UDC 517.941

MATHEMATICS

G. S. ZAĬTSEVA

ON A MULTIPOINT BOUNDARY-VALUE PROBLEM

(Presented by Academician A. Yu. Ishlinskii, 13 XII 1966)

Consider the equation

\[ x^{(n)}+p_1(t)x^{(n-1)}+\cdots+p_n(t)x=0 \qquad (a\le t\le b), \tag{1} \]

where \(p_1(t),\ldots,p_n(t)\) are real continuous functions on the interval \([a,b]\),

\[ |p_i(t)|\le L_i,\qquad i=1,2,\ldots,n \qquad (a\le t\le b), \]

and the multipoint boundary-value problem

\[ x(a_i)=A_{i,1},\quad x'(a_i)=A_{i,2},\ldots,x^{(r_i-1)}(a_i)=A_{i,r_i}, \]

\[ i=1,2,\ldots,m\quad (2\le m\le n,\ r_1+r_2+\cdots+r_m=n), \tag{2} \]

\[ a\le a_1<a_2<\cdots<a_m\le b. \]

A number of works \((^{1-5})\) are devoted to estimating the length of the interval \([a,b]\) on which problem (1)—(2) has a unique solution. The following are known: the Vallée-Poussin estimate \((^1)\)

\[ \sum_{l=1}^{n} L_k\frac{(b-a)^k}{k!}\le 1; \tag{3} \]

the estimate of A. Yu. Levin \((^2)\)

\[ \sum_{k=1}^{n}\frac{L_k(b-a)^k}{2^k k\left[\frac{k-1}{2}\right]!\left[\frac{k}{2}\right]!}\le 1, \tag{4} \]

and, finally, Nehari’s integral estimate \((^3)\)

\[ \sum_{k=0}^{n-1}\left(\frac{b-a}{2}\right)^k\int_a^b |p_{k+1}(t)|\,dt\le 2. \tag{5} \]

Let us note that, for a fixed value of \((b-a)\), conditions (3)—(5) express the requirement of smallness of the coefficients \(L_i\), or respectively

\[ \int_a^b |p_i(t)|\,dt,\qquad i=1,2,\ldots,n. \]

In the present paper each of the estimates (3)—(5) is sharpened. Conditions are obtained under which the coefficient \(p_1(t)\) may assume arbitrarily large values on the interval \([a,b]\) owing to the smallness of the remaining coefficients, something not allowed by inequalities (3)—(5).

Theorem 1. Problem (1)—(2), for arbitrary \(A_{i,k}\), has a unique solution on the interval \([a,b]\) if

\[ \sum_{k=2}^{n} L_k\frac{(b-a)^k}{k!}\exp\{L_1(b-a)\}\le 1. \tag{6} \]

Estimate (6) sharpens estimate (3). Indeed, let us rewrite (3) and (6), respectively, in the form

\[ \sum_{k=2}^{n} L_k\frac{(b-a)^k}{k!}\le 1-L_1(b-a), \tag{7} \]

\[ \sum_{k=2}^{n} L_k \frac{(b-a)^k}{k!} \leqslant \exp\{-L_1(b-a)\}. \tag{8} \]

The right-hand side of inequality (8), for any value of \((b-a)\), is greater than the right-hand side of inequality (7). Therefore inequality (6) determines a larger interval of solvability of problem (1)—(2) than inequality (3).

A consequence of Theorem 1 is the assertion that, for the equation

\[ x^{(n)}+p_1(t)x^{(n-1)}=0 \]

problem (1)—(2) is solvable on any interval \([a,b]\), where the function \(p_1(t)\) is continuous.

Theorem 2. Problem (1)—(2), for arbitrary \(A_{i,k}\), has a unique solution on the interval \([a,b]\), if

\[ \sum_{k=2}^{n} \frac{L_k(b-a)^k} {2^k k\left[\frac{k-1}{2}\right]!\left[\frac{k}{2}\right]!} \exp\left\{\frac{L_1}{2}(b-a)\right\} \leqslant 1. \tag{9} \]

Inequality (9) determines a larger interval of solvability of problem (1)—(2) than inequality (4).

Theorem 3. Problem (1)—(2), for arbitrary \(A_{ik}\), has a unique solution on the interval \([a,b]\), if

\[ \sum_{k=1}^{n-1} \frac{(b-a)^k\int_a^b |p_{k+1}(t)|\,dt} {2^k k\left[\frac{k-1}{2}\right]!\left[\frac{k}{2}\right]!} \exp\left\{\frac{1}{2}\int_a^b |p_1(t)|\,dt\right\} \leqslant 2. \tag{10} \]

Inequality (10) is a strengthening of inequality (5).

Corollary. If a nontrivial solution of the equation

\[ x''+p_1(t)x'+p_2(t)x=0 \tag{11} \]

has two zeros on the interval \([a,b]\), then

\[ (b-a)\int_a^b |p_2(t)|\,dt \exp\left\{\frac{1}{2}\int_a^b |p_1(t)|\,dt\right\}>4. \tag{12} \]

If in equation (11) \(p_1(t)=0\), then inequality (12) passes into Lyapunov’s inequality

\[ (b-a)\int_a^b |p_2(t)|\,dt>4. \]

Therefore Theorem 3 may be regarded as a generalization of Lyapunov’s inequality for an equation of order \(n\).

In conclusion we give the proof of Theorem 3. Suppose the contrary: let inequality (10) hold and, at the same time, let some nontrivial solution \(x(t)\) of equation (1) have on the interval \([a,b]\) at least \(n\) zeros. Then on the interval \([a,b]\) there will be found a system of points

\[ a\leqslant a_1\leqslant a_2\leqslant \cdots \leqslant a_{n-1}\leqslant c\leqslant b_{n-1}\leqslant \cdots \leqslant b_1\leqslant b, \]

at which

\[ x(a_1)=x'(a_2)=\cdots=x^{(n-2)}(a_{n-1})=x^{(n-1)}(c)=\cdots=x(b_1)=0. \]

Let

\[ \sup_{a\leq t\leq c}|x^{(n-1)}(t)|=|x^{(n-1)}(\alpha)|=\mu \qquad (a\leqslant \alpha\leqslant c). \]

Then on the interval \([a,c]\) the estimate (2) holds:

\[ \left|x^{(n-k-1)}(t)\right| \leqslant \mu\, \frac{(c-a)^k} {k\left[\frac{k-1}{2}\right]!\left[\frac{k}{2}\right]!}, \qquad k=1,2,\ldots,n-1. \tag{13} \]

We rewrite equation (1) in the form

\[ \left\{x^{(n-1)}(t)\exp\left[\int^t p_1(\xi)\,d\xi\right]\right\}' = -\sum_{k=1}^{n-1}p_{k+1}(t)x^{(n-k-1)}(t) \exp\left[\int^t p_1(\xi)\,d\xi\right]. \]

Integrating this equation from \(a\) to \(c\) and applying estimate (13), we obtain the inequality

\[ 1< \sum_{k=1}^{n-1} \frac{(c-a)^k\displaystyle\int_a^c |p_{k+1}(t)|\,dt} {k\left[\frac{k-1}{2}\right]!\left[\frac{k}{2}\right]!} \exp\int_a^c |p_1(\xi)|\,d\xi . \tag{14} \]

Similarly, for the interval \([c,b]\) we have

\[ 1< \sum_{k=1}^{n-1} \frac{(b-c)^k\displaystyle\int_c^b |p_{k+1}(t)|\,dt} {k\left[\frac{k-1}{2}\right]!\left[\frac{k}{2}\right]!} \exp\int_c^b |p_1(\xi)|\,d\xi . \tag{15} \]

Denote the \(k\)-th term of the sums appearing on the right-hand sides of inequalities (14) and (15), respectively, by \(A_k^{1+k}\), \(B_k^{1+k}\), and introduce the quantity

\[ \chi=\frac12\int_a^b |p_1(t)|\,dt-\int_a^c |p_1(t)|\,dt . \]

Then

\[ \frac{\displaystyle\int_a^b |p_{k+1}(t)|\,dt} {k\left[\frac{k-1}{2}\right]!\left[\frac{k}{2}\right]!} \exp\left\{\frac12\int_a^b |p_1(t)|\,dt\right\} = \frac{\left[A_k\exp\frac{\chi}{1+k}\right]^{1+k}}{(c-a)^k} + \]

\[ + \frac{\left[B_k\exp\left(-\frac{\chi}{1+k}\right)\right]^{1+k}}{(b-c)^k} \ge \frac{\left[A_k\exp\frac{\chi}{1+k} + B_k\exp\left(-\frac{\chi}{1+k}\right)\right]^{1+k}}{(b-a)^k} \]

or

\[ \sum_{k=1}^{n-1} \frac{(b-a)^k\displaystyle\int_a^b |p_{k+1}(t)|\,dt} {2^k k\left[\frac{k-1}{2}\right]!\left[\frac{k}{2}\right]!} \exp\left\{\frac12\int_a^b |p_1(t)|\,dt\right\} \ge \]

\[ \ge \sum_{k=1}^{n-1} 2^{-k} \left[ A_k\exp\frac{\chi}{1+k} + B_k\exp\left(-\frac{\chi}{1+k}\right) \right]^{1+k} >2 \tag{16} \]

under conditions (14) and (15), i.e., under the condition that

\[ \sum_{k=1}^{n-1}A_k^{1+k}>1, \qquad \sum_{k=1}^{n-1}B_k^{1+k}>1 . \]

Inequality (16) contradicts inequality (10). The theorem is proved.

Bauman Higher Technical School

Received
10 XII 1966

REFERENCES

1. Ch. J. de la Vallée-Poussin, J. math. pures et appl., 9, No. 8, 125 (1929).
2. A. Yu. Levin, Matem. sborn., 64, No. 3, 396 (1964).
3. Z. Nehari, Studies in Mathematical Analysis and Related Topics, 1962, p. 256.
4. A. Yu. Levin, DAN, 153, No. 6, 1257 (1963).
5. G. A. Bessmertnykh, A. Yu. Levin, DAN, 144, No. 3, 471 (1962).