Abstract

The equation 𝐿𝑒=𝑓, where 𝐿=𝐴+𝐡, with 𝐴 being a K-positive definite operator and 𝐡 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 π‘₯π‘œ, where 𝐿 is FrΓ©chet differentiable and an iterative scheme which converges strongly to the unique solution of this equation.

1. Introduction

Let π»π‘œ be a dense subspace of a Hilbert space, 𝐻. An operator 𝑇 with domain 𝐷(𝑇)βŠ‡π»π‘œ is said to be continuously π»π‘œ-invertible if the range of 𝑇, 𝑅(𝑇) with 𝑇 considered as an operator restricted to π»π‘œ 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)βŠ†π·(𝐾) and a constant 𝛼>0 such thatβŸ¨π΄π‘’,πΎπ‘’βŸ©β‰₯𝛼‖𝐾𝑒‖2,π‘’βˆˆπ·(𝐴).(1.1) Then 𝐴 is called K-positive definite (see, e.g., [1]). If 𝐾=𝐼 (the identity operator on 𝐻), then (1.1) reduces to βŸ¨π΄π‘’,π‘’βŸ©β‰₯𝛼‖𝑒‖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 (when K = 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. If 𝐴 is a K-pd operator and 𝐷(𝐴)=𝐷(𝐾), then there exists a constant 𝛼>0 such that, for all π‘’βˆˆπ·(𝐾), ‖𝐴𝑒‖≀𝛼‖𝐾𝑒‖.(1.2) Furthermore, the operator 𝐴 is closed, 𝑅(𝐴)=𝐻, and the equation 𝐴𝑒=𝑓, π‘“βˆˆπ», 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 FrΓ©chet differentiable operators. A new notionβ€”the asymptotically K-pd operatorsβ€”was also introduced and studied in certain Banach spaces. We proved, among others, the following theorem.

Theorem 1.2 (see [6]). Suppose that 𝑋 is a real uniformly smooth Banach space. Suppose that 𝐴 is an asymptotically K-positive definite operator defined in a neighborhood π‘ˆ(π‘₯π‘œ) of a real uniformly smooth Banach space, 𝑋. Define the sequence {π‘₯𝑛} by π‘₯π‘œβˆˆπ‘ˆ(π‘₯π‘œ), π‘₯𝑛+1=π‘₯𝑛+π‘Ÿπ‘›, 𝑛β‰₯0, π‘Ÿπ‘›=πΎβˆ’1π‘¦βˆ’πΎβˆ’1𝐴π‘₯𝑛, π‘¦βˆˆπ‘…(𝐴). Then {π‘₯𝑛} converges strongly to the unique solution of 𝐴π‘₯=π‘¦βˆˆπ‘ˆ(π‘₯π‘œ).

In this paper, we consider the composed equation (𝐴+𝐡)𝑒=𝑓,(1.3) where 𝐴 is K-pd and 𝐡 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

Let 𝐸 be a real normed linear space with dual πΈβˆ—. We denote by 𝐽 the normalized duality mapping from 𝐸 to 2πΈβˆ— defined by 𝐽π‘₯=π‘“βˆˆπΈβˆ—βˆΆβŸ¨π‘₯,π‘“βŸ©=β€–π‘₯β€–2=‖𝑓‖2ξ€Ύ,(2.1) where βŸ¨β‹…,β‹…βŸ© denotes the generalized duality pairing. It is well known that if πΈβˆ— is strictly convex then 𝐽 is single valued and if 𝐸 is uniformly smooth (equivalently if πΈβˆ— 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. Let 𝐸 be a real Banach space, and let 𝐽 be the normalized duality map on 𝐸. Then for any given π‘₯,π‘¦βˆˆπΈ, the following inequality holds: β€–π‘₯+𝑦‖2≀‖π‘₯β€–2+2βŸ¨π‘¦,𝑗(π‘₯+𝑦)⟩,βˆ€π‘—(π‘₯+𝑦)∈𝐽(π‘₯+𝑦).(2.2)

3. Main Result

Let 𝐸 be an arbitrary Banach space and 𝐴 a K-positive definite operator defined in a dense domain 𝐷(𝐴)βŠ†πΈ. Let 𝐡 be a linear unbounded operator such that 𝐷(𝐡)βŠ‡π·(𝐴). We prove that the equation 𝐿𝑒=𝑓,(3.1) where 𝐿=𝐴+𝐡, has a unique solution and construct an iterative scheme that converges to the unique solution of this equation. Let𝐿𝑒=(𝐴+𝐡)𝑒=𝑓.(3.2) Multiplying both sides of (3.2) by π΄βˆ’1, we have𝑒+𝑇𝑒=𝑔,(3.3) where 𝑇=π΄βˆ’1𝐡, 𝑔=π΄βˆ’1𝑓. Since 𝐴 is continuously invertible, the operator 𝑇=π΄βˆ’1𝐡 is completely continuous. Hence 𝑇 is locally lipschitzian and accretive. It follows that (3.3) has a unique solution (see [7]).

If 𝐴=𝐡, then 𝐿=𝐴+𝐡=2𝐴. In this case βŸ¨πΏπ‘’,πΎπ‘’βŸ©=2βŸ¨π΄π‘’,πΎπ‘’βŸ©β‰₯2𝛼‖𝐾𝑒‖2=𝛽‖𝐾𝑒‖2. Thus 𝐿 is K-positive definite and so the equation 𝐿𝑒=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≠𝐡, then let 𝐸=𝑙2, for instance, and define π΄βˆΆπ‘™2→𝑙2 by 𝐴π‘₯=(π‘Žπ‘₯1,π‘Žπ‘₯2,π‘Žπ‘₯3,…) for π‘₯=(π‘₯1,π‘₯2,π‘₯3,…)βˆˆπ‘™2 and π‘Ž>0. Let 𝐾=𝐼, the identity operator, then βˆ‘βŸ¨π΄π‘₯,π‘₯⟩=π‘Žβˆžπ‘–=1π‘₯2𝑖=π‘Žβ€–π‘₯β€–2>(1/2)π‘Žβ€–π‘₯β€–2. Thus 𝐴 is K-positive definite. Let 𝐡 be any linear operator; in particular, let π΅βˆΆπ‘™2→𝑙2 be defined by 𝐡π‘₯=(0,π‘₯1,π‘₯2,π‘₯3,…). Then by (3.2) and (3.3), the equation 𝐿𝑒=𝑓, where 𝐿=𝐴+𝐡, 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 that 𝐸, π‘Œ are Banach spaces and πΏβˆΆπΈβ†’π‘Œ is such that 𝐿 has uniformly continuous FrΓ©chet derivatives in a neighborhood of some point π‘’π‘œ of 𝐸. Then if 𝐿′(π‘’π‘œ) is a linear homeomorphism of 𝐸 onto π‘Œ, then 𝐿 is a local homeomorphism of a neighborhood π‘ˆ(π‘’π‘œ) of π‘’π‘œ to a neighborhood 𝐿(π‘’π‘œ).

Proof. For a sketch of proof of this theorem, see [6].
By mimicking the proof of Theorem 3.1 of [6], we get that, if β€–π‘”βˆ’πΏπ‘’π‘œβ€– is sufficiently small, 𝐿𝑒=𝑔 has a unique solution 𝑒=π‘’π‘œ+πœŒβˆ—, where πœŒβˆ— is the limit of the sequence πœŒπ‘œ=0, πœŒπ‘›+1=π‘„πœŒπ‘›, where 𝑄 is a contraction mapping of a sphere 𝑆(0,πœ–) in 𝐸 into itself, for someπœ– sufficiently small. It follows that the sequence 𝑒𝑛=π‘’π‘œ+πœŒπ‘› converges to π‘’π‘œ+πœŒβˆ—, the unique solution of 𝐿𝑒=𝑔 in π‘ˆ(π‘’π‘œ). Now 𝑒𝑛=π‘’π‘œ+πœŒπ‘›=π‘’π‘œ+π‘„πœŒπ‘›βˆ’1=π‘’π‘œ+ξ€ΊπΏξ…žξ€·π‘’π‘œξ€Έξ€»βˆ’1ξ€Ίξ€·π‘’π‘”βˆ’πΏπ‘œξ€Έξ€·π‘’βˆ’π‘…π‘œ,πœŒπ‘›βˆ’1ξ€»fromTaylorξ…žstheorem=π‘’π‘œ+ξ€ΊπΏξ…žξ€·π‘’π‘œξ€Έξ€»βˆ’1𝑔+πΏξ…žξ€·π‘’π‘œξ€ΈπœŒπ‘›βˆ’1ξ€·π‘’βˆ’πΏπ‘œ+πœŒπ‘›βˆ’1ξ€Έξ€»=π‘’π‘œ+πœŒπ‘›βˆ’1+ξ€ΊπΏξ…žξ€·π‘’π‘œξ€Έξ€»βˆ’1ξ€Ίξ€·π‘’π‘”βˆ’πΏπ‘›βˆ’1ξ€Έξ€»=π‘’π‘›βˆ’1+ξ€ΊπΏξ…žξ€·π‘’π‘œξ€Έξ€»βˆ’1ξ€Ίπ‘”βˆ’πΏπ‘’π‘›βˆ’1ξ€».(3.4) Hence 𝑒𝑛+1=𝑒𝑛+ξ€ΊπΏξ…žξ€·π‘’π‘œξ€Έξ€»βˆ’1ξ€Ίπ‘”βˆ’πΏπ‘’π‘›ξ€».(3.5)

Special Cases
(1)If 𝐡=𝐼, then (3.5) becomes 𝑒𝑛+1=𝑒𝑛+ξ€Ίπ΄ξ…žξ€·π‘’π‘œξ€Έξ€»βˆ’1ξ€Ίπ‘”βˆ’π΄π‘’π‘›+𝑒𝑛.(3.6)(2)If 𝐡=0, then we have Corollary 3.2 of [6]. For the case 𝐡=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. Let 𝐸 be a Banach space, and let 𝐴 be a linear unbounded operator defined on a dense domain 𝐷(𝐴)βŠ‚πΈ. The operator 𝐴 is called asymptotically K-positive definite if there exist a continuously 𝐷(𝐴)-invertible closed linear operator 𝐾 with 𝐷(𝐾)βŠ‡π·(𝐴)βŠ‡π‘…(𝐴) and a constant 𝑐>0 such that, for 𝑗(𝐾𝑒)∈𝐽(𝐾𝑒), ξ«πΎπ‘›βˆ’1𝐴𝑒,𝑗(𝐾𝑛𝑒)β‰₯π‘π‘˜π‘›β€–πΎπ‘›π‘’β€–2,π‘’βˆˆπ·(𝐴),(3.7) where {π‘˜π‘›} is a real sequence such that π‘˜π‘›β‰₯1, limπ‘›β†’βˆžπ‘˜π‘›=1.

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

Theorem 3.3. Let 𝐸 be a real Banach space. Suppose that 𝐴 is an asymptotically K-positive definite operator defined in a neighborhood π‘ˆ(π‘₯π‘œ) of a real Banach space, 𝐸. Define the sequence π‘₯𝑛 by π‘₯π‘œβˆˆπ·(𝐴), π‘₯𝑛+1=π‘₯𝑛+π‘Ÿπ‘›, 𝑛β‰₯0, π‘Ÿπ‘›=πΎβˆ’1π‘“βˆ’πΎβˆ’1π΄π‘Ÿπ‘›, π‘“βˆˆπ‘…(𝐴). Then π‘₯𝑛 converges strongly to the unique solution of 𝐴π‘₯=𝑓.

Proof. By the linearity of K we have πΎπ‘Ÿπ‘›+1=πΎπ‘Ÿπ‘›βˆ’π΄π‘Ÿπ‘›.(3.8) Using Lemma 2.1 and Definition 3.2, we obtain β€–β€–πΎπ‘›π‘Ÿπ‘›+1β€–β€–2=β€–β€–πΎπ‘›π‘Ÿπ‘›βˆ’πΎπ‘›βˆ’1π΄π‘Ÿπ‘›β€–β€–2β‰€β€–β€–πΎπ‘›π‘Ÿπ‘›β€–β€–2ξ«πΎβˆ’2π‘›βˆ’1π΄π‘Ÿπ‘›ξ€·πΎ,π‘—π‘›π‘Ÿπ‘›βˆ’πΎπ‘›βˆ’1π΄π‘Ÿπ‘›β‰€β€–β€–πΎξ€Έξ¬π‘›π‘Ÿπ‘›β€–β€–2βˆ’2π‘π‘˜π‘›β€–β€–πΎπ‘›π‘Ÿπ‘›+1β€–β€–2.(3.9) It follows that ξ€·1+2π‘π‘˜π‘›ξ€Έβ€–β€–πΎπ‘›π‘Ÿπ‘›+1β€–β€–2β‰€β€–β€–πΎπ‘›π‘Ÿπ‘›β€–β€–2(3.10) or β€–β€–πΎπ‘›π‘Ÿπ‘›+1β€–β€–2≀1+2π‘π‘˜π‘›ξ€Έβˆ’1β€–β€–πΎπ‘›π‘Ÿπ‘›β€–β€–2.(3.11) The last inequality shows that the sequence πΎπ‘Ÿπ‘› is monotonically decreasing and hence converges to a real number 𝛿β‰₯0. Hence limπ‘›β†’βˆžβ€–πΎπ‘›π‘Ÿπ‘›β€–=0. Since 𝐾 is continuously invertible, then π‘Ÿπ‘›β†’0, and since 𝐴 has a bounded inverse, we have that π‘₯π‘›β†’π΄βˆ’1𝑓, the unique solution of 𝐴π‘₯=𝑓, π‘“βˆˆπΈ.

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

Lemma 3.4 (Alber-Guerre [8]). Let {πœ†π‘˜} and {π›Ύπ‘˜} be sequences of nonnegative numbers, and let {π›Όπ‘˜} be a sequence of positive numbers satisfying the condition βˆ‘βˆž1{π›Όπ‘˜}=∞ and 𝛾𝑛/𝛼𝑛→0, as π‘›β†’βˆž. Let the recursive inequality πœ†π‘›+1β‰€πœ†π‘›βˆ’π›Όπ‘›πœ™ξ€·πœ†π‘›ξ€Έ+𝛾𝑛,𝑛=1,2,…(3.12) be given where πœ™(πœ†) is a continuous and nondecreasing function from β„œ+β†’β„œ+ such that it is positive on β„œ+βˆ’{0}, πœ™(0)=0, limπ‘‘β†’βˆžπœ™(𝑑)=∞. Then πœ†π‘›β†’0, as π‘›β†’βˆž.

Theorem 3.5. Suppose that 𝐸 is a real Banach space and 𝐴 is an asymptotically K-positive definite operator defined in a neighbourhood π‘ˆ(π‘₯0) of a real Banach space, 𝐸. Suppose that 𝐴 is FreΔ‡het differentiable. Define the sequence {π‘₯𝑛} by π‘₯0βˆˆπ‘ˆ(π‘₯0), π‘₯𝑛+1=π‘₯𝑛+π‘Ÿπ‘›, 𝑛β‰₯0, π‘Ÿπ‘›=πΎβˆ’1π‘¦βˆ’πΎβˆ’1𝐴π‘₯𝑛, π‘¦βˆˆπ‘…(𝐴), and π‘₯𝑛+1βˆ’π‘₯𝑛→0, as π‘›β†’βˆž. Then {π‘₯𝑛} converges strongly to the unique solution of the equation 𝐴π‘₯=π‘¦βˆˆπ‘ˆ(π‘₯0).

Proof. By the linearity of 𝐾 we have πΎπ‘Ÿπ‘›+1=πΎπ‘Ÿπ‘›βˆ’π΄π‘Ÿπ‘›. Using Lemma 2.1 and the definition of an asymptotically K-positive definite operator, we obtain β€–β€–πΎπ‘›π‘Ÿπ‘›+1β€–β€–2β‰€β€–β€–πΎπ‘›π‘Ÿπ‘›βˆ’πΎπ‘›βˆ’1π΄π‘Ÿπ‘›β€–β€–2β‰€β€–β€–πΎπ‘›π‘Ÿπ‘›β€–β€–2ξ«πΎβˆ’2π‘›βˆ’1π΄π‘Ÿπ‘›ξ€·πΎ,π‘—π‘›π‘Ÿπ‘›+1β‰€β€–β€–πΎξ€Έξ¬π‘›π‘Ÿπ‘›β€–β€–2ξ«πΎβˆ’2π‘›βˆ’1π΄π‘Ÿπ‘›ξ€·πΎ,π‘—π‘›π‘Ÿπ‘›ξ«πΎξ€Έξ¬βˆ’2π‘›βˆ’1π΄π‘Ÿπ‘›ξ€·πΎ,π‘—π‘›π‘Ÿπ‘›+1ξ€·πΎβˆ’π‘—π‘›π‘Ÿπ‘›β‰€β€–β€–πΎξ€Έξ¬π‘›π‘Ÿπ‘›β€–β€–2βˆ’2π‘π‘˜π‘›β€–β€–πΎπ‘›π‘Ÿπ‘›β€–β€–2ξ«πΎβˆ’2π‘›βˆ’1π΄π‘Ÿπ‘›ξ€·πΎ,π‘—π‘›π‘Ÿπ‘›+1ξ€Έξ€·πΎβˆ’π‘—π‘›π‘Ÿπ‘›β‰€β€–β€–πΎξ€Έξ¬π‘›π‘Ÿπ‘›β€–β€–2βˆ’2π‘π‘˜π‘›β€–β€–πΎπ‘›π‘Ÿπ‘›β€–β€–2‖‖𝐾+2π‘›βˆ’1π΄π‘Ÿπ‘›β€–β€–β€–β€–π‘—ξ€·πΎπ‘›π‘Ÿπ‘›+1ξ€Έξ€·πΎβˆ’π‘—π‘›π‘Ÿπ‘›ξ€Έβ€–β€–.(3.13) Now, πΎπ‘›π‘Ÿπ‘›+1βˆ’πΎπ‘›π‘Ÿπ‘›=πΎπ‘›ξ€·π‘Ÿπ‘›+1βˆ’π‘Ÿπ‘›ξ€Έ=πΎπ‘›πΎβˆ’1𝐴π‘₯𝑛+1βˆ’π‘₯𝑛.(3.14) Since π‘₯𝑛+1βˆ’π‘₯𝑛→0 and 𝑗 is uniformly continuous, it follows that ‖𝑗(πΎπ‘›π‘Ÿπ‘›+1)βˆ’π‘—(πΎπ‘›π‘Ÿπ‘›)β€–β†’0 as π‘›β†’βˆž. Since 𝐴 is FrΓ©chet differentiable, then β€–πΎπ‘›βˆ’1π΄π‘Ÿπ‘›β€– is necessarily bounded in π‘ˆ(π‘₯0), whence β€–β€–πΎπ‘›π‘Ÿπ‘›+1β€–β€–2β‰€β€–β€–πΎπ‘›π‘Ÿπ‘›β€–β€–2βˆ’2π‘π‘˜π‘›β€–β€–πΎπ‘›π‘Ÿπ‘›β€–β€–2+π‘œ(π‘Ÿ).(3.15) We invoke Alber-Guerre lemma, Lemma 3.4, with πœ™(𝑑)=𝑑 and πœ†π‘›=β€–πΎπ‘›π‘Ÿπ‘›β€–2. Thus β€–πΎπ‘›π‘Ÿπ‘›β€–β†’0 as π‘›β†’βˆž. Since 𝐾 has a bounded inverse; then π‘Ÿπ‘›β†’0 as π‘›β†’βˆž, that is, 𝐴π‘₯𝑛→𝑦. Hence π‘₯π‘›β†’π΄βˆ’1𝑦, the unique solution of 𝐴π‘₯=𝑦 in π‘ˆ(π‘₯0).

Acknowledgment

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