Mathematics

A. Yu. LEVIN

ON CERTAIN ESTIMATES OF A DIFFERENTIABLE FUNCTION

(Presented by Academician A. N. Kolmogorov, 20 XII 1960)

1. In the present note two estimates* are given for a real differentiable function satisfying certain conditions, and an application of one of them to a multipoint boundary-value problem is indicated.

Theorem 1. Let \(x(t)\) be continuously differentiable \(n\) times on the interval \([a,b]\) and satisfy the conditions

\[ x(a_1)=x'(a_2)=\cdots=x^{(n-1)}(a_n)=0, \tag{1} \]

where \(a_1,a_2,\ldots,a_n\) are certain points of the interval \([a,b]\). Then on \([a,b]\) the estimate

\[ |x(t)| \leq C_n (b-a)^n \max_{a\leq t\leq b} |x^{(n)}(t)|,\qquad a\leq t\leq b, \tag{2} \]

holds, where the numbers \(C_1,C_2,\ldots\) are determined from the expansion

\[ \tg t+\sec t=1+\sum_{k=1}^{\infty} C_k t^k \quad \left(|t|<\frac{\pi}{2}\right). \tag{3} \]

In accordance with (3), the values \(C_k\) can be expressed in terms of the Bernoulli numbers \(B_k\) \((B_1=1/6,\ B_2=1/30,\ldots)\) and the Euler numbers \(E_k\) \((E_1=1,\ E_2=5,\ldots)\) as follows \((n=1,2,\ldots)\):

\[ C_{2n-1}=\frac{2^{2n}(2^{2n}-1)B_n}{(2n)!},\qquad C_{2n}=\frac{E_n}{(2n)!}. \]

Theorem 2. Let \(x(t)\) satisfy the conditions (1), and suppose that one of the inequalities

\[ (a\leq) \ a_2\leq a_3\leq \cdots \leq a_n \ (\leq b), \]

\[ (a\leq) \ a_n\leq a_{n-1}\leq \cdots \leq a_2 \ (\leq b). \]

is fulfilled. Then on \([a,b]\) the estimate

\[ |x(t)| \leq \frac{1}{\,n\left[\frac{n-1}{2}\right]!\left[\frac{n}{2}\right]!}\, (b-a)^n \max_{a\leq t\leq b} |x^{(n)}(t)|, \qquad a\leq t\leq b. \tag{4} \]

holds.

The values of the constants in the estimates (2), (4) cannot be improved.

2. Let the multipoint boundary-value problem be considered:

\[ x^{(n)}+p_1(t)x^{(n-1)}+\cdots+p_n(t)x=f(t), \tag{5} \]

\[ x(a_1)=A_1,\quad x(a_2)=A_2,\ldots,\quad x(a_n)=A_n, \tag{6} \]

\[ a\leq a_1<a_2<\cdots<a_n\leq b, \]

where \(p_1(t),\ldots,p_n(t), f(t)\) are continuous on \([a,b]\). One may also not exclude the case of coincidence of several \(a_i\); in that case, at one point values are prescribed both for \(x(t)\) and for its successive derivatives (in accordance with the multiplicity of \(a_i\)).

* The first of them, as has become clear, was obtained by S. N. Bernstein as early as 1910.

Set

\[ h=a_n-a_1,\qquad \max_{a_1\le t\le a_n}|p_i(t)|=P_i\quad (i=1,2,\ldots,n). \]

Vallée-Poussin (¹) (see also (²)) indicated the following sufficient condition for the solvability of problem (5)—(6):

\[ \sum_{k=1}^{n}\frac{1}{k!}\,P_k h^k \leqslant 1. \tag{7} \]

The author (³) (and, independently of him, V. G. Maz’ya) found a less restrictive condition than (7) for the solvability of problem (5)—(6):

\[ \sum_{k=1}^{n-1}\frac{1}{k!}\,P_k h^k+ \frac{(n-1)^{\,n-1}}{n^n n!}\,P_n h^n \leqslant 1. \tag{8} \]

With the aid of Theorem 2 the following proposition can be proved.

Theorem 3. In order that problem (5)—(6) be solvable for arbitrary \(A_1,A_2,\ldots,A_n\) and arbitrary continuous \(f(t)\), it is sufficient that the inequality

\[ \sum_{k=1}^{n} \frac{1}{2^k k\left[\frac{k-1}{2}\right]!\left[\frac{k}{2}\right]!}\, P_k h^k \leqslant 1 \tag{9} \]

hold.

From the relation

\[ \frac{1}{2^k k\left[\frac{k-1}{2}\right]!\left[\frac{k}{2}\right]!} = \frac{C_{k-1}^{\left[\frac{k}{2}\right]}}{2^k}\,\frac{1}{k!} \leqslant \frac{1}{2}\,\frac{1}{k!} \]

it follows that in inequality (9) the coefficients of all powers of \(h\) are smaller than the corresponding coefficients in inequality (7).

It is easy to verify that conditions (8) and (9) are (for \(n>2\)) mutually independent.

1. With the aid of the same Theorem 2 one can obtain a strengthening of Vallée-Poussin’s uniqueness theorem (see (²)) for a nonlinear boundary-value problem. Namely, the following proposition is valid:

Theorem 4. Let the right-hand side of the equation

\[ x^{(n)}=f(t,x,x',\ldots,x^{(n-1)}) \tag{10} \]

satisfy the condition

\[ |f(t,v_0,v_1,\ldots,v_{n-1})-f(t,u_0,u_1,\ldots,u_{n-1})| \leqslant \sum_{k=0}^{n-1} P_{n-k}|v_k-u_k| \]

and, in addition, suppose that one of inequalities (8), (9) is fulfilled. Then equation (10) has no more than one solution satisfying conditions (6).

The author expresses his sincere gratitude to his adviser M. A. Krasnosel’skii.

Received
17 XII 1960

References

¹ Ch. J. De la Vallée Poussin, J. de Math. pures et appl. (9), 8 (1929).
² J. Sansone, Ordinary Differential Equations, IL, 1953.
³ A. Yu. Levin, Scientific Reports of the Higher School. Physics and Mathematics, No. 5 (1958).
⁴ S. N. Bernstein, Collected Works, 2, article 100, 1954, p. 497.