Abstract
Full Text
Reports of the Academy of Sciences of the USSR
1962, Volume 142, No. 4
MATHEMATICS
A. F. ANDREEV
A UNIQUENESS THEOREM FOR A NORMAL FROMMER DOMAIN OF THE SECOND TYPE
(Presented by Academician V. I. Smirnov on 3 X 1961)
In studying the behavior of integral curves of a first-order differential equation in a neighborhood of a singular point, one often has to deal with the so-called distinction problems for Frommer normal domains \((^1,^2)\). In the present note we shall consider the distinction problem for a normal domain of the second type (a domain \(N_2\)). The formulation of this problem is apparent from the following Hartman–Wintner theorem \((^3)\).
Theorem on the existence of an \(O\)-curve in \(N_2\). Let, in the equation
\[ \alpha(r)\frac{d\varphi}{dr}=\Psi(r,\varphi) \tag{1} \]
1) the function \(\Psi(r,\varphi)\) be defined and continuous in \(r,\varphi\) in the domain
\[ 0<r\leq \rho,\qquad -\delta\leq \varphi\leq \delta, \tag{N} \]
where \(\rho\) and \(\delta\) are positive numbers; \(\dfrac{1}{\Psi(r,\varphi)}\) is bounded for sufficiently small \(r\) outside any fixed neighborhood of the point \(O(0,0)\);
2) \(\Psi(r,-\delta)>0,\ \Psi(r,\delta)<0\) for \(0<r\leq \rho\);
3) the function \(\alpha(r)\) be defined, continuous, and positive for \(0<r\leq \rho\),
\[ \int_0^\rho \frac{dr}{\alpha(r)}=+\infty . \]
Then there exists at least one solution \(\varphi(r)\) of equation (1), which is defined on the interval \(0<r\leq \rho\), and every such solution has the property \(\varphi(r)\to 0\) as \(r\to 0\) (i.e., it is an \(O\)-curve of equation (1)).
Under the conditions of the Hartman–Wintner theorem, the domain \((N)\) is, for equation (1), a Frommer normal domain of the second type (a domain \(N_2\)). In it equation (1) has: a) either a unique \(O\)-curve, b) or an infinite set of \(O\)-curves. Thus there arises the problem of distinguishing these two possible arrangements of the integral curves of equation (1) in the domain \(N_2\). We note that if \(\Psi(r,\varphi)\) in the domain \((N)\) does not increase with increasing \(\varphi\) for each fixed \(r\), then there exists a unique \(O\)-curve (Peano’s criterion). In the general case the problem remains open. Lohn \((^2,^4)\), R. E. Vinograd and D. M. Grobman \((^5)\), A. F. Andreev \((^6,^7)\), I. S. Kukles \((^8,^9)\), and D. M. Gruz \((^{10})\) gave a series of sufficient criteria for uniqueness of the \(O\)-curve in \(N_2\). These criteria partially overlap, but do not completely cover one another (there are errors in the papers of Kukles and Gruz; see \((^{11})\)). Below we shall prove a sufficient uniqueness criterion that includes, as special cases, the criteria of Lohn and of Vinograd–Grobman, as well as the criteria of Kukles and Gruz (corrected).
We shall consider the equation
\[ \alpha(r)\frac{d\varphi}{dr}=\Phi(\varphi)(1+\beta(r,\varphi))+\psi(r,\varphi)\equiv \Psi(r,\varphi) \tag{2} \]
under the following assumption.
Condition A. 1) The function \(\alpha(r)\) is defined, continuous, and positive for \(0<r\leq \rho_1\); \(\rho_1\) is a constant,
\[ \int_0^{\rho_1}\frac{dr}{\alpha(r)}=+\infty. \]
2) The function \(\Phi(\varphi)\) is defined and continuous on the interval \([-\delta_1,\delta_1]\), \(\delta_1>0\) a constant; \(\Phi(0)=0\); for \(\varphi\ne 0\), \(\varphi\Phi(\varphi)<0\), \(|\Phi(\varphi)|\geq a|\varphi|^k\), where \(a\) and \(k\) are positive numbers.
3) The function \(\beta(r,\varphi)\) is defined in the region
\[ 0<r\leq \rho_1,\qquad -\delta_1\leq \varphi\leq \delta_1, \tag{3} \]
is continuous there with respect to \(r,\varphi\), and satisfies the inequality \(|\beta(r,\varphi)|\leq \varepsilon<1\), where \(\varepsilon\) is a constant.
4) The function \(\psi(r,\varphi)\) is defined in the region (3) and is continuous there with respect to \(r,\varphi\); \(\psi(r,\varphi)\to 0\) as \(r\to 0\), uniformly with respect to \(\varphi\in[-\delta_1,\delta_1]\).
Under these conditions, for sufficiently small \(\rho\) and \(\delta\), the region \((N)\) will be, for equation (2), a region \(N_2\), since in it the conditions of the Hartman–Wintner theorem are fulfilled. Consequently, equation (2) has in the region \((N)\) at least one \(O\)-curve.
Uniqueness theorem for an \(O\)-curve in \(N_2\). Let, for equation (2):
1) condition A be fulfilled;
2)
\[ \frac{\psi(r,\varphi)}{\omega^\sigma(r)}\to 0 \quad \text{as } r\to 0 \tag{4} \]
uniformly with respect to \(\varphi\in[-\delta,\delta]\), where \(\omega(r)\) is a fixed function of class \(C^1\) on \((0,\rho]\); \(\omega(r)>0\); \(\omega'(r)>0\) for \(0<r\leq \rho\); \(\omega(r)\to 0\) as \(r\to 0\); \(\sigma>0\) is a fixed number*;
3) in the region \(0<r\leq \rho,\ |\varphi|\leq u_0\omega^{\sigma/k}(r)\), where \(u_0>0\) is any small number, for \(\varphi_2>\varphi_1\)
\[ \Psi(r,\varphi_2)-\Psi(r,\varphi_1)\leq \frac{\sigma}{k}\Lambda(r)(\varphi_2-\varphi_1), \tag{5} \]
where the function \(\Lambda(r)\) is continuous on \((0,\rho]\) and satisfies the inequality
\[ \Lambda(r)\leq \frac{\alpha(r)\omega'(r)}{\omega(r)}. \]
Then equation (2) has a unique \(O\)-curve in the region \((N)\).
Proof. Dividing both sides of equation (2) by \(\omega^\sigma(r)\) and taking into account that, for \(|\varphi|\geq u_0\omega^{\sigma/k}(r)\), where \(u_0>0\) is any fixed number,
\[ \frac{|\Phi(\varphi)|}{\omega^\sigma(r)}\geq a\frac{|\varphi|^k}{\omega^\sigma(r)}\geq au_0^k, \]
we find that in the region
\[ 0<r\leq \rho,\qquad |\varphi|\geq u_0\omega^{\sigma/k}(r) \tag{6} \]
for any integral curve \(\varphi=\varphi(r)\) of equation (2), \(\varphi(r)\varphi'(r)<0\), if \(\rho\) is sufficiently small. Consequently, from the region (6) integral curves cannot approach the point \(O(0,0)\).
\[ \text{* From the assumptions concerning }\psi(r,\varphi)\text{ it follows that such functions }\omega(r)\text{ satisfying condition (4) exist.} \]
To study equation (2) in the domain
\[ 0<r\leqslant \rho,\qquad |\varphi|\leqslant u_0\omega^{\sigma/k}(r), \]
we transform it by the substitution
\[ \varphi=u\omega^{\sigma/k}(r). \]
In doing so we obtain the equation
\[ \frac{\omega(r)}{\omega'(r)}=u'=U(r,u)\equiv \frac{\Phi\left(u\omega^{\sigma/k}(r)\right)\left(1+\beta\left(r,u\omega^{\sigma/k}(r)\right)\right)+\psi\left(r,u\omega^{\sigma/k}(r)\right)} {\alpha(r)\omega^{\sigma/k-1}(r)\omega'(r)} -\frac{\sigma}{k}u, \tag{7} \]
which must be studied in the domain
\[ 0<r\leqslant \rho,\qquad |u|\leqslant u_0 \tag{8} \]
(where \(u_0>0\) is an arbitrarily small number); namely, it is necessary to show that in this domain it has a unique solution \(u(r)\), defined for all \(r\in(0,\rho]\).
But for equation (7) in the domain (8) all the conditions of the Hartman–Wintner theorem are fulfilled. In particular, for any number \(\varepsilon_0\) \((0<\varepsilon_0<u_0)\) one can indicate a number \(\rho_0>0\) such that in each of the two domains \(0<r\leqslant \rho_0,\ \varepsilon_0\leqslant |u|\leqslant u_0\) both terms on the right-hand side of equation (7) have the same sign, opposite to the sign of \(u\). Consequently, in these domains \(|U(r,u)|\geqslant \frac{\sigma}{k}|u|\geqslant \frac{\sigma}{k}\varepsilon_0\), i.e. \(\frac{1}{U(r,u)}\) is bounded. According to the Hartman–Wintner theorem, any solution of equation (7) which lies in the domain (8) and is defined for all \(r\in(0,\rho]\) has the property \(u(r)\to0\) as \(r\to0\). But equation (7) has in the domain (8) a unique solution possessing this property, since for it in this domain the conditions of Peano’s criterion are fulfilled: for \(u_2>u_1\)
\[ U(r,u_2)-U(r,u_1)\leqslant \left[\frac{\sigma}{k}\Lambda(r)-\frac{\omega(r)}{\alpha(r)\omega'(r)}-\frac{\sigma}{k}\right](u_2-u_1)\leqslant 0. \]
The theorem is proved.
In the formulation of the theorem there occur the functions \(\omega(r)\) and \(\Lambda(r)\). The first characterizes the smallness of the function \(\psi(r,\varphi)\) as \(r\to0\), the second the smallness of the Lipschitz coefficient of the right-hand side. As follows from the theorem, these two functions are organically connected with each other. Choosing various functions \(\omega(r)\), we find that equation (2) will have in the domain \((N)\) a unique \(O\)-curve for the following functions \(\Lambda(r)\):
| \(\omega(r)\) | \(o(1)\) | \(\dfrac{1}{\ln|\ln r|}\) | \(\dfrac{1}{|\ln r|}\) | \(r\) | \(e^{-1/r}\) |
| \(\Lambda(r)\) | \(\dfrac{\alpha(r)\omega'(r)}{\omega(r)}\) | \(\dfrac{\alpha(r)}{r|\ln r|\ln|\ln r|}\) | \(\dfrac{\alpha(r)}{r|\ln r|}\) | \(\dfrac{\alpha(r)}{r}\) | \(\dfrac{\alpha(r)}{r^2}\) |
From the table it is seen, in particular, that the smallness as \(r\to0\) of the Lipschitz coefficient of the right-hand side is a necessary condition for the uniqueness of the \(O\)-curve of equation (2) in the domain \(N_2\) only when the order of smallness of the function \(\psi(r,\varphi)\) is not high. If the function \(\psi(r,\varphi)\) has a sufficiently high order of smallness as \(r\to0\), then uniqueness is preserved even with an unbounded Lipschitz coefficient.
In works \((^{2-9})\) the problem of uniqueness of the \(O\)-curve in \(N_2\) was considered, as a rule, for the equation
\[ r\frac{d\varphi}{dr}=-a\varphi^k(1+\varepsilon(\varphi))+\psi(r,\varphi), \tag{9} \]
where \(k \geqslant 1\) is an odd number; \(a\) is a positive number; \(\varepsilon(\varphi)\) is an analytic function of \(\varphi\) in a neighborhood of the point \(\varphi=0\); \(\varepsilon(0)=0\); \(\psi(r,\varphi)\) satisfies condition A. This equation is a special case of equation (2). Its right-hand side will satisfy condition (5) if the function \(\psi(r,\varphi)\) satisfies this condition.
Applying our uniqueness theorem to equation (9), we obtain from it, for \(\omega(r)=O(1)\), the Lonn lemma \({}^{(2,4)}\); for \(\omega(r)=r\), the Vinograd–Grobman theorem \({}^{(5)}\); for
\[ \omega(r)=\frac{1}{|\ln r|} \]
and for
\[ \omega(r)= \frac{1}{ \ln \frac{1}{r}\, \ln \ln \frac{1}{r}\,\ldots\, \underbrace{\ln \ln \cdots \ln}_{n}\frac{1}{r} } \]
an amended version for two theorems of Kukles \({}^{(8,9)}\) (see also \({}^{(11)}\)).
Applying our theorem to the equation
\[ r\,\frac{d\varphi}{dr} = -a\varphi^k(1+\beta(r,\varphi))+\psi(r,\varphi), \]
where the number \(k \geqslant 1\) is odd; the number \(a>0\); \(\beta(r,\varphi)\) and \(\psi(r,\varphi)\) satisfy condition A, we obtain from it, for \(\omega(r)=r\), an amended version of one of Gruz’s theorems \({}^{(10)}\) (see also \({}^{(11)}\)).
Leningrad Branch
of the V. A. Steklov Mathematical Institute
Academy of Sciences of the USSR
Received
27 IX 1961
REFERENCES
\({}^{1}\) M. Frommer, UMN, No. 9, 212 (1941).
\({}^{2}\) V. V. Nemytskii, V. V. Stepanov, Qualitative Theory of Differential Equations, Moscow–Leningrad, 1949.
\({}^{3}\) R. Hartman, A. Wintner, Am. J. Math., 68, No. 2, 304 (1946).
\({}^{4}\) E. R. Lonn, Math. Zs., 44, No. 4, 515 (1938).
\({}^{5}\) R. E. Vinograd, D. M. Grobman, UMN, 12, No. 5, 193 (1957).
\({}^{6}\) A. F. Andreev, Vestn. Leningrad Univ., No. 13, 84 (1958).
\({}^{7}\) A. F. Andreev, Vestn. Leningrad Univ., No. 7, 18 (1959).
\({}^{8}\) I. S. Kukles, UMN, 14, No. 5, 151 (1959).
\({}^{9}\) I. S. Kukles, DAN, 128, No. 2, 241 (1959).
\({}^{10}\) D. M. Gruz, Tr. Uzbek. Univ., No. 78, 25 (1958).
\({}^{11}\) A. F. Andreev, Proceedings of the Fourth All-Union Mathematical Congress, 1961 (in press).