ON CONDITIONS FOR THE EXISTENCE OF THE GREEN’S FUNCTION OF A BOUNDARY VALUE PROBLEM

V. A. Churikov

UDC [[unclear: number cut off]]

The article considers conditions for the existence of the Green’s function of the boundary-value problem

\[ L[y]\equiv y^{(n)}(x)-\sum_{k=0}^{n-1} p_k(x)y^{(k)}(x)=f(x), \]

\[ l_i[y]=\sum_{k=0}^{n-1}\left[\alpha_i^{(k)}y^{(k)}(a)+\beta_i^{(k)}y^{(k)}(b)\right]=0 \]

\[ (i=0,\ 1,\ \ldots,\ n-1;\quad n\geqslant 2), \]

where \(p_{n-1}(x),\ldots,p_0(x), f(x)\) are continuous functions; \(\alpha_i^{(k)}, \beta_i^{(k)}\) are const[[unclear: word cut off]].

We introduce the following notation and definitions.

1. \(\{u_k(x)\}\) is a fundamental system of solutions of the equation \(L[y]=0\).

2. \(W(x)\) is the Wronskian of the system \(\{u_k(x)\}\).

3. \(K(x,s)\) is the Cauchy function of the operation \(L[y]\):

\[ K(x,s)=\frac{1}{W(s)} \begin{vmatrix} u_0(s) & u_1(s) & \cdots & u_{n-1}(s)\\ \cdot & \cdot & \cdots & \cdot\\ u_0^{(n-2)}(s) & u_1^{(n-2)}(s) & \cdots & u_{n-1}^{(n-2)}(s)\\ u_0(x) & u_1(x) & \cdots & u_{n-1}(x) \end{vmatrix}. \]

4.

\[ A= \begin{vmatrix} \alpha_0^{(0)} & \alpha_0^{(1)} & \cdots & \alpha_0^{(n-1)}\\ \alpha_1^{(0)} & \alpha_1^{(1)} & \cdots & \alpha_1^{(n-1)}\\ \cdot & \cdot & \cdots & \cdot\\ \alpha_{n-1}^{(0)} & \alpha_{n-1}^{(1)} & \cdots & \alpha_{n-1}^{(n-1)} \end{vmatrix}, \qquad B= \begin{vmatrix} \beta_0^{(0)} & \beta_0^{(1)} & \cdots & \beta_0^{(n-1)}\\ \beta_1^{(0)} & \beta_1^{(1)} & \cdots & \beta_1^{(n-1)}\\ \cdot & \cdot & \cdots & \cdot\\ \beta_{n-1}^{(0)} & \beta_{n-1}^{(1)} & \cdots & \beta_{n-1}^{(n-1)} \end{vmatrix}, \]

\[ D= \begin{vmatrix} \alpha_0^{(0)}+\beta_0^{(0)} & \alpha_0^{(1)}+\beta_0^{(1)} & \cdots & \alpha_0^{(n-1)}+\beta_0^{(n-1)}\\ \alpha_1^{(0)}+\beta_1^{(0)} & \alpha_1^{(1)}+\beta_1^{(1)} & \cdots & \alpha_1^{(n-1)}+\beta_1^{(n-1)}\\ \cdot & \cdot & \cdots & \cdot\\ [[\text{unclear: lower part of determinant cut off}]] \end{vmatrix}. \]

1. \(B^{(j)}_i\bigl(D^{(j)}_i\bigr)\) \((i,j=0,1,\ldots,n-1)\) is the determinant obtained from the determinant \(B(D)\) by replacing in it the \((i+1)\)-st column by the column \((\alpha^{(j)}_0,\ldots,\alpha^{(j)}_{n-1})'\).

2. \(A^{(k_1),(k_2),\ldots,(k_m)}_{i_1,i_2,\ldots,i_m}\) \((2\le m\le n;\quad 0\le i_1<i_2<\cdots<i_m\le n-1;\quad 0\le k_1<k_2<\cdots<k_m\le n-1)\) is the determinant obtained from the determinant \(A\) by replacing in it the \((i_1+1)\)-st, \((i_2+1)\)-st, ..., \((i_m+1)\)-st columns respectively by the columns \((\beta^{(k_1)}_0,\ldots,\beta^{(k_1)}_{n-1})'\), \((\beta^{(k_2)}_0,\ldots,\beta^{(k_2)}_{n-1})'\), ..., \((\beta^{(k_m)}_0,\ldots,\beta^{(k_m)}_{n-1})'\).

3. \(B^{(k_1),(k_2),\ldots,(k_m)}_{i_1,i_2,\ldots,i_m}\) \((2\le m\le n;\quad 0\le i_1<i_2<\cdots<i_m\le n-1;\quad 0\le k_1<k_2<\cdots<k_m\le n-1)\) is the determinant obtained from the determinant \(B\) by replacing in it the \((i_1+1)\)-st, \((i_2+1)\)-st, ..., \((i_m+1)\)-st columns respectively by the columns \((\alpha^{(k_1)}_0,\ldots,\alpha^{(k_1)}_{n-1})'\), \((\alpha^{(k_2)}_0,\ldots,\alpha^{(k_2)}_{n-1})'\), ..., \((\alpha^{(k_m)}_0,\ldots,\alpha^{(k_m)}_{n-1})'\).

4. The interval \([a,\lambda)\) \(((\lambda,b])\) will be called a right (left) precritical interval for problem (1), (2), if for any \(b\in(a,\lambda)\) (for any \(a\in(\lambda,b)\)) problem (1), (2) has a unique solution.

5. The interval \((\alpha,\beta)\) is called precritical for problem (1), (2), if for any \(a,b\in(\alpha,\beta)\) \((a<b)\) problem (1), (2) has a unique solution.

6. The interval \([a,\beta]\) is called a nonoscillation interval for the equation \(L[y]=0\), if every nontrivial solution of the equation \(L[y]=0\) has on \([a,\beta]\) no more than \(n-1\) zeros.

§ 1. For the unique solvability of problem (1), (2) it is necessary and sufficient that its Green’s function exist, i.e., that the determinant

\[ D(a,b)= \left| \begin{array}{cccc} l_0[u_0] & \cdot & \cdots & l_0[u_{n-1}]\\ \cdot & \cdot & \cdots & \cdot\\ l_{n-1}[u_0] & \cdot & \cdots & l_{n-1}[u_{n-1}] \end{array} \right| \ne 0. \tag{3} \]

Below we consider conditions under which there exist precritical intervals of problem (1), (2).

Theorem 1. For problem (1), (2) there exists a right (left) precritical interval if one of the following conditions is satisfied:

1) \(D\ne0\), or

\[ \text{2) } D=0,\ \text{but }\sum_{i=0}^{n-1} C_i^{(n-1)}p_i(a)+\sum_{i=1}^{n-1} C_i^{(i-1)}\ne0 \quad \left( D=0,\ \text{but }\sum_{i=0}^{n-1} D_i^{(n-1)}p_i(b)+ \sum_{i=1}^{n-1} D_i^{(i-1)}\ne0 \right). \]

Proof. We shall prove the theorem in the case of a right precritical interval. Put

\[ u_i^{(j)}(a)=\delta_{i,j}= \begin{cases} 0, & \text{if } i\ne j,\\ 1, & \text{if } i=j \end{cases} \quad (i,j=0,1,\ldots,n-1), \]

then

\[ D(a,b)= \left| \begin{array}{cccc} \alpha_0^{(0)}+\displaystyle\sum_{k=0}^{n-1}\beta_0^{(k)}u_0^{(k)}(b) & \cdot & \cdots & \alpha_0^{(n-1)}+\displaystyle\sum_{k=0}^{n-1}\beta_0^{(k)}u_{n-1}^{(k)}(b) \\[1.2em] \cdot & \cdot & \cdots & \cdot \\[1.2em] \alpha_{n-1}^{(0)}+\displaystyle\sum_{k=0}^{n-1}\beta_{n-1}^{(k)}u_0^{(k)}(b) & \cdot & \cdots & \alpha_{n-1}^{(n-1)}+\displaystyle\sum_{k=0}^{n-1}\beta_{n-1}^{(k)}u_{n-1}^{(k)}(b) \end{array} \right|. \tag{4} \]

We shall regard \(D(a,b)\) as a function of \(b\) for fixed \(a\). It is obvious that \(D(a,b)\) is a continuously differentiable function of \(b\). If \(D(a,a)=D\ne 0\), then also \(D(a,b)\ne 0\) in some interval \([a,\lambda_1)\). Differentiating the determinant (4), we obtain

\[ D'(a,a)=\sum_{i=0}^{n-1} C_i^{(n-1)}p_i(a)+\sum_{i=1}^{n-1} C_i^{(i-1)}. \]

Let \(D=0\), but \(D'(a,a)\ne 0\). Then from Lagrange’s formula it follows that \(D(a,b)\ne 0\) in some interval \((a,\lambda_2)\). Put \(\lambda=\min[\lambda_1,\lambda_2]\). The interval \([a,\lambda)\) is the right subcritical interval of problem (1), (2).

Remark. For the case of a left subcritical interval, the proof is analogous to that given above. In this case we fix \(b\) and set \(u_i^{(j)}(b)=\delta_{i,j}\). The formula for \(D(a,b)\) is obtained from determinant (4) by replacing in it \(\alpha_i^{(j)}\) by \(\beta_i^{(j)}\) and \(u_i^{(j)}(b)\) by \(u_i^{(j)}(a)\) \((i,j=0,1,\ldots,n-1)\).

From Theorem 1 there follows immediately

Corollary 1. Problem (1), (2) has a right (left) subcritical interval for every operation \(L[y]\) and arbitrary \(a\) (arbitrary \(b\)), if one of the following conditions is satisfied:

1) \(D\ne 0\), or

2) \(D=0,\ C_i^{(n-1)}=0\ \ (i=0,1,\ldots,n-1)\), but
\[ \sum_{i=1}^{n-1} C_i^{(i-1)}\ne 0 \]
\[ (D=0,\quad D_i^{(n-1)}=0\ \ (i=0,1,\ldots,n-1),\quad \text{but}\quad \sum_{i=1}^{n-1} D_i^{(i-1)}\ne 0). \]

When restrictions are imposed on the coefficients of the boundary conditions, a further selection of conditions for the existence of a right (left) subcritical interval is possible.

Put:

1) for fixed \(a\) and \(u_i^{(j)}(a)=\delta_{i,j}\),

\[ \widetilde D_1^{(k)}(a,b)=\frac{d^kD(a,b)}{db^k}\qquad (k=2,3,\ldots,n), \]

if all determinants of the form \(A_i^{(n-1)}=\ldots=A_i^{(n-k+1)}=0,\ A_{i_1,i_2,\ldots,i_m}^{(k_1),(k_2),\ldots,(n-1)}=\ldots=A_{i_1,i_2,\ldots,i_{m-k+2}}^{(k_1),(k_2),\ldots,(n-k+1)}=0\) \((m=2,3,\ldots,n)\);

2) for fixed \(b\) and \(u_i^{(j)}(b)=\delta_{i,j}\),

\[ \widetilde D_2^{(k)}(a,b)=\frac{d^kD(a,b)}{da^k}\qquad (k=2,3,\ldots,n), \]

if all determinants of the form \(B_i^{(n-1)}=\ldots=B_i^{(n-k+1)}=0,\quad B_{i_1,i_2,\ldots,i_m}^{(k_1),(k_2),\ldots,(n-1)}=\ldots=B_{i_1,i_2,\ldots,i_{m-k+2}}^{(k_1),(k_2),\ldots,(n-k+1)}=0\) \((m=2,3,\ldots,n)\).

Theorem 2. Problem (1), (2) has a right subcritical interval if one of the following conditions is satisfied:

1) \(D\ne 0\), or

2) \(D=0\), but
\[ D'(a,a)=\sum_{i=0}^{n-1} C_i^{(n-1)}p_i(a)+\sum_{i=1}^{n-1} C_i^{(i-1)}\ne 0, \]
or

3) \(D=0,\ D'(a,a)=0\), all determinants of the form \(A_i^{(n-1)}=0,\ A_{i_1,i_2,\ldots,i_m}^{(k_1),(k_2),\ldots,(n-1)}=0\) \((m=2,3,\ldots,n)\), but \(\widetilde D_1''(a,a)\ne 0\), or

4) \(D=0,\quad D'(a,a)=0,\quad \tilde D_1''(a,a)=0,\ldots,\tilde D_1^{(k-1)}(a,a)=0\), all determinants of the form
\[ A_i^{(n-1)}=\cdots=A_i^{(n-k+1)}=0,\qquad A_{i_1,i_2,\ldots,i_m}^{(k_1),(k_2),\ldots,(n-1)}=\cdots \]
\[ = A_{i_1,i_2,\ldots,i_{m-k+2}}^{(k_1),(k_2),\ldots,(n-k+1)}=0 \quad (m=2,3,\ldots,n), \]
but \(\tilde D_1^{(k)}(a,a)\ne 0\) \((k=3,4,\ldots,n)\).

Theorem 2′. Problem (1), (2) has a left subcritical interval if one of the following conditions is satisfied:

1) \(D\ne 0\), or

2) \(D=0\), but
\[ D'(b,b)=\sum_{i=0}^{n-1}D_i^{(n-1)}p_i(b)+\sum_{i=1}^{n-1}D_i^{(i-1)}\ne 0, \]
or

3) \(D=0,\ D'(b,b)=0\), all determinants of the form
\[ B_i^{(n-1)}=0,\qquad B_{i_1,i_2,\ldots,i_m}^{(k_1),(k_2),\ldots,(n-1)}=0 \quad (m=2,3,\ldots,n), \]
but \(\tilde D_2''(b,b)\ne 0\), or

4) \(D=0,\quad D'(b,b)=0,\quad \tilde D_2''(b,b)=0,\ldots,\tilde D_2^{(k-1)}(b,b)=0\), all determinants of the form
\[ B_i^{(n-1)}=\cdots=B_i^{(n-k+1)}=0,\qquad B_{i_1,i_2,\ldots,i_m}^{(k_1),(k_2),\ldots,(n-1)}=\cdots \]
\[ = B_{i_1,i_2,\ldots,i_{m-k+2}}^{(k_1),(k_2),\ldots,(n-k+1)}=0 \quad (m=2,3,\ldots,n), \]
but \(\tilde D_2^{(k)}(b,b)\ne 0\) \((k=3,4,\ldots,n)\).

Proof. It is enough to prove only the third and fourth assertions of the theorem, since the first two follow from theorem (1). We shall prove the fourth assertion (the third is proved analogously).

Expanding determinant (4) into a sum of determinants, we obtain
\[ \begin{aligned} D(a,b)=&\,A+\sum_{i=0}^{n-1}A_i^{(0)}u_i(b)+\sum_{i=0}^{n-1}A_i^{(1)}u_i'(b)+\cdots+\sum_{i=0}^{n-1}A_i^{(n-1)}u_i^{(n-1)}(b)+\\ &+\sum_{\substack{0\le k_1<k_2\le n-1\\ 0\le i_1<i_2\le n-1}} A_{i_1,i_2}^{(k_1),(k_2)} \begin{vmatrix} u_{i_1}^{(k_1)}(b) & u_{i_2}^{(k_1)}(b)\\ u_{i_1}^{(k_2)}(b) & u_{i_2}^{(k_2)}(b) \end{vmatrix} +\cdots+\\ &+\sum_{\substack{0\le k_1<k_2<\cdots<k_m\le n-1\\ 0\le i_1<i_2<\cdots<i_m\le n-1}} A_{i_1,i_2,\ldots,i_m}^{(k_1),(k_2),\ldots,(k_m)} \begin{vmatrix} u_{i_1}^{(k_1)}(b) & u_{i_2}^{(k_1)}(b) & \cdots & u_{i_m}^{(k_1)}(b)\\ u_{i_1}^{(k_2)}(b) & u_{i_2}^{(k_2)}(b) & \cdots & u_{i_m}^{(k_2)}(b)\\ \cdot & \cdot & \cdots & \cdot\\ u_{i_1}^{(k_m)}(b) & u_{i_2}^{(k_m)}(b) & \cdots & u_{i_m}^{(k_m)}(b) \end{vmatrix} +\cdots+\\ &+A_{0,1,\ldots,n-1}^{(0),(1),\ldots,(n-1)}W(b). \end{aligned} \tag{5} \]

If all determinants
\[ A_i^{(n-1)}=\cdots=A_i^{(n-k+1)}=0,\qquad A_{i_1,i_2,\ldots,i_m}^{(k_1),(k_2),\ldots,(n-1)}= \]
\[ =\cdots=A_{i_1,i_2,\ldots,i_{m-k+2}}^{(k_1),(k_2),\ldots,(n-k+1)}=0 \quad (m=2,3,\ldots,n), \]
then from formula (5) it is clear that \(D(a,b)\) does not depend on \(u_i^{(n-r+1)}(b)\) \((i=0,1,\ldots,n-1;\ r=1,2,\ldots,k)\). Hence, for fixed \(a\), \(D(a,b)\) may be regarded as a \(k\)-times continuously differentiable function of \(b\). If condition 4 of theorem 2 is fulfilled, then, using Taylor’s formula, we obtain
\[ D(a,b)=\frac{1}{k!}\tilde D_1^{(k)}[a,a+\theta(b-a)](b-a)^k \qquad (0<\theta<1). \]

Consequently, if \(\tilde D_1^{(k)}(a,a)\ne 0\), then in some interval \((a,\lambda_k)\) also \(D(a,b)\ne 0\), which proves the theorem.

Remark. The proof of Theorem \(2'\) is analogous to the proof of Theorem 2. In this case we fix \(b\) and put \(u_i^{(j)}(b)=\delta_{i,j}\); then the expansion \(D(a,b)\) is obtained from formula (5) by replacing \(A\) by \(B\), \(A_i^{(j)}\) by \(B_i^{(j)}\), \(A_{i_1,i_2,\ldots,i_m}^{(k_1),(k_2),\ldots,(k_m)}\) by \(B_{i_1,i_2,\ldots,i_m}^{(k_1),(k_2),\ldots,(k_m)}\) \((2\le m\le n)\), and \(u_i^{(j)}(b)\) by \(u_i^{(j)}(a)\) \((i,j=0,1,\ldots,n-1)\).

Let us note some corollaries that follow directly from Theorems 2 and \(2'\).

Corollary 2. For problem (1), (2), when \(n=2\), there exists a right subcritical interval if one of the following conditions is satisfied:

1) \(D\ne 0\);

2) \(D=0\), but
\[ A_1^{(0)}+\bigl[A_1^{(1)}+A_{0,1}^{(0),(1)}\bigr]p_1(a)+A_0^{(1)}p_0(a)\ne 0; \]

3) \(A_0^{(1)}=A_1^{(1)}=A_1^{(0)}=A_{0,1}^{(0),(1)}=0,\quad A+A_0^{(0)}=0,\) but \(A_0^{(0)}p_0(a)\ne 0.\)

Corollary \(2'\). For problem (1), (2), when \(n=2\), there exists a left subcritical interval if one of the following conditions is satisfied:

1) \(D\ne 0\);

2) \(D=0\), but
\[ B_1^{(0)}+\bigl[B_1^{(1)}+B_{0,1}^{(0),(1)}\bigr]p_1(b)+B_0^{(1)}p_0(b)\ne 0; \]

3) \(B_0^{(1)}=B_1^{(1)}=B_1^{(0)}=B_{0,1}^{(0),(1)}=0,\quad B+B_0^{(0)}=0,\) but \(B_0^{(0)}p_0(b)\ne 0.\)

Corollary 3. For problem (1), (2), when \(n=3\), there exists a right subcritical interval if one of the following conditions is satisfied:

1) \(D\ne 0\);

2) \(D=0\), but
\[ \sum_{i=0}^{2} C_i^{(2)}p_i(a)+\sum_{i=1}^{2} C_i^{(i-1)}\ne 0; \]

3)
\[ A+A_0^{(0)}+A_1^{(1)}+A_{0,1}^{(0),(1)}=0,\qquad A_1^{(0)}+A_2^{(1)}+A_{0,2}^{(0),(1)}=0, \]
\[ A_i^{(2)}=0\quad (i=0,1,2),\qquad A_{i_1,i_2}^{(k_1),(k_2)}=0\quad (k=0,1;\; 0\le i_1<i_2\le 2), \]
\[ A_{0,1,2}^{(0),(1),(2)}=0,\quad \text{but}\quad A_2^{(0)}+A_{1,2}^{(0),(1)}+\bigl[A_2^{(1)}+A_{0,2}^{(0),(1)}\bigr]p_2(a)+ \]
\[ +\bigl[A_1^{(1)}+A_{0,1}^{(0),(1)}\bigr]p_1(a)+A_0^{(1)}p_0(a)\ne 0; \]

4) \(A+A_0^{(0)}=0\), all determinants of the form \(A_i^{(k)}\), \(A_{i_1,i_2}^{(k_1),(k_2)}\), \(A_{0,1,2}^{(0),(1),(2)}\) are equal to zero, except \(A_0^{(0)}\), but \(A_0^{(0)}p_0(a)\ne 0.\)

Remark. The assertion for a left subcritical interval of problem (1), (2), when \(n=3\), is obtained from Corollary 3 by replacing the symbols \(A\) by \(B\), \(C_i^{(j)}\) by \(D_i^{(j)}\), \(A_{i_1,i_2,\ldots,i_m}^{(k_1),(k_2),\ldots,(k_m)}\) by \(B_{i_1,i_2,\ldots,i_m}^{(k_1),(k_2),\ldots,(k_m)}\) \((1\le m\le 3)\), and \(p_i(a)\) by \(p_i(b)\).

For the proof of Corollaries 2 and 3 it is enough to compute \(\widetilde D_1'(a,a)\) for \(n=2\), and \(\widetilde D_2''(a,a)\), \(\widetilde D_1'''(a,a)\) for \(n=3\). These computations are conveniently made by formula (5).

In conclusion of this section we consider the question of the existence of a subcritical interval for problem (1), (2).

Theorem 3. If \(D\ne 0\), or \(D=0\), but
\[ \sum_{k=0}^{n-1}\bigl[C_k^{(n-1)}-D_k^{(n-1)}\bigr]p_k(c)+ \sum_{k=1}^{n-1}\bigl[C_k^{(k-1)}-D_k^{(k-1)}\bigr]\ne 0, \]
then there exists a neighborhood of the point \(c\in(-\infty,+\infty)\) which is a subcritical interval of problem (1), (2).

Proof. Choose a fundamental system of solutions of the equation \(L[y]=0\) so that \(u_i^{(j)}(c)=\delta_{i,j}\); then determinant (3) has the form

\[ D(a,b)= \left| \begin{array}{ccc} \displaystyle \sum_{k=0}^{n-1}\left[\alpha_0^{(k)}u_0^{(k)}(a)+\beta_0^{(k)}u_0^{(k)}(b)\right] & \ldots & \displaystyle \sum_{k=0}^{n-1}\left[\alpha_0^{(k)}u_{n-1}^{(k)}(a)+\beta_0^{(k)}u_{n-1}^{(k)}(b)\right] \\[2.2em] \cdot & \ldots & \cdot \\[1.8em] \displaystyle \sum_{k=0}^{n-1}\left[\alpha_{n-1}^{(k)}u_0^{(k)}(a)+\beta_{n-1}^{(k)}u_0^{(k)}(b)\right] & \ldots & \displaystyle \sum_{k=0}^{n-1}\left[\alpha_{n-1}^{(k)}u_{n-1}^{(k)}(a)+\beta_{n-1}^{(k)}u_{n-1}^{(k)}(b)\right] \end{array} \right|. \]

We shall consider \(D(a,b)\) as a function of the two variables \(a\) and \(b\). If \(D(c,c)=D\ne0\), then also \(D(a,b)\ne0\) in some neighborhood of the point \(c\in(-\infty,+\infty)\). Let \(D(c,c)=0\). Then, taking into account that \(D(a,b)\) has continuous partial derivatives with respect to \(a\) and \(b\), we obtain

\[ \begin{aligned} D(a,b) &=\frac{\partial}{\partial a}D(c,c)(a-c)+\frac{\partial}{\partial b}D(c,c)(b-c)+\alpha(a-c)+\beta(b-c) \\ &=\left\{\sum_{k=0}^{n-1}\left[C_k^{(n-1)}-D_k^{(n-1)}\right]p_k(c) +\sum_{k=1}^{n-1}\left[C_k^{(k-1)}-D_k^{(k-1)}\right]\right\}(b-a) \\ &\quad+(\beta-\alpha)(b-a), \end{aligned} \]

where \(\beta-\alpha=0(b-a)\).

Theorem 3 directly implies

Corollary 4. If \(D\ne0\) or \(D=0,\ C_k^{(n-1)}=D_k^{(n-1)}\) \((k=0,1,\ldots,n-1)\), but
\[ \sum_{k=1}^{n-1}\left[C_k^{(k-1)}-D_k^{(k-1)}\right]\ne0, \]
then for every operation \(L[y]\) there exists a neighborhood of the point \(c\in(-\infty,+\infty)\) which is a subcritical interval of problem (1), (2).

§ 2. In this paragraph we consider the question of estimating the length of the right (left) subcritical interval and the subcritical interval of problem (1), (2).

Let \(q_k\) \((k=0,1,\ldots,n-1)\) be some system of numbers. Put \(q=\{q_0,q_1,\ldots,q_{n-1}\}\) and denote the Green’s function for the operation
\[ y^{(n)}(x)-\sum_{k=0}^{n-1}q_k y^{(k)}(x) \]
with boundary conditions (2) by: a) \(G^1[a;q,x,s,b]\), if \(a\) is fixed and \(b\) is a parameter; b) \(G^1(a,q,x,s;b]\), if \(b\) is fixed and \(a\) is a parameter; c) \(G^1(a,q,x,s,b)\), if \(a\) and \(b\) are parameters.

The following theorems hold.

Theorem 4. If

\[ \sum_{m=0}^{n-1} \sup_{b\in[a,a+h]} \left[ \sup_{x,s\in[a,b]} \left| \frac{\partial^m}{\partial x^m}G^1[a;q,x,s,b] \right| \right] \int_a^{a+h}|p_m(s)-q_m|\,ds<1 \]

\[ \left( \sum_{m=0}^{n-1} \sup_{a\in[b-h,b]} \left[ \sup_{x,s\in[a,b]} \left| \frac{\partial^m}{\partial x^m}G^1(a,q,x,s;b] \right| \right] \int_{b-h}^{b}|p_m(s)-q_m|\,ds<1 \right), \]

then \([a,a+h]\) is a right (\([b-h,b]\) is a left) subcritical interval of problem (1), (2).

Theorem 4′. If

\[ \sum_{m=0}^{n-1} \sup_{a,b\in[\alpha,\beta]} \left[ \sup_{x,s\in[a,b]} \left| \frac{\partial^m}{\partial x^m}G^1(a,q,x,s,b) \right| \right] \int_{\alpha}^{\beta}|p_m(s)-q_m|\,ds<1. \]

then \([\alpha,\beta]\) is a subcritical interval of problem (1), (2).

The proof of Theorems 4 and \(4'\) is analogous and follows from [1] (see also [5, 6]).

Remark. It is expedient to take as the quantities \(q_k\) the values \(p_k(a)\) in the case of a right subcritical interval and \(p_k(b)\) in the case of a left subcritical interval. In all cases one may put
\[ q_k=\lim_{b\to a}\frac{1}{b-a}\int_a^b p_k(s)\,ds, \]
if the limit is finite.

However, arbitrariness in the choice of the quantities \(q_k\) may be used to simplify the construction of the function \(G^1\). In particular, in many cases it is sufficient to take \(q_0=\cdots=q_{n-1}=0\).

Theorem 5. If the determinants
\[ A\gg 0,\qquad A_{n-1}^{(0)}\gg 0,\qquad A_{n-2,n-1}^{(0),(1)}\gg 0,\ldots,\qquad A_{0,1,\ldots,n-1}^{(0),(1),\ldots,(n-1)}\gg 0 \]
\[ \left(A\ll 0,\qquad A_{n-1}^{(0)}\ll 0,\qquad A_{n-2,n-1}^{(0),(1)}\ll 0,\ldots,\qquad A_{0,1,\ldots,n-1}^{(0),(1),\ldots,(n-1)}\ll 0\right) \]
and at least one of them is not equal to zero, while all the remaining determinants of the form
\[ A_{i_1,i_2,\ldots,i_m}^{(k_1),(k_2),\ldots,(k_m)}=0 \qquad (m=1,2,\ldots,n-1), \]
then the maximal right subcritical interval \([a,r_+(a))\) of problem (1), (2) is not smaller than the interval of nonoscillation.

Theorem \(5'\). If the determinants
\[ B\gg 0,\qquad B_{n-1}^{(0)}\gg 0,\qquad B_{n-2,n-1}^{(0),(1)}\gg 0,\ldots,\qquad B_{0,1,\ldots,n-1}^{(0),(1),\ldots,(n-1)}\gg 0 \]
\[ \left(B\ll 0,\qquad B_{n-1}^{(0)}\ll 0,\qquad B_{n-2,n-1}^{(0),(1)}\ll 0,\ldots,\qquad B_{0,1,\ldots,n-1}^{(0),(1),\ldots,(n-1)}\ll 0\right) \]
and at least one of them is not equal to zero, while all the remaining determinants of the form
\[ B_{i_1,i_2,\ldots,i_m}^{(k_1),(k_2),\ldots,(k_m)}=0 \qquad (m=1,2,\ldots,n-1), \]
then the maximal left subcritical interval \((r_-(b),b]\) of problem (1), (2) is not smaller than the interval of nonoscillation.

Proof. We shall prove Theorem 5. It follows from [4] that, in order that the interval \([\alpha,\beta]\) be an interval of nonoscillation, it is necessary and sufficient that in it the following do not vanish:
\[ u_{n-1}(x),\quad \left| \begin{matrix} u_{n-2}(x)&u_{n-1}(x)\\ u'_{n-2}(x)&u'_{n-1}(x) \end{matrix} \right|,\ldots,\quad W(x). \]

From formula (5) it is seen that, if the conditions of Theorem 5 are fulfilled, then on the interval of nonoscillation \(D(a,b)\ne 0\). Theorem \(5'\) is proved analogously.

For an estimate of the length of the interval of nonoscillation, see [2, 3].

Denote by \((a,a+h_j^+)\) \(\bigl((b-h_j^-,b)\bigr)\) the interval on which the \(j\)-th derivative of the solution of the problem
\[ L[y]=-A\sum_{k=0}^{j}p_k(x)\frac{(x-a)^{j-k}}{(j-k)!},\qquad y^{(k)}(a)=A_k^{(j)}\quad (k\ne j;\ k=0,1,\ldots,n-1), \]
\[ y^{(j)}(a)=A+A_j^{(j)} \tag{6} \]
\[ \left( \begin{aligned} L[y]&=-B\sum_{k=0}^{j}p_k(x)\frac{(x-b)^{j-k}}{(j-k)!},\qquad y^{(k)}(b)=B_k^{(j)} \\ &(k\ne j;\ k=0,1,\ldots,n-1),\qquad y^{(j)}(b)=B+B_j^{(j)} \end{aligned} \right) \]
does not vanish, and by \((a,a+H_j^+)\) \(\bigl((b-H_j^-,b)\bigr)\) the interval on which

\[ A_j^{(n-1)}u_j^{(n-1)}(x)+A_j^{(n-2)}u_j^{(n-2)}(x)+\cdots+A_j^{(0)}u_j(x)+A \tag{7} \]

\[ \left(B_j^{(n-1)}u_j^{(n-1)}(x)+B_j^{(n-2)}u_j^{(n-2)}(x)+\cdots+B_j^{(0)}u_j(x)+B\right) \tag{8} \]

does not vanish. By \(u_k(x)\) \((k=0,1,\ldots,n-1)\) is denoted a fundamental system of solutions of the equation \(L[y]=0\), satisfying the condition \(u_i^{(j)}(a)=\delta_{i,j}\) for operation (7) and \(u_i^{(j)}(b)=\delta_{i,j}\) for operation (8).

Theorem 6. If \(A\ne0\), and all \(A_i^{(k)}=0\) \((i,k=0,1,\ldots,n-1)\), except, perhaps, \(A_i^{(j)}\) \((i=0,1,\ldots,n-1)\) (except, perhaps, \(A_j^{(k)}\) \((k=0,1,\ldots,n-1)\)), then \([a,a+h_j^+)\) \(([a,a+H_j^+))\) is a right Dokritsky interval of problem (1), (2).

Theorem 6′. If \(B\ne0\), and all \(B_i^{(k)}=0\) \((i,k=0,1,\ldots,n-1)\), except, perhaps, \(B_i^{(j)}\) \((i=0,1,\ldots,n-1)\) (except, perhaps, \(B_j^{(k)}\) \((k=0,1,\ldots,n-1)\)), then \((b-h_j^-,b]\) \(((b-H_j^-,b])\) is a left Dokritsky interval of problem (1), (2).

Proof. We shall prove Theorem 6. The solution of problem (1), (2) can be sought in the form

\[ y(x)=\sum_{k=0}^{n-1}c_k u_k(x)+\int_a^x K(x,s)f(s)\,ds. \tag{9} \]

Putting \(u_i^{(j)}(a)=\delta_{i,j}\) and substituting \(y(x)\) from formula (9) into the boundary conditions (2), we obtain the system of algebraic equations

\[ \sum_{k=0}^{n-1}\alpha_i^{(k)}c_k+ \sum_{m=0}^{n-1}\beta_i^{(m)} \left[ \sum_{k=0}^{n-1}c_k u_k^{(m)}(b)+ \int_a^b K_x^{(m)}(b,s)f(s)\,ds \right]=0 \quad (i=0,1,\ldots,n-1). \tag{10} \]

Since \(A\ne0\), system (10) can be written in the form

\[ c_i=-\frac{1}{A} \left\{ \sum_{m=0}^{n-1}A_i^{(m)} \left[ \sum_{k=0}^{n-1}c_k u_k^{(m)}(b)+ \int_a^b K_x^{(m)}(b,s)f(s)\,ds \right] \right\} \quad (i=0,1,\ldots,n-1). \tag{11} \]

If the determinant

\[ \Delta= \left| \begin{array}{cccc} 1+\dfrac{1}{A}\displaystyle\sum_{k=0}^{n-1}A_0^{(k)}u_0^{(k)}(b) & \dfrac{1}{A}\displaystyle\sum_{k=0}^{n-1}A_0^{(k)}u_1^{(k)}(b) & \cdots & \dfrac{1}{A}\displaystyle\sum_{k=0}^{n-1}A_0^{(k)}u_{n-1}^{(k)}(b) \\[2.2em] \dfrac{1}{A}\displaystyle\sum_{k=0}^{n-1}A_1^{(k)}u_0^{(k)}(b) & 1+\dfrac{1}{A}\displaystyle\sum_{k=0}^{n-1}A_1^{(k)}u_1^{(k)}(b) & \cdots & \dfrac{1}{A}\displaystyle\sum_{k=0}^{n-1}A_1^{(k)}u_{n-1}^{(k)}(b) \\[2.2em] \cdot & \cdot & \cdot & \cdot \\[1.2em] \dfrac{1}{A}\displaystyle\sum_{k=0}^{n-1}A_{n-1}^{(k)}u_0^{(k)}(b) & \dfrac{1}{A}\displaystyle\sum_{k=0}^{n-1}A_{n-1}^{(k)}u_1^{(k)}(b) & \cdots & 1+\dfrac{1}{A}\displaystyle\sum_{k=0}^{n-1}A_{n-1}^{(k)}u_{n-1}^{(k)}(b) \end{array} \right| \ne0, \tag{12} \]

then system (11) has the unique solution \(c_k\) \((k=0,1,\ldots,n-1)\), and then problem (1), (2) also has a unique solution. From determinant (12) it is seen that, if all \(A_i^{(k)}=0\) \((i,k=0,1,\ldots,n-1)\), except \(A_i^{(j)}\) \((i=0,1,\ldots,n-1)\), then

\[ \Delta=\Delta_1(b)=1+\frac{1}{A}\sum_{i=0}^{n-1} A_i^{(j)}u_i^{(j)}(b), \]

but

\[ A+\sum_{i=0}^{n-1} A_i^{(j)}u_i^{(j)}(x) \]

is the \(j\)-th derivative of the solution of problem (6). If all

\[ A_i^{(k)}=0 \quad (i,k=0,1,\ldots,n-1), \]

except \(A_j^{(k)}\) \((k=0,1,\ldots,n-1)\), then from determinant (12) it is also easy to note that

\[ \Delta=\Delta_2(b)=1+\frac{1}{A}\sum_{k=0}^{n-1} A_j^{(k)}u_j^{(k)}(b), \]

which proves the theorem.
Theorem \(6'\) is proved similarly.

Let us note the conditions for the existence of the intervals \([a,a+h_j^+)\) and \([a,a+H_j^+)\).

Theorem 7. If one of the following conditions is satisfied:

1) \(A+A_j^{(j)}\ne 0,\quad (j=0,1,\ldots,n-1)\);

2) \(A+A_j^{(j)}=0\), but at least one determinant \(A_i^{(j)}\ne 0\) \((j+1\le i\le n-1;\ j=0,1,\ldots,n-2)\);

3) \(A+A_j^{(j)}=0\), all \(A_i^{(j)}=0\) \((j+1\le i\le n-1;\ j=0,1,\ldots,n-2)\), but

\[ \sum_{k=0}^{j} A_k^{(j)}p_k(a)\ne 0; \]

4) \(A+A_{n-1}^{(n-1)}=0\), but

\[ \sum_{k=0}^{n-1} A_k^{(n-1)}p_k(a)\ne 0, \]

then there exists an interval

\[ [a,a+h_j^+). \]

Theorem \(7'\). If one of the following conditions is satisfied:

1) \(A+A_j^{(j)}\ne 0\);

2) \(A+A_j^{(j)}=0\), but \(A_j^{(j-1)}+A_j^{(n-1)}p_j(a)\ne 0\);

3) \(A+A_j^{(j)}=0,\quad A_j^{(j-1)}=\cdots=A_j^{(j-k+1)}=0,\quad A_j^{(n-1)}=\cdots=A_j^{(n-k+1)}=0\), but
\(A_j^{(j-k)}+A_j^{(n-k)}p_j(a)\ne 0\) \((k=2,3,\ldots,j)\);

4) \(A+A_j^{(j)}=0,\quad A_j^{(0)}=\cdots=A_j^{(j-1)}=0,\quad A_j^{(n-j)}=\cdots=A_j^{(n-1)}=0\), but
\(p_j(a)\ne 0\), and at least one determinant \(A_j^{(k)}\ne 0\) \((k=j,j+1,\ldots,n-j-1)\),

then there exists an interval \([a,a+H_j^+)\).

The theorems are proved by successive differentiation of \(\Delta_1(b)\), \(\Delta_2(b)\) and passage to the limit as \(b\to a\).

I take this opportunity to thank the participants of the Izhevsk mathematical seminar for valuable comments.

References

1. Azbelev N. V., Tsalyuk Z. B. DAN SSSR, 156, No. 2, 239—242, 1964.
2. Bessmertnykh G. A., Levin A. Yu. DAN SSSR, 144, No. 3, 471—474, 1962.
3. Levin A. Yu. DAN SSSR, 138, No. 1, 1961.
4. Mammana G. Math. Zeitschr., 33, 186—231, 1931.
5. Churikov V. A. Differential Equations, 1, No. 7, 933—945, 1965.
6. Churikov V. A. Differential Equations, 2, No. 4, 463—478, 1966.

Received by the editors October 5, 1965
Izhevsk Mechanical Institute