Abstract
The equation , where , with being a K-positive definite operator and 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 , where is FrΓ©chet differentiable and an iterative scheme which converges strongly to the unique solution of this equation.
1. Introduction
Let be a dense subspace of a Hilbert space, . An operator with domain is said to be continuously -invertible if the range of , with considered as an operator restricted to is dense in and has a bounded inverse on . Let be a complex and separable Hilbert space, and let be a linear unbounded operator defined on a dense domain in with the property that there exist a continuously -invertible closed linear operator with and a constant such that Then is called K-positive definite (see, e.g., [1]). If (the identity operator on ), then (1.1) reduces to , and in this case is called positive definite. Positive definite operators have been studied by various authors (see, e.g., [1β4]). 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 is a K-pd operator and , then there exists a constant such that, for all , Furthermore, the operator is closed, , and the equation , , 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 notionβthe asymptotically K-pd operatorsβwas also introduced and studied in certain Banach spaces. We proved, among others, the following theorem.
Theorem 1.2 (see [6]). Suppose that is a real uniformly smooth Banach space. Suppose that is an asymptotically K-positive definite operator defined in a neighborhood of a real uniformly smooth Banach space, . Define the sequence by , , , , . Then converges strongly to the unique solution of .
In this paper, we consider the composed equation where is K-pd and is some linear operator in a Banach space . 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 be a real normed linear space with dual . We denote by the normalized duality mapping from to defined by where denotes the generalized duality pairing. It is well known that if is strictly convex then is single valued and if is uniformly smooth (equivalently if is uniformly convex) then is uniformly continuous on bounded subsets of . We will denote the single-valued duality mapping by .
Lemma 2.1. Let be a real Banach space, and let be the normalized duality map on . Then for any given , the following inequality holds:
3. Main Result
Let be an arbitrary Banach space and a K-positive definite operator defined in a dense domain . Let be a linear unbounded operator such that . We prove that the equation where , has a unique solution and construct an iterative scheme that converges to the unique solution of this equation. Let Multiplying both sides of (3.2) by , we have where , . Since is continuously invertible, the operator is completely continuous. Hence is locally lipschitzian and accretive. It follows that (3.3) has a unique solution (see [7]).
If , then . In this case . Thus is K-positive definite and so the equation has a unique solution (see [5]). Examples of such are all positive operators when and are all invertible operators when . If , then let , for instance, and define by for and . Let , the identity operator, then . Thus is K-positive definite. Let be any linear operator; in particular, let be defined by . Then by (3.2) and (3.3), the equation , where , 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 , are Banach spaces and is such that has uniformly continuous FrΓ©chet derivatives in a neighborhood of some point of . Then if is a linear homeomorphism of onto , then is a local homeomorphism of a neighborhood of to a neighborhood .
Proof. For a sketch of proof of this theorem, see [6].
By mimicking the proof of Theorem of [6], we get that, if is sufficiently small, has a unique solution , where is the limit of the sequence , , where is a contraction mapping of a sphere in into itself, for some sufficiently small. It follows that the sequence converges to , the unique solution of in . Now
Hence
Special Cases
()If , then (3.5) becomes ()If , then we have Corollary of [6]. For the case , 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 be a Banach space, and let be a linear unbounded operator defined on a dense domain . The operator is called asymptotically K-positive definite if there exist a continuously -invertible closed linear operator with and a constant such that, for , where is a real sequence such that , .
We now prove the following theorem for an asymptotically K-positive definite operator equation in an arbitrary Banach space, .
Theorem 3.3. Let be a real Banach space. Suppose that is an asymptotically K-positive definite operator defined in a neighborhood of a real Banach space, . Define the sequence by , , , , . Then converges strongly to the unique solution of .
Proof. By the linearity of K we have Using Lemma 2.1 and Definition 3.2, we obtain It follows that or The last inequality shows that the sequence is monotonically decreasing and hence converges to a real number . Hence . Since is continuously invertible, then , and since has a bounded inverse, we have that , the unique solution of , .
Our next result is a generalization of Theorem of Chidume and Aneke [6] to an arbitrary real Banach space.
Lemma 3.4 (Alber-Guerre [8]). Let and be sequences of nonnegative numbers, and let be a sequence of positive numbers satisfying the condition and , as . Let the recursive inequality be given where is a continuous and nondecreasing function from such that it is positive on , , . Then , as .
Theorem 3.5. Suppose that is a real Banach space and is an asymptotically K-positive definite operator defined in a neighbourhood of a real Banach space, . Suppose that is FreΔhet differentiable. Define the sequence by , , , , , and , as . Then converges strongly to the unique solution of the equation .
Proof. By the linearity of we have . Using Lemma 2.1 and the definition of an asymptotically K-positive definite operator, we obtain Now, Since and is uniformly continuous, it follows that as . Since is FrΓ©chet differentiable, then is necessarily bounded in , whence We invoke Alber-Guerre lemma, Lemma 3.4, with and . Thus as . Since has a bounded inverse; then as , that is, . Hence , the unique solution of in .
Acknowledgment
S. J. Aneke would like to thank the referee for his comments and suggestions, which helped to improve the manuscript.