Abstract
Full Text
MATHEMATICS
A. E. GEL′MAN
ON MAJORANT EQUATIONS
(Presented by Academician V. I. Smirnov, 4 III 1961)
In the monographs \((^{1,2})\) L. V. Kantorovich, introducing the notion of a majorant of an operator, investigated an operator equation of the form
\[ x = U(x) \]
in \(B_R\)- and \(B\)-spaces, establishing a number of important results concerning its solution (in \((^2)\) the operator \(U(x)\) is assumed to be continuously differentiable).
In the present work the notion of a strict majorant of an operator is introduced, making it possible to formulate a new criterion for the existence of a solution for an operator equation of the form
\[ y = \Omega(y) \]
in complete metric spaces, and also to establish for it a number of quantitative results; here no conditions more restrictive than those imposed in Banach’s principle are placed on the operator \(\Omega\).
It is shown that for one class of metric spaces (containing the class of \(B\)-spaces) it is possible effectively to construct, in a definite sense, the best of the strict majorants. The results obtained in this case in principle cannot be achieved by applying Banach’s principle.
We shall use the following notation: \(Y\) is a complete metric space; \(\Omega\) is an operator defined on \(Y\), with \(\Omega(Y) \subset Y\); \(S_R(y_0)\) is the set of elements of \(Y\) satisfying the condition \(\rho(y,y_0) < R\) \((R>0)\); \(F(x)\) is a real function of a real argument, defined on \([0,R)\).
Definition 1. We shall call a function \(F(x)\) a strict majorant of the operator \(\Omega(y)\) in \(S_R(y_0)\) if: 1) it is increasing and convex; 2) \(\rho[y_0,\Omega(y_0)] \leq F(0)\); 3) from \(y_1 \in S_R(y_0)\), \(y_2 \in S_R(y_0)\), and \(y_1 \ne y_2\) it follows that
\[ \frac{\rho[\Omega(y_2),\Omega(y_1)]}{\rho(y_2,y_1)} \leq \frac{F[\rho(y_0,y_2)] - F[\rho(y_0,y_1)]}{\rho(y_0,y_2) - \rho(y_0,y_1)} \; * . \]
It can be shown that, in order that the operator \(\Omega(y)\) have in \(S_R(y_0)\) a strict majorant, it is necessary and sufficient that in \(S_R(y_0)\) it satisfy a Lipschitz condition.
Definition 2. Let \(M(y_0,\Omega)\) be the set of all positive numbers \(r\) having the property that in \(S_r(y_0)\) there lies only one solution of the equation
\[ y = \Omega(y). \tag{1} \]
* If \(\rho(y_0,y_2)=\rho(y_0,y_1)=x_0\), then the right-hand side of the inequality should be understood as the upper derivative of \(F(x)\) at the point \(x_0\).
Obviously, \(M(y_0,\Omega)\) is an interval (possibly empty). We shall call the equation
\[ x=F(x) \tag{2} \]
\(y_0\)-strictly majorant for equation (1), if \(M(0,F)\subset M(y_0,\Omega)\).*
It is clear from the definition that if equation (2) is \(y_0\)-strictly majorant for equation (1) and \(M(0,F)\) is nonempty, i.e. \(M(0,F)=(a,b)\), where \(a<b\), then: 1) \(a\) is the smallest root of equation (2), and \(b\) is either the second largest root (if \(b<R\)), or the right endpoint of the interval on which the function \(F(x)\) is defined; 2) equation (1) has a solution \(y\) satisfying the inequality \(\rho(y_0,y)\leq a\); 3) if equation (1) has solutions different from \(y\), then all of them lie outside \(S_b(y_0)\).
Theorem 1. Let \(F(x)\) be a strict majorant of the operator \(\Omega(y)\) in \(S_R(y_0)\). Then the equation \(x=F(x)\) is \(y_0\)-strictly majorant for the equation \(y=\Omega(y)\).
It is clear from the theorem that knowledge of a strict majorant of the operator \(\Omega(y)\) gives important information about the solution of the equation \(y=\Omega(y)\). We shall give one property of strict majorants.
Let \(\bar R\) be the greatest of the radii of the spheres \(S_R(y_0)\) in which the operator \(\Omega(y)\) satisfies the Lipschitz condition. Introduce the functions \(\hat f(x)\) and \(\widetilde F(x)\), defined on \([0,\bar R)\):
\[ \hat f(x)= \sup_{\substack{\rho(y_2,y_0)<x\\ \rho(y_1,y_0)<x\\ y_1\ne y_2}} \frac{\rho[\Omega(y_2),\Omega(y_1)]}{\rho(y_2,y_1)}, \qquad \widetilde F(x)=\rho[y_0,\Omega(y_0)]+\int_0^x \hat f(t)\,dt . \]
The following simple statement holds.
Proposition. If \(F(x)\) is a strict majorant of the operator \(\Omega(y)\) on \(S_R(y_0)\), then on the interval \([0,R)\) the inequality
\[ F(x)\geq \widetilde F(x) \]
holds.
In this connection it is important to study the question: is the function \(\widetilde F(x)\) always a strict majorant of the operator \(\Omega(y)\)? We do not have an answer to this question in the case when \(Y\) is an arbitrary complete metric space. It is possible, however, to indicate a class of spaces in which this question can be answered affirmatively. The results obtained in this direction are set forth below.
Definition. We shall call a complete metric space \(Y\) an A-space (after A. D. Aleksandrov, who first considered such objects in the monograph \((^3)\)) if it has the following properties:
-
Any two points of this space \(y_1\) and \(y_2\) can be joined by a curve** \(L(y_1,y_2)\), all points of which also belong to \(Y\), and whose length is equal to \(\rho(y_1,y_2)\) (there exists a shortest path).
-
From \(y\in L(y_1,y_2)\) and \(y_0\in Y\) it follows that
\[ \rho(y_0,y)-\rho(y_0,y_1) \leq \frac{\rho(y,y_1)}{\rho(y_2,y_1)} \,[\,\rho(y_2,y_0)-\rho(y_1,y_0)\,], \]
i.e. the metric \(\rho(y_1,y_2)\) is convex.***
* If the smallest root \(x_1\) of equation (2) is multiple (i.e. \(x_1\) is a minimum point of the function \(F(x)-x\)), then in defining \(M(0,F)\) we count it as two equal roots. In this case, obviously, \(M(0,F)\) is empty.
** We understand here the terms “curve” and “length of a curve” in the sense of A. D. Aleksandrov \((^3)\).
*** We give a different definition of convexity of the metric than in \((^3)\).
It is easy to see that \(B\)-spaces are a special case of \(A\)-spaces.
Theorem 2. If \(Y\) is an \(A\)-space, then \(\widetilde F(x)\) is a strict majorant of the operator \(\Omega(y)\).
In cases where \(f(x)\) is not constant, \(\widetilde F(x)\) is a strict majorant of the operator \(\Omega(y)\), \(M(0,\widetilde F)\) is nonempty, and \(b<R\), the estimates obtained for the solution of equation (1) and for the domain of uniqueness cannot be obtained with the aid of Banach’s principle (this is seen, for example, from the fact that \(M(0,\widetilde F)\) contains points where \(f(x)>1\)).
In conclusion we note that Theorems 1 and 2, for the sake of brevity, have been formulated here in a somewhat weakened version.
Leningrad Electrotechnical Institute
named after V. I. Ulyanov (Lenin)
Received
23 II 1961
REFERENCES
- L. V. Kantorovich, B. Z. Vulikh, A. G. Pinsker, Functional Analysis in Semiordered Spaces, Moscow–Leningrad, 1950.
- L. V. Kantorovich, G. P. Akilov, Functional Analysis in Normed Spaces, Moscow, 1959.
- A. D. Aleksandrov, Intrinsic Geometry of Convex Surfaces, Moscow–Leningrad, 1948.