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

Authors: S. J. Aneke
Journal: International Journal of Mathematics and Mathematical Sciences
 (2010)

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## Abstract

The equationLu=f, where L=A+B, with A being 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 Petryshyn (1962), as well as Lax and Milgram  (1954). Furthermore,
an application of the inverse function theorem provides simultaneously a general solution to this
equation in some neighborhood of a point xo, where L is Fréchet differentiable and an iterative scheme which converges strongly to the unique solution of this equation.


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## Body

## 1. Introduction

LetHo be a dense  subspace of a Hilbert space, H. An operator T with domain D(T)⊇Ho is said to be continuously Ho-invertible if the range of T, R(T) with T considered as an operator restricted to Ho is dense in H and T has a bounded inverse on R(T). Let H be a complex and separable Hilbert space, and let A be a linear unbounded operator defined on a dense domain D(A) in H with the property that there exist a continuously D(A)-invertible closed linear operator K with D(A)⊆D(K) and a constant α>0 such that(1.1)〈Au,Ku〉≥α∥Ku∥2,u∈D(A).
Then A is called K-positive definite (see, e.g., [1]). If K=I (the identity operator on H), then (1.1) reduces to 〈Au,u〉≥α∥u∥2, and in this case A is called positive definite. Positive definite operators have been studied by various authors (see, 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 operators (whenK = A) as its subclass.The class of K-positive definite operators was first studied by   Petryshyn, who proved, interalia, the following theorem (see [1]).Theorem 1.1.
IfA is a K-pd operator and D(A)=D(K), then there exists a constant α>0 such that, for all u∈D(K),
(1.2)∥Au∥≤α∥Ku∥.
Furthermore, the operator A is closed, R(A)=H, and the equation Au=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 Fréchet 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 1.2 (see [6]).
Suppose thatX is a real uniformly smooth Banach space. Suppose that A is an asymptotically K-positive definite operator defined in a neighborhood U(xo) of a real uniformly smooth Banach space, X. Define the sequence {xn} by xo∈U(xo), xn+1=xn+rn, n≥0, rn=K-1y-K-1Axn, y∈R(A). Then {xn} converges strongly to the unique solution of Ax=y∈U(xo).In this paper, we consider the composed equation(1.3)(A+B)u=f,
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 Petryshyn [1] and a host of other authors.


## 2. Preliminaries

LetE be a real normed linear space with dual E*. We denote by J the normalized duality mapping from E to 2E* defined by (2.1)Jx={f∈E*:〈x,f〉=∥x∥2=∥f∥2},
where 〈·,·〉 denotes the generalized duality pairing. It is well known that if E* is strictly convex then J is single valued and if E is uniformly smooth (equivalently if E* is uniformly convex) then J is uniformly continuous on bounded subsets of E. We will denote the single-valued duality mapping by j.Lemma 2.1.
LetE be a real Banach space, and let J be the normalized duality map on E. Then for any given x,y∈E, the following inequality holds:
(2.2)∥x+y∥2≤∥x∥2+2〈y,j(x+y)〉,∀j(x+y)∈J(x+y).


## 3. Main Result

LetE be an arbitrary Banach space and A a K-positive definite operator defined in a dense domain D(A)⊆E. Let B be a linear unbounded operator such that D(B)⊇D(A). We prove that the equation (3.1)Lu=f,
where L=A+B, has a unique solution and construct an iterative scheme that converges to the unique solution of this equation. Let(3.2)Lu=(A+B)u=f.
Multiplying both sides of (3.2) by A-1, we have(3.3)u+Tu=g,
where T=A-1B, g=A-1f. Since A is continuously invertible, the operator T=A-1B is completely continuous. Hence T is locally lipschitzian and accretive. It follows that (3.3) has a unique solution (see [7]).IfA=B, then L=A+B=2A. In this case 〈Lu,Ku〉=2〈Au,Ku〉≥2α∥Ku∥2=β∥Ku∥2. Thus L is K-positive definite and so the equation Lu=f has a unique solution (see [5]). Examples of such A are all positive operators when K=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,…) for x=(x1,x2,x3,…)∈l2 and a>0. Let K=I, the identity operator, then 〈Ax,x〉=a∑i=1∞xi2=a∥x∥2>(1/2)a∥x∥2. Thus A is K-positive definite. Let B be any linear operator; in particular, let B:l2→l2 be defined by Bx=(0,x1,x2,x3,…). Then by (3.2) and (3.3), the equation Lu=f, where L=A+B, has a unique solution.Next we derive the solution to   (3.2)  from the inverse function theorem and construct an iterative scheme which converges to the unique solution of this equation.Theorem 3.1 (the inverse function theorem).
Suppose thatE, Y are Banach spaces and L:E→Y is such that L has uniformly continuous Fréchet derivatives in a neighborhood of some point uo of E. Then if L'(uo) is a linear homeomorphism of E onto Y, then L is a local homeomorphism of a neighborhood U(uo) of uo to a neighborhood L(uo).Proof.
For a sketch of proof of this theorem, see [6].
By mimicking the proof of Theorem3.1 of [6], we get that, if ∥g-Luo∥ is sufficiently small, Lu=g has a unique solution u=uo+ρ*, where ρ* is the limit of the sequence ρo=0, ρn+1=Qρn, where Q is a contraction mapping of a sphere S(0,ϵ) in E into itself, for someϵ sufficiently small. It follows that the sequence un=uo+ρn converges to uo+ρ*, the unique solution of Lu=g in U(uo). Now
(3.4)un=uo+ρn=uo+Qρn-1=uo+[L′(uo)]-1[g-L(uo)-R(uo,ρn-1]fromTaylor′stheorem=uo+[L′(uo)]-1[g+L′(uo)ρn-1-L(uo+ρn-1)]=uo+ρn-1+[L′(uo)]-1[g-L(un-1)]=un-1+[L′(uo)]-1[g-Lun-1].
Hence
(3.5)un+1=un+[L′(uo)]-1[g-Lun].Special Cases
(1)
IfB=I, then (3.5) becomes(3.6)un+1=un+[A′(uo)]-1[g-Aun+un].(2)
IfB=0, then we have Corollary 3.2 of [6]. For the case B=0, we prove the following theorem for an asymptotically K-positive definite operator. Recall (see [6], page 606) the definition of an asymptotically K-pd operator. For simplicity and ease of reference, we repeat the definition.Definition 3.2.
LetE be a Banach space, and let A be a linear unbounded operator defined on a dense domain D(A)⊂E. The operator A is called asymptotically K-positive definite if there exist a continuously D(A)-invertible closed linear operator K with D(K)⊇D(A)⊇R(A) and a constant c>0 such that, for j(Ku)∈J(Ku),
(3.7)〈Kn-1Au,j(Knu)〉≥ckn∥Knu∥2,u∈D(A),
where {kn} is a real sequence such that kn≥1, limn→∞kn=1.We now prove the following theorem for an asymptotically K-positive definite operator equation in an arbitrary Banach space,E.Theorem 3.3.
LetE be a real Banach space. Suppose that A is an asymptotically K-positive definite operator defined in a neighborhood U(xo) of a real Banach space, E. Define the sequence xn by xo∈D(A), xn+1=xn+rn, n≥0, rn=K-1f-K-1Arn, f∈R(A). Then xn converges strongly to the unique solution of Ax=f.Proof.
By the linearity ofKwe have
(3.8)Krn+1=Krn-Arn.
Using Lemma 2.1 and Definition 3.2, we obtain
(3.9)∥Knrn+1∥2=∥Knrn-Kn-1Arn∥2≤∥Knrn∥2-2〈Kn-1Arn,j(Knrn-Kn-1Arn)〉≤∥Knrn∥2-2ckn∥Knrn+1∥2.
It follows that
(3.10)(1+2ckn)∥Knrn+1∥2≤∥Knrn∥2
or
(3.11)∥Knrn+1∥2≤(1+2ckn)-1∥Knrn∥2.
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, then rn→0, and since A has a bounded inverse, we have that xn→A-1f, the unique solution of Ax=f, f∈E.Our next result is a generalization of Theorem3.6 of Chidume and Aneke [6] to an arbitrary real Banach space.Lemma 3.4 (Alber-Guerre [8]).
Let{λk} and {γk} be sequences of nonnegative numbers, and  let {αk} be a sequence of positive numbers satisfying the condition ∑1∞{αk}=∞ and γn/αn→0, as n→∞. Let the recursive inequality
(3.12)λn+1≤λn-αnϕ(λn)+γn,n=1,2,…
be given where ϕ(λ) is a continuous and nondecreasing function from ℜ+→ℜ+ such that it is positive on ℜ+-{0}, ϕ(0)=0, limt→∞ϕ(t)=∞. Then λn→0, as n→∞.Theorem 3.5.
Suppose thatE is a real Banach space and A is an asymptotically K-positive definite operator defined in a neighbourhood U(x0) of a real Banach space, E. Suppose that A is Frećhet differentiable. Define the sequence {xn} by x0∈U(x0), xn+1=xn+rn, n≥0, rn=K-1y-K-1Axn, y∈R(A), and xn+1-xn→0, as n→∞. Then {xn} converges strongly to the unique solution of the equation Ax=y∈U(x0).Proof.
By the linearity ofK we have Krn+1=Krn-Arn. Using Lemma 2.1 and the definition of an asymptotically K-positive definite operator, we obtain
(3.13)∥Knrn+1∥2≤∥Knrn-Kn-1Arn∥2≤∥Knrn∥2-2〈Kn-1Arn,j(Knrn+1)〉≤∥Knrn∥2-2〈Kn-1Arn,j(Knrn)〉-2〈Kn-1Arn,j(Knrn+1-j(Knrn)〉≤∥Knrn∥2-2ckn∥Knrn∥2-2〈Kn-1Arn,j(Knrn+1)-j(Knrn)〉≤∥Knrn∥2-2ckn∥Knrn∥2+2∥Kn-1Arn∥∥j(Knrn+1)-j(Knrn)∥.
Now,
(3.14)Knrn+1-Knrn=Kn(rn+1-rn)=KnK-1A(xn+1-xn).
Since xn+1-xn→0 and j is uniformly continuous, it follows that ∥j(Knrn+1)-j(Knrn)∥→0 as n→∞. Since A is Fréchet differentiable, then ∥Kn-1Arn∥ is necessarily bounded in U(x0), whence
(3.15)∥Knrn+1∥2≤∥Knrn∥2-2ckn∥Knrn∥2+o(r).
We invoke Alber-Guerre lemma, Lemma 3.4, with ϕ(t)=t and λn=∥Knrn∥2. Thus ∥Knrn∥→0 as n→∞. Since K has a bounded inverse; then rn→0 as n→∞, that is, Axn→y. Hence xn→A-1y, the unique solution of Ax=y in U(x0).



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