators contains, among others, the class of positive definite operators and also contains the class of invertible operatorswhen KAas its subclass.
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−1y−K−1Axn,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
Jx
f ∈E∗: x, f
x2 f2
, 2.1
where ·,· denotes the generalized duality pairing. It is well known that if E∗ is strictly convex thenJis single valued and ifEis uniformly smoothequivalently ifE∗is uniformly convexthenJis uniformly continuous on bounded subsets ofE. We will denote the single- valued duality mapping byj.
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 ≤ x22 y, j
xy , ∀j
xy
∈J xy
. 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−1B, g A−1f. SinceA is continuously invertible, the operatorT A−1Bis 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 ∞i1x2i 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, ρn1 Qρn, whereQis a contraction mapping of a sphereS0, inEinto itself, for some
sufficiently small. It follows that the sequenceun uoρn converges touoρ∗, the unique solution ofLuginUuo. Now
unuoρnuoQρn−1 uo
Luo−1
g−Luo−R
uo, ρn−1
from Taylors theorem uo
Luo−1
gLuoρn−1−L
uoρn−1 uoρn−1
Luo−1
g−Lun−1 un−1
Luo−1
g−Lun−1 .
3.4
Hence
un1un
Luo−1
g−Lun
. 3.5
Special Cases
1IfBI, then3.5becomes
un1un
Auo−1
g−Aunun
. 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−1Au, jKnu
≥cknKnu2, 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 ∈ DA,xn1 xnrn,n≥ 0,rn K−1f−K−1Arn,f ∈RA. Thenxnconverges strongly to the unique solution ofAxf.
Proof. By the linearity of K we have
Krn1Krn−Arn. 3.8
UsingLemma 2.1andDefinition 3.2, we obtain
Knrn12Knrn−Kn−1Arn2
≤ Knrn2−2
Kn−1Arn, j
Knrn−Kn−1Arn
≤ Knrn2−2cknKnrn12.
3.9
It follows that
12cknKnrn12 ≤ Knrn2 3.10
or
Knrn12≤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, y∈RA, andxn1−xn → 0, asn → ∞. Then{xn}converges strongly to the unique solution of the equationAxy∈Ux0.
Proof. By the linearity ofKwe haveKrn1 Krn−Arn. UsingLemma 2.1and the definition of an asymptotically K-positive definite operator, we obtain
Knrn12≤Knrn−Kn−1Arn2
≤ Knrn2−2
Kn−1Arn, jKnrn1
≤ Knrn2−2
Kn−1Arn, jKnrn
−2
Kn−1Arn, j
Knrn1−jKnrn
≤ Knrn2−2cknKnrn2−2
Kn−1Arn, jKnrn1−jKnrn
≤ Knrn2−2cknKnrn22Kn−1ArnjKnrn1−jKnrn.
3.13
Now,
Knrn1−KnrnKnrn1−rn KnK−1Axn1−xn. 3.14 Sincexn1−xn → 0 andjis uniformly continuous, it follows thatjKnrn1−jKnrn → 0 asn → ∞. SinceAis Fr´echet differentiable, thenKn−1Arnis necessarily bounded inUx0, whence
Knrn12≤ 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.
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