Mathematics

A. Yu. Levin

ON LINEAR DIFFERENTIAL EQUATIONS OF THE SECOND ORDER

(Presented by Academician I. N. Vekua, 11 VII 1963)

1. We first note one simple comparison theorem. As is known, in the case of nonoscillation of the equation \(\ddot{x}+q(t)x=0\) (i.e., in the case when its nontrivial solutions have a finite number of zeros on \((0,\infty)\)), there exists a solution \(x(t)\), unique up to a factor, such that

\[ \int_{t_0}^{\infty} x^{-2}(t)\,dt=\infty \]

(\(t_0\) is chosen to the right of the zeros of \(x(t)\)); this solution is naturally called minimal. We shall call a fundamental system \(x_1(t), x_2(t)\) a proper fundamental system (abbreviated p.f.s.) if \(x_2(t)\) is a minimal solution and \(x_1(t), x_2(t)>0\) for \(t\geq t_0\). Consider the equations

\[ \ddot{x}+q(t)x=0, \tag{1} \]

\[ \ddot{y}+r(t)y=0. \tag{2} \]

Theorem 1. Let \(q(t)\leq r(t)\) \((t\geq t_1)\), and let equation (2) be nonoscillatory. Then the p.f.s.’s \(x_1(t), x_2(t)\) and \(y_1(t), y_2(t)\) for (1) and (2), respectively, satisfy the conditions:

\[ x_1(t)\geq C_1y_1(t),\qquad x_2(t)\leq C_2y_2(t)\qquad (C_1,C_2>0,\ t\geq t_0). \]

Examples: If \(q(t)\leq -a^2\) \((a>0,\ t\geq t_1)\), then any p.f.s. is such that \(x_1(t)\geq C_1e^{at}\), \(x_2(t)\leq C_2e^{-at}\) \((t\geq t_0)\); if \(q'(t)\leq k(1-k)t^{-2}\) \((k>1/2,\ t\geq t_1)\), then any p.f.s. is such that \(x_1(t)\geq C_1t^k\), \(x_2(t)\leq C_2t^{1-k}\) \((t\geq t_0)\), and so on. With the help of Theorem 1 it is also easy to show that if \(q(t)\) is positive and nondecreasing for \(t\geq t_1\), then any p.f.s. for the equation \(\ddot{x}-q(t)x=0\) is such that

\[ x_1(t)\geq C_1\exp\left(\int_{t_0}^{t}\sqrt{q(s)}\,ds\right),\qquad x_2(t)\geq C_2\exp\left(-\int_{t_0}^{t}\sqrt{q(s)}\,ds\right)\quad (t\geq t_0). \tag{3} \]

1. Consider the antiperiodic boundary-value problem

\[ \ddot{x}+[q(t)+\lambda]x=0,\qquad x(a)=-x(b),\qquad \dot{x}(a)=-\dot{x}(b). \tag{4} \]

Theorem 2. The smallest eigenvalue \(\lambda_1\) of the boundary-value problem (4) satisfies the inequality\(^*\)

\[ \lambda_1\leq \frac{\pi^2}{(b-a)^2}-\frac{I}{b-a},\qquad \text{where } I=\int_a^b q(t)\,dt. \tag{5} \]

\(^*\) As M. G. Krein informed the author, he has also obtained some analogous estimates.

Estimate (5) is attained for \(q(t)\equiv \mathrm{const}\); it shows, among other things, that in the case \(q(t)\geqslant 0\), using the single characteristic \(I\), one can give a two-sided estimate for the first positive eigenvalue \(\mu_1\) of the boundary-value problem
\(\ddot{x}+\mu q(t)x=0,\ x(a)=-x(b),\ \dot{x}(a)=-\dot{x}(b)\):

\[ 4<\mu_1(b-a)I\leqslant \pi^2 \]

(note that, for example, for the periodic problem or the problem \(x(a)=x(b)=0\), such an estimate is impossible).

1. For many questions (estimates of eigenvalues, applicability of Chaplygin’s theorem, etc.) conditions are of interest under which every nontrivial solution of the equation

\[ \ddot{x}+p(t)\dot{x}+q(t)x=0 \tag{6} \]

has no more than one zero on the given interval \([a,b]\). In this connection various characteristics of the coefficients may be used; the criterion given below uses the norms of the coefficients in \(L_1[a,b]\), and Sturm’s theorem makes it possible, instead of the norm of \(q(t)\), to restrict oneself to the norm of the function
\(q_+(t)=\max\{0,q(t)\}\).

Theorem 3. If the inequality

\[ I_1 e^{I_2/2}\leqslant \frac{4}{b-a},\qquad \text{where } I_1=\int_a^b q_+(t)\,dt,\qquad I_2=\int_a^b |p(t)|\,dt, \]

is satisfied, then every nontrivial solution of equation (6) has no more than one zero on \([a,b]\).

The formulated criterion is unimprovable in the characteristics \(I_1, I_2\).

1. Put

\[ I(p)=\int^\infty \exp\left(-\int^t p(s)\,ds\right)\,dt,\qquad K(p,q)=\int^\infty q(t)\exp\left(\int^t p(s)\,ds\right)\,dt \]

(the lower limits may be taken to be arbitrary finite ones, since we shall be interested only in questions of convergence). Sturm’s theorem shows that a decrease of the “attraction” \(q(t)\) preserves nonoscillation, i.e., from the nonoscillation of (6) follows the nonoscillation of the equation

\[ \ddot{x}+p_1(t)\dot{x}+q_1(t)x=0, \tag{7} \]

if \(p_1(t)\equiv p(t)\), \(q_1(t)\leqslant q(t)\). The natural question is how, in this respect, a change of the “friction” \(p(t)\) affects the matter.

For the case \(q(t)\geqslant 0\) the answer is given by the following propositions.

Theorem 4. Let \(q(t)\geqslant 0\). If \(I(p)<\infty\) \((=\infty)\), then an increase (decrease) of \(p(t)\) preserves nonoscillation. More precisely, if \(q_1(t)\leqslant q(t)\), then the nonoscillation of (6) implies the nonoscillation of (7) in either of the following cases: a) \(I(p)<\infty,\ p_1(t)\geqslant p(t)\); b) \(I(p)=\infty,\ p_1(t)\leqslant p(t)\).

Theorem 5. Let \(q(t)\geqslant 0,\ I(p)<\infty,\ K(p,q)<\infty\). Then both an increase and a decrease of \(p(t)\) preserve nonoscillation, i.e., if \(q_1(t)\leqslant q(t)\) and either of the inequalities \(p_1(t)\geqslant p(t)\), \(p_1(t)\leqslant p(t)\) is satisfied, then the nonoscillation of (6) implies the nonoscillation of (7).

We note that, in particular, \(I(p)=\infty\) if \(p(t)\leqslant t^{-1}\), and \(I(p)<\infty\) if \(p(t)\geqslant (1+\varepsilon)t^{-1}\) \((\varepsilon>0)\) for sufficiently large \(t\).

1. Consider the equation with a parameter

\[ \ddot{x}+\alpha p(t)\dot{x}+q(t)x=0,\qquad (p(t),q(t)\geqslant 0,\ -\infty<\alpha<\infty). \tag{8} \]

Denote by \(\alpha_0\) (\(\leqslant \infty\)) the infimum of those degrees for which the function

\[ h(t)=\exp\left(-\int^t p(s)\,ds\right) \]

is summable on \((0,\infty)\). Theorem 4 leads, in particular, to the following “extrapolation” assertion: in order that equation (6) be nonoscillatory for every \(\alpha\), it is necessary and sufficient that it be nonoscillatory in some neighborhood of the point \(\alpha=\alpha_0\). It is curious that the point of “greatest oscillation” \(\alpha_0\) is determined by the single function \(p(t)\), and moreover in a quite effective way.

Theorem 6. In the case of nonoscillation of (6), the condition \(I(p)<\infty\) is necessary, and, when \(q(t)\geqslant 0\), sufficient for the boundedness of all solutions of (6) on \((0,\infty)\).

This proposition is a generalization of a close result from \((^2)\). Comparing Theorems 4 and 6 leads to the following comparison theorem.

Theorem 7. Let \(0\leqslant q_1(t)\leqslant q(t)\), \(p_1(t)\geqslant p(t)\). If the solutions of (6) do not oscillate and are bounded for \(t\geqslant 0\), then the solutions of (7) also do not oscillate and are bounded for \(t\geqslant 0\).

1. Let the coefficients of the equation

\[ \ddot{x}+p(t)\dot{x}+q(t)x=f(t) \tag{9} \]

satisfy the conditions

\[ 0\leqslant m\leqslant q(t)\leqslant M,\qquad p(t)\geqslant l>0\qquad (t\geqslant t_0). \tag{10} \]

As the following propositions show, some of the results formulated in \((^2)\) for equation (6) extend to the case of the nonhomogeneous equation (9).

Below we use the function \(\psi(l,m,M)\) defined in \((^2)\) and the transcendental constant \(h_0\approx 3.046\).

Theorem 8. Let conditions (10) be fulfilled, and suppose that

\[ \psi(l,m,M)<1. \tag{11} \]

Then for any \(f(t)=O(1)\) all solutions of (9), together with their derivatives, are bounded on \((0,\infty)\).

Theorem 9. Let conditions (10), (11) be fulfilled and, in addition,

\[ \int^\infty dt\int^t \exp\left(-\int_s^t p(\tau)\,d\tau\right)\,ds=\infty. \tag{12} \]

Then for any \(f(t)=O(1)\) all solutions of (9), together with their derivatives, tend to zero as \(t\to\infty\).

Of course, first derivatives are meant. Condition (12), first used by Opial in \((^1)\), is a restriction, in a certain sense, on the growth of \(p(t)\) as \(t\to\infty\); it is not difficult to show that, in particular, it is fulfilled when \(p(t)=O(t)\).

1. We now consider, instead of (10), the less restrictive conditions

\[ 0\leqslant q(t)\leqslant M,\qquad p(t)\geqslant l>0\qquad (t\geqslant t_0). \tag{13} \]

Theorem 10. Let conditions (13) be fulfilled, and suppose that

\[ M<h_0l^2. \tag{14} \]

Then for any \(f(t)=O[q,t]\) all solutions of (9), together with their derivatives, are bounded on \((0,\infty)\).

Theorem 11. Suppose that conditions (13), (14) are satisfied and, moreover,

\[ \int^\infty dt \int^t q(s)\exp\left(-\int_s^t(-p(\tau)\,d\tau)\right)\,ds=\infty. \tag{15} \]

Then for any \(f(t)=o[q(t)]\) all solutions of (9), together with their derivatives, tend to zero as \(t\to\infty\).

The notation \(f(t)=O[q(t)]\) and \(f(t)=o[q(t)]\) means, respectively, that \(|f(t)|\leq Cq(t)\) \((t\geq t_1)\) and \(f(t)=\varepsilon(t)q(t)\), \(\varepsilon(t)\to 0\) as \(t\to\infty\). We note that condition (15) (considered earlier in \((^2,^3)\)) is satisfied, in particular, if \(q(t)\geq C_1t^{-r}\), \(p(t)\geq C_2t^{1-r}\) \((C_1,C_2>0,\ 0\leq r\leq 1,\ t\geq t_0)\).

1. Theorems 8–11 cannot be improved with respect to the characteristics \(l,m,M\). Condition (11) in Theorems 8 and 9 may be replaced by the simpler-to-check (but more restrictive) conditions \(\sqrt M-\sqrt m\leq \pi l/2\) or \(M\leq h_0l^2\). Analogous results can also be obtained for certain nonlinear equations; in particular, Theorems 8–11 remain valid for the equation
   \[ \ddot x+p(t,x,\dot x)\dot x+q(t,x,\dot x)x=f(t,x,\dot x), \]
   if the functions \(p,q,f\) satisfy the corresponding requirements uniformly in \(x,\dot x\).

The proof of Theorems 8–11 is based on reducing the questions under consideration to a certain optimal-control problem; in solving the latter, the mechanical interpretation of (9) as the equation of one-dimensional oscillations with variable friction coefficient \(p(t)\), variable attraction coefficient \(q(t)\), and disturbing force \(f(t)\) is heuristically useful.

The author takes this opportunity to express his gratitude to M. A. Krasnosel’skii for his constant attention.

Received
6 VII 1963

REFERENCES

\(^1\) Z. Opial, Ann. polon. math., 6, No. 2 (1959).
\(^2\) A. Yu. Levin, DAN, 141, No. 6 (1961).
\(^3\) N. Gheorghiu, Anal. Ști. Univ. Cuza, Iași, 7, No. 1 (1961).