UDC 513.83

MATHEMATICS

A. D. GORBUNOV

ON THE WELL-POSEDNESS AND PSEUDO-WELL-POSEDNESS OF PROBLEMS

(Presented by Academician Yu. N. Rabotnov on 24 IX 1966)

1. Let a mapping \(A\) be given from a topological space \(R_1\) into a topological space \(R_2\):
\(y=Ax\), \(x\) ranges over \(P_A\), \(y\) ranges over \(Q_A\) \((P_A\subset R_1,\ Q_A\subset R_2)\).
Let the equation

\[ Ax=g,\qquad g\in R_2 \tag{1} \]

be considered with respect to \(x\); \(g\) is an arbitrary fixed element.

As usual \((^{1,2})\), we shall say that problem (1) is well-posed or correctly formulated if the following conditions are satisfied:

I. The equality \(Q_A=R_2\) holds and the space \(R_2\) is complete.
II. The mapping \(A\) is one-to-one.
III. The mapping \(A^{-1}\) is continuous.

If at least one of the listed conditions is violated, we shall say that problem (1) is ill-posed or incorrectly formulated.

2. Let a translation \(T\) be given from a topological space \(R_1\) into a topological space \(R_2\): \(\eta=Tx\), \(x\) ranges over \(P_T\), and \(Q_T\) is the union of the sets \(\eta\). (It is assumed that the topological spaces occurring in the article are once and for all equipped with neighborhoods and generalized neighborhoods; moreover, in metric spaces, spheres of points are taken as neighborhoods, and spheres of sets as generalized neighborhoods.)

We shall say that the translation \(T\) is partially continuous of the \(i\)-th type \((i=0,1,2,3,4,5,6,7)\) at \(x=x_0\), if \(x_0\in P_T\) and there exists a sequence \(W_T^{(m)}(x_0)\), \(m=1,2,\ldots\), of subsets of the set \(P_T\) such that \(x_0\in W_T^{(m)}(x_0)\), \(\overline{P}_T=\bigcup_m W_T^{(m)}(x_0)\), and for every \(m\) the translation
\(\eta=Tx\), \(x\) ranges over \(W^{(m)}(x_0)\), \(\eta\subset Q_T\), is continuous of the \(i\)-th type at the point \(x_0\).

We shall say that the translation \(T\) is partially continuous of the \(i\)-th type \((i=0,1,2,3,4,5,6,7)\), if it is partially continuous at every point of the set \(P_T\) and if, for every point \(x_0\in P_T\) and every sequence of points \(x_k\), \(k=1,2,\ldots\), of the set \(P_T\) converging in \(R_1\) to \(x_0\), the sequence of sets \(W_T^{(m)}(x_k)\), \(k=1,2,\ldots\), converges in \(R_1\) of the seventh type to the set \(W_T^{(m)}(x_0)\) \((m=1,2,\ldots)\) \((^3)\).

It is easy to see that the concept of partial continuity of the \(i\)-th type generalizes the concept of continuity in various directions. Thus, if the translation \(T\) is partially continuous of the \(i\)-th type at the point \(x_0\) and \(W_T^{(m)}(x_0)=P_T\), then the translation \(T\) is continuous of the \(i\)-th type at this point. If, however, the translation \(T\) is partially continuous of the \(i\)-th type and one-to-one, then it represents a partially continuous mapping.

We shall say that the translation \(T\) is strictly partially discontinuous of the \(i\)-th type if it is partially continuous of the \(i\)-th type, but is not continuous of the \(i\)-th type \((i=0,1,2,3,4,5,6,7)\).

3. In the case where problem (1) is ill-posed, let us introduce the following notions.

A translation $\xi = Bg$, where $g$ ranges over $R_2$, $\xi \subset R_1$, will be called a generalized solution of problem (1) if $Bg = A^{-1}g$ when $g$ ranges over $Q_A$. A generalized solution will be called complete if the translation $B$ is complete.

A translation $\xi = B^*g$, where $g$ ranges over $R_2$, $\xi \subset R_1$, will be called a partial generalized solution of problem (1) if $B^*g \subset A^{-1}g$ when $g$ ranges over $Q_A$.

A mapping $x = B_1g$, where $g$ ranges over $R_2$, $x \in R_1$, will be called a partial generalized solution of problem (1) if $B_1g \subset A^{-1}g$ when $g$ ranges over $Q_A$.

We shall call problem (1) pseudo-well-posed, or pseudo-correctly posed, if the following conditions are satisfied:

A. The space $R_2$ is complete.
B. There exists at least one generalized solution $\xi = Bg$, where $g$ ranges over $R_2$, $\xi \subset R_1$, of problem (1).
C. The translation $B$ is partially continuous of zero type.

We shall call problem (1) partially pseudo-well-posed if, in the conditions of the preceding definition, the translation $B$ constitutes a partial generalized solution of problem (1).

We shall call problem (1) partially well-posed if the following conditions are satisfied:

A′. The space $R_2$ is complete.
B′. There exists at least one partial generalized solution $x = B_1g$, where $g$ ranges over $R_2$, $x \in R_1$, of problem (1).
C′. The mapping $B_1$ is partially continuous.

It is clear that the notions of pseudo-well-posedness, partial pseudo-well-posedness, and partial well-posedness generalize, in different directions, the notion of well-posedness. Thus every well-posed problem is pseudo-well-posed, partially pseudo-well-posed, and partially well-posed.

If, in the conditions of the definition of pseudo-well-posedness (partial pseudo-well-posedness), continuity of the $i$-th type $(i = 0, 1, 2, 3, 4, 5, 6, 7)$ holds, then we shall say that problem (1) is pseudo-well-posed (partially pseudo-well-posed) of the $i$-th type.

If, in the conditions of the definition of pseudo-well-posedness (partial pseudo-well-posedness), partial discontinuity of the $i$-th type $(i = 0, 1, 2, 3, 4, 5, 6, 7)$ holds, then we shall say that problem (1) is pseudo-well-posed (partially pseudo-well-posed) of the $(\alpha+i)$-th type, where $\alpha$ is some transfinite $\geq 0$.

If, in the conditions of the definition of pseudo-well-posedness (partial pseudo-well-posedness), purely partial discontinuity of the $i$-th type $(i = 0, 1, 2, 3, 4, 5, 6, 7)$ holds, then we shall say that problem (1) is pseudo-well-posed (partially pseudo-well-posed) purely of the $(\alpha+i)$-th type, in order to emphasize that $\alpha > 0$.

If, in the conditions of the definition of pseudo-well-posedness, the translation $B$ is single-valued, we shall say that problem (1) is pseudo-well-posed with uniqueness.

Theorem 1. If problem (1) is pseudo-well-posed of the $(\alpha+7)$-th type, then the translation $A^{-1}$ admits a unique continuous extension of the $(\alpha+7)$-th type to the set $\overline{Q}_A$.

It follows from the theorem that if the set $Q_A$ is everywhere dense in $R_2$ and problem (1) is pseudo-well-posed of the $(\alpha+7)$-th type with respect to some complete generalized solution, then the latter is determined in a unique way.

A generalized solution of problem (1) will be called minimal if it constitutes a minimal extension of the translation $A^{-1}$.*

* An extension $\tilde B$ of a translation $B$ to the set $P_{\tilde B}$ is called minimal if the translation $\tilde B$ on the set $P_{\tilde B} \setminus P_B$ is single-valued.

Theorem 2. If problem (1) is pseudowell-posed of type \((a+i)\) \((i=0, 1, 2, 3, 4, 5, 6, 7)\) with respect to some minimal generalized solution, then the translation \(A^{-1}\) admits a unique minimal discontinuous continuation of type \((a+i)\) to the set \(\overline{Q}_A\).

It follows from the last theorem that if the set \(Q_A\) is everywhere dense in \(R_2\) and problem (1) is pseudowell-posed of type \((a+i)\) \((i=0, 1, 2, 3, 4, 5, 6, 7)\) with respect to some minimal generalized solution, then the latter is determined uniquely.

1. Let us dwell on some examples.

Example 1. Suppose two real Euclidean spaces are given: \(R_1\) of dimension \(n\) and \(R_2\) of dimension \(m\). Let a real matrix \(A=[a_{ik}]\) of size \((m\times n)\), of rank \(r\), be given. Consider the equation

\[ Ax=g,\qquad g\in R_2 \tag{2} \]

with respect to \(x\); \(g\) is an arbitrary given element. If \(r=n=m\), then problem (2), as is well known, is well-posed. In the other cases it is, for one reason or another, ill-posed. We define a generalized solution of problem (2) as follows. Consider the functional

\[ \Phi_g x=\|Ax-g\|^2,\qquad x \text{ ranges over } P_A \tag{3} \]

and denote by \(Bg\) the set of elements minimizing it. This set is always nonempty \((^3)\). Since for every \(g\in Q_A\) the equality \(Bg=A^{-1}g\) holds, the translation \(B\) represents a generalized solution of problem (3).

Theorem 3. If \(r=n<m\), then problem (3) is pseudowell-posed with uniqueness, and the mapping \(B\) is continuous.

Theorem 4*. If \(r<n\le m\) or \(r\le m<n\), then problem (3) is pseudowell-posed of the 7th type.

Example 2. Let \(H\) be a real Hilbert space and let \(A\) be a distributive operator (a mapping from \(H\) into \(H\)) with real eigenvalues and with a system of orthonormal eigenvectors complete in the subspace \(P_A\). Consider the equation

\[ Ax=g,\qquad g\in H \tag{4} \]

with respect to \(x\); \(g\) is an arbitrary fixed element.

It is easy to see that problem (4) is ill-posed except in the case when \(A\) is a topological mapping of the space \(H\) onto itself.

To define a generalized solution of (4), consider the functional

\[ \Phi_g x=\|\tilde A x-g\|^2,\qquad x \text{ ranges over } P_{\tilde A}, \tag{5} \]

where \(\tilde A\) is the minimal distributive continuation of the operator \(A\) with closed range. We denote by \(Bg\) the set of elements minimizing the functional (5) \((^4)\). Since for every \(g\in Q_A\), \(Bg=A^{-1}g\), \(B\) constitutes a generalized solution of problem (4).

Theorem 5. If zero is neither an eigenvalue of the operator \(A\) nor a limit point of its eigenvalues, then problem (4) is pseudowell-posed with uniqueness, and the mapping \(B\) is continuous.

Theorem 6. If zero, being a limit point of the eigenvalues of the operator \(A\), is not an eigenvalue of it, then problem (4) is pseudowell-posed with uniqueness, and the mapping \(B\) is purely partially discontinuous.

Theorem 7. If zero, being an eigenvalue of the operator \(A\), is not a limit point of its eigenvalues, then problem (4) is pseudowell-posed of the seventh type.

Theorem 8. If zero is both an eigenvalue of the operator \(A\) and a limit point of its eigenvalues, then problem (4) is pseudowell-posed of pure type \((a+7)\).

Moscow State University
named after M. V. Lomonosov

Received
16 IX 1966

CITED LITERATURE

\(^{1}\) I. G. Petrovskii, Lectures on Partial Differential Equations, Moscow–Leningrad, 1950, pp. 80, 81.
\(^{2}\) A. N. Tikhonov, DAN, 153, No. 1 (1964).
\(^{3}\) A. D. Gorbunov, DAN, 174, No. 1 (1966).
\(^{4}\) V. I. Smirnov, A Course of Higher Mathematics, Moscow, 1959, pp. 393–395.

* The condition of Theorem 4 should be read as follows: if the mapping \(B\) of the space \(R_1\) onto \(R_2\) is open and the translation \(B^{-1}\) is finite-valued.