ON EXISTENCE, UNIQUENESS, AND ESTIMATES OF THE SOLUTION OF A CERTAIN BOUNDARY-VALUE PROBLEM

V. A. CHURIKOV

The paper considers the boundary-value problem

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

\[ l_i[y]=\sum_{k=0}^{n-1}\alpha_i^k y^{(k)}(a)+\beta_i^{(0)}y(b)=0 \qquad (i=0,1,\ldots,n-1;\ n\geq 2) \tag{2} \]

with continuous \(p_{n-1}(x),\ldots,p_0(x), f(x)\) and constants \(\alpha_i^{(k)}, \beta_i^{(0)}\).

The solution of problem (1), (2) may be sought in the form

\[ y=\sum_{k=0}^{n-1}c_k u_k(x)+\int_a^x K(x,s)f(s)\,ds, \tag{3} \]

where \(u_k(x)\) is a fundamental system of solutions of the equation \(L[y]=0\), satisfying the conditions:

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

and \(K(x,s)\) is the Cauchy function of the operation \(L[y]\) [5].

Substituting (3) into (2), we obtain

\[ \sum_{k=0}^{n-1}\left[\alpha_i^{(n-k-1)}+\beta_i^{(0)}u_k(b)\right]c_k = -\beta_i^{(0)}\int_a^b K(b,s)f(s)\,ds \tag{4} \]

\[ (i=0,1,\ldots,n-1). \]

If the determinant of this system

\[ D(a,b)= \left| \begin{array}{cccc} \alpha_0^{(n-1)}+\beta_0^{(0)}u_0(b) & \ldots & \alpha_0^{(0)}+\beta_0^{(0)}u_{n-1}(b)\\ \cdot & \cdot & \cdot\\ \alpha_{n-1}^{(n-1)}+\beta_{n-1}^{(0)}u_0(b) & \ldots & \alpha_{n-1}^{(0)}+\beta_{n-1}^{(0)}u_{n-1}(b) \end{array} \right| \ne 0, \tag{5} \]

then problem (1), (2) has a unique solution.

Let us introduce the notation:

a)

\[ A= \begin{vmatrix} \alpha_0^{(n-1)} & \alpha_0^{(n-2)} & \ldots & \alpha_0^{(0)}\\ \alpha_1^{(n-1)} & \alpha_1^{(n-2)} & \ldots & \alpha_1^{(0)}\\ \cdot & \cdot & \cdot & \cdot\\ \alpha_{n-1}^{(n-1)} & \alpha_{n-1}^{(n-2)} & \ldots & \alpha_{n-1}^{(0)} \end{vmatrix}, \qquad B= \begin{pmatrix} \beta_0^{(0)}\\ \beta_1^{(0)}\\ \vdots\\ \beta_{n-1}^{(0)} \end{pmatrix}, \]

\(A_i\) \((i=0,1,\ldots,n-1)\) is the determinant obtained from the determinant \(A\) by replacing its \((i+1)\)-st column by the column \(B\).

b)

\[ \widetilde K(x,s)= \begin{cases} K(x,s), & \text{for } a\leq s<x\leq b,\\ 0, & \text{for } a\leq x\leq s\leq b. \end{cases} \]

Expanding the determinant (5) into a sum of determinants and determining \(c_k\) from the system (4), we obtain

\[ D(a,b)=A+\sum_{k=0}^{n-1} A_k u_k(b),\qquad c_k=-\frac{A_k}{D(a,b)}\int_a^b K(b,s)f(s)\,ds \]

\[ (k=0,1,\ldots,n-1). \tag{6} \]

Substituting these values of \(c_k\) into formula (3), we shall have

\[ y=\int_a^b G(x,s)f(s)\,ds, \tag{7} \]

where

\[ G(x,s)=\widetilde K(x,s)- \frac{\displaystyle\sum_{i=0}^{n-1} A_i u_i(x)} {\displaystyle A+\sum_{i=0}^{n-1} A_i u_i(b)}\,K(b,s) \tag{8} \]

is the Green’s function of problem (1), (2).

§ 1. On the existence and uniqueness of the solution

The conditions for unique solvability of problem (1), (2) are equivalent to the conditions for the existence of the Green’s function of this problem and are expressed by the following theorem.

Theorem 1. For every operation \(L[y]\) there exists an interval \([a,\lambda)\) such that, for any \(b\in(a,\lambda)\), the Green’s function \(G(x,s)\) of problem (1), (2), for \(x,s\in(a,b)\), exists if and only if:

a) \(A+A_{n-1}\ne0\) and at least one determinant \(A_i\ne0\) \((i=0,1,\ldots,n-1)\), or

b) \(A+A_{n-1}=0\), but at least one determinant \(A_i\ne0\) \((i=0,1,\ldots,n-2)\).

Proof. Fix the point \(a\). Then \(D(a,b)\) will be a continuous function of \(b\). Consider all cases in which \(D(a,b)\ne0\). In this situation one of two possibilities may occur: a) either \(D(a,a)=A+A_{n-1}\ne0\), b) or \(A+A_{n-1}=0\).

a). Let \(A+A_{n-1}\ne0\). Then in some neighborhood \((a,\rho(a))\) of the point \(a\), also \(D(a,b)\ne0\). In condition a) of Theorem 1, the requirement that at least one determinant \(A_i\) \((i=0,1,\ldots,n-1)\) be nonzero is needed so that problem (1), (2) does not become a Cauchy problem.

Indeed, suppose that all \(A_i=0\) \((i=0,1,\ldots,n-1)\), and \(A\ne0\); then the system of algebraic equations

\[ \sum_{k=1}^{n} \alpha_j^{(n-k)} x_{k-1}=\beta_j^{(0)} \quad (j=0,1,\ldots,n-1) \]

has the unique solution

\[ x_j=\frac{A_j}{A}=0 \quad (j=0,1,\ldots,n-1), \]

but this is possible only when the system is homogeneous, i.e.

\[ \beta_0^{(0)}=\beta_1^{(0)}=\ldots=\beta_{n-1}^{(0)}=0. \]

b). Let now \(A+A_{n-1}=0\) and let at least one determinant \(A_k\ne 0\), \((k=0,1,\ldots,n-2)\). Denote by \(A_j\) the last \(A_k\ne 0\) and by \((a,\rho_j(a))\) \((j=0,1,\ldots,n-1)\) the interval in which \(u_j(b)\ne 0\). In the interval \((a,r_j(a))\) we have

\[ D(a,b)=A_{n-1}[u_{n-1}(b)-1]+\sum_{k=0}^{j} A_k u_k(b)= \]

\[ = u_j(b)\left\{A_{n-1}\frac{u_{n-1}(b)-1}{u_j(b)} +\sum_{k=0}^{j-1} A_k\frac{u_k(b)}{u_j(b)}+A_j\right\}. \]

Since \(u_j(b)>0\), it follows that

\[ \operatorname{sign} D(a,b)=\operatorname{sign} A_j, \tag{9} \]

for the quantities \(u_{n-1}(b)-1\), \(u_k(b)\) \((k=0,1,\ldots,j-1;\ j\le n-2)\) are infinitely small of higher order in comparison with \(u_j(b)\) as \(b\to a\). Consequently, \(D(a,b)\ne 0\) in some neighborhood \((a,\rho_j(a))\subset(a,r_j(a))\).

If \(A+A_{n-1}=0\) and all \(A_i=0\) \((i=0,1,\ldots,n-2)\), then one of two cases may occur: 1) either \(A_{n-1}=-A=0\), then problem (1), (2) has no unique solution; 2) or \(A_{n-1}=-A\ne 0\), then

\[ D(a,b)=A_{n-1}[u_{n-1}(b)-1] \]

and the existence of an interval \([a,\lambda)\) depends on the operation \(L[y]\). Thus, for example, if \(p_0(x)=0\), then problem (1), (2) has no unique solution for any \(a\) and \(b\) \((a<b)\). On the other hand, in case 2) for any operation \(L[y]\) for which the function \(u_{n-1}(x)-1\ne 0\) and has no accumulation of zeros at the point \(a\), there is a unique solution of problem (1), (2) if \(b\) belongs to some neighborhood of the point \(a\).

Denote \(\lambda=\min[\rho,\rho_j]\).

The theorem is proved.

The interval \([a,\lambda)\), spoken of in Theorem 1, will be called subcritical for problem (1), (2). Since \(D(a,x)\) is the solution of the problem

\[ L[z]=-p_0(x)\cdot A,\quad z(a)=A+A_{n-1},\quad z^{(k)}(a)=A_{n-k-1}\ (k=1,\ldots,n-1), \tag{10} \]

it follows that

Theorem 2. Under the conditions of Theorem 1, the interval \([a,\lambda)\) is subcritical for problem (1), (2) if in the interval \((a,\lambda)\) the solution of problem (10) does not vanish.

Thus, estimating the subcritical interval of problem (1), (2) reduces to estimating the interval of sign-constancy of the solution of problem (10).

Estimates of the disconjugacy intervals of boundary value problems can be obtained from the ideas of the works [2, 9].

Thus, for example, problem (1), (2) (for \(f(x)\equiv 0\)) reduces to the system of integral equations:
\[ y_m(x)=\sum_{k=0}^{n-1}\int_a^b \frac{\partial^m}{\partial x^m}G_0(x,s,a,b)p_k(s)y_k(s)\,ds \quad (m=0,1,\ldots,n-1), \tag{11} \]
where \(G_0(x,s,a,b)\) is the Green’s function for the equation \(y^{(n)}=0\) with boundary conditions (2). Denote by
\[ \widetilde K_0^{(m)}(x,s)= \begin{cases} \dfrac{(x-s)^{\,n-m-1}}{(n-m-1)!}, & \text{for } a\le s<x\le b,\\[6pt] 0, & \text{for } a\le x\le s\le b, \end{cases} \quad (m=0,1,\ldots,n-1), \]
where
\[ K_0^{(m)}(x,s)=\frac{(x-s)^{\,n-m-1}}{(n-m-1)!} \]
is the \(m\)-th derivative of the Cauchy function of the operation \(L[y]\equiv y^{(n)}\).

Using Theorem 4 of [2] for system (11), we arrive at the following lower estimate for the disconjugacy interval of problem (1), (2).

Theorem 3. If the conditions of Theorem 1 are satisfied and the inequality
\[ \sum_{m=0}^{n-1}\sup_{b\in(a,a+h)} \left[ \sup_{x,s\in(a,b)} \left| \widetilde K_0^{(m)}(x,s)- \frac{ \displaystyle \sum_{i=0}^{n-m-1}\frac{A_i(x-a)^{\,n-m-i-1}}{(n-m-i-1)!} }{ \displaystyle A+\sum_{i=0}^{n-1}\frac{A_i(b-a)^{\,n-i-1}}{(n-i-1)!} } \frac{(b-s)^{\,n-1}}{(n-1)!} \right| \right] \times \int_a^{a+h}|p_m(s)|\,ds<1, \tag{12} \]
then \([a,a+h)\) is a disconjugacy interval for problem (1), (2).

Let us note some special cases of problem (1), (2). If all elements of the \((i+1)\)-st column of the determinant \(A\) are equal to zero, but the determinant \(A_i\ne 0\) \((i=0,1,\ldots,n-1)\), then problem (1), (2) turns into the problem
\[ L[y]=f(x),\quad y(a)=y'(a)=\ldots=y^{(n-2)}(a)=y(b)=0 \quad \text{for } i=0, \tag{13} \]
\[ L[y]=f(x),\quad y'(a)=\ldots=y^{(n-i-2)}(a)=y^{(n-i)}(a)=\ldots=y^{(n-1)}(a)=y(b)=0 \]
\[ \text{for } i=1,2,\ldots,n-2, \tag{14} \]
\[ L[y]=f(x),\quad y'(a)=y''(a)=\ldots=y^{(n-1)}(a)=y(b)=0 \quad \text{for } i=n-1. \tag{15} \]

From formula (6) and the definition of \(A_i\) it is clear that, for problems (13)—(15),
\[ D(a,b)=A_i u_i(b)\quad (i=0,1,\ldots,n-1), \]
and, consequently, the interval \([a,r_i(a))\) is disconjugate for these problems.

Let
\[ [a,r(a))=\bigcap_{i=0}^{n-1}[a,r_i(a)). \]
Then, if \(A\ge 0\) \((A\le 0)\), \(A_i\ge 0\) \((A_i\le 0)\)

and at least one \(A_i \ne 0\) \((i=0,1,\ldots,n-1)\), the interval \([a,r(a))\) is subcritical for problem (1), (2).

Let us consider comparison theorems which may be useful in estimating the subcritical interval.

Suppose that the boundary-value problem is given by

\[ L_1[y]\equiv y^{(n)}-p_0(x)y=f(x),\qquad l_i[y]=0\quad (i=0,1,\ldots,n-1) \tag{16} \]

with the boundary conditions (2), and the Cauchy problems

\[ L_1[z]\equiv z^{(n)}-p_0(x)z=-p_0(x)A,\quad z(a)=A+A_{n-1},\quad z^{(k)}(a)=A_{n-k-1} \]

and

\[ (k=1,\ldots,n-1) \tag{17} \]

\[ \overline{L}_1[\overline{y}]\equiv \overline{y}^{(n)}-\overline{p}_0(x)\overline{y} =-\overline{p}_0(x)A,\quad \overline{y}(a)=A+A_{n-1},\quad \overline{y}^{(k)}(a)=A_{n-k-1} \]

\[ (k=1,\ldots,n-1), \tag{18} \]

where \(p_0(x)\geq \overline{p}_0(x)\).

Denote by \(K_1(x,s)\) and \(\overline{K}_1(x,s)\) the Cauchy functions, respectively, for the operators \(L_1[y]\) and \(\overline{L}_1[\overline{y}]\). From the positivity criterion for the Cauchy function (see [4], [5]) it follows that if in \(\Delta(a<s<x<b)\) the function \(\overline{K}_1(x,s)>0\), then in this triangle also \(K_1(x,s)>0\).

Let \(\overline{K}_1(x,s)>0\) in \(\Delta(a<s<x<\overline{h}_1(a))\). Denote by \((a,\overline{Y}_1(a))\) the interval in which the solution of problem (18) satisfies \(\overline{y}(x)\ne 0\), and

\[ T_1(a)=\min[\overline{h}_1(a),\overline{Y}_1(a)], \]

then the following holds.

Theorem 4. Under the assumptions of Theorem 1, the interval \([a,T_1(a))\) is subcritical for problem (16).

Proof. Subtracting (18) from (17), we obtain

\[ (z-\overline{y})^{(n)}=p_0(x)[z-\overline{y}]+[p_0(x)-\overline{p}_0(x)]\overline{y}, \]

\[ z^{(k)}(a)-\overline{y}^{(k)}(a)=0\quad (k=0,1,\ldots,n-1). \]

Applying the Cauchy formula to the last problem, we shall have

\[ z(x)-\overline{y}(x)=\int_a^x K_1(x,s)[p_0(s)-\overline{p}_0(s)]\overline{y}(s)\,ds. \]

It is seen from this that on the interval \((a,T(a))\), for \(\overline{y}>0\) also \(z>\overline{y}>0\), and for \(\overline{y}<0\) also \(z<\overline{y}<0\); but then, by Theorem 2, problem (16) has a unique solution for any \(b\in(a,T_1(a))\).

Denote the negative part of the coefficient \(p_0(x)\) by \(p_-(x)\) and put \(\overline{p}_0(x)=p_-(x)=\min[0,p_0(x)]\). Then from Theorems 3 and 4 there follows immediately

Corollary 1. Let the assumptions of Theorem 1 be fulfilled. Then the interval \([a,a+h)\) will be subcritical for problem (16), if \([a,a+h)\subset [a,h_1(a))\) and

\[ \sup_{b\in(a,a+h)} \left[ \sup_{x,s\in(a,b)} \left| \widetilde{K}_0(x,s) - \frac{ \displaystyle \sum_{i=0}^{n-1} A_i\frac{(x-a)^{\,n-i-1}}{(n-i-1)!} }{ \displaystyle A+\sum_{i=0}^{n-1} A_i\frac{(b-a)^{\,n-i-1}}{(n-i-1)!} } \right| \times \]

\[ \times \frac{(b-s)^{\,n-1}}{(n-1)!} \left\|\int_a^{a+h}|p_{-}(s)|\,ds\right\|<1. \tag{19} \]

According to Theorem 3 of [6], if

\[ \frac{(b-a)^{\,n-1}}{4^{\,n-1}(n-1)!}\int_a^b |p_{-}(s)|\,ds<1, \tag{20} \]

then \(K_1(x,s)>0\) in \(\Delta(a<s<x<b)\), where \(K_1(x,s)\) is the Cauchy function of the operation \(L_1[y]\). Since \(K(x,a)=u_0(x)\), when inequality (20) is satisfied, \(h=b-a\) is a subcritical interval of problem (13).

Using the above-mentioned criterion for positivity of the function \(K_1(x,s)\), from Corollary 1 we obtain the following estimates of subcritical intervals for problems (14), (15).

Corollary 2. If the inequality

\[ \frac{i}{n-i-1} \left[\frac{n-i-1}{n-1}\right]^{\frac{n-1}{i}} \frac{h^{\,n-1}}{(n-1)!}\int_a^{a+h}|p_{-}(s)|\,ds<1, \]

holds, where \(i\) is a fixed number from the set \(\{1,2,\ldots,n-2\}\), then \([a,a+h)\) is a subcritical interval of problem (14) for the corresponding value of \(i\).

Corollary 3. If the inequality

\[ \frac{h^{\,n-1}}{(n-1)!}\int_a^{a+h}|p_{-}(s)|\,ds<1 \]

holds, then \([a,a+h)\) is a subcritical interval of problem (15).

Putting \(p_0(x)=-\alpha^2\) \((\alpha=\mathrm{const}>0)\), we apply Theorem 4 to problem (16) for \(n=2\).

Corollary 4. Let the conditions of Theorem 1 be satisfied and let \(p_0(x)\ge -\alpha^2\). Then the subcritical interval \([a,a+h)\) of problem (16) for \(n=2\) satisfies the following relations:

a) if \(A_0^2+\alpha^2 A_1^2\ne 0\), \(A\ne0\) and

\[ |A|>\frac{\sqrt{A_0^2+\alpha^2 A_1^2}}{\alpha}, \]

then

\[ [a,a+h)=\left[a,a+\frac{\pi}{\alpha}\right); \]

b) if \(A_0^2+\alpha^2 A_1^2\ne0\) and

\[ |A|\le \frac{\sqrt{A_0^2+\alpha^2 A_1^2}}{\alpha}, \]

then

\[ [a,a+h)=[a,a+h_1), \]

where

\[ 0<h_1=\min\left\{\frac{\pi}{\alpha},\, \left|\arctg\frac{A_0}{\alpha A_1} +\arccos\frac{\alpha A}{\sqrt{A_0^2+\alpha^2 A_1^2}} +2\pi m\right|\right\}, \]

and

\[ m=0,\ \pm1,\ \pm2,\ldots \]

Proof. For the operation \(\bar L_1[\bar y] \equiv \bar y''+\alpha^2\bar y\) we have:

\[ \bar u_0(x)=\frac{1}{\alpha}\sin \alpha(x-a), \qquad \bar u_1(x)=\cos \alpha(x-a), \qquad \bar K_1(x,s)=\frac{1}{\alpha}\sin \alpha(x-s), \]

therefore,

\[ [a,\bar h_1(a))=\left[a,a+\frac{\pi}{\alpha}\right), \]

\[ D(a,b)=\frac{A_0}{\alpha}\sin \alpha(b-a)+A_1\cos \alpha(b-a)+A= \]

\[ =\frac{\sqrt{A_0^2+\alpha^2 A_1^2}}{\alpha}\cos \alpha(b-a-\varphi)+A, \]

where

\[ \tg \alpha\varphi=\frac{A_0}{\alpha A_1}. \]

Considering the conditions under which \(D(a,b)\ne 0\), \(b-a>0\), and applying Theorem 4, we obtain the assertion of the corollary.

Remark. \(D(a,b)\ne 0\) also in the case \(A_0=0\), \(A_1=0\), \(A\ne 0\). Let us recall that in this case we have the Cauchy problem.

Corollary 5. Suppose the conditions of Theorem 1 are satisfied and \(p_0(x)\ge \alpha^2\). Then the right-extended interval \([a,a+h)\) of problem (16) for \(n=2\) satisfies the following relations:

a) if \(A_0^2-\alpha^2 A_1^2=0\), \(A\ne 0\), then \([a,a+h)=[a,+\infty)\);

b) if \(A_0^2-\alpha^2 A_1^2\ne 0\) and \(h_1>0\), then \([a,a+h)=[a,a+h_1)\), where

\[ h_1=\frac{1}{\alpha}\left[-\operatorname{Arcth}\frac{\alpha A_1}{A_0} +\operatorname{Arcsh}\left(-\frac{\alpha A}{\sqrt{A_0^2-\alpha^2 A_1^2}}\right)\right]; \]

c) if \(A_0^2-\alpha^2 A_1^2\ne 0\) and \(h_1\le 0\), then \([a,a+h)=[a,+\infty)\).

Proof. For the operation \(\bar L_1[\bar y]\equiv \bar y''-\alpha^2\bar y\) we have:

\[ \bar u_0(x)=\frac{1}{\alpha}\operatorname{sh}\alpha(x-a), \qquad \bar u_1(x)=\operatorname{ch}\alpha(x-a), \qquad \bar K_1(x,s)=\frac{1}{\alpha}\operatorname{sh}\alpha(x-s), \]

therefore,

\[ [a,\bar h_1(a))=[a,+\infty), \]

\[ D(a,b)=\frac{A_0}{\alpha}\operatorname{sh}\alpha(b-a)+A_1\operatorname{ch}\alpha(b-a)+A= \]

\[ =\frac{\sqrt{A_0^2-\alpha^2 A_1^2}}{\alpha}\operatorname{sh}\alpha(b-a+\varphi)+A, \]

where

\[ \operatorname{th}\alpha\varphi=\frac{A_1\alpha}{A_0}. \]

Considering the conditions under which \(D(a,b)\ne 0\), \(b-a>0\), and applying Theorem 4, we obtain the assertion of the corollary.

Let us consider one more comparison theorem. For problem (1), (2) we associate the Cauchy problem

\[ \overline{L}[\overline{y}] \equiv \overline{y}_x^{(n)} - \sum_{k=0}^{n-1} \overline{p}_k(x)\,\overline{y}^{(k)}(x) = -Ap_0(x), \]

\[ \overline{y}(a)=A+A_{n-1},\qquad \overline{y}^{(k)}(a)=A_{n-k-1}\quad (k=1,\ldots,n-1), \tag{21} \]

where

\[ p_k(x)\geq \overline{p}_k(x)\quad (k=0,1,\ldots,n-1). \]

Let \(\overline{K}(x,s)>0\) in \(\Delta(a<s<x<\overline{h}(a))\), where \(\overline{K}(x,s)\) is the Cauchy function of the operation (21). Denote by \((a,\overline{Y}(a))\) and \((a,\overline{Y}^{i}(a))\) the intervals in which, respectively, the solution and the \(i\)-th derivative of the solution of problem (21) do not vanish, where \(i\) are the indices of the coefficients \(p_k(x)\ne 0\) \((k\ne 0)\), and let \((a,\overline{\overline{Y}}(a))\) be the intersection of all \((a,\overline{Y}^{i}(a))\). Put

\[ T(a)=\min[\overline{h}(a),\overline{Y}(a),\overline{\overline{Y}}(a)]. \]

Theorem 5. If \(A+A_{n-1}\geq 0,\ A_{n-2}>0\) \((A+A_{n-1}\leq 0,\ A_{n-2}<0)\) and all \(A_{n-i-1}>0\) \((A_{n-i-1}<0)\), where \(i\) are the indices of the coefficients \(p_k(x)\ne 0\) \((k\ne 0)\), then the interval \([a,T(a))\) is subcritical for problem (1), (2).

Proof. Subtracting (21) from (10) and applying Cauchy’s formula, we obtain

\[ z-\overline{y} = \int_a^x K(x,s) \sum_{k=0}^{n-1} [p_k(s)-\overline{p}_k(s)]\,\overline{y}^{(k)}(s)\,ds. \]

The conditions of the theorem ensure the existence of the intervals \((a,\overline{Y}(a))\), \((a,\overline{Y}^{i}(a))\) and the same sign of \(\overline{y}\) and \(\overline{y}^{(i)}\) on the corresponding intervals. But then on the interval \((a,T(a))\) we shall have: either \(\overline{y}>0\) and \(z-\overline{y}>0\), or \(\overline{y}<0\) and \(z-\overline{y}<0\), and by virtue of Theorem 2 the interval \([a,T(a))\) is subcritical for problem (1), (2).

Remark. For an estimate of the “height” \(\overline{h}(a)\) of the triangle \((a<s<x<\overline{h}(a))\), in which the Cauchy function is positive, see [4, 5]. From the ideas of [9] one can obtain estimates of the intervals in which the \(i\)-th derivative of the solution of problem (21) does not vanish.

§ 2. On estimates of the solution of the boundary value problem

Estimates of the solution of problem (1), (2) are given by us on the basis of the theorem on a differential inequality [1, 3, 8]. If \(G(x,s)\) is the Green’s function of problem (1), (2), \(u\) is the solution of this problem, and \(z\) is some function satisfying the conditions (2), then, provided the signs of the Green’s function and of the residual \(\varphi=L[z]-f\) are preserved, the difference

\[ z-u=\int_a^b G(x,s)\varphi(s)\,ds \]

also preserves its sign. Thus, Chaplygin’s theorem on a differential inequality for problem (1), (2) is valid when the sign of the Green’s function is preserved. Below we give conditions for preservation of the sign of the function \(G(x,s)\).

Lemma 1. Under the hypotheses of Theorem 1, for any operation \(L[y]\) there exists an interval \([a,\mu)\) such that, for any \(b \in (a,\mu)\), the Green’s function \(G(x,s)\) of problem (1), (2) exists and preserves its sign in the triangle \((a<x\leq s<b<\mu)\).

Moreover, we have:

a) if
\[ A+A_{n-1}\ne 0 \quad \text{and} \quad A_{n-1}\ne 0, \]
then
\[ \operatorname*{sign}_{a<x\leq s<b<\mu} G(x,s) = -\operatorname{sign}\frac{A_{n-1}}{A+A_{n-1}}; \]

b) if
\[ A\ne 0,\quad A_j\ne 0,\quad \text{but}\quad A_{j+1}=\cdots=A_{n-1}=0 \quad (j=0,1,\ldots,n-2), \]
then
\[ \operatorname*{sign}_{a<x\leq s<b<\mu} G(x,s) = -\operatorname{sign}\frac{A_j}{A}; \]

c) if
\[ A+A_{n-1}=0, \]
but at least one determinant \(A_i\ne 0\) \((i=0,1,\ldots,n-2)\), then

1) for \(A=A_{n-1}=0\),
\[ G(x,s)<0 \qquad (a<x\leq s<b<\mu), \]

2) for \(A_{n-1}\ne 0\),
\[ \operatorname*{sign}_{a<x\leq s<b<\mu} G(x,s) = -\operatorname{sign}\frac{A_{n-1}}{A_j}, \]
where \(A_j\) is the last \(A_i\ne 0\) \((i=0,1,\ldots,n-2)\).

Proof. In the triangle under consideration,
\[ G(x,s)= - \frac{\displaystyle\sum_{i=0}^{n-1} A_i u_i(x)} {\displaystyle A+\sum_{i=0}^{n-1} A_i u_i(b)} K(b,s). \tag{22} \]

a) From formula (22), for \(x=a,\ s=a\), we have:
\[ G(a,a) = -\frac{A_{n-1}}{D(a,b)}K(b,a) = -A_{n-1}\frac{u_0(b)}{D(a,b)}. \]

If \(A+A_{n-1}=D(a,a)\ne 0\) and \(A_{n-1}\ne 0\), then
\[ \operatorname*{sign}_{a<x\leq s<b<\mu_1} G(x,s) = \operatorname{sign}G(a,a) = -\operatorname{sign}\frac{A_{n-1}}{D(a,b)} = \]
\[ = -\operatorname{sign}\frac{A_{n-1}}{D(a,a)} = -\operatorname{sign}\frac{A_{n-1}}{A+A_{n-1}}. \]

b) If \(A\ne 0,\ A_j\ne 0\), but
\[ A_{j+1}=\cdots=A_{n-1}=0 \quad (j=0,1,\ldots,n-2), \]
then
\[ \operatorname*{sign}_{a<x\leq s<b<\mu_2} G(x,s) = -\operatorname{sign} \frac{\displaystyle\sum_{k=0}^{j} A_k u_k(x)} {D(a,b)} \]
\[ = -\operatorname{sign}\frac{A_j}{D(a,a)} = -\operatorname{sign}\frac{A_j}{A}. \]

c) Let
\[ A=A_{n-1}=0,\quad A_j\ne 0,\quad A_{j+1}=\cdots=A_{n-2}=0 \quad (j=0,1,\ldots,n-2), \]
then
\[ \operatorname*{sign}_{a<x\leq s<b<\mu_3} G(x,s) = -\operatorname{sign}\frac{A_j}{A_j} = -\operatorname{sign}1,\qquad G(x,s)<0. \]

If \(A=-A_{n-1}\ne 0,\ A_j\ne 0,\ A_{j+1}=\ldots=A_{n-2}=0\) \((j=0,1,\ldots,n-2)\), then

\[ G(x,s)=-\frac{K(b,s)}{D(a,b)} \left[\sum_{k=0}^{j} A_k u_k(x)+A_{n-1}u_{n-1}(x)\right], \]

where

\[ D(a,b)=A_{n-1}[u_{n-1}(b)-1]+\sum_{k=0}^{j} A_k u_k(b). \]

On the basis of (9)

\[ \operatorname{sign} D(a,b)=\operatorname{sign} A_j \]

and, consequently,

\[ \operatorname*{sign}_{a<x\le s<b<\mu_4} G(x,s) = -\operatorname{sign}\frac{A_{n-1}}{A_j}. \]

Denote by \([a,\mu)=\bigcap_{i=0}^{4}[a,\mu_i)\).

The lemma is proved.

Consider the question of the size of the side of the square in which the Green’s function preserves its sign.

Case I. \(G(x,s)>0\).

Let

\[ L[u]=0,\quad u(a)=A_{n-1},\quad u^{(k)}(a)=A_{n-k-1}\quad (k=1,\ldots,n-1). \tag{23} \]

Denote by \((a,h(a))\) the height of \(\Delta(a<s<x<h(a))\), in which \(K(x,s)>0\), where \(K(x,s)\) is the Cauchy function of the operation \(L[y]\); by \((a,Y(a))\) and \((a,U(a))\) the intervals in which the solutions of the problems (10) and (23), respectively, do not vanish. Put

\[ R(a)=\min[h(a),Y(a),U(a)]. \]

Theorem 6. Under the conditions of Theorem 1 and Lemma 1, under which \(G(x,s)>0\) in \(\Delta(a<x\le s<b<\mu)\), the Green’s function of problem (1), (2) exists and is positive in the square \(x,s\in(a,b)\subset[a,R(a))\).

Proof. From Lemma 1 it is easy to see that if \(G(x,s)>0\) in \(\Delta(a<x\le s<b)\), then \(A\ne 0\). Since

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

are solutions of the problems (10) and (23), respectively, for \(x\in(a,R(a))\) these sums preserve their sign. In view of the fact that \(K(x,s)>0\) in \(\Delta(a<s<x<b)\) for any \(b\in(a,R(a))\), so also \(K(b,s)\) for \(b\in(a,R(a))\).

From the proof of Lemma 1 it is seen that in \(\Delta(a<x\le s<b)\)

\[ \operatorname{sign} G(x,s)= \operatorname{sign}\left[ \frac{\displaystyle\sum_{i=0}^{n-1} A_i u_i(x)} {\displaystyle\sum_{i=0}^{n-1} A_i u_i(b)+A} \right], \]

if in this triangle \(G(x,s)\) preserves its sign. Therefore, when any conditions of Lemma 1 ensuring positivity are fulfilled—

ness of \(G(x,s)\) in \(\Delta(a<x\le s<b<\mu)\), the Green’s function of problem (1), (2) exists and is positive in \(\Delta(a<x\le s<b)\) for any \(b\in(a,R(a))\). But then \(G(x,s)>0\) also in \(\Delta(a<s<x<b)\) for any \(b\in(a,R(a))\), since \(K(x,s)>0\) in this triangle.

Let us state a comparison theorem for problem (16). Let

\[ \overline{K}_1(x,s)>0 \quad \text{in } \Delta(a<s<x<\overline{h}_1(a)), \]

where \(\overline{K}_1(x,s)\) is the Cauchy function of operator (18). Denote by \((a,\overline{Y}_1(a))\) and \((a,\overline{U}_1(a))\) the intervals in which, respectively, the solutions of problem (18) for \(A\ne0\) and \(A=0\) do not vanish,

\[ \overline{R}_1(a)=\min[\overline{h}_1(a),\overline{Y}_1(a),\overline{U}_1(a)]. \]

Theorem 7. Under the assumptions of Theorem 1 and Lemma 1, under which \(G(x,s)>0\) in \(\Delta(a<x\le s<b<\mu)\), the Green’s function of problem (16) exists and is positive in the square

\[ x,\ s\in(a,b)\subset [a,\overline{R}_1(a)). \]

Proof. Let \((a,Y_1(a))\) and \((a,U_1(a))\) denote the intervals in which, respectively, the solutions of problem (17) for \(A\ne0\) and \(A=0\) do not vanish. By Theorem 4,

\[ (a,\overline{Y}_1(a))\subset(a,Y_1(a)),\quad (a,\overline{U}_1(a))\subset(a,U_1(a)). \]

By the positivity criterion for the Cauchy function,

\[ (a,\overline{h}_1(a))\subset(a,h_1(a)). \]

But then \((a,\overline{R}_1(a))\subset(a,R_1(a))\), and the validity of Theorem 7 follows from Theorem 6.

II. The case \(G(x,s)<0\). Remark. Here we consider operators \(L[y]\) for which there exists \(\Delta(a<s<x<h'(a))\), in which

\[ K'_x(x,s)>0 \quad \text{and} \quad K'_s(x,s)<0. \]

Introduce the notation:

\[ 1.\quad \varphi(x)=\sqrt{u_0(x)}, \]

where \(u_0(x)\) is the solution of the problem

\[ L[y]=0,\quad y^{(k)}(a)=0 \quad (k=0,1,\ldots,n-2),\quad y^{n-1}(a)=1; \]

\[ 2.\quad \psi(x)= \frac{u(x)\sqrt{u_0^*(x)}}{\displaystyle\sum_{i=0}^{n-1} A_i u_i(b)+A}, \]

where \(u(x)\) is the solution of problem (23), \(u_0^*(x)\) is the solution of the problem \(L^*[y]=0\),

\[ y^{(k)}(b)=0 \quad (k=0,1,\ldots,n-2),\quad y^{(n-1)}(b)=1 \]

considered for \(a\le x\le b\); \(L^*[y]=0\) is the equation adjoint to \(L[y]=0\);

1. \((a,h'(a))\) is the height of \(\Delta(a<s<x<h'(a))\), in which

\[ K'_x(x,s)>0 \quad \text{and} \quad K'_s(x,s)<0; \]

1. \((a,P(a))\) is the interval in which the graphs of the functions \(\varphi(x)\) and \(\psi(x)\) have no common point;

1. \(Q(a)=\min [P(a),\, h'(a),\, U(a),\, Y(a)]\),

where \((a,Y(a))\) and \((a,U(a))\) are intervals in which the solutions of problems (10) and (23), respectively, do not vanish.

Theorem 8. Under the conditions of Theorem 1 and Lemma 1, for which \(G(x,s)<0\) in \(\Delta(a<x\leq s<b<\mu)\), the Green’s function of problem (1), (2) exists and is negative in the square \(x,s\in(a,b)\subset [a,Q(a))\).

Proof. Suppose the conditions of Lemma 1 are satisfied, under which \(G(x,s)<0\) in \(\Delta(a<x\leq s<b<\mu)\); then

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

do not vanish and have the same signs. Obviously, one may put

\[ \mu=\min [h'(a),\, U(a),\, Y(a)], \]

since \(K(b,s)>0\) in \(\Delta(a<s<x<h'(a))\).

Let us now consider \(G(x,s)\) in \(\Delta(a<s<x<b)\) with height not exceeding \((a,Q(a))\). Since \(K'_x(x,s)>0\) and \(K'_s(x,s)<0\), the inequalities hold

\[ K(x,s)\leq K(x,a), \qquad K(x,s)\leq K(b,s), \qquad K(x,s)\leq \sqrt{K(x,a)K(b,s)}, \]

therefore,

\[ G(x,s)\leq \sqrt{K(b,s)} \left[ \sqrt{K(x,a)} - \frac{\displaystyle\sum_{i=0}^{n-1} A_i u_i(x)} {\displaystyle\sum_{i=0}^{n-1} A_i u_i(b)+A} \sqrt{K(b,s)} \right] < \]

\[ <\sqrt{K(b,s)} \left[ \sqrt{u_0(x)} - \frac{u(x)\sqrt{u_0^*(x)}} {\displaystyle\sum_{i=0}^{n-1} A_i u_i(b)+A} \right] = \sqrt{K(b,s)}[\varphi(x)-\psi(x)] = \Phi(x,s). \]

Under the conditions of Theorem 1, \(\varphi(x)\) is an infinitesimal of higher order compared with \(\psi(x)\) as \(x\to a\); therefore, in a sufficiently small \(\Delta(a<s<x<b<\nu)\),

\[ \operatorname{sign}\Phi(x,s) = -\operatorname{sign}\psi(x) = -\operatorname{sign} \frac{\displaystyle\sum_{i=0}^{n-1} A_i u_i(x)} {\displaystyle\sum_{i=0}^{n-1} A_i u_i(b)+A} \]

and, by what was said above, \(\Phi(x,s)<0\) in \(\Delta(a<s<x<b<\nu)\). Obviously, one may put \((a,\nu)=(a,Q(a))\). But then in \(\Delta(a<s<x<b<Q(a))\) also \(G(x,s)<0\).

The theorem is proved.

In conclusion, let us consider an existence theorem for the nonlinear boundary-value problem

\[ N[y]\equiv y^{(n)}-f(x,y)=0, \tag{24} \]

\[ \sum_{k=0}^{n-1}\alpha_i^{(k)}y^{(k)}(a)+\beta_i^{(0)}y(b)=0 \quad (i=0,1,\ldots,n-1). \tag{25} \]

We assume that \(f(x,y)\) is continuous and satisfies a Lipschitz condition with respect to \(y\) in the domain \(R:\ a\le x\le b,\ a_1\le y\le b_1\). Thus, for \(N[y]\) the conditions \(L_1\) and \(L_2\) of N. V. Azbelev are simultaneously fulfilled, consisting in the existence of such linear operations

\[ \widetilde L_j[y]\equiv y^{(n)}-\widetilde p_j(x)y \quad (j=1,2), \]

that for any ordered pair \(y_1\ge y_2\) we have:

\[ \widetilde L_1[y_1-y_2]\ge N[y_1]-N[y_2]\ge \widetilde L_2[y_1-y_2]. \]

In this case

\[ f(x,y)=\widetilde p_j(x)y+M_j(x,y)\quad (j=1,2), \]

where \(M_1\) does not decrease and \(M_2\) does not increase with respect to \(y\). Denote by \(\widetilde G_j(x,s)\) the Green’s function for the equation \(\widetilde L_j[y]=0\) with conditions (2).

Theorem 9. Let \(\widetilde G_1(x,s)>0\) \((\widetilde G_2(x,s)<0)\) in the square \(x,s\in(a,b)\). Suppose further that a pair of functions \(z(x)\) and \(v(x)\), \(n\) times continuously differentiable on \([a,b]\), \((z(x)\le v(x))\), whose graphs lie in the domain \(R\), satisfy the boundary conditions (25) and, for \(x\in[a,b]\), the inequalities \(N[z]\ge 0\), \(N[v]\le 0\).

Then: a) there exists a solution of problem (24), (25); b) this solution can be obtained as the limit of monotonically convergent approximations of Chaplygin-method type, determined by the rule

\[ u_{i+1}=\omega+\int_a^b \widetilde G_j(x,s)\{\widetilde L_j[u_i(s)]-N[u_i(s)]\}\,ds,\quad j=1\ (j=2), \]

where \(u_0=z\), or \(u_0=v\), and \(\omega\) is the solution of \(\widetilde L_j[y]=0\), satisfying the boundary conditions (25).

Proof. Problem (24), (25) corresponds to the integral equation

\[ y(x)=\int_a^b \widetilde G_j(x,s)\{f(s,y(s))-\widetilde p_j(s,y(s))\}\,ds,\quad j=1\ (j=2). \]

If the conditions of Theorem 9 are fulfilled, then the Tarski—Birkhoff—Kantorovich theorem is applicable (see [7]), whence the assertion of the theorem follows.

References

1. Azbelev N. V., Tsaljuk Z. B. On integral and differential inequalities. Proceedings of the Fourth All-Union Mathematical Congress, 1961, vol. 2. L., “Nauka”, 1964, pp. 384—391.
2. Azbelev N. V., Tsaljuk Z. B. DAN SSSR, 156, No. 2, 1964, pp. 239—242.
3. Azbelev N. V., Tsaljuk Z. B., Khokhryakov A. Ya. Mat. sb., 59, No. 1, 1962, pp. 125—143.
4. Azbelev N. V., Tsaljuk Z. B. Ukr. matem. zhurnal, 10, No. 1, 1958, pp. 3—11.
5. Azbelev N. V., Smolin I. M., Tsaljuk Z. B. DAN SSSR, 135, 1960, pp. 511—514.
6. Levin A. Yu. DAN SSSR, 148, No. 3, 1963, pp. 512—515.
7. Kantorovich L. V., Vulikh B. Z., Pinsker A. G. Functional Analysis in Partially Ordered Spaces, GITTL. M.—L., 1950, p. 464.
8. Pak S. A. Abstract of Cand. diss., Voronezh State University, 1964.
9. Pak S. A. Sibirskii matem. zhurnal, III, No. 4, 1962, pp. 569—574.

Received by the editors
January 28, 1965

Izhevsk Mechanical Institute