BRIEF COMMUNICATIONS

ON CONDITIONS FOR BOUNDEDNESS OF THE DERIVATIVES OF BOUNDED SOLUTIONS OF ORDINARY DIFFERENTIAL EQUATIONS

A. Ya. Lepin, A. D. Myshkis

For the equation \(y''=f(x,y,y')\), the condition of S. N. Bernstein is well known (see [1], p. 192)

\[ |f(x,y,y')|\leq Ay'^2+B, \]

which guarantees the boundedness of \(y'\) and \(y''\) if \(y\) is bounded. Recently several papers [2—5] have appeared in which this theorem is developed in various directions. In particular, a result close to the limiting one was obtained by Yu. A. Klokov. Here we shall show that the results obtained in these directions, as well as some stronger assertions (with the exception of Bernstein’s original theorem!) are immediate consequences of one general theorem. This theorem, in turn, follows immediately from the Kolmogorov–Gorny inequalities (see, for example, [6], p. 393) for any \(n\)-times continuously differentiable function \(\psi(x)\) defined for \(a\leq x\leq b\):

\[ \max_{[a,b]}|\psi^{(i)}(x)| \leq a_{ni}\left[\max_{[a,b]}|\psi(x)|\right]^{\frac{n-i}{n}} \left[ \max\left\{\max_{[a,b]}|\psi^{(n)}(x)|,\; n!(b-a)^{-n}\max_{[a,b]}|\psi(x)|\right\} \right]^{\frac{i}{n}} \tag{1} \]

\[ (i=1,\ldots,n-1), \]

where \(a_{ni}>0\) are absolute constants. We shall also give examples indicating the sharpness of these assertions. We note that it would be interesting to carry these assertions over to a system of first-order equations for which an a priori estimate is given for some function of the coordinates of the solution.

All quantities involved will be assumed real.

1. Theorem. Let an \(n\geq 2\) times continuously differentiable function \(y(x)\) \((0\leq x<\infty)\) satisfy, for all sufficiently large \(x\), the inequality

\[ |y^{(n)}(x)|\leq \varphi\left(x,\max_{0\leq s\leq x}|y(s)|,\ldots,\max_{0\leq s\leq x}|y^{(n-1)}(s)|\right), \tag{2} \]

where the function \(\varphi(x,y_1,\ldots,y_n)\) is defined for \(x\geq x_0\), \(y_1\geq 0,\ldots,y_n\geq 0\), is nondecreasing in \(y_1,\ldots,y_n\), and for some \(A>0\) satisfies, for all sufficiently large \(x,y_2,\ldots,y_n\), the inequality

\[ \varphi(x,a,y_2,\ldots,y_n)\leq A\left(y_2^n+y_3^{\frac n2}+\cdots+y_n^{\frac n{\,n-1\,}}\right); \tag{3} \]

here the constant \(a\) is fixed and so small that

\[ A\left(a_{n1}^n\alpha^{\,n-1} +a_{n2}^{\frac n2}\alpha^{\frac{n-2}{2}} +\cdots +a_{n,n-1}^{\frac n{\,n-1\,}}\alpha^{\frac1{\,n-1\,}}\right)<1. \tag{4} \]

Suppose, in addition, that \(|y(x)|\leq \alpha\) \((0\leq x<\infty)\). Then for \(0\leq x<\infty\) all derivatives \(y',\ldots,y^{(n)}\) are bounded.

Proof. From (1) it follows first of all that it is enough to prove boundedness of \(y^{(n)}\). We shall assume that \(x>\sqrt[n]{\alpha n!}\). Then, if \(\max_{0\leq s\leq x}|y^{(n)}(s)|\) is sufficiently large, it follows from (2), (1), and (3) that ...

\[ |y^{(n)}(x)|\leq \varphi\left(x,\alpha,a_{n1}\alpha^{\frac{n-1}{n}} \left[\max_{0\leq s\leq x}|y^{(n)}(x)|\right]^{\frac1n}, \ldots,a_{n,n-1}\alpha^{\frac1n}\times\right. \]

\[ \left.\times \left[\max_{0\leq s\leq x}|y^{(n)}(x)|\right]^{\frac{n-1}{n}}\right) \leq A\left(a_{n1}^{\,n}\alpha^{n-1}+\cdots+ a_{n,n-1}^{\frac{n}{n-1}}\alpha^{\frac1{n-1}}\right) \max_{0\leq s\leq x}|y^{(n)}(x)|. \]

Hence, by virtue of (4), the assertion of the theorem follows.

Let us give several corollaries of the theorem and remarks on it.

1. From the theorem there immediately follows the following extension of Bernstein’s theorem to equations of higher order. Suppose the equation considered is

\[ y^{(n)}=f\left(x,y,y',\ldots,y^{(n-1)}\right) \qquad (x_0\leq x<\infty), \tag{5} \]

and, for some \(y_0>0,\ A>0,\ B>0\), the right-hand side is continuous for \(|y|\leq y_0\) and satisfies the inequality

\[ \left|f\left(x,y,y',\ldots,y^{(n-1)}\right)\right| \leq A\left(|y'|^n+|y''|^{\frac n2}+\cdots+ |y^{(n-1)}|^{\frac n{n-1}}\right)+B. \tag{6} \]

Then every solution of equation (5), defined for \(x_0\leq x<\infty\) and bounded in absolute value by a sufficiently small constant \(\alpha\) (this constant must not exceed \(y_0\) and must satisfy inequality (4)), also has bounded derivatives \(y',\ldots,y^{(n)}\). Moreover, an analysis of the proof of the theorem shows that on any interval \(x\geq \bar x=\mathrm{const}>x_0\) the estimate of the derivatives depends only on \(A,\ B,\ \alpha\), and \(\bar x\); hence it follows that a solution satisfying the a priori estimate \(|y|\leq \alpha\) can be continued from any finite interval to the whole half-axis \(x_0\leq x<\infty\).

Analogous assertions are valid for differential equations with retarded argument, for which one can derive an estimate of the form (2) with a function \(\varphi\) possessing the properties described above.

1. The formulation given in item 2 may seem unnatural, since it contains the requirement not only of boundedness (as in Bernstein’s theorem), but also of sufficient smallness of the solution. It is remarkable, however, that for \(n\geq 3\) this requirement is essential. This is seen from the example of the function \(y=\sin \dfrac{x^2}{2}\), satisfying a differential equation of the form

\[ y^{(n)}=a(x)\left(|y'|^n+\left|y''-\cos\frac{x^2}{2}\right|^{\frac n2}\right), \]

where

\[ \sup_{1<x<\infty}|a(x)|=a<\infty. \]

Here the right-hand side, in absolute value, does not exceed
\(a\left(|y'|^n+2^{\frac n2}|y''|^{\frac n2}\right)+2^{\frac n2}a\), and nevertheless

\[ \sup_{[1,\infty)}|y(x)|=1,\qquad \sup_{[1,\infty)}|y'(x)|=\sup_{[1,\infty)}|y''(x)|=\cdots=\infty. \]

In an analogous way, the function \(y=\sin \dfrac1x\) on a small interval \((-h,0)\) satisfies an equation of the form

\[ y^{(n)}=b(x)\left(|y'|^{(n)}+|y''|^{\frac n2}\right), \qquad \text{where } \sup |b(x)|<\infty. \]

Here the solution is bounded, but cannot be continued through the point \(x=0\).

It is of interest to find broad classes of equations above the second order for which the indicated requirement of sufficient smallness of the solution can be replaced by the requirement of its boundedness.

1. Since in the proof of the theorem we relied only on inequality (1), the theorem and its corollaries are valid for functions with values in any normed space in which analogous inequalities hold. In particular, this is true for functions with values in a finite-dimensional space (and hence for systems of differential—

differential equations) in any norm. At the same time, the remark of item 3 applies to systems not only of order higher than the second, but also of the second order. This is seen from the example of the pair of functions \(y_1=\sin \dfrac{x^2}{2}\), \(y_2=\cos \dfrac{x^2}{2}\), satisfying the system of equations

\[ y_1''=y_2-(y_1'^2+y_2'^2)\sin\frac{x^2}{2},\qquad y_2''=-y_1-(y_1'^2+y_2'^2)\cos\frac{x^2}{2}. \]

1. The requirement that the solution be bounded turns out to be sufficient if the requirements on the function \(\varphi\) are somewhat strengthened. Namely, let us replace (3) by the following condition: for every fixed \(\alpha>0\), as \(x\to\infty\), \(y_2\to\infty,\ldots,y_n\to\infty\), one has

\[ \varphi(x,\alpha,y_2,\ldots,y_n)=o\left(y_2^n+\cdots+y_n^{\frac{n}{n-1}}\right). \tag{7} \]

Then a bounded function \(y(x)\) satisfying estimate (2) has all derivatives up to order \(n\) inclusive bounded. This follows immediately from the fact that, for any \(\alpha\), inequality (4) can be satisfied by decreasing \(A\).

Under more restrictive assumptions this assertion appeared in the papers of M. A. Rutman [2]

\[ (\varphi=A[|y_2|+\cdots+|y_n|]+B) \]

and Yu. A. Klokov [3]

\[ \left(\varphi=A\left[|y_2|^{\,n-\varepsilon}+\cdots+|y_n|^{\frac{n}{n-1}-\varepsilon}\right]+B;\quad \varepsilon=\mathrm{const}>0\right), \]

and, for systems of second order with \(\varphi=A|y'|^{2-\varepsilon}+B\), in the paper of Opial [4].

To verify condition (7) one may use the equivalent condition: for every fixed \(\alpha>0\) and as \(x\to\infty\), \(y\to\infty\), one must have

\[ \varphi(x,\alpha,y,y^2,\ldots,y^{n-1})\,y^{-n}\to 0. \]

(To verify condition (3), one may use the equivalent condition of boundedness of the left-hand side for all sufficiently large \(x,y\).)

1. Let the function \(y(x)\) under consideration satisfy \(y(x)\to 0\) as \(x\to\infty\). Then, shifting the origin of coordinates sufficiently far to the right, we see that if inequality (3) is fulfilled for some \(\alpha>0\), \(A>0\), then \(y^{(n)}\) is bounded and therefore, by virtue of inequality (1), all derivatives \(y',\ldots,y^{(n-1)}\) tend to zero as \(x\to\infty\). (This assertion, for the solution of the scalar equation (5) under condition (6), was proved by Klokov [3], as became known to us after the main results of this paper had been obtained.) Making the change of variable \(y=y_0+z\) (\(y_0=\mathrm{const}\)), we see that an analogous result is valid if \(y(x)\) has a finite limit as \(x\to\infty\); however, in this case it is necessary to require that, for any fixed \(\alpha>0\), as \(x\to\infty\), \(y_2\to\infty,\ldots,y_n\to\infty\), the condition

\[ \varphi(x,\alpha,y_2,\ldots,y_n)=O\left(y_2^n+\cdots+y_n^{\frac{n}{n-1}}\right) \]

be satisfied.

For autonomous systems of second order this was proved by Klokov [5].

This remark is automatically transferred to equations (and systems) of the form (5), as well as to equations with retarded argument, for which one can obtain the inequality

\[ |y^{(n)}(x)|\leq \varphi\left(x,\max_{x-\Delta(x)\leq s\leq x}|y(s)|,\ldots, \max_{x-\Delta(x)\leq s\leq x}|y^{(n-1)}(s)|\right), \]

where \(x-\Delta(x)\to\infty\) as \(x\to\infty\).

1. Analyzing the proof of the theorem, it is easy to obtain not only the fact of boundedness of the derivatives indicated in it, but also explicit estimates for these derivatives. Under the assumptions of remark 5 one can estimate from above the derivative of order \(n\) from the beginning and at the end of the solution, whose estimate is prescribed (for this it is necessary to choose \(A(x)\) according to \(\alpha(x)=\max_{0\leq s\leq x}|y(s)|\)). The growth of the lower derivatives in this case can be estimated from inequalities (1).

Let us note in conclusion that, since inequalities (1) are also valid for a function \(\psi(x)\) whose derivative of order \(n-1\) satisfies the Lipschitz condition, the theorem (in which inequality (2) must hold almost everywhere), as well as all its consequences, are also valid for functions \(y(x)\) satisfying this requirement.

References

1. S. N. Bernstein, Collected Works, III, Academy of Sciences of the USSR, 1960.

2. M. A. Rutman, in: Investigations on Contemporary Problems of Constructive Function Theory, Fizmatgiz, 1961.

3. Yu. A. Klokov, Boundary-Value Problems with a Condition at Infinity for Equations, Riga, 1963.

4. Z. Opial, “Ann. polon. math.”, 10, No. 1, 73–79, 1961.

5. Yu. A. Klokov, Vestnik of Moscow University, 5, 197–204, 1959.

6. G. H. Hardy, J. E. Littlewood, G. Pólya, Inequalities, IL, 1948.