MATHEMATICS

A. Yu. LEVIN

ON A COMPARISON PRINCIPLE FOR SECOND-ORDER DIFFERENTIAL EQUATIONS

(Presented by Academician I. G. Petrovskii, 22 VI 1960)

1. Let two equations be given
   \[ x_1''+\varphi_1(t)x_1=0, \tag{1} \]
   \[ x_2''+\varphi_2(t)x_2=0, \tag{2} \]
   where the coefficients \(\varphi_1(t)\), \(\varphi_2(t)\) are assumed summable on the interval \([a,b]\) (here equations (1), (2), as well as all differential equations encountered below, are to be understood as being satisfied almost everywhere on \([a,b]\)). Speaking of solutions of equations (1), (2), we shall everywhere in what follows have in mind nontrivial solutions.

2. Theorem 1. Let \(x_1(t)\) not vanish on \([a,b]\). Then:

1) if on \([a,b]\) the inequality
\[ -\frac{x_1'(a)}{x_1(a)}+\int_a^t \varphi_1(\tau)\,d\tau > \left| -\frac{x_2'(a)}{x_2(a)}+\int_a^t \varphi_2(\tau)\,d\tau \right|, \qquad a\leq t\leq b, \tag{3} \]
holds, then \(x_2(t)\) does not vanish on \([a,b]\) and
\[ -\frac{x_1'(t)}{x_1(t)} > \left| \frac{x_2'(t)}{x_2(t)} \right|, \qquad a\leq t\leq b; \tag{4} \]

2) if on \([a,b]\) the inequality
\[ \frac{x_1'(b)}{x_1(b)}+\int_t^b \varphi_1(\tau)\,d\tau > \left| \frac{x_2'(b)}{x_2(b)}+\int_t^b \varphi_2(\tau)\,d\tau \right|, \qquad a\leq t\leq b, \tag{5} \]
holds, then \(x_2(t)\) does not vanish on \([a,b]\) and
\[ \frac{x_1'(t)}{x_1(t)} > \left| \frac{x_2'(t)}{x_2(t)} \right|, \qquad a\leq t\leq b. \tag{6} \]

If (3) (respectively, (5)) is fulfilled with the sign of a non-strict inequality, then (4) (respectively, (6)) is also valid with the sign of a non-strict inequality.

Proof. Assertion 2) follows from 1) with the aid of the change
\(\widetilde{x}_i(t)=x_i(a+b-t)\), \(i=1,2\). It is therefore sufficient to prove assertion 1). The substitution
\[ z_i(t)=-\frac{x_i'(t)}{x_i(t)},\qquad i=1,2, \tag{7} \]
transforms, as is known, equations (1), (2) into the Riccati equations
\[ z_1'=z_1^2+\varphi_1(t), \tag{8} \]
\[ z_2'=z_2^2+\varphi_2(t). \tag{9} \]

The functions \(z_i(t)\) are, evidently, continuous everywhere except at the zeros of \(x_i(t)\), which are points of infinite discontinuity for the corresponding \(z_i(t)\). Since \(x_1(t)\) does not vanish on \([a,b]\), \(z_1(t)\) is continuous on \([a,b]\) and, consequently, (8) is equivalent to the integral equation

\[ z_1(t)=z_1(a)+\int_a^t z_1^2(\tau)\,d\tau+\int_a^t \varphi_1(\tau)\,d\tau, \tag{10} \]

whence, taking (3) into account, in particular it follows that

\[ z_1(t)\geq z_1(a)+\int_a^t \varphi_1(\tau)\,d\tau>0,\qquad a\leq t\leq b. \tag{11} \]

Let us now turn to the behavior of the function \(z_2(t)\) on the interval \([a,b]\). From condition (3) it follows (for \(t=a\)) that

\[ |z_2(a)|<z_1(a). \tag{12} \]

Therefore \(z_2(t)\) is continuous at the point \(t=a\), and hence also on some interval \([a,c]\), \(a<c\leq b\). On this interval (9) can be rewritten in the form of the integral equation

\[ z_2(t)=z_2(a)+\int_a^t z_2^2(\tau)\,d\tau+\int_a^t \varphi_2(\tau)\,d\tau,\qquad a\leq t\leq c. \tag{13} \]

We shall prove that on \([a,c]\) the inequality

\[ |z_2(t)|<z_1(t),\qquad a\leq t\leq c \tag{14} \]

holds. Indeed, suppose the contrary. Then two cases may occur:

a) For some \(t=t_0\), \(a<t_0\leq c\),

\[ z_2(t_0)\leq -z_1(t_0). \tag{15} \]

b) For some \(t=t_0\), \(a<t_0\leq c\),

\[ z_2(t_0)\geq z_1(t_0). \tag{16} \]

We show that neither of these cases can occur:

a) Taking (13), (3), and (11) into account, we find that for any \(t_0\) in \([a,c]\)

\[ z_2(t_0)=z_2(a)+\int_a^{t_0} z_2^2(\tau)\,d\tau+\int_a^{t_0}\varphi_2(\tau)\,d\tau\geq z_2(a)+\int_a^{t_0}\varphi_2(\tau)\,d\tau> \]

\[ > -z_1(a)-\int_a^{t_0}\varphi_1(\tau)\,d\tau\geq -z_1(t_0). \tag{17} \]

b) Suppose (16) holds. Then, by virtue of the continuity of \(z_1(t)\), \(z_2(t)\) on \([a,c]\) and relation (12), there exists a point \(t=t_1\), \(a<t_1\leq t_0\), such that

\[ z_2(t_1)=z_1(t_1); \tag{18} \]

\[ z_2(t)<z_1(t)\quad \text{for } a\leq t<t_1. \tag{19} \]

Relations (17) and (19) show that

\[ |z_2(t)|<z_1(t),\qquad a\leq t<t_1, \]

whence

\[ \int_a^{t_1} z_2^2(\tau)\,d\tau<\int_a^{t_1} z_1^2(\tau)\,d\tau. \tag{20} \]

Taking (13), (3), (20), and (10) into account, we obtain

\[ z_2(t_1)=z_2(a)+\int_a^{t_1}\varphi_2(\tau)\,d\tau+\int_a^{t_1}z_2^2(\tau)\,d\tau< \]

\[ <z_1(a)+\int_a^{t_1}\varphi_1(\tau)\,d\tau+\int_a^{t_1}z_1^2(\tau)\,d\tau=z_1(t_1), \]

which contradicts relation (18).

We have thus shown that (14) is fulfilled on any interval \([a,c]\subset [a,b]\) of continuity of \(z_2(t)\). But, since \(z_1(t)\) is bounded on \([a,b]\), while \(z_2(t)\) can have only infinite discontinuities, it follows that \(z_2(t)\) is continuous everywhere on \([a,b]\). Therefore inequality (14) holds on the whole segment \([a,b]\). Assertion 1) is proved.

We have proved Theorem 1 in the part concerning strict inequalities. The case of non-strict inequalities is obtained from this by a limiting passage, whose validity is due to the uniqueness of the solution of the Riccati equation passing through a given point.

Remark 1. With respect to the signs of the inequalities, the theorem can be sharpened in the following way. If (3) (respectively (5)) is fulfilled on \([a,b]\) with the sign \(\geq\), but for some \(t=t_0,\ a\leq t_0\leq b\), the sign \(>\) holds, then (4) (respectively (6)) is also fulfilled with the sign \(\geq\), and for all \(t\) from the interval \([t_0,b]\) (respectively \([a,t_0]\)) the sign \(>\) holds.

Remark 2. The requirements (3), (5) can be weakened if additional characteristics of the functions \(\varphi_1(t), \varphi_2(t)\), connected with repeated integration, are brought into consideration. For example, (3) can be weakened if, in addition to the functions

\[ \Phi_i(t)=-\frac{x_i'(a)}{x_i(a)}+\int_a^t \varphi_i(\tau)d\tau,\qquad i=1,2, \tag{21} \]

one introduces into consideration the functions

\[ \psi_i(t)=\int_a^t \Phi_i^2(\tau)d\tau,\qquad i=1,2. \]

Because of lack of space we shall not dwell on this.

3. Corollary. In case 1), on \((a,b]\) the inequality

\[ \frac{x_2'(t)}{x_2(t)}-\frac{x_1'(t)}{x_1(t)}>\Phi_1(t)-\Phi_2(t),\qquad a<t\leq b, \]

holds, where \(\Phi_i(t)\) are given by formula (21). For the proof it suffices to compare (4), (10), and (13).

Similarly, in case 2), on \([a,b)\) the inequality

\[ \frac{x_1'(t)}{x_1(t)}-\frac{x_2'(t)}{x_2(t)}> \frac{x_1'(b)}{x_1(b)}-\frac{x_2'(b)}{x_2(b)} +\int_t^b[\varphi_1(\tau)-\varphi_2(\tau)]\,d\tau,\qquad a\leq t<b. \]

holds.

4. Introduce the notation

\[ \int_{(\alpha,\beta)} f(\tau)d\tau = \int_{\min\{\alpha,\beta\}}^{\max\{\alpha,\beta\}} f(\tau)d\tau. \]

It follows from Theorem 1 that:

Theorem 2. Let a solution \(x_2(t)\) of equation (2) satisfy the conditions

\[ x_2(a)=x_2(b)=x_2'(c)=0,\qquad a<c<b . \tag{22} \]

Further, suppose that for the coefficients of equations (1), (2) on \([a,b]\) the inequality

\[ \int_{(c,t)} \varphi_1(\tau)d\tau \geq \left|\int_{(c,t)} \varphi_2(\tau)d\tau\right|,\qquad a\leq t\leq b . \tag{23} \]

is fulfilled. Then every solution of equation (1) has at least one zero in the interval \([a,b]\).

Proof. Consider some nonzero solution \(x_1(t)\) of equation (1) satisfying the condition \(x_1'(c)=0\). It is easy to see that \(x_1(t)\) has at least one zero in each of the half-intervals \([a,c)\) and \((c,b]\). Indeed, if \(x_1(t)\) had no zeros, for example, in \((c,b]\), then we would be in the conditions of part 1) of Theorem 1, whence it would follow that \(x_2(t)\) has no zeros in \((c,b]\). But this is certainly false, since \(x_2(b)=0\). Similarly, using part 2) of Theorem 1, it is established that \(x_1(t)\) vanishes in \([a,c)\).

Thus \(x_1(t)\) has at least two zeros on \([a,b]\). But, by the zero-separation theorem, in this case every solution of equation (1) has at least one zero on \([a,b]\). The theorem is proved.

1. The results obtained above make it possible to establish comparison theorems for second-order equations in self-adjoint form:

\[ \frac{d}{dt}\left(k_1(t)\frac{dx_1}{dt}\right)+q_1(t)x_1=0, \tag{24} \]

\[ \frac{d}{dt}\left(k_2(t)\frac{dx_2}{dt}\right)+q_2(t)x_2=0 \tag{25} \]

(as usual, it is assumed that \(k_1(t),\,k_2(t)>0\)).

Here we shall confine ourselves to the formulation of the following result:

Theorem 3. Let \(q_1(t), q_2(t)\geq 0\). Suppose, further, that a nonzero solution \(x_2(t)\) of equation (25) satisfies conditions (22), with \(a\) and \(b\) being adjacent zeros.

In order that every solution of equation (24) vanish on \([a,b]\), it is sufficient that the following conditions be fulfilled:

\[ \int_a^t \frac{d\tau}{k_1(\tau)} \geq \int_a^t \frac{d\tau}{k_2(\tau)},\qquad \int_t^c q_1(\tau)d\tau \geq \int_t^c q_2(\tau)d\tau \qquad \text{for } a\leq t\leq c; \]

\[ \int_t^b \frac{d\tau}{k_1(\tau)} \geq \int_t^b \frac{d\tau}{k_2(\tau)},\qquad \int_c^t q_1(\tau)d\tau \geq \int_c^t q_2(\tau)d\tau \qquad \text{for } c\leq t\leq b. \]

Theorem 3 may be regarded as a strengthening of the classical Sturm comparison theorem for nonnegative \(q_1(t), q_2(t)\).

I express my sincere gratitude to my supervisor M. A. Krasnosel’skii.

Voronezh State
University

Received
18 VI 1960