The stable computation of values of unbounded operators is one of the most important problems in computational mathematics. Indeed, let A be a linear operator from X into Y with domain and range , where X and Y are normed spaces and A is unbounded, that is, there exists a sequence of elements , such that as . Let and . We put , where is an arbitrarily small number. Let . Then
while may be arbitrarily small.
Therefore, the problem of computing values of an operator in the considered case is unstable . Moreover, if we bear in mind arbitrarily -approximation to the element in X, that is the elements with , we can see that the values of the operator A may not even be defined on the elements , that is, and if , it may happen as , since the operator A is unbounded.
In the case, where A is a closed densely defined unbounded linear operator from a Hilbert space X into a Hilbert space Y, V. A. Morozov has studied a stable method for approximating the value when only approximate data is available . This method takes as an approximation to the element , where minimizes the parametric functional
He shows that, if as , in such a way that , then as . Moreover, the order of convergence results for have been established  - .
In the another case, where A is a monotone operator from a real strictly convex reflexive Banach space X into its dual , an approximation to is the element , where is the unique solution of the equation
where is the dual mapping in X  . Then the sequence for , in the norm of , to a generalized value of the operator A at .
We now assume that both the operator A and are only given approximately by and , which satisfy
where is also an operator from X into Y. We should approximate values of A when we are given the approximations and . Until now, this problem is still an open problem.
In this paper we shall be concerned with the construction of a stable method of computing values of the operator A for the perturbations (2).
2. The Stable Method of Computing Values of Closed Densely Defined Unbounded Linear Operators
In this section, we assume that is a closed densely defined unbounded linear operator from a Hilbert space X into a Hilbert space Y with domain and . is called an exact data.
Instead of the exact data , we have an approximation , which satisfies (1.2), where is also a closed densely defined unbounded linear operator from X into Y with domain .
First, we define the regularization functional
where is called the regularization parameter, .
We shall take as an approximation to the element , where minimizes the regularization functional over .
Theorem 2.1.  For any the minimization problem (1) has a unique solution
To establish the convergence of (3), it will be convenient to reformulate (3) as
are known to be bounded everywhere defined linear operators and is a self-adjoint with spectrum ( , p. 38).
To further simplify the presentation, we introduce the functions
We then have
We also denote
The following lemma will be used in the proof of Theorem 2.2.
Lemma 2.1. Under the stated assumption, we obtain
Proof. We denote
Since is a closed densely defined linear operator then and are complementary orthogonal subspaces of the Hilbert space ( , p. 307). Hence, for any , we have the uniquely determined decomposition
Therefore, and . Because of the uniqueness of decomposition (7), x is uniquely determined by z, and so the everywhere defined inverse exists.
In a similar way as above, the everywhere defined inverse exists. It follows from (8) that
that means . Moreover, are bouned operators and
( , p. 308).
Theorem 2.2. If and , and , as , then converges to .
Proof. Let . Then . Since (Lemma 2.1) and , we have
Since and , for all , we obtain
On the other hand we have
It follows from (9) that
The theorem is proved.
We shall call the approximate values of the operator A at .
3. The Stable Method of Computing Values of Hemi-Continuous Monotone Operators
Let X be a real strictly convex reflexive Banach space with the dual be an E- space. Suppose that is a hemi-continuous monotone operator from X into with domain (possibly multi-valued) and y is a given element in . We consider the following three problems
1) To solve the equation
2) To solve the variational inequality
3) To compute values of the operator A at in X with given approximately.
These problems are important objects of investigation in the theory unstable problems. In    -  a class of monotone operators was singled out and, as an approximate method, the operator-regularization method was used.
As it is known , a solution of (1) is understood to be an element such that if A is a single-valued, and if A is a maximal monotone (possibly multi-valued). If A is an arbitrary monotone operator, we follow  and understand a solution of (1) to be an element such that
where values of the linear functional at .
We shall call a generalized solution of Equation (1). We note that, if A is hemi-continuous and is open or everywhere dense in X, or if A is maximal monotone, then a generalized solution coincides with the corresponding solution , and (3) is equivalent to the inclusion .
We now deal with the stable method of computing values of the operator A at when only the approximations as in (2) are given, where is also a hemi-continuous monotone operator from X into with domain .
We denote the set values of A at
In we consider the set
and we call the set of generalized values of A at . It is easy to show that .
Lemma 3.1.  The set is convex and closed in , moreover, there is a unique element such that
Under the above hypotheses, there exist the dual mappings
being strictly monotone, single-valued, homogeneous, hemi-continuous and such that
(see    ).
We consider the equation
The following theorem asserts the existence and uniqueness of generalized solution of (4).
Theorem 3.1. Under hypotheses as above, Equation (4) has a unique solution , for any .
Proof. Let be the maximal monotone extension of (such an extension exists by virtue of Zorn’s lemma). Therefore, the operator is maximal monotone  and Browder’s theorem  implies that Equation (4) has a unique solution , i.e., . In view of the preceding remark, this follows that
Thus, coincides with the generalized solution of Equation (2). Therefore, (2) has a unique solution , for any . We now consider the sequence
The uniqueness of implies that is uniquely determined. It is easy to show that .
is call approximate value of A at for the given approximation .
Theorem 3.2. Under the stated assumption, if , , as , then the sequence converges to the generalized value of the operator A at .
Proof. By applying the dual mapping to (5), we obtain
Let denote the set of generalized values of at , i.e.
By using  we obtain . It follows from (6) that
It is easy to show that and hence
It follows from (7) and (8), that
It follows from (9), that
In view of preceding remark and (2) we obtain
Since is an E- space and from (10) and by using  we see that the sequence converges to as , , .
The theorem is proved.
As a simple concrete example of this type of approximation, consider differentiation in . That is, the operator A is defined on , the Sobolev space of functions possessing a weak derivative in by
For a given data function and a given data operator is defined on possessing a weak derivative in , by satisfying
The stabilized approximate derivative (3) is easily seen (using Fourier transform analysis) to be given by
where the kernel is given by
Then in (2) is the approximate value of the operator A at for this method.
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 Abramov, A. and Gaipova, A.N. (1972) On the Solvability of Certain Equations Containing Monotonic Discontinuous Transformations. USSR Computational Mathematics and Mathematical Physics, 12, 320-324.