UDC 517.917

GENERALIZED EMDEN–FOWLER EQUATION

T. V. STEPANOVA

§ 1. STATEMENT OF THE PROBLEM

In important physical applications, for example in astrophysics and in atomic physics, one often encounters the so-called Emden–Fowler equation, which can be written in the form

\[ \ddot{x}+bt^\alpha x^n=0, \tag{1} \]

where \(a, n, b\) are real constants.

The study of the asymptotic behavior (as \(t\to+\infty\)) of solutions of this equation has been the subject of extensive literature (see, for example, [1]). L. A. Beklemisheva, in [2], studies the asymptotic behavior of solutions of the equation

\[ \ddot{x}+\sum_{k=1}^{s} b_k t^{\alpha_k}(1+o_k(t))x^{n_k}=0, \tag{2} \]

which is a generalization of equation (1). Concerning the quantities entering equation (2), the following assumptions are made: \(\alpha_k, b_k\) are arbitrary real numbers, \(n_k\) are irreducible rational fractions with integer odd denominators, \(0\le n_1<n_2<\cdots<n_s\), and the functions \(o_k(t)\) are defined together with their derivatives on the half-line \(t\ge T>0\) and satisfy conditions of the form

\[ |o_k(t)|+t|o'_k(t)|<At^{-c}\qquad (k=1,2,\ldots,s), \]

where \(A,c\) are positive constants. Without loss of generality we may assume that the terms in the sum entering equation (2) are arranged in strictly increasing powers of \(x\).

In what follows, following [2], we shall use the following definitions and assumptions. A rational number \(n\) with odd denominator will be called even or odd according to the parity of its numerator. All functions considered are assumed continuous for \(t\ge T>0\). Solutions of equation (2) are assumed continuous with a continuous first derivative.

A solution is called singular if, for some finite value \(t_1\ge T\), it vanishes together with its derivative and for \(t>t_1\) is identically zero.

A solution of equation (2), if it is nonsingular, extendable to all \(t\ge T\), and as \(t\to+\infty\) grows (tends to zero) no faster than some power function\(^*\), will be called ordinary.

\(^*\) That is, for \(t>T\) the solution \(x(t)\) satisfies the inequality \(|x(t)|<t^k\), where \(k\) is some constant (respectively, for \(t>T\), \(|x(t)|>t^{-k}\), where \(k\) is some positive constant).

The results of L. A. Beklemisheva show that ordinary solutions of equation (2), as a rule, admit the following representation:

\[ x=t^\omega y(t), \]

where \(y(t)\) satisfies either the conditions

\[ \lim_{t\to+\infty}\bigl(y(t)t^{-\varepsilon}\bigr)=0,\qquad \lim_{t\to+\infty}\lvert y(t)t^\varepsilon\rvert=+\infty, \]

or the conditions

\[ \lim_{t\to+\infty}\bigl(y(t)\ln^{-\varepsilon}t\bigr)=0,\qquad \lim_{t\to+\infty}\lvert y(t)\ln^\varepsilon t\rvert=+\infty \]

for arbitrary \(\varepsilon>0\). She calls the numbers \(\omega\) regimes.

L. A. Beklemisheva finds the possible regimes (their number is finite; as a rule, they are the abscissas of the vertices of the broken line

\[ \eta=\eta(\omega)\equiv \max_k \eta_k(\omega),\qquad \eta_0(\omega)=0, \]

\[ \eta_k(\omega)\equiv a_k+2+\omega(n_k-1)\quad (k=1,2,\ldots,s)), \]

and then, for each possible regime \(\omega\), passes to the so-called reduced equation, namely, applies the following substitutions to equation (2):

1) if \(\eta(\omega)>0\), the substitution is made

\[ x=t^\omega y,\qquad \tau=t^\mu,\qquad \mu=\frac12\bigl(a_k+2+\omega(n_k-1)\bigr), \tag{3_1} \]

2) if \(\eta(\omega)=0\), the substitution is made

\[ x=t^\omega y,\qquad t=e^\tau . \tag{3_2} \]

As a result of the substitution \((3_1)\), one obtains a reduced equation of the form

\[ \mu^2\ddot y+\frac{d}{\tau}\dot y+\sum_{i=1}^{p} b_{k_i}y^{n_{k_i}} = -\sum_{k\ne k_i} b_k\tau^{-\beta_k}\bigl(1+o_k(\tau^{1/\mu})\bigr)y^{n_k} - \sum_{i=1}^{p} o_{k_i}(\tau^{1/\mu})y^{n_{k_i}} -\omega(\omega-1)\tau^{-2}y . \tag{4_1} \]

As a result of the substitution \((3_2)\), one obtains a reduced equation of the form

\[ \ddot y+(2\omega-1)\dot y+\bigl(b_0+\omega(\omega-1)\bigr)y+\sum_{i=1}^{p} b_{k_i}y^{n_{k_i}} = \]

\[ = -\sum_{k\ne k_i,\,0} b_k e^{-\beta_k\tau}\bigl(1+o_k(e^\tau)\bigr)y^{n_k} -\sum_{i=1}^{p} b_{k_i}o_{k_i}(e^\tau)y^{n_k} -o_0(e^\tau)y . \tag{4_2} \]

L. A. Beklemisheva classifies the solutions of the reduced equation corresponding to one chosen regime into three groups: tending to zero, bounded but not tending to zero, and unbounded.

The solutions of equation (2) corresponding to the reduced solutions of these groups she calls, respectively, inexact solutions of the first type, exact solutions, and inexact solutions of the second type, and she studies the question of the existence of solutions of each class.

In studying possible nonexact solutions of the first type, the reduced equations can be divided into 31 groups. These will be 19 groups given in the table of L. A. Beklemisheva (see below), and another 12 groups differing from groups 4–7, 12–19 of the right column of this table only by the parity of \(n_{k_1}\). The reduced equations can be divided into 32 groups with respect to possible nonexact solutions of the second type. These are, first, cases 1–20 of the left column of the mentioned table. Secondly, 12 cases differing from cases 9–20 of the left column only by the parity of \(n_{k_p}\) *).

There are also \(12+12\) cases corresponding to even \(n_{k_1}\) and \(n_{k_p}\), in all respects repeating cases 9–20 and 4–7, 12–19. Indeed, for the left column (nonexact solution of the second type) there are 12 even cases. In [2] 13 cases are mentioned, apparently because in case 4 of the left column, alongside odd \(n_{k_1}\), it is proposed to consider also even \(n_{k_1}\). But this contradicts Definition 2 on p. 227 of [2], where the regime \(\omega_{02}\) is defined as the abscissa of the point of intersection of the line \(\eta=2-4\omega\) with the polygonal line \(\eta=\eta(\omega)\), under the condition that this point is nonangular, lies above the \(\omega\)-axis on the \(k\)-th segment of the polygonal line, and at the intersection point \(\eta(\omega)=\eta_k(\omega)\), \(n_k\) is odd and \(b_k>0\). If, according to this definition, \(n_k\) is considered odd, then in case 4 \(n_{k_1}\) cannot be considered even, and there will be 12 even cases in all.

The same applies to the right column. 13 cases would result if in case 3 of the right column one could consider \(n_{k_1}\) even, but, according to Definition 2, p. 227 of [2], \(n_{k_1}\) is odd, and therefore 12 cases result.

Results concerning even \(n_{k_1}\) and \(n_{k_p}\) are not formulated separately, since the study of each even case reduces to the study of two odd ones.

Table of L. A. Beklemisheva **)

*) L. A. Beklemisheva distinguishes 33 groups for nonexact solutions of the first type and 30 groups for nonexact solutions of the second type. Such a classification seems to us inexact.

) In cases marked ) and ) on p. 834, changes have been introduced.

***) In [2], on p. 232, \(n_{k_p}\) is odd. In fact, \(n_{k_p}<1\) and is odd. Indeed, the point \((\omega,0)\) is the left end of a segment of the polygonal line lying on the \(\omega\)-axis. At this point the axis \(\eta=0\) intersects with the lines \(\eta_k(\omega)=(n_k-1)\omega+a_k+2\). Since to the right of the point \((\omega,0)\) the polygonal line coincides with the \(\omega\)-axis and since

\[ \eta=\eta(\omega)=\max_k\bigl(0,(n_k-1)\omega+a_k+2\bigr), \]

to the right of \((\omega,0)\) \((n_k-1)\omega+a_k+2<0\), and to the left \((n_k-1)\omega+a_k+2>0\). Therefore the angular coefficient of the lines \((n_k-1)\) intersecting at the point \((\omega,0)\) with the \(\omega\)-axis will be \(<0\), i.e. \(n_k-1<0\). The smallest of these numbers is \(n_{k_p}-1<0\), i.e. \(n_{k_p}<1\).

1. \(2\omega-1 \geqslant 0,\quad b'_{k_p}>0.\)
   △10. \(2\omega-1<0,\quad b'_{k_p}>0.\)
2. \(2\omega-1<0,\quad b'_{k_p}<0.\)
   △12. \(2\omega-1 \geqslant 0,\quad b'_{k_p}<0.\)

Equation \((38_2)\); \(\omega\) is the right endpoint of the polygonal-line segment lying on the \(\omega\)-axis, \(n_{k_p}\) is odd.

1. \(2\omega-1>0,\quad b'_{k_p}>0.\)
   △ 14. \(2\omega-1<0,\quad b'_{k_p}>0.\)
2. \(2\omega-1\leqslant 0,\quad b'_{k_p}<0.\)
3. \(2\omega-1>0,\quad b'_{k_p}<0.\)

Equation \((38_1)\); \(\omega\) is the abscissa of an angular point of the polygonal line lying above the \(\omega\)-axis.

1. \(a\geqslant 0,\quad b'_{k_p}>0.\)
   △18. \(a<0,\quad b'_{k_p}>0.\)
2. \(a\leqslant 0,\quad b'_{k_p}<0.\)
3. \(a>0,\quad b'_{k_p}<0.\)

Equation \((38_2)\); \(\omega\) is the right endpoint of the polygonal-line segment lying on the \(\omega\)-axis.

△ 8. \(2\omega-1>0,\quad b_0+\omega(\omega-1)>0.\)
9. \(2\omega-1\leqslant 0,\quad b_0+\omega(\omega-1)>0.\)
10. \(2\omega-1<0,\quad b_0+\omega(\omega-1)<0.\)
11. \(2\omega-1\geqslant 0,\quad b_0+\omega(\omega-1)<0.\)
\(n_{k_1}>1\) is odd, \(b_0+\omega(\omega-1)=0.\)
△12. \(2\omega-1>0,\quad b_{k_1}>0.\)
13. \(2\omega-1\leqslant 0,\quad b_{k_1}>0.\)
△14*). \(2\omega-1\leqslant 0,\quad b_{k_1}<0.\)
15*). \(2\omega-1>0,\quad b_{k_1}<0.\)

Equation \((38_1)\); \(\omega\) is the abscissa of an angular point of the polygonal line lying above the \(\omega\)-axis.

△16. \(a>0,\quad b_{k_1}>0.\)
17. \(a\leqslant 0,\quad b_{k_1}>0.\)
18. \(a<0,\quad b_{k_1}<0.\)
19. \(a>0,\quad b_{k_1}<0.\)

In cases 1, 3, 4, 5, 7, 8, 9, 11, 13, 15, 16, 17, 19, 20 of the left column, nonexact solutions of the second type are absent.

In case 2 of the left column, nonexact solutions of the second type may exist. Such solutions have the form

\[ x(t)=At^{\frac12}\ln t\,(1+o(1)). \]

In cases 1, 2, 3, 5, 6, 7, 9, 10, 11, 13, 15, 17, 18, 19 of the right column, nonexact solutions of the first type are absent.

The cases marked by the symbol △ (6, 10, 12, 14, 18 of the left column and 4, 8, 12, 14, 16 of the right column) were not investigated in [2].

The aim of the present work is to investigate these latter cases.

§ 2. INVESTIGATION OF EQUATION (2) IN CASES NOT CONSIDERED IN ARTICLE [2]

Consider, for example, case 12 of the right column. The transformed equation in this case has the form

\[ \ddot y+(2\omega-1)\dot y+P(y)=\varphi(y,\tau), \tag{5} \]

* ) In [2] these cases have the form
14. \(2\omega-1<0,\quad b_{k_1}<0.\)
15. \(2\omega-1\geqslant 0,\quad b_{k_1}<0.\)

In the same work, on p. 233, the author asserts that in case 15 of the right column one can exclude the transformed solutions tending to zero. It is not difficult to show that in case 15 of the right column, when \(2\omega-1=0\), there may be transformed solutions tending to zero. In order that in case 15 of the right column there be no transformed solutions tending to zero, it is expedient to consider the case \(2\omega-1=0\) together with case 14 of the right column, and in case 15 of the right column to impose the condition \(2\omega-1>0\).

where

\[ P(y)=b_{k_1}y^{n_{k_1}}\bigl(1+o_1(y)\bigr),\quad o_1(y)\to0,\quad \text{as } y\to0, \]

and \(n_{k_1}>1\) is odd, \(b_0+\omega(\omega-1)=0,\quad 2\omega-1>0,\quad b_{k_1}>0\).

Along with equation (5), we shall consider the equations

\[ \ddot y+(2\omega-1)\dot y+P(y)=0, \tag{6} \]

\[ (2\omega-1)\dot y+P_1(y)=0, \tag{7} \]

where

\[ P_1(y)=\frac{P(y)}{1-\dfrac{1}{a^2}P'(y)} = P(y)\left[1+\frac{P'}{a^2}+\left(\frac{P'}{a^2}\right)^2+\cdots\right], \]

\[ a=2\omega-1>0. \]

The solutions of equation (7), not identically equal to 0, tend to zero as \(\tau\to+\infty\) and have the asymptotic form

\[ y=B\tau^{-\frac{1}{n_{k_1}-1}}\bigl(1+\theta(\tau)\bigr), \tag{8} \]

where

\[ B=\left[\frac{b_{k_1}}{(2\omega-1)(n_{k_1}-1)}\right]^{\frac{1}{n_{k_1}-1}}, \quad \theta(\tau)\to0,\quad \text{as } \tau\to+\infty . \]

Let \(y_0(\tau)\) be a fixed solution of the form (8) of equation (7).
We turn to equation (6).
Consider the integral equation

\[ y(\tau)=-\frac{\dot y_0(\tau_0)}{a}e^{a(\tau_0-\tau)} + \int_{\tau}^{+\infty} \left( e^{-a\tau_1} \int_{\tau_0}^{\tau_1} P(y)e^{a\tau_2}\,d\tau_2 \right)d\tau_1, \tag{9} \]

where \(\tau_0\) is any fixed number, \(a=2\omega-1\).

The solution of the integral equation (9), \(y(\tau)\), \(\tau\ge \tau_0\), is a solution of the differential equation (6); moreover, this solution satisfies the conditions

\[ y(+\infty)=0,\quad \dot y(\tau_0)=\dot y_0(\tau_0). \]

We solve equation (9) by the method of successive approximations. As the zeroth approximation we take the solution \(y_0(\tau)\). The subsequent approximations are defined by the recurrence formula

\[ y_{n+1} = -\frac{\dot y_0(\tau_0)}{a}e^{a(\tau_0-\tau)} + \int_{\tau}^{+\infty} e^{-a\tau_1}\,d\tau_1 \int_{\tau_0}^{\tau_1} P\bigl(y_n(\tau)\bigr)e^{a\tau_2}\,d\tau_2, \quad n=0,1,2,\ldots . \]

It is not difficult to prove that on some fixed interval \([\tau_1,+\infty)\) all the approximations exist, where \(\tau_1\) is determined with the aid of the following conditions:

1. \(\tau_1\ge \tau_0\), where \(\tau_0\) is a sufficiently large number such that for \(\tau\ge\tau_0\) the first approximation exists.

2. For \(\tau\ge\tau_1\), for any function \(y(\tau)\) satisfying the inequality

\[ |y(\tau)-y_0(\tau)|<\frac{D_1}{1-\beta_1}\frac{y_0(\tau)}{\tau^2}, \]

one can write the estimate

\[ \left|P'\bigl(y(\tau)\bigr)\right|<n_{k_1}b_{k_1}y_0^{\,n_{k_1}-1}(\tau)(1+\varepsilon), \]

where \(0<\varepsilon<1\) is chosen so that

\[ \left(\frac{1+\varepsilon}{1-\varepsilon}\right)^2 \frac{n_{k_1}}{2n_{k_1}-1}=\beta_1<1. \]

It is not difficult to establish that, for \(\tau\geq \tau_1\), the approximations satisfy the inequalities

\[ |y_{k+1}(\tau)-y_k(\tau)|<\beta_1^k\,\frac{D_1\dot y_0(\tau)}{\tau^2}, \]

\[ |y_{k+1}(\tau)-y_0(\tau)|< \frac{D_1}{1-\beta_1}\,\frac{y_0(\tau)}{\tau^2} \]

for all \(k=0,1,2,\ldots\).

The series \(y_0(\tau)+(y_1(\tau)-y_0(\tau))+(y_2(\tau)-y_1(\tau))+\cdots\) is majorized by the series

\[ y_0(\tau)+y_0(\tau)\frac{D_1}{\tau^2} +y_0(\tau)\frac{D_1\beta_1}{\tau^2} +y_0(\tau)\frac{D_1\beta_1^2}{\tau^2}+\cdots \]

\[ = y_0(\tau)\left[1+\frac{D_1}{\tau^2}\frac{1}{1-\beta_1}\right], \]

which converges uniformly on the interval \([\tau_1,+\infty)\), since

\[ \frac{y_0(\tau)D_1\beta_1^n}{\tau^2} < \left(\frac{y_0(\tau_1)}{\tau_1^2}D_1\right)\beta_1^n. \]

Therefore the series \(y_0+(y_1-y_0)+\cdots\) also converges on \([\tau_1,+\infty)\) absolutely and uniformly to a certain function \(y(\tau)\), which can be represented in the form

\[ y(\tau)=y_0(\tau)\left(1+\frac{D_1}{\tau^2}\frac{\theta_2}{1-\beta_1}\right), \]

where \(|\theta_2|<1\). The \(n\)-th partial sum of the series is \(y_n\); hence it has been proved that \(y_n(\tau)\to y(\tau)\) as \(n\to\infty\), uniformly on \([\tau_1,+\infty)\).

It is then easy to show that the limiting function is a solution of equation (9). It has the same asymptotic form as \(y_0(\tau)\), i.e.,

\[ y(\tau)=B\tau^{-\frac{1}{n_{k_1}-1}}\bigl(1+\tilde\theta(\tau)\bigr), \tag{10} \]

where

\[ \tilde\theta(\tau)\to 0,\quad \text{as } \tau\to +\infty,\qquad B=\left[\frac{b_{k_1}}{(2\omega-1)(n_{k_1}-1)}\right]^{\frac{1}{n_{k_1}-1}}. \]

Thus, there exist solutions of equation (6) of the form (10).

Let \(y_0(\tau)\) be a fixed solution of the form (10) of equation (6). Consider equation (5). We shall denote by \(\tau_0\) a number such that, for \(\tau\geq \tau_0\), the successive approximations of the equation exist and converge--

of (6), i.e., on \([\tau_0,+\infty)\) there exists a solution of equation (6) having the same asymptotic form as the solution of equation (7), not identically equal to 0.

For the function \(\varphi(y,\tau)\), for sufficiently large \(\tau\) the estimates
\[ |\varphi(y,\tau)|<Ae^{-c\tau}(1+|y|^n),\qquad \left|\frac{\partial\varphi}{\partial y}\right|<A_1e^{-c\tau}\frac{(1+|y|^n)}{|y|} \]
are valid.

Let \(\tau_1\) be the number starting from which these estimates hold. Take \(\widetilde{\tau}_0>\max(\tau_0,\tau_1)\). The integral equation corresponding to equation (5) has the form
\[ y(\tau)=-\frac{\dot y_0(\widetilde{\tau}_0)}{a}e^{a(\widetilde{\tau}_0-\tau)} +\int_\tau^{+\infty} e^{-a\tau_1}\,d\tau_1 \int_{\widetilde{\tau}_0}^{\tau_1} e^{a\tau_2}[P(y(\tau))-\varphi(y,\tau)]\,d\tau_2, \tag{11} \]
where \(a=2\omega-1\).

Put \(y(+\infty)=0,\ \dot y(\widetilde{\tau}_0)=\dot y_0(\widetilde{\tau}_0)\). We solve the integral equation (11) by the method of successive approximations, taking as the zeroth approximation the just chosen solution \(y_0(\tau)\). We define the following approximations by the recurrence formula
\[ y_{n+1}=y_0(\tau)+\int_\tau^{+\infty} e^{-a\tau_1}\,d\tau_1 \int_{\widetilde{\tau}_0}^{\tau_1} e^{a\tau_2}\,[P(y_n(\tau))- \]
\[ -P(y_0(\tau))-\varphi(y_n,\tau)]\,d\tau_2,\qquad n=0,1,2,\ldots \]

On some interval \([\widetilde{\tau}_0,+\infty)\), where \(\widetilde{\tau}_0\) is determined by the conditions:

1) for \(\tau\geqslant\widetilde{\tau}_0\) the first approximation exists;

2) for \(\tau\geqslant\widetilde{\tau}_0\), for any function \(y(\tau)\) satisfying the inequality
\[ |y(\tau)-y_0(\tau)|<2De^{-c\tau}, \]
one has
\[ |P'_y-\varphi'_y|<\frac{(a-c)c}{2}, \]
all approximations exist and satisfy the inequalities
\[ |y_k(\tau)-y_0(\tau)|<2De^{-c\tau},\qquad k=1,\ldots,n;\ \tau\geqslant\widetilde{\tau}_0, \]
\[ |y_{k+1}(\tau)-y_k(\tau)|< \left(\frac12\right)^k De^{-c\tau},\qquad k=0,1,\ldots,n-1;\ \tau\geqslant\widetilde{\tau}_0. \]

The successive approximations \(y_n(\tau)\) converge on \([\widetilde{\tau}_0,+\infty)\) absolutely and uniformly to a certain function \(y(\tau)\), which can be represented in the form:
\[ y(\tau)=y_0(\tau)+2D\theta(\tau)e^{-c\tau}. \]

Further, it is not difficult to prove that the limiting function \(y(\tau)\) is a solution of the corresponding integral equation, and therefore also a solution of the differential equation (5). This solution has the same asymptotic

asymptotic form as \(y_0(\tau)\), i.e.

\[ y(\tau)=B\tau^{-\frac{1}{n_{k_1}-1}}(1+\theta(\tau)), \]

where \(\theta(\tau)\to 0\) as \(\tau\to +\infty\).

Thus, in case 12 of the right column there exist inexact solutions of the first type of equation (2). Such solutions have the form

\[ x(t)=Bt^\omega(\ln t)^{-\frac{1}{n_{k_1}-1}}(1+o(1)). \]

Acting by the same method, one can also establish the following. In case 14 of the right column \((2\omega-1<0)\) there exist inexact solutions of the first type of equation (2). Such solutions have the form

\[ x(t)=Bt^\omega(\ln t)^{-\frac{1}{n_{k_1}-1}}(1+o(1)). \]

In case 14 of the right column \((2\omega-1=0)\), differential equation (2) has inexact solutions of the first type, which have the form

\[ x(t)=B_1t^\omega(\ln t)^{-\frac{2}{n_{k_1}-1}}(1+o(1)), \qquad B_1=\left[\frac{2(n_{k_1}+1)}{-b_{k_1}(n_{k_1}-1)^2}\right]^{\frac{1}{n_{k_1}-1}}. \]

In cases 10, 12 of the left column \((2\omega-1>0)\), differential equation (2) has inexact solutions of the second type, which have the form

\[ x(t)=B_2t^\omega(\ln t)^{\frac{1}{1-n_{k_p}}}(1+o(1)), \qquad B_2=\left[\frac{b_{k_p}(1-n_{k_p})}{1-2\omega}\right]^{\frac{1}{1-n_{k_p}}}. \]

In case 12 of the left column \((2\omega-1=0)\), differential equation (2) has inexact solutions of the second type, which have the form

\[ x(t)=B_3t^\omega(\ln t)^{\frac{2}{1-n_{k_p}}}(1+o(1)), \qquad B_3=\left[\frac{(1-n_{k_p})^2|b_{k_p}|}{2(1+n_{k_p})}\right]^{\frac{1}{1-n_{k_p}}}. \]

It remains to consider cases 4, 8, 16 of the right column and 6, 14, 18 of the left column. We have arrived at the following results.

In cases 4, 8, 16 of the right column there are no inexact solutions of the first type.

In cases 6, 14, 18 of the left column there are no inexact solutions of the second type.

The proofs in the last 6 cases were obtained by contradiction. Assuming the existence of solutions of the given class, it is not difficult to verify that their properties contradict their belonging to this class.

I am deeply grateful to my scientific adviser A. F. Andreev for his assistance and guidance in carrying out the present work.

References

1. Bellman R. The theory of stability of solutions of differential equations. Moscow, 1955.
2. Beklemisheva L. A. Mat. sb., 56 (98), No. 2, 1962, pp. 207—236.

Received by the editors
December 4, 1965

Leningrad State University
named after A. A. Zhdanov