MATHEMATICS

A. F. ANDREEV

A STRENGTHENING OF THE UNIQUENESS THEOREM FOR AN \(O\)-CURVE IN \(N_2\)

(Presented by Academician V. I. Smirnov on 2 IV 1962)

Consider the equation

\[ \alpha(r)\frac{d\varphi}{dr}=\Phi(r,\varphi)+\psi(r,\varphi)\equiv \Psi(r,\varphi) \tag{1} \]

under the following assumption.

Condition A. 1) The function \(\alpha(r)\) is continuous and positive on the interval \((0,r_1]\), \(r_1>0\) is constant,

\[ \int_0^{r_1}\frac{dr}{\alpha(r)}=+\infty; \]

2) the functions \(\Phi(r,\varphi)\) and \(\psi(r,\varphi)\) are continuous in the domain (\(\varphi_1>0\) is constant)

\[ 0<r\leq r_1,\qquad -\varphi_1\leq \varphi\leq \varphi_1; \tag{2} \]

\(\Phi(r,0)\equiv 0\); for \(\varphi\ne 0\), \(\varphi\Phi(r,\varphi)<0\); \(|\Phi(r,\varphi)|\geq a|\varphi|^k\), \(a>0\), \(k>0\) are constants; \(\psi(r,\varphi)\to 0\) as \(r\to 0\), uniformly with respect to \(\varphi\in[-\varphi_1,\varphi_1]\).

Under this condition the domain

\[ 0<r\leq r_0,\qquad -\varphi_0\leq \varphi\leq \varphi_0, \tag{3} \]

if the constants \(r_0>0\) and \(\varphi_0>0\) are sufficiently small, will be a normal Frommer domain of the second type (an \(N_2\)-domain) for equation (1), and for it there arises the problem of uniqueness of the \(O\)-curve (integral curve of equation (1) adjoining the point \(r=0,\varphi=0\) from this domain).

In note \((^1)\) a uniqueness theorem for an \(O\)-curve in \(N_2\) was proved for an equation close to (1). The following theorem contains much less restrictive uniqueness conditions (including for the equation considered in (1)).

Theorem. Suppose that for equation (1):

1) condition A is satisfied;

2)

\[ \frac{\psi(r,\varphi)}{\omega^\sigma(r)}\to 0 \]

as \(r\to 0\), uniformly with respect to \(\varphi\in[-\varphi_0,\varphi_0]\), where \(\omega(r)\) is a fixed function of class \(C^1\) on \((0,r_0]\); \(\omega(r)>0\), \(\omega'(r)>0\) for \(r\in[0,r_0]\); \(\omega(r)\to 0\) as \(r\to 0\); \(\sigma>0\) is a fixed number;

3) there exists a number \(u_0>0\) such that in the subdomain \(|\varphi|\leq u_0\omega^{\sigma/k}(r)\) of the domain (3), for \(\bar{\varphi}>\bar{\bar{\varphi}}\),

\[ \Psi(r,\bar{\varphi})-\Psi(r,\bar{\bar{\varphi}}) \leq \frac{\sigma}{k}\Lambda(r)(\bar{\varphi}-\bar{\bar{\varphi}}), \]

where \(\Lambda(r)\) is continuous on \((0,r_0]\) and satisfies the inequality (\(M\) is constant, \(r\) is any point of \((0,r_0]\)),

\[ \int_r^{r_0}\left(\frac{\Lambda(r)}{\alpha(r)}-\frac{\omega'(r)}{\omega(r)}\right)\,dr\leq M<+\infty. \]

Then equation (1) has a unique \(O\)-curve in the domain (3).

Proof. Analogously to how this was done in \((^{1})\), it is easy to show that the problem under consideration is equivalent to the problem of uniqueness of the \(O\)-curve of the equation

\[ \frac{\omega(r)}{\omega'(r)}u' = \frac{\Psi\left(r,u\,\omega^{\sigma/k}(r)\right)} {\alpha(r)\omega^{\sigma/k-1}(r)\omega'(r)} -\frac{\sigma}{k}u \equiv U(r,u) \tag{4} \]

in the domain

\[ 0<r\le r_0,\qquad -u_0\le u\le u_0 . \tag{5} \]

Let \(u=u_1(r)\) be an \(O\)-curve of equation (4) from the domain (5). Putting \(u=u_1(r)+v\) in (4), we obtain

\[ \frac{\omega(r)}{\omega'(r)}v' = U(r,u_1(r)+v)-U(r,u_1(r)). \tag{6} \]

We shall show that this equation has no solutions \(v(r)\) possessing the property \(v(r)\ge 0\) for \(0<r\le r_0\), \(v(r)\ne 0\), \(v(r)\to 0\) as \(r\to 0\). This, evidently, will prove the theorem.

According to condition 3) of the theorem, in the domain

\[ 0<r\le r_0,\qquad 0<v\le u_0-u_1(r) \tag{7} \]

the inequality

\[ U(r,u_1(r)+v)-U(r,u_1(r)) \le \frac{\sigma}{k} \left( \frac{\Lambda(r)\omega(r)}{\alpha(r)\omega'(r)}-1 \right)v \]

is satisfied, i.e., for any solution of equation (6),

\[ v' \le \frac{\sigma}{k} \left( \frac{\Lambda(r)}{\alpha(r)} - \frac{\omega'(r)}{\omega(r)} \right)v . \]

Consequently, any solution \(v(r)\) of equation (6) with initial data \((r^*,v^*)\) from the domain (7) satisfies the inequality

\[ v(r)\ge v^* \exp \left[ -\frac{\sigma}{k} \int_r^{r^*} \left( \frac{\Lambda(r)}{\alpha(r)} - \frac{\omega'(r)}{\omega(r)} \right)\,dr \right], \qquad r\in(0,r^*], \]

i.e., \(v(r)\) does not tend to zero as \(r\to 0\). The theorem is proved.

It is easy to verify that the present theorem includes, as special cases, many uniqueness theorems for an \(O\)-curve in \(N_2\) obtained earlier by other authors (the results of Lonn, Vinograd and Grobman, Kukles, Gruz \((^{2-6})\)).

Leningrad Branch
of the V. A. Steklov Mathematical Institute
Academy of Sciences of the USSR

Received
28 III 1962

CITED LITERATURE

\(^{1*}\) A. F. Andreev, DAN, 142, No. 4, 754 (1962).
\(^{2}\) V. V. Nemytskii, V. V. Stepanov, Qualitative Theory of Differential Equations, M.–L., 1949, p. 115.
\(^{3}\) R. E. Vinograd, D. M. Grobman, UMN, 12, No. 5, 193 (1957).
\(^{4}\) I. S. Kukles, UMN, 14, No. 5, 151 (1959).
\(^{5}\) I. S. Kukles, DAN, 128, No. 2, 241 (1959).
\(^{6}\) D. M. Gruz, Tr. Uzbeksk. univ., No. 78, 25 (1958).

* In paper (1) I allowed an inaccuracy: the Lonn lemma \((^{2})\) follows from the theorem of that paper only in the case when, in the conditions of the latter, the coefficient of Lipschitz \(\lambda(r)\) may be considered comparable from below with \(\omega(r)\), \(r\). The theorem of the present article includes Lonn’s lemma completely (as a very special case).