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UDC 517.941
MATHEMATICS
G. S. ZAITSEVA
ON SOME CRITERIA FOR NONOSCILLATION OF LINEAR DIFFERENTIAL OPERATORS
(Presented by Academician A. Yu. Ishlinskii, May 15, 1967)
The interval \([a,b]\) is an interval of nonoscillation of the operator
\[ L(x) \equiv x^{(n)}+p_1(t)x^{(n-1)}+\cdots+p_n(t)x, \]
if every nontrivial solution of the equation \(L[x]=0\) has on the interval \([a,b]\) no more than \((n-1)\) zeros (zeros are counted according to their multiplicities).
- Theorem 1. Let \(|p_i(t)|\leq L_i,\ i=1,2,\ldots,n\) \((a\leq t\leq b)\). If the inequality
\[ \sum_{k=2}^{n} \frac{L_k}{\left[\frac{k-1}{2}\right]!\left[\frac{k}{2}\right]!L_1^k} \int_{0}^{L_1(b-a)/2}\tau^{k-1}e^\tau\,d\tau \leq 1, \tag{1} \]
is satisfied, then the interval \([a,b]\) is an interval of nonoscillation of the operator
\[ L(x)=x^{(n)}+p_1(t)x^{(n-1)}+\cdots+p_n(t)x. \]
Inequality (1) determines a larger interval of nonoscillation than inequality (5)
\[ \sum_{k=2}^{n} \frac{L_k(b-a)^k}{k\left[\frac{k-1}{2}\right]!\left[\frac{k}{2}\right]!2^k} \exp\left\{L_1\frac{b-a}{2}\right\}\leq 1, \tag{2} \]
and therefore refines the estimates \((^{1-2,5})\).
In the proof of Theorem 1 the following is used.
Theorem 2. Let an \(n\)-times differentiable function satisfy the conditions
\[ x(a_1)=x'(a_2)=\cdots=x^{(n-1)}(a_n)=0 \qquad (a\leq a_1\leq a_2\leq\cdots\leq a_n\leq b), \]
\[ |x^{(n)}+p_1(t)x^{(n-1)}|\leq \mu,\qquad |p_1(t)|\leq L_1 \qquad (a\leq t\leq b). \]
Then the inequality holds
\[ |x(t)|<\mu \int_{0}^{L_1(b-a)} \tau^{n-1}e^\tau\,d\tau \bigg/ L_1^n\left[\frac{n-1}{2}\right]!\left[\frac{n}{2}\right]!. \]
Lemma. If \(x(t)\) on \([a,b]\) is a solution of the equation
\[ x^{(n)}+p_1(t)x^{(n-1)}=f(t), \]
\[ |f(t)|\leq \mu,\qquad |p_1(t)|\leq L_1 \qquad (a\leq t\leq b), \]
\[ x^{(n-1)}(c)=x^{(n-2)}(c)=\cdots=x^{(k)}(c)=0 \qquad (a\leq c\leq b), \]
\[ k=0,1,\ldots,n-1, \]
then
\[ |x^{(k)}(t)|\leq \frac{\mu}{L_1^{\,n-k}} \left\{ \exp[L_1|t-c|]-1-L_1|t-c|-\cdots- \frac{[L_1|t-c|]^{\,n-k-1}}{(n-k-1)!} \right\}. \]
- Let us note that inequalities (1) and (2) allow the coefficient \(L_1\) to take arbitrarily large values at the expense of the smallness of the remaining co-
coefficients. The following theorem does not depend on the existing criteria \((1\text{--}5)\), but it shows that the coefficient \(p_2(t)\) may assume arbitrarily large negative values in absolute value, owing to the smallness of the remaining coefficients.
Theorem 3. Let
\[ |p_i(t)| \leq L_i,\qquad i=1,3,4,\ldots,n, \]
\[ -L_2 \leq p_2(t) \leq L_2^+ \qquad (a \leq t \leq b). \]
The interval \([a,b]\) of length \(b-a < 2\min[h_1,h_2]\) is an interval of nonoscillation of the operator
\[ L(x)=x^{(n)}+p_1x^{(n-1)}+\cdots+p_nx, \]
if \(h_1,h_2\) are, respectively, the first positive roots of the equations
\[ \sum_{k=3}^{n} \frac{L_k h_1^k}{(k-2)\left[\frac{k-3}{2}\right]!\left[\frac{k-2}{2}\right]!} =F_1(h_1), \qquad \sum_{k=3}^{n} \frac{L_k h_2^k}{(k-2)\left[\frac{k-3}{2}\right]!\left[\frac{k-2}{2}\right]!} =F_2(h_2), \]
where
\[ F_1(h)= \frac{L_2}{\left[\operatorname{ch}\eta h-\frac{L_1}{2\eta}\operatorname{sh}\eta h\right]\exp\frac{L_1}{2}h-1}, \qquad \eta=\sqrt{\frac{L_1^2}{4}+L_2}; \]
\[ F_2(h)= \begin{cases} \displaystyle \frac{ L_2^+\left[\operatorname{ch}\beta_1 h-\frac{L_1}{2\beta_1}\operatorname{sh}\beta_1 h\right]\exp\frac{L_1}{2}h }{ 1-\left[\operatorname{ch}\beta_1 h-\frac{L_1}{2\beta_1}\operatorname{sh}\beta_1 h\right]\exp\frac{L_1}{2}h }, & \text{if } \beta_1^2=\dfrac{L_1^2}{4}-L_2^+>0, \\[2.2ex] \displaystyle \frac{ L_2^+\left[\cos\beta_2 h-\frac{L_1}{2\beta_2}\sin\beta_2 h\right]\exp\frac{L_1}{2}h }{ 1-\left[\cos\beta_2 h-\frac{L_1}{2\beta_2}\sin\beta_2 h\right]\exp\frac{L_1}{2}h }, & \text{if } \beta_2^2=L_2^+-\dfrac{L_1^2}{4}>0, \\[2.2ex] \displaystyle \frac{ L_2^+\left[1-\sqrt{L_2^+}\,h\right]\exp\sqrt{L_2^+}\,h }{ 1-\left[1-\sqrt{L_2^+}\,h\right]\exp\sqrt{L_2^+}\,h }, & \text{if } L_2^+=\dfrac{L_1^2}{4}, \\[2.2ex] \displaystyle \frac{L_1^2}{\exp L_1h-1-L_1h}, & \text{if } L_2^+=0, \\[2.2ex] \displaystyle \frac{2}{h^2}, & \text{if } L_2^+=L_1=0. \end{cases} \tag{3} \]
For \(n=2\) the stated nonoscillation criterion coincides with the criterion, unimprovable in the characteristics \(L_1,L_2^+\) \((6)\).
In the proof of Theorem 3 the following is used.
Lemma. Let the interval \([a,b]\) be an interval of nonoscillation of the operator
\[ L[x]=x''+p_1x'+p_2x, \]
\[ |p_1(t)|\leq L_1,\qquad p_2(t)\leq L_2^+\qquad (a\leq t\leq b), \]
and let the function \(v(t)\) be a solution of the problem
\[ Lv=1,\qquad v(a)=v(b)=0. \]
Then
\[ |v(t)|\leq \frac{1}{F_2[(b-a)/2]}\qquad (a\leq t\leq b), \]
where \(F_2(h)\) is defined by the inequalities (3).
Bauman Higher Technical School
Received
15 V 1967
CITED LITERATURE
\({}^{1}\) D. Zh. Sansone, Ordinary Differential Equations, Moscow–Leningrad, 1953.
\({}^{2}\) A. Yu. Levin, Mathematical Collection, 64 (106), 3, 396 (1964).
\({}^{3}\) G. A. Bessmertnykh, A. Yu. Levin, DAN, 144, No. 3, 471 (1962).
\({}^{4}\) A. Yu. Levin, DAN, 153, No. 6 (1963).
\({}^{5}\) G. S. Zaitseva, DAN, 176, No. 4 (1967).
\({}^{6}\) H. Erpheser, Math. Zs., 61, No. 4 (1955).