UDC 517.948

MATHEMATICS

V. P. TANANA

ILL-POSED PROBLEMS AND GEOMETRIES OF BANACH SPACES

(Presented by Academician A. N. Tikhonov on 8 I 1970)

1. Statement of the problem. Let \(X\) be a reflexive space; \(Y\) a separable locally convex space, and \(A\) a linear continuous operator mapping \(X\) into \(Y\), such that there exists an inverse operator \(A^{-1}\), which, generally speaking, is not continuous. Consider the operator equation of the first kind

\[ Ax = y_0,\qquad x \in X,\qquad y_0 \in R(A) \subset Y. \tag{1} \]

Suppose that \(y_0\) is not known to us, and instead of it there is given a basis of a filter of neighborhoods \(\{V_\delta\}\) of the point \(y_0\), where each \(V_\delta\) is a convex closed set: \(V_\delta=\overline{\operatorname{co}}(V_\delta)\).

It is required, from the given \(V_\delta\), to find an approximate solution \(x_\delta\) of equation (1) such that \(x_\delta \to x_0\), where \(x_0\) is the exact solution of equation (1), i.e. \(x_0=A^{-1}y_0\).

2. The residual method. The idea of a stable solution of an operator equation of the first kind

\[ \bar A[x]=u_0\qquad (x\in X,\quad u_0\in U), \tag{2} \]

where \(\bar A[x]\) is a continuous operator from \(X\) into \(U\), satisfying the uniqueness condition: \(\bar A[x_1]\ne \bar A[x_2]\) if \(x_1\ne x_2\); \(X,U\) are linear normed spaces, was first put forward by A. N. Tikhonov \(\left({}^{1-3}\right)\). Its basis was the additional assumption that the exact solution \(x_0\): \(\bar A[x_0]=u_0\) of equation (2) belongs to some class of well-posedness (stabilization) \(m\), \(m\subset X\).

In \(\left({}^{3}\right)\), for example, for solving equation (2) the idea of a compact embedding is developed. This idea may be interpreted, as was done by V. K. Ivanov in \(\left({}^{5}\right)\) for equation (2), as follows: it was assumed that there is a Hilbert space \(Z\), which is mapped into \(X\) by means of a linear completely continuous operator \(B\). It was further assumed that the exact solution \(x_0\) of equation (2) belongs to the range \(R(B)\) of the operator \(B\) in the space \(X\): \(x_0\in R(B)\). The set \(R(B)\), \(R(B)\subset X\), in this case plays the role of the stabilization class in the space \(X\), and, generally speaking, \(R(B)\ne X\). The residual method for solving equation (2) (see \(\left({}^{5}\right)\)) consisted in finding an element \(z_\delta\) with minimal norm on the set
\(\Omega_\delta:\ \Omega_\delta=\{z:\ \|Cz-u_\delta\|\le \delta\}\), where \(C=\bar A B\) is a continuous operator acting from \(Z\) into \(U\), \(u_\delta\in U\), \(\|u_0-u_\delta\|\le \delta\).

Let us note that if the operator \(\bar A\), in addition, is linear, then for a stable solution of equation (2) it is sufficient to have merely a continuous embedding, i.e. the embedding operator \(B\), mapping \(Z\) into \(X\), may be assumed linear and continuous (see \(\left({}^{13,14}\right)\)). In connection with the idea of a continuous embedding, the case \(Z=X\), and \(B\) the identity operator on \(Z\), is of interest. In this case the stabilization class \(R(B)\) is maximal in the space \(X\), or, in other words, \(R(B)=X\). This circumstance is very convenient, since the necessity of checking whether the exact solution \(x_0\) of equation (2) belongs to the stabilization class \(R(B)\) is eliminated.

In the work of V. A. Morozov, for example (see \(\left({}^{13}\right)\)), the space \(X\) is assumed to be Hilbert, and the stabilization class to coincide with the whole \(X\),

and \(\overline A\) is a linear continuous operator mapping \(X\) into a linear normed space \(U\), and under these assumptions a stable solution of equation (2) is given by the method of A. N. Tikhonov. However, even the assumption that the spaces \(X\) are Hilbert spaces is inessential (see \((^{14},\,^{16})\)). It is enough that \(X\) satisfy the Efimov–Stechkin condition (see \((^{17})\)). An example of an Efimov–Stechkin space may be, for instance, the space \(L_p\) \((p>1)\).

A natural question arises: in which spaces may one suppose the class of stabilization \(R(B)\) to coincide with the whole space \(X\)? This note is devoted to the solution of this question. Therefore, in what follows we shall consider only the case \(Z=X\) and \(B\) is the identity operator on \(X\). Then the residual method, as applied to the solution of the problem posed in Sec. 1, consists in finding a point of minimum of the functional

\[ K[x]=\|x\| \tag{3} \]

under the condition

\[ Ax\in V_\delta . \tag{4} \]

Since \(X\) is reflexive, the point of minimum exists (see \((^{10})\)). This point is unique if \(X\) is strictly convex and reflexive (see \((^{10})\)). Therefore, in what follows we shall always assume \(X\) to be strictly convex and reflexive. The points of minimum will be called solutions of the problem posed in Sec. 1, obtained by the residual method, and will be denoted by \(x_\delta\). In what follows, the problem posed in Sec. 1 will be denoted by \(\theta\): \(\theta=\theta[X,Y,x_0,A,\{V_\delta[Ax_0]\}]\), where \(X\in\mathfrak X\), \(\mathfrak X\) is the class of reflexive strictly convex spaces; \(Y\in\mathfrak Y\), \(\mathfrak Y\) is the class of separable locally convex spaces; \(x_0\in X\); \(A\in\mathcal L(X,Y)\), \(\mathcal L(X,Y)\) is the space of linear continuous one-to-one operators acting from \(X\) into \(Y\); \(\{V_\delta[Ax_0]\}\in\mathfrak B\), \(\mathfrak B\) is the totality of all such bases of the filter of neighborhoods of the point \(Ax_0\in Y\), each of which consists only of convex closed sets of the space \(Y\). By \(\Theta_X\) we shall denote the set of problems \(\theta\) for which \(X\)—a reflexive strictly convex space—is fixed, whereas the remaining components of the problem \(\theta\) range over the corresponding domains of definition.

3. Basic definitions.

Definition 1. The residual method will be called stable for the problem \(\theta\) if the solutions \(x_\delta\) of this problem, obtained by the residual method, converge strongly to the exact solution \(x_0\) of equation (1).

Remark. In this definition all components of the problem \(\theta\) are regarded as fixed.

Definition 2. The residual method will be called stable on the space \(X\) if it is stable for every problem \(\theta\): \(\theta\in\Theta_X\).

Definition 3. We shall say that the space \(X\) satisfies condition (A1) if from the fact that \(\{x_n\}\subset X\), \(x_0\in X\),

\[ x_n \xrightarrow{\mathrm{w}} x_0 \quad\text{and}\quad \|x_n\|=\inf_{x\in \overline{\operatorname{co}}(x_n,x_{n+1},\ldots)} \|x\| \]

it follows that \(x_n\to x_0\).

Definition 4. We shall say that the space \(X\) satisfies condition (A2) if from the fact that \(\{x_n\}\subset X\), \(x_0\in X\), \(x_n\xrightarrow{\mathrm{w}}x_0\), \(\|x_n\|=1\) and

\[ \lim_{n\to\infty}\sup_{k\ge n}\rho(x_k,H_n)=0, \]

where \(H_n=\{x:f_n(x)=1,\ f_n\in X^*,\ \|f_n\|=1\}\) is a closed hyperplane supporting the ball \(s=\{x:\|x\|\le 1\}\) at the point \(x_n\), it follows that \(x_n\to x_0\).

Definition 5. We shall say that the space \(X\) satisfies condition (A3) if from the fact that \(\{x_n\}\subset X\), \(x_0\in X\), \(\|x_n\|=1\),

\[ x_n \xrightarrow{\mathrm{w}} x_0 \quad\text{and}\quad \lim_{n\to\infty}\rho(x_0,H_n)=0, \]

where \(H_n\) is a closed hyperplane supporting the ball \(s=\{x:\|x\|\le 1\}\) at the point \(x_n\), it follows that \(x_n\to x_0\).

Definition 6. We shall say that the space \(X\) satisfies condition (A4) if from the fact that \(\{x_n\}\subset X\), \(x_0\in X\), \(\|x_n\|=\|x_0\|=1\), and \(x_n \xrightarrow{\mathrm{w}} x_0\), it follows that \(x_n\to x_0\).

Definition 7. The space \(X\) is called an \(E\)-space (see (9)) if it is reflexively strictly convex and satisfies condition (A4). \(E\)-spaces are the largest known class of Banach spaces for which the operator \(P\) of metric projection onto a closed convex set is well-posed in the sense of Hadamard (see (6)).

4. Main theorems.

Theorem 1. Let \(X\) be a separable reflexive strictly convex space. In order that the residual method be stable on the space \(X\), it is necessary that \(X\) satisfy condition (A1).

Theorem 2. Let \(X\) be a reflexive strictly convex space. In order that the residual method be stable on the space \(X\), it is sufficient that \(X\) satisfy condition (A4).

Theorem 3. Let \(X\) be a reflexive strictly convex space; then conditions (A1), (A2), (A3), (A4) on the space \(X\) are equivalent.

Theorem 4. Let \(X\) be a separable reflexive strictly convex space. In order that the residual method be stable on the space \(X\), it is necessary and sufficient that \(X\) be an \(E\)-space.

5. Finite-dimensional approximations of problem \(\theta\). Let \(X\) be a separable \(E\)-space and

\[ X_1 \subseteq X_2 \subseteq \ldots \subseteq X_n \subseteq \ldots \subseteq X \tag{5} \]

be an increasing chain of finite-dimensional subspaces of the space \(X\) such that

\[ \overline{\bigcup_{n=1}^{\infty} X_n}=X. \tag{6} \]

The existence of such an increasing chain follows easily from the separability of \(X\). We shall solve the problem of finding a point of minimum of the functional

\[ K[x]=\|x\| \tag{7} \]

under the conditions

\[ x\in X_n,\qquad Ax\in V_\delta. \tag{8} \]

From the fact that \(X\) is an \(E\)-space, there follows the existence and uniqueness of the point of minimum. Denote this point by \(x_\delta^n\).

Theorem 5. The sequence of elements \(\{x_\delta^n\}\) converges strongly to the approximate solution \(x_\delta\) of problem \(\theta\):

\[ x_\delta^n\to x_\delta \quad \text{as } n\to\infty . \tag{9} \]

Ural State University
named after A. M. Gorky
Sverdlovsk

Received
25 VIII 1969

CITED LITERATURE

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