UDC 517.919

ON STABILITY IN THE LARGE OF SOLUTIONS OF SYSTEMS OF SECOND-ORDER DIFFERENTIAL EQUATIONS

Yu. N. Bibikov

This paper considers nonautonomous systems of second-order differential equations for which generalized Routh–Hurwitz conditions are satisfied.

In § 1, systems having a zero solution are considered, and the question is studied of the extent to which the fulfillment of the Routh–Hurwitz conditions guarantees stability in the large of the zero solution. Sufficient conditions for stability are given.

The results obtained in § 1 are applied in § 2 to the investigation of the question of convergence (i.e., of the convergence of all solutions as \(t \to +\infty\)) in systems without a zero solution.

§ 1. STABILITY OF THE ZERO SOLUTION

Consider the nonautonomous system of second-order differential equations, which we write in the form

\[ \begin{aligned} \frac{dx}{dt} &= ax + h_2(x,y,t)y,\\ \frac{dy}{dt} &= h_1(x,y,t)x + by, \end{aligned} \tag{1} \]

where \(a\) and \(b\) are certain constants.

Assume that the functions \(h_1(x,y,t)\) and \(h_2(x,y,t)\) are continuous jointly in the variables throughout the whole space \(x,y,t\) (with the possible exception of the line \(x=y=0\)) and

\[ \lim_{x,y\to 0} h_1(x,y,t)x = \lim_{x,y\to 0} h_2(x,y,t)y = 0. \]

Let, moreover, the generalized Routh–Hurwitz conditions hold:

\[ a+b<0,\qquad ab-h_1(x,y,t)h_2(x,y,t)>0 \tag{2} \]

for all \(x\), \(y\), and \(t\).

N. N. Krasovskii showed [1, 2] that if in system (1) \(h_1(x,y,t)\equiv h_1(x)\), \(h_2(x,y,t)\equiv h_2(y)\), then under our assumptions the zero solution of the system is stable in the large. In the general case this is not true. For example, consider a positive periodic function \(p(t)\) for which the zero solution of the equation

\[ \frac{d^2 x}{dt^2}+p(t)x=0, \]

and hence also the system

\[ \frac{dx}{dt}=y, \]

\[ \frac{dy}{dt}=-p(t)x \]

is unstable, with one of the multipliers having modulus greater than 1. Such functions exist [3]. Consequently, for small in absolute value and negative \(a\) and \(b\), the zero solution of the system

\[ \frac{dx}{dt}=ax+y, \]

\[ \frac{dy}{dt}=-p(t)x+by, \]

for which inequalities (2) are satisfied with \(h_2=1,\quad h_1=-p(t)\), is also unstable.

Below we give conditions under which the Routh–Hurwitz conditions turn out to be sufficient conditions for stability in the large of the zero solution of system (1). The investigation is carried out by means of spirals composed of smooth curves having the property that the trajectories of solutions of the system intersect them in the direction toward the origin. Spirals of this kind were also used in [1].

In what follows, instead of “trajectories of solutions” we shall simply say “solutions.”

Let first \(ab>0\), i.e. \(a<0,\ b<0\). Inequalities (2) in this case are equivalent to the following [1]:

\[ c_1\leq h_1(x,y,t)\leq c_1^*,\qquad c_2\leq h_2(x,y,t)\leq c_2^*, \tag{3} \]

where \(c_1c_2=c_1^*c_2^*=ab\) \((c_1\leq 0,\ c_2\leq 0,\ c_1^*\geq 0,\ c_2^*\geq 0)\), and 1) in (3) one of the two left-hand inequalities and one of the two right-hand inequalities is necessarily strict, 2) if one of the numbers in the pairs \(c_1,c_2\) or \(c_1^*,c_2^*\) is equal to zero, then the other is regarded as equal to \(\pm\infty\). We note that among the numbers \(c_i,c_i^*\) \((i=1,2)\), those equal to zero occur only when at least one of the functions \(h_1(x,y,t)\) or \(h_2(x,y,t)\) assumes values of one sign (or values equal to zero).

Consider the case where among the numbers \(c_i,c_i^*\) there are no zeros and, consequently, no values equal to \(\pm\infty\).

Taking inequalities (3) into account, it is not difficult to verify [4] that, by virtue of system (1),

\[ \frac{d(x^2)}{dt}<0 \]

in the sectors

\[ -\frac{a}{c_2}x<y<-\frac{c_1^*}{b}x,\qquad x>0 \]

and

\[ -\frac{c_1^*}{b}x<y<-\frac{a}{c_2}x,\qquad x<0, \]

and

\[ \frac{d(y^2)}{dt}<0 \]

in the sectors

\[ -\frac{c_2}{a}\,y<x<-\frac{b}{c_1^*}\,y,\quad y>0 \]

and

\[ -\frac{b}{c_1^*}\,y<x<-\frac{c_2}{a}\,y,\quad y<0. \]

Therefore, using segments of the lines \(x=\mathrm{const}\) and \(y=\mathrm{const}\) in the corresponding sectors, one can construct a spiral that is crossed by the solutions of system (1) in the direction toward the origin (at corner points tangency is possible).

Let, for definiteness, \(c_1^*>-c_1\). Then, as the polar angle decreases, this spiral winds toward the origin, and as the polar angle increases, it unwinds to infinity. If a solution, starting from some moment of time, does not fall on the positive semiaxis of ordinates, then the segments of the spirals between successive intersections of this semiaxis play the role of level lines of a Lyapunov function, and therefore this solution tends to the origin as \(t\to+\infty\). If, however, the solution falls on the positive semiaxis of ordinates an infinite number of times as \(t\to+\infty\), then two cases are possible:

1) The polar angle of the points of intersection does not increase. In this case the solution also tends to the origin.

2) The polar angle of some point of intersection has increased by \(2\pi\) in comparison with the polar angle of the preceding point of intersection.

Let us investigate this case. Construct a continuous curve \(\Gamma\), beginning and ending on the positive semiaxis of ordinates and composed of integral curves of the following differential equations:

\[ \text{A)}\qquad \frac{dy}{dx}=\frac{c_1x+by}{ax+c_2y} \tag{4} \]

in the sectors \(0\leq x\leq -\dfrac{c_2}{a}y,\ y<0\) and \(-\dfrac{c_2}{a}y\leq x\leq 0,\ y>0\). Since

\[ c_1c_2=ab, \]

equation (4) has the form

\[ \frac{dy}{dx}=\frac{b}{c_2}. \]

\[ \text{B)}\qquad \frac{dy}{dx}=\frac{c_1^*x+by}{ax+c_2^*y} \tag{5} \]

in the sectors \(0\leq y\leq -\dfrac{c_1^*}{b}x,\ x>0\) and \(-\dfrac{c_1^*}{b}x\leq y\leq 0,\ x<0\). Since

\[ c_1^*c_2^*=ab, \]

equation (5) has the form

\[ \frac{dy}{dx}=\frac{b}{c_2^*}. \]

\[ \text{C)}\qquad \frac{dy}{dx}=\frac{c_1^*x+by}{ax+c_2y} \tag{6} \]

in all the remaining sectors of the plane.

These equations are chosen so that the solutions of system (1) cross their integral curves in the direction toward the origin or are tangent to them, with tangency possible only for the integral curves of equation (6). The latter follows from Remark 1 to inequalities (3).

The general solution of equation (4) is

\[ y-\frac{b}{c_2}\,x=C. \]

(here and below \(C\) is an arbitrary constant), equations (5) give

\[ y-\frac{b}{c_2^*}x=C, \]

and equations (6) give

\[ x\left[-c_2u^2+(b-a)u+c_1^*\right]^{1/2}\times \]

\[ \times \exp\left\{-\frac{a+b}{\sqrt{\Delta}}\operatorname{arctg} \frac{-2c_2u+b-a}{\sqrt{\Delta}}\right\}=C, \]

where

\[ u=\frac{y}{x},\qquad \Delta=-(a-b)^2-4c_1^*c_2>0. \]

We shall not need the integral curves of equation (6) for the case \(\Delta\leq 0\).

Computing the total derivatives with respect to \(t\) of these functions by virtue of system (1), one can verify the validity of the assertion made above.

Take a point \((0,y_1)\), \(y_1>0\), and draw through it the curve \(\Gamma\) in the direction of increasing polar angle. Let \((0,y_2)\) be the next point after \((0,y_1)\) at which \(\Gamma\) intersects the positive semiaxis of ordinates. If \(y_2<y_1\), then the spiral composed of the curves \(\Gamma\) winds toward the origin as the polar angle increases without bound. Hence every solution for which the polar angle of the points of intersection with the positive semiaxis of ordinates increases by \(2\pi\) an infinite number of times also tends to the origin as \(t\to+\infty\).

It follows from what has been said that, in the case under consideration, a sufficient condition for stability in the large of the zero solution of system (1) is the validity of the inequality

\[ y_2<y_1. \tag{7} \]

Let \((0,-y_0)\) be the point at which \(\Gamma\) intersects the negative semiaxis of ordinates. Taking into account that the arcs of the curve \(\Gamma\) are given by the same equations in sectors symmetric with respect to the origin, we obtain that inequality (7) is equivalent to the following:

\[ y_2<y_0. \tag{8} \]

Let \((x_0,0)\) be the point at which \(\Gamma\) intersects the positive semiaxis of abscissas. More or less laborious computations give

\[ y_2=-\frac{x_0\sqrt{c_1^*(ab-c_1^*c_2)}}{(a+b)\sqrt{-c_2}}\times \]

\[ \times \exp\left\{\frac{a+b}{\sqrt{\Delta}}\left[ \frac{\pi}{2}-\operatorname{arctg} \frac{2c_1^*c_2+b(b-a)}{b\sqrt{\Delta}} \right]\right\}, \]

\[ y_0=\frac{(a+b)\sqrt{c_1^*}\,x_0}{c_2\sqrt{c_1^*-c_1}}\times \]

\[ \times \exp\left\{\frac{a+b}{\sqrt{\Delta}}\left( \operatorname{arctg}\frac{a+b}{\sqrt{\Delta}} +\operatorname{arctg}\frac{a-b}{\sqrt{\Delta}} \right)\right\}. \]

From (8) we obtain

\[ ab-c_1^*c_2<(a+b)^2\exp\left\{\frac{a+b}{\sqrt{\Delta}}\left[ \operatorname{arctg}\frac{a+b}{\sqrt{\Delta}}+ \right.\right. \]

\[ \left.\left. +\operatorname{arctg}\frac{a-b}{\sqrt{\Delta}} +\operatorname{arctg}\frac{2c_1^*c_2+b(b-a)}{b\sqrt{\Delta}} -\frac{\pi}{2}\right]\right\}. \tag{9} \]

By elementary transformations, inequality (9) is reduced to the form

\[ ab-c_1^*c_2<(a+b)^2\exp\left\{\frac{2(a+b)}{\sqrt{\Delta}}\operatorname{arctg}\frac{a+b}{\sqrt{\Delta}}\right\},\qquad \Delta>0. \tag{10} \]

It is clear that, instead of the integral curves of equation (6), in order to construct \(\Gamma\) we may use segments of the straight lines \(x=\mathrm{const}\) and \(y=\mathrm{const}\), which are also intersected by the solutions of system (1) in the direction of the origin. Inequality (8) in this case will give an obviously weaker result than (10); but with its help the case \(\Delta\leq 0\) is investigated. We have

\[ y_2=-\frac{c_1^*x_0}{a+b},\qquad y_0=\frac{(a+b)x_0}{c_2}. \]

From (8) we obtain

\[ -c_1^*c_2<(a+b)^2. \tag{11} \]

But if \(\Delta\leq 0\), then

\[ -c_1^*c_2\leq \frac14(a-b)^2<(a+b)^2, \]

i.e. in this case stability takes place.

The case \(c_1^*>-c_1\) is studied analogously; in this case the same result is obtained if \(c_1^*c_2\) is replaced by \(c_2^*c_1\). If, however, \(c_1^*=-c_1\), then closed curves (rectangles) can be constructed from segments of the straight lines \(x=\mathrm{const}\) and \(y=\mathrm{const}\), whence the stability of the zero solution follows.

We note that if \(c_1^*>-c_1\), then \(-c_1^*c_2>-c_2^*c_1\), while if \(c_1^*<-c_1\), then \(-c_1^*c_2<-c_2^*c_1\).

Put

\[ A=\max\{-c_1^*c_2,\,-c_2^*c_1\}. \tag{12} \]

Then we obtain the theorem.

Theorem 1. Suppose that in system (1) \(ab>0\) and the inequalities (2) hold. Then the zero solution of the system is stable in the large if either of the inequalities holds:

\[ 1)\ \Delta=4A-(a-b)^2\leq 0, \]

\[ 2)\ ab+A<(a+b)^2\exp\left\{\frac{2(a+b)}{\sqrt{\Delta}}\operatorname{arctg}\frac{a+b}{\sqrt{\Delta}}\right\}, \tag{13} \]

where \(A\) is defined by formula (12).

Remark 1. Inequality (11) gives a weaker, but simpler, sufficient condition for stability:

\[ A<(a+b)^2. \tag{14} \]

Remark 2. It was noted earlier that the integral curves of equations (4) and (5) in the corresponding sectors are intersected by the solutions of system (1) strictly in the direction of the origin (tangencies excluded). Using this circumstance, it is not difficult to prove that inequality (7) (and hence also (8)) can be replaced by a non-strict one. Therefore, in inequalities (13) and (14) the inequality signs may be replaced by non-strict ones.

Remark 3. Inequality (13) cannot be strengthened if one uses only the fact that the admissible regions of values of the functions \(h_1(x,y,t)\) and \(h_2(x,y,t)\) are determined by inequalities (3).

Let us now suppose that the conditions of Theorem 1 are not satisfied (such a case arises whenever there are zeros among the numbers \(c_i,c_i^*\)).

If \(h_1(x,y,t)\) and \(h_2(x,y,t)\) are bounded, then \(A \geqslant \sup |h_1h_2|\) (here and below in § 1, \(\sup\) and \(\inf\) denote the exact bounds of the functions for all \(x,y,t\)). Then from Theorem 1, taking Remark 2 into account, it follows that

Theorem \(1'\). Let \(ab>0\) in system (1), and let inequalities (2) hold. Then the zero solution of the system is stable in the large if either of the following inequalities holds:

\[ \begin{gathered} 1)\quad \Delta' = 4\sup |h_1h_2| - (a-b)^2 \leqslant 0,\\ 2)\quad ab+\sup |h_1h_2| \leqslant (a+b)^2 \exp\left\{\frac{2(a+b)}{\sqrt{\Delta'}}\operatorname{arctg}\frac{a+b}{\sqrt{\Delta'}}\right\}. \end{gathered} \tag{13'} \]

Remark. If the range of values of the functions \(h_1\) and \(h_2\) coincides with the domain determined by inequalities (3), then Theorem \(1'\) coincides with Theorem 1 and condition \((13')\) cannot be improved. If, however, the range of values of the functions \(h_1\) and \(h_2\) is contained in the domain determined by inequalities (3), then inequality \((13')\) can be strengthened. For this purpose, in the differential equations one should put

\[ c_1=\inf h_1,\qquad c_1^*=\sup h_1, \]

\[ c_2=\inf h_2,\qquad c_2^*=\sup h_2. \]

In this case either \(c_1c_2<ab\), or \(c_1^*c_2^*<ab\), or both hold. Therefore equations (4) and (5) are not simplified as in the case considered, and the inequality to which inequality (8) reduces will, generally speaking, be more cumbersome than \((13')\), but more accurate.

Using the remark just made, we shall obtain sufficient conditions for stability in the large of the zero solution for a number of special forms of system (1).

Suppose that among the numbers \(c_i, c_i^*\) from inequalities (3) there are some equal to zero. We shall restrict ourselves to the case of two zeros. This is possible in two cases: 1) one of the functions \(h_1\) or \(h_2\) is identically equal to zero, 2) the functions \(h_1\) and \(h_2\) do not take values of the same sign.

In the first case, a sufficient condition for stability is the boundedness of the nonzero function. This follows directly from consideration of system (1).

Let us consider the second case. Let \(h_1\geqslant 0,\ h_2\leqslant 0\) (the case \(h_1\leqslant 0,\ h_2\geqslant 0\) reduces to the case under consideration by the substitution \(x_1=-x,\ y_1=y\)). In accordance with the remark to Theorem \(1'\), put

\[ c_1=0,\qquad c_1^*=\sup h_1, \]

\[ c_2=\inf h_2,\qquad c_2^*=0. \]

All the arguments carried out in the proof of Theorem 1 remain valid also in the case under consideration. Here equation (4) takes the form

\[ \frac{dx}{dy}=\frac{ax}{by}+\frac{c_2}{b}. \]

Integrating this linear equation, we find its general solution in the form

\[ x=Cy^{a/b}+\frac{c_2y}{b-a}\qquad (a\ne b), \]

analogously, equation (5) takes the form

\[ \frac{dy}{dx}=\frac{by}{ax}+\frac{c_1^*}{a}. \]

Its general solution is

\[ y=Cx^{b/a}+\frac{c_1^*x}{a-b}\qquad (a\ne b). \]

Since, by assumption, \(h_2\leqslant 0\), we have

\[ \left.\frac{dx}{dt}\right|_{x=0}=h_2y\leqslant 0 \quad \text{for } y>0. \]

Therefore the polar angle of successive intersections of the solutions with the positive semiaxis of ordinates can only increase. Consequently, inequality (8) also gives a sufficient condition for stability in the case under consideration. Taking into account that \(-c_1^*c_2^*=\sup |h_1h_2|\), we obtain the inequality

\[ ab+\sup |h_1h_2|\leqslant ab\left(\frac{a}{b}\right)^{\frac{a+b}{a-b}} \exp\left\{\frac{2(a+b)}{\sqrt{\Delta'}}\operatorname{arctg} \frac{a+b}{\sqrt{\Delta'}}\right\}. \tag{15} \]

Here, as in (13), \(\Delta'=4\sup |h_1h_2|-(a-b)^2\).

Thus, if the functions \(h_1(x,y,t)\) and \(h_2(x,y,t)\) do not take values of one sign, then inequality (13′) in Theorem \(1'\) may be replaced by the stronger inequality 15.

If \(a=b\), then in (15) one must pass to the limit as \(a-b\to 0\). Since in this case \(\Delta'=4\sup |h_1h_2|\), (15) takes the form

\[ \alpha\leqslant a^2\left[ \exp\left\{2+\frac{2a}{\sqrt{\alpha}}\operatorname{arctg}\frac{a}{\sqrt{\alpha}}\right\}-1 \right], \]

where \(\alpha=\sup |h_1h_2|\).

Let now \(ab=0\). It follows from (2) that in this case the functions \(h_1(x,y,t)\) and \(h_2(x,y,t)\) have opposite signs. Hence a sufficient condition for stability is given by inequality (15), where either \(a=0\) or \(b=0\). For definiteness let \(b=0\). Then from (15) we obtain

\[ \sup |h_1h_2|\leqslant a^2\exp\left\{\frac{2a}{\sqrt{\Delta'}}\operatorname{arctg}\frac{a}{\sqrt{\Delta'}}\right\}, \tag{16} \]

where \(\Delta'=4\sup |h_1h_2|-a^2>0\).

An important special case of system (1) is the case \(b=0\), \(|h_2(x,y,t)|\equiv 1\). In accordance with the remark to Theorem \(1'\), set (if \(h_1(x,y,t)>0\)) \(c_1=0,\ c_1^*=\sup h_1,\ c_2=c_2^*=-1\). Then equation (4) reduces to \(y'=0\), and equations (5) and (6) coincide and take the form

\[ \frac{dy}{dx}=\frac{c_1^*x}{ax-y}. \tag{17} \]

The general solution of equation (17) has the form

\[ x(u^2-au+c_1^*)^{1/2} \exp\left\{-\frac{a}{\sqrt{\Delta}}\operatorname{arctg} \frac{2u-a}{\sqrt{\Delta}}\right\}=C, \tag{18} \]

where \(u=\dfrac{y}{x}\), \(\Delta=4c_1^*-a^2\). The curve \(\Gamma\) in the right half-plane is composed of a segment of the straight line \(y=\mathrm{const}\) for \(y\leq ax\) and an arc of the curve (18) for \(y\geq ax\). From inequality (8), taking into account Remark 2 to Theorem 1, we obtain

\[ c_1^* \leq a^2 \exp\left\{\frac{a}{\sqrt{\Delta}}\left(2\operatorname{arctg}\frac{a}{\sqrt{\Delta}}-\pi\right)\right\}, \tag{19} \]

where \(\Delta\) may be regarded as a positive quantity.

Instead of system (1) for \(a<0,\ b=0\), let us now consider the more general system

\[ \begin{aligned} \frac{dx}{dt} &= f(x,y,t)x+h_2(x,y,t)y,\\ \frac{dy}{dt} &= h_1(x,y,t)x, \end{aligned} \tag{20} \]

where \(f(x,y,t)\leq a<0,\quad h_1(x,y,t)h_2(x,y,t)<0\).

We shall prove that inequalities (16) and (19) give sufficient conditions for stability in the large of the zero solution of system (20). For this it is enough to show that the arcs of the curves of which \(\Gamma\) is composed are intersected by the solutions of system (20) in the direction toward the origin. Let these curves be given by the equation \(V(x,y)=C>0\), and let \(V'_{(1)}\) denote the total derivative of \(V\) with respect to \(t\) in virtue of system (1), while \(V'_{(20)}\) denotes, respectively, the derivative in virtue of system (20). Then we have

\[ V'_{(20)}=V'_{(1)}+\frac{\partial V}{\partial x}(f-a)x. \]

It is not difficult to verify that, for the arcs of the curve \(\Gamma\),

\[ \frac{\partial V}{\partial x}x\geq 0. \]

Therefore \(V'_{(20)}\leq V'_{(1)}\), which was required to be proved. From what has been said it follows:

Theorem 2. Let, in system (20), \(f(x,y,t)\leq a<0,\ h_1(x,y,t)h_2(x,y,t)<0\). Then the zero solution of system (20) is stable in the large if either of the inequalities holds:

\[ 1)\quad \Delta'=4\sup |h_1h_2|-a^2\leq 0, \]

\[ 2)\quad \sup |h_1h_2|\leq a^2\exp\left\{\frac{2a}{\sqrt{\Delta'}}\operatorname{arctg}\frac{a}{\sqrt{\Delta'}}\right\}. \]

Moreover, if \(|h_2(x,y,t)|\equiv 1\), then for stability in the large of the zero solution it is sufficient that either of the inequalities be satisfied:

\[ 1)\quad \Delta'=4\sup |h_1|-a^2\leq 0, \]

\[ 2)\quad \sup |h_1|\leq a^2\exp\left\{\frac{a}{\sqrt{\Delta}}\left(2\operatorname{arctg}\frac{a}{\sqrt{\Delta'}}-\pi\right)\right\}. \]

In conclusion of the paragraph, we note that the case \(ab<0\) can be studied with the aid of analogous reasoning. A simple sufficient condition for stability in the large for this case is given in [5].

§ 2. CONVERGENCE IN SYSTEMS WITH DISSIPATION

Consider the system

\[ \begin{aligned} \frac{dx}{dt} &= ax+f_2(y)+P(t),\\ \frac{dy}{dt} &= f_1(x)+by+Q(t), \end{aligned} \tag{21} \]

where \(f'_1(x)\) and \(f'_2(y)\) are differentiable, while \(P(t)\) and \(Q(t)\) are continuous and bounded. Suppose further that

\[ a+b<0,\qquad ab-f'_1(x)f'_2(y)>\varepsilon(x)+\varepsilon(y), \tag{22} \]

where \(\varepsilon(z)>0\) and \(\varepsilon(z)z\to \pm\infty\) as \(z\to \pm\infty\).

Under these assumptions (and even under weaker ones) system (21) is dissipative [6], i.e., there exists a constant \(r>0\) such that, for all solutions of system (21),

\[ \varlimsup_{t\to+\infty}|x(t)|<r,\qquad \varlimsup_{t\to+\infty}|y(t)|<r . \]

Denote by \(R\) the square \(|x|<r,\ |y|<r\).

Let \(x(t), y(t)\) and \(x_1(t), y_1(t)\) be any two solutions of system (21). Put \(v=x-x_1,\ w=y-y_1\). Then we obtain

\[ \begin{aligned} \frac{dv}{dt} &= av+f_2(y_1+w)-f_2(y_1),\\ \frac{dw}{dt} &= f_1(x_1+v)-f_1(x_1)+bw . \end{aligned} \tag{23} \]

System (23) can be written in the form

\[ \begin{aligned} \frac{dv}{dt} &= av+f'_2(y_1+\xi w)\,w,\\ \frac{dw}{dt} &= f'_1(x_1+\eta v)\,v+bw \qquad (0\leq \xi;\ \eta\leq 1). \end{aligned} \tag{24} \]

System (24) is a system of type (1), and therefore, in order to formulate a theorem on convergence in system (21), we can use Theorem \(1'\).

Theorem 3. Suppose that in system (21) \(ab>0\) and inequalities (22) hold. Then all solutions of system (21) converge as \(t\to+\infty\), i.e.,
\(x_1(t)-x(t)\to 0,\ y_1(t)-y(t)\to 0\) as \(t\to+\infty\), if either of the inequalities

\[ 1)\quad \Delta=4\sup_R |f'_1(x)f'_2(y)|-(a-b)^2\leq 0, \]

\[ 2)\quad ab+\sup_R |f'_1(x)f'_2(y)| \leq (a+b)^2 \exp\left\{ \frac{2(a+b)}{\sqrt{\Delta}}\operatorname{arctg}\frac{a+b}{\sqrt{\Delta}} \right\}. \]

Remark 1. Inequality (14) gives the following sufficient condition for convergence:

\[ \sup_R |f_1'(x) f_2'(y)| \leq (a+b)^2. \]

Remark 2. Using the remark to Theorem \(1'\), one can obtain a stronger sufficient condition for convergence in system (21). In particular, if \(f_1'(x)\) and \(f_2'(y)\) do not take values of one sign in \(R\), one may use inequality (15).

In conclusion, let us consider the equation

\[ \frac{d^2 x}{dt^2} + f(x)\frac{dx}{dt} + g(x) = p(t), \tag{25} \]

where \(f(x)>0\) and is continuous; \(g(x)\) is continuous, strictly increasing, and changes sign; \(p(t)\) is continuous and bounded.

The question of convergence in this equation reduces to the question of stability in the large of the zero solution of the system

\[ \frac{dv}{dt} = w - F(\varphi+v) + F(\varphi), \]

\[ \frac{dw}{dt} = -g(\varphi+v) + g(\varphi), \]

where
\[ F(x)=\int_0^x f(\xi)\,d\xi, \]
and \(\varphi(t)\) is an arbitrary solution of equation (25). It is known [7] that, under our assumptions,

\[ \varlimsup_{t\to+\infty} |\varphi(t)| < r > 0. \]

It follows from Theorem 2 that convergence takes place in equation (25) if either

\[ \Delta = 4B-a^2 \leq 0, \]

or

\[ a^2 > B \exp\left\{-\frac{a}{\sqrt{\Delta}}\left(2\operatorname{arctg}\frac{a}{\sqrt{\Delta}}+\pi\right)\right\}, \tag{26} \]

where

\[ a=\inf_{|x|<r} f(x);\qquad B=\sup_{\substack{|x_1|<r,\ |x_2|<r}} \frac{g(x_1)-g(x_2)}{x_1-x_2} \quad (x_1\ne x_2). \]

Opial [8] again obtained the strict inequality (26) by the same method, but under the assumption that \(g(x)\) is continuously differentiable.

References

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2. N. N. Krasovskii, Some Problems in the Theory of Stability of Motion. Moscow, Fizmatgiz, 1959.
3. L. Cesari, Asymptotic Behavior and Stability of Solutions of Ordinary Differential Equations. Moscow, “Mir,” 1964.
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Received by the editors
November 26, 1965

Leningrad State University
named after A. A. Zhdanov