On a Hartman—Wintner Problem

A. F. Andreev

The present note is devoted to the study of a problem whose formulation follows from the following theorem.

Hartman—Wintner Theorem [1]. Suppose that in the equation

\[ a(r)\frac{d\varphi}{dr}=\Psi(r,\varphi) \tag{1} \]

1) the function \(\Psi(r,\varphi)\) is defined and continuous in \(r,\varphi\) in the domain

\[ 0<r\le r_0,\qquad -\varphi_0\le \varphi\le \varphi_0, \tag{2} \]

where \(r_0\) and \(\varphi_0\) are positive numbers; \(\dfrac{1}{\Psi(r,\varphi)}\) is bounded for \(0<r\le \varepsilon\), \(\theta\le |\varphi|\le \varphi_0\), where \(\theta>0\) is arbitrarily small, and \(\varepsilon\) is sufficiently small;

2) \(\Psi(r,-\varphi_0)>0\), \(\Psi(r,\varphi_0)<0\) for \(0<r\le r_0\);

3) the function \(a(r)\) is defined, continuous, and positive for \(0<r\le r_0\),

\[ \int_0^{r_0}\frac{dr}{a(r)}=+\infty. \]

Then there exists at least one solution \(\varphi=\varphi(r)\) of equation (1), which is defined on the interval \(0<r\le r_0\), 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 assumptions of the Hartman—Wintner theorem, equation (1) has in the domain (2) either a) a unique \(O\)-curve, or b) an infinite set of \(O\)-curves. Many works have been devoted to the problem of distinguishing these two possibilities for special cases of equation (1). A survey of works published before 1961 is contained in the article [2]. In recent years this question has been treated in the articles [3], [4], [5]. In the present note the problem of uniqueness of the \(O\)-curve of equation (1) in the domain (2) is considered in a rather general case.

Lemma. Suppose that for equation (1)

1) the assumptions of the Hartman—Wintner theorem are satisfied,

2) in the domain (2) (or in any of its subdomains of the same form), for \(\overline{\varphi}>\underline{\varphi}\),

\[ \Psi(r,\overline{\varphi})-\Psi(r,\underline{\varphi}) \le \lambda(r)(\overline{\varphi}-\underline{\varphi}), \]

where \(\lambda(r)\) is continuous for \(r\in(0,r_0]\),

\[ \int_r^{r_0}\frac{\lambda(r)}{\alpha(r)}\,dr \le M < +\infty, \]

\(M\) is a constant, \(r'\) is any point of \((0,r_0]\).

Then equation (1) has in the domain (2) a unique \(O\)-curve.

This lemma is a direct generalization of Lonn’s lemma ([6], p. 115) and can be proved by the same method.

In order to make stronger assertions possible, we must make additional assumptions concerning the structure of the function \(\Psi(r,\varphi)\).

Consider the equation

\[ \alpha(r)\frac{d\varphi}{dr}=\Phi(r,\varphi)+\psi(r,\varphi)=\Psi(r,\varphi) \tag{3} \]

under the following assumptions.

Condition A. 1) The functions \(\Phi(r,\varphi)\) and \(\psi(r,\varphi)\) are continuous jointly in the variables \(r,\varphi\) in the domain (2).

2) \(\Phi(r,0)\equiv 0\); for \(\varphi\ne 0\), \(\varphi\Phi(r,\varphi)<0\),

\[ \frac{\Phi(r,\varphi)}{\gamma(\varphi)} \ge 1, \]

where \(\gamma(\varphi)\) is continuous for \(|\varphi|<\Delta\) (the constant \(\Delta>\varphi_0\)) and twice differentiable for \(0<|\varphi|<\Delta\), with \(\gamma(0)=0\), \(|\gamma(\varphi)|\to+\infty\) as \(|\varphi|\to\Delta\), \(\gamma'(\varphi)<0\), \(\gamma''(\varphi)\ne 0\) for \(0<|\varphi|<\Delta\) (to simplify the subsequent exposition we shall suppose the function \(\gamma(\varphi)\) to be odd).

3) \(\psi(r,\varphi)\to 0\) as \(r\to 0\) uniformly with respect to \(\varphi\in[-\varphi_0,\varphi_0]\) (and, consequently, there exists a function \(\omega(r)\) of class \(C^1\) on \((0,r_0]\), possessing the properties: \(\omega(r)>0\), \(\omega'(r)>0\) for \(r>0\), \(\omega(r)\to 0\) as \(r\to 0\),

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

as \(r\to 0\) uniformly with respect to \(\varphi\in[-\varphi_0,\varphi_0]\)).

4) The function \(\alpha(r)\) is continuous and positive on the interval \((0,r_0]\),

\[ \int_0^{r_0}\frac{dr}{\alpha(r)}=+\infty. \]

Under condition A, equation (3) in the domain

\[ 0<r\le \rho,\qquad -\delta\le \varphi\le \delta, \tag{4} \]

where \(\rho\) and \(\delta\) are sufficiently small positive numbers, satisfies the hypotheses of the Hartman–Wintner theorem and, consequently, for it there arises the problem of uniqueness of the \(O\)-curve in the domain (4).

To investigate this problem, pass from the variables \(r,\varphi\) to the variables \(r,\varphi_1\) by the formula

\[ \gamma(\varphi)=\gamma(\varphi_1)\omega(r). \tag{5} \]

Then the domain (4) of the \(r,\varphi\)-plane will correspond in the \(r,\varphi_1\)-plane to the domain

\[ 0<r\le \rho,\qquad |\varphi_1|\le \varphi_{1\delta}(r), \tag{6} \]

where \(\varphi_{1\delta}(r)\) is continuous and positive for \(0<r\le \rho\), and \(\varphi_{1\delta}(r)\to\Delta\) as \(r\to 0\). Domains of the \(r,\varphi_1\)-plane

\[ 0<r\le \rho,\qquad |\varphi_1|\ge \delta_1, \tag{7} \]

\(\delta_1\) is a constant, \(0<\delta_1<\Delta\), will have as their images in the \(r,\varphi\)-plane the domains

\[ 0<r\leq \rho,\qquad |\varphi|\geq \varphi_{\delta_1}(r), \tag{8} \]

\(\varphi_{\delta_1}(r)\) is continuous and positive for \(0<r\leq \rho\), \(\varphi_{\delta_1}(r)\to 0\) as \(r\to 0\).

Let \(\delta_1\) be any number from \((0,\Delta)\). Write equation (3) in the form

\[ a(r)\frac{d\varphi}{dr} = \omega(r)\left[ \frac{\Phi(r,\varphi)}{\gamma(\varphi)} - \frac{\gamma(\varphi)}{\omega(r)} + \frac{\psi(r,\varphi)}{\omega(r)} \right] \equiv \Psi(r,\varphi). \]

It is seen from this that, if \(\rho\) is sufficiently small, then in the domains (8), for any integral curve \(\varphi=\varphi(r)\) of equation (3), \(\varphi(r)\varphi'(r)<0\), and therefore the \(O\)-curves of equation (3) are entirely located in the domain

\[ 0<r\leq \rho,\qquad |\varphi|\leq \varphi_{\delta_1}(r), \tag{9} \]

where \(\delta_1>0\) is arbitrarily small, \(\rho=\rho(\delta_1)\) sufficiently small.

To study the question of the existence of \(O\)-curves of equation (3) in the domain (9), we transform it by means of the substitution (5). In doing so we obtain the equation

\[ \frac{\omega(r)}{\omega'(r)}\varphi_1' = - \frac{\gamma(\varphi_1)}{\gamma'(\varphi_1)} + \frac{\Psi(r,\varphi)\gamma'(\varphi)} {a(r)\omega'(r)\gamma'(\varphi_1)} \equiv \Psi_1(r,\varphi_1) \tag{10} \]

(where \(\varphi=\gamma^{-1}(\gamma(\varphi_1)\omega(r))\)), which must be studied in the domain

\[ 0<r\leq \rho,\qquad -\delta_1\leq \varphi_1\leq \delta_1 \tag{4\(_1\)} \]

(\(0<\delta_1<\Delta\) arbitrarily small, \(\rho\) sufficiently small), namely, one must investigate the question of the number of solutions of equation (10) lying in the domain (4\(_1\)) for all \(r\), \(0<r\leq \rho_0\).

Introduce the notation:

\[ a_1(r)=\frac{\omega(r)}{\omega'(r)},\qquad \Phi_1(\varphi_1)=- \frac{\gamma(\varphi_1)}{\gamma'(\varphi_1)},\qquad \beta(r,\varphi_1)=\frac{\gamma'(\varphi)}{\gamma'(\varphi_1)}, \]

\[ \psi_1(r,\varphi_1)= \frac{\Psi(r,\varphi)\beta(r,\varphi_1)} {a(r)\omega'(r)}. \]

Then equation (10) is written in the form

\[ a_1(r)\frac{d\varphi_1}{dr} = \Phi_1(\varphi_1)+\psi_1(r,\varphi_1) \equiv \Psi_1(r,\varphi_1). \tag{3\(_1\)} \]

From the assumptions concerning \(\gamma(\varphi)\) it follows that, for \(\varphi_1\ne 0\),

\[ \varphi_1\Phi_1(\varphi_1)<0,\qquad \beta(r,\varphi_1)>0. \]

Therefore equation (3\(_1\)) satisfies in the domain (4\(_1\)) all the conditions of the Hartman—Wintner theorem\(^1\), with the possible exception of one: the continuity of the right-hand side may be violated at \(\varphi_1=0\), since the function \(\beta(r,\varphi_1)\) may have a nonremovable discontinuity on the line \(\varphi_1=0\).

\(^1\) \(\dfrac{1}{\Psi_1(r,\varphi_1)}\) is bounded in the domain \(\theta_1\leq |\varphi_1|\leq \delta_1,\ 0<r\leq \varepsilon_1,\ \theta_1>0\) arbitrary, \(\varepsilon_1\) sufficiently small, since in this domain the function \(\psi_1(r,\varphi_1)\) takes values of the same sign as the function \(\Phi_1(\varphi_1)\).

Examples.

\[ \begin{gathered} 1.\quad \gamma(\varphi)= \begin{cases} -\dfrac{1}{\ln|\varphi|}, & -1<\varphi\leq 0,\\[4pt] 0, & \varphi=0,\\[4pt] \dfrac{1}{\ln\varphi}, & 0<\varphi<1. \end{cases} \end{gathered} \]

Here

\[ \beta(r,\varphi_1)=\omega^2(r)|\varphi_1|^{\,1-\frac{1}{\omega(r)}} . \]

1. \(\gamma(\varphi)=(-\operatorname{sign}\varphi)a|\varphi|^k,\quad a>0,\quad k>0\) are constants, \(-\infty<\varphi<+\infty\). Here

\[ \beta(r,\varphi_1)=\omega^{\frac{k-1}{k}}(r). \]

\[ \begin{gathered} 3.\quad \gamma(\varphi)= \begin{cases} e^{\frac{1}{\varphi}}, & -\dfrac{\Delta}{2}\leq \varphi<0,\\[4pt] 0, & \varphi=0,\\[4pt] -e^{-\frac{1}{\varphi}}, & 0<\varphi\leq \dfrac{\Delta}{2}, \end{cases} \end{gathered} \]

\(\Delta\) is a constant; on the interval \(\dfrac{\Delta}{2}\leq |\varphi|<\Delta\), \(\gamma(\varphi)\) is continued in such a way that it satisfies condition A. Here

\[ \beta(r,\varphi_1)=\omega(r)\bigl(1+|\varphi_1\ln\omega(r)|\bigr)^2. \]

Remark. We are now interested in the behavior of the solutions of equation \((3_1)\) in the domain \((4_1)\), where \(\delta_1\) and \(\rho\) are arbitrarily small, or, equivalently, the behavior of the solutions of equation (3) in the domain (9). Therefore, we may take into account only small values of the variables \(\varphi\) and \(\varphi_1\). Consequently, the function \(\beta(r,\varphi_1)\) in the domain \((4_1)\) with sufficiently small \(\delta_1\) and \(\rho\) is determined by the specification of the function \(\gamma(\varphi)\) for small \(\varphi\) (it does not depend on the way in which \(\gamma(\varphi)\) is continued to the entire interval \((-\Delta,\Delta)\)).

Example 1 shows that the function \(\beta(r,\varphi_1)\) may indeed turn out to be discontinuous at \(\varphi_1=0\). Examples 2 and 3, which are of much greater interest, show that in many important cases the function \(\beta(r,\varphi_1)\) is continuous in the domain \((4_1)\).

Condition B. The function \(\beta(r,\varphi_1)\) is continuous in the domain \((4_1)\). Then equation \((3_1)\) satisfies, in the domain \((4_1)\), all the conditions of the Hartman—Wintner theorem, and the problem under consideration reduces to the problem of uniqueness of the \(O\)-curve of equation \((3_1)\) in the domain \((4_1)\), where \(\delta_1>0\) is arbitrarily small and \(\rho\) is sufficiently small.

Condition C. There exists a number \(\delta_1>0\) such that, for \(|\varphi_1|\leq \delta_1\),

\[ \Phi'_1(\varphi_1)\leq -g<0, \]

where \(g\) is a constant.

Under this condition one can formulate the following criterion for uniqueness of the \(O\)-curve of equation (3) in the domain (4).

Theorem 1. Let, for equation (3),

1) conditions A, B, and C be satisfied,

2) a) in the plane \(r,\varphi\), in the domain \(0<r\leq \rho,\ 0<|\gamma(\varphi)|\leq |\gamma(\delta_1)|\omega(r)\),

\[ \Psi'_\varphi(r,\varphi)\leq \lambda(r), \]

b) in the plane \(r,\varphi_1\), in the domain \(0<r\leq \rho,\ 0<|\varphi_1|\leq \delta_1\),

\[ \Psi\bigl(r,\gamma^{-1}(\gamma(\varphi_1)\omega(r))\bigr)\beta'_{\varphi_1}(r,\varphi_1)\leq \mu(r)\omega(r), \]

where \(\lambda(r)\) and \(\mu(r)\) are continuous on \((0,\rho]\), and there exists a constant \(M\) such that

\[ \int_r^\rho \frac{\lambda(r)+\mu(r)}{\alpha(r)}\,dr \le M - g\ln\omega(r),\quad r\in(0,\rho]. \]

Then equation (3) has a unique \(O\)-curve in the domain (4).

Proof. Taking into account the equalities

\[ \frac{\partial\varphi}{\partial\varphi_1}=\frac{\omega(r)}{\beta(r,\varphi_1)},\qquad \beta(r,0)=\lim_{\varphi_1\to 0}\beta(r,\varphi_1) \]

and condition 2) of the theorem, we see that in the domain \((4_1)\), for \(\bar{\bar{\varphi}}_1>\bar{\varphi}_1\), the estimate

\[ \Psi_1(r,\bar{\bar{\varphi}}_1)-\Psi_1(r,\bar{\varphi}_1) \le \left( -g+\frac{[\lambda(r)+\mu(r)]\omega(r)}{\alpha(r)\omega'(r)} \right) (\bar{\bar{\varphi}}-\bar{\varphi}) \]

holds. If \(\bar{\bar{\varphi}}_1\bar{\varphi}_1>0\), the validity of this estimate is obvious. If, however, \(\bar{\varphi}_1<0<\bar{\bar{\varphi}}_1\), then it follows from the equality

\[ \Psi_1(r,\bar{\bar{\varphi}}_1)-\Psi_1(r,\bar{\varphi}_1) = [\Psi_1(r,\bar{\bar{\varphi}}_1)-\Psi_1(r,0)] + [\Psi_1(r,0)-\Psi_1(r,\bar{\varphi}_1)]. \]

Hence we conclude that equation \((3_1)\) satisfies, in the domain \((4_1)\), the second condition of the lemma. This, on the basis of what was said above, proves the theorem.

Let us note an important special case of Theorem 1. Suppose equation (3) satisfies condition A with \(\gamma(\varphi)=-(\operatorname{sign}\varphi)a|\varphi|^k\), \(a>0\), \(k>0\) constants. Then the substitution (5), which transforms equation (3) to the form \((3_1)\), takes the form \(\varphi=\varphi_1\omega^{1/k}(r)\), and in equation \((3_1)\)

\[ \Phi_1(\varphi_1)=-\frac{\varphi_1}{k},\qquad \beta(r,\varphi_1)=\omega^{\frac{k-1}{k}}(r), \]

i.e., condition 1) of Theorem 1 is fulfilled, while condition 2) assumes the simpler form: in the domain \(0<r\le\rho\), \(|\varphi|\le\delta_1\omega^{1/k}(r)\), the inequality \(\Psi'_\varphi(r,\varphi)\le\lambda(r)\) holds, where \(\lambda(r)\in C\) on \((0,\rho]\), and

\[ k\int_r^\rho \frac{\lambda(r)}{\alpha(r)}\,dr+\ln\omega(r)\le M<+\infty, \]

\(r\) is arbitrary in \((0,\rho]\), \(M\) is a constant. In other words, with such a choice of the function \(\gamma(\varphi)\) we obtain\(^1\) from Theorem 1 the author’s theorem from [4], which in turn contains a number of theorems of similar type obtained earlier by other authors.

Let us now suppose that condition C is not fulfilled. In this case it is sometimes possible to solve the problem by applying Theorem 1, with the special choice of the function \(\gamma(\varphi)\) noted above, to equation \((3_1)\).

\(^1\) True, in [4] the Lipschitz condition on \(\Psi(r,\varphi)\) is imposed directly (without the assumption of the existence of \(\Psi'_\varphi(r,\varphi)\)).

Suppose, for example, that equation (3) satisfies condition A with the function \(\gamma(\varphi)\) indicated in Example 3. Then \(\Phi_1(\varphi)=-(\operatorname{sign}\varphi)\varphi^2\) and, consequently, Theorem 1 is not applicable to equation (3) (condition C is violated). However, equation \((3_1)\) in this case has the form

\[ \frac{\omega(r)}{\omega'(r)}\,\varphi_1' = -(\operatorname{sign}\varphi_1)\varphi_1^2+\psi_1(r,\varphi_1), \]

\[ \psi_1(r,\varphi_1)= \frac{\omega(r)\Psi(r,\varphi)}{a(r)\omega'(r)} \left(\frac{\varphi_1}{\varphi}\right)^2, \qquad \varphi=\varphi_1[1+|\varphi_1\ln\omega(r)|]^{-1}. \]

Under certain additional assumptions it proves possible to apply Theorem 1 to it.

In conclusion, we make several remarks concerning the results of the paper [5]. In it the author considers the equation1

\[ x\frac{dy}{dx}=-Ay^kP(y)+f(x,y) \tag{1} \]

[\(k>1\) odd, \(A>0\) constant (one may take \(A=\dfrac{1}{k-1}\)), \(P(y)\) and \(f(x,y)\) are differentiable in a neighborhood of \((0,0)\), with the possible exception of points of the \(y\)-axis, and \(P(0)=1,\ f(0,y)\equiv0\)] and proves for it five theorems distinguishing configurations a) and b). There are errors in that paper. Below we note those which, it seems to me, cannot be corrected without additional assumptions concerning the functions \(P(y)\) or \(f(x,y)\).

First, the formulation of the problem is imprecise: the condition \(f(x,y)\to0\) as \(x\to0\), \(|y|\le y_0\), \(y_0>0\) constant, is absent; without it the region \(0<x\le x_0,\ |y|\le y_0\) may fail to be a normal region of the second type, however small \(x_0\) and \(y_0\) may be.

Second, strictly speaking, all five theorems are imprecise. The main error consists in the fact that the author, imposing conditions on \(f_y'(x,y)\), imposes no restrictions on \(P'(y)\). There are other errors as well. Let us dwell briefly on each theorem separately.

The assertion of Theorem 1 is true when \(P(y)\equiv1\). If, however, \(P(y)\not\equiv1\), then in the proof of the first part of the theorem (uniqueness) the inequality \(R_u'(x,u)<0\) (p. 73, line 9), and in the proof of the second part of the theorem (non-uniqueness) the inequalities

\[ \frac{\partial R(x,u)}{\partial u}>\frac{dR_0(u)}{du} \]

(p. 75, line 5) do not hold without additional assumptions concerning \(P(y)\). For the validity of the first part of the theorem it is sufficient, for example, to assume that

\[ kP(y)+yP'(y)\ge0 \quad\text{for } |y|\le y_0, \]

and for the validity of the second part, that

\[ kP(y)+yP'(y)\le k(1+\varepsilon), \qquad 0<\varepsilon<\frac{h}{kp+h}. \]

The assertion of Theorem 2 for \(P(y)\equiv 1\) is also valid. For \(P(y)\not\equiv 1\) it will be valid if, for example, the following additional conditions are introduced. For the first part:

a) in the case when \(f'_y(x,u\lambda(u))\le \dfrac{x\lambda'}{\lambda}\) for \(0<u<1\):

\[ kP(y)+yP'(y)\ge k,\qquad |y|\le y_0; \]

b) in the case when \(f'_y(x,u\lambda(x))\ge \dfrac{x\lambda'}{\lambda}\) for \(0<u<1\):

\[ k\le kP(y)+yP'(y)\le k(1+h_1),\qquad |y|\le y_0,\quad 0\le h_1<k-2. \]

For the second part:

\[ kP(y)+yP'(y)\le k(1+h_1),\qquad |y|\le y_0,\quad h_1<h. \]

The assertion of Theorem 3 for \(P(y)\equiv 1\) is valid, although the proof requires a substantial refinement. For \(P(y)\not\equiv 1\), additional assumptions of the type indicated for Theorems 1 and 2 are required.

Theorem 4 is inaccurate already for \(P(y)\equiv 1\) (the issue concerns the proof; I have not constructed counterexamples). The author partly relies on Theorem 2, which assumes the constancy of sign of the function

\[ f'_y(x,y)-\frac{x\lambda'(x)}{\lambda(x)},\quad 0<x<x_0,\quad 0<y<\lambda(x), \]

which, under the conditions of Theorem 4, may fail to hold. In the case when \(p(x)\to 0\) as \(x\to 0\), the author relies on Theorem 1; but from condition (13) there follows in this case only the inequality

\[ f'_y(x,y)<\frac{1+h}{(k-1)|\ln x|},\quad 0<x<x_0,\quad |y|\le y_0, \]

which is weaker than in the conditions of Theorem 1.

Theorem 5 is also false already for \(P(y)\equiv 1\) (the issue again concerns the proof). In order for the author’s proof to go through, it is necessary to make the following additional assumptions: 1)

\[ \left(\frac{\lambda(x)}{\lambda_1(x)}\right)^k\ge 1-h, \]

where \(h>0\) is the constant appearing in the conditions of the theorem, 2)

\[ \int_0^{x_0}\frac{\lambda_1^k(x)}{x}\,dx=+\infty. \]

The proof here proceeds according to the same scheme as in Theorem 2. Here the condition

\[ \left(\frac{\lambda}{\lambda_1}\right)^k\ge 1-h \]

ensures the inequality \(r_1(x,u)\ge \varepsilon>0\) for \(u\le 0\), while the divergence of the integral is the cornerstone of the proof of Theorem 2. Taking these additional conditions into account, there can no longer be any question of including in Theorem 5 my theorem from paper [3].

The attentive reader will find the answers to all the accusations directed against me, expressed in [5], in my paper [2].

References

1. Hartman P., Wintner A. On the asymptotic behavior of the solutions of a nonlinear differential equation. Amer. J. Math., 68, No. 2, 1946.

2. Andreev A. F. On Frommer’s distinction problems. Proceedings of the Fourth All-Union Mathematical Congress, vol. II, 1964, pp. 393–401.

3. Andreev A. F. Dokl. Akad. Nauk SSSR, 142, No. 4, 1962, pp. 754–757.

4. Andreev A. F. Dokl. Akad. Nauk SSSR, 146, No. 1, 1962, pp. 9–10.

5. Kukles I. S. On uniqueness conditions for a normal domain of the second kind. Proceedings of Samarkand State University named after A. Navoi. New series, issue 119, 1962, pp. 71–85.

6. Nemytskii V. V. and Stepanov V. V. Qualitative Theory of Differential Equations. GITTL, 1949.

Received by the editors
November 3, 1964

Leningrad Institute of Precision Mechanics
and Optics

1. Below, throughout, references to formulas and page numbers refer to the paper [5]. The notation of that paper is also preserved in full. ↩