UDC 517.941.9

ON THE GREEN MATRIX FOR A CLASS OF LINEAR TWO-POINT BOUNDARY VALUE PROBLEMS

M. I. URBANOVICH

In the present article we consider sufficient (and in certain cases also necessary) conditions for the existence and sign-constancy of the Green matrix of the boundary value problem

\[ \frac{dy}{dt}+A(t)y=f(t), \qquad My(\alpha)+Ny(\beta)=0, \tag{1} \]

where

\[ y=\{y_1,\ldots,y_n\}, \qquad f(t)=\{f_1(t),\ldots,f_n(t)\}, \qquad o=\{0,\ldots,0\} \]

are \(n\)-dimensional vectors (column matrices); \(M,\ N,\ A(t)\) are \(n\times n\) matrices; \(A(t)\) and \(f(t)\) are continuous on \([\alpha,\beta]\)*). The elements of all matrices are assumed to be real. The term “solution” is used everywhere in the classical sense.

The main results of the work concern the case when \(A(t), M, N\) are upper (lower) triangular matrices.

Notation and definitions used. 1. \(Y(t)\) is some fundamental (nonsingular) solution of the matrix equation

\[ \frac{dY}{dt}+A(t)Y=O, \]

where \(O\) is the zero \(n\times n\) matrix; \(K(t,s)\) is the Cauchy matrix of this equation, i.e. its solution with respect to the argument \(t\) for which \(K(s,s)=E\). Obviously, \(K(t,s)=Y(t)Y^{-1}(s)\).

1. \(\Gamma(t,s)\) is the Green matrix [2] of problem (1).

2. Inequalities between matrices will be understood in the sense that the analogous inequalities hold for the corresponding elements of these matrices.

3. The matrix \(A=\|a_{ik}\|_1^n\) will be called upper (lower) triangular if \(a_{ik}=0\) for \(i>k\) (respectively, for \(i<k\)).

4. Following [1], the interval \([\alpha,T)\), \(T>\alpha\), will be called an interval of unique solvability of problem (1) if, for fixed \(\alpha\), this problem has a unique solution for every \(\beta\in(\alpha,T)\), whatever the continuous vector-function \(f(t)\).

By \([\alpha,T_{\max})\) we shall denote the maximal interval of unique solvability of problem (1).

\[ \text{*) The case } M=-N=E, \]
where \(E\) is the identity matrix, was considered in [1]. There also, using matrices of Green-matrix type, a necessary and sufficient criterion was given for unique solvability of problem (1) and for sign-constancy of its Green matrix. In the present work, however, we shall not rely on that criterion.

§ 1. In this section we single out those facts which are the starting point for the consideration of the triangular problem (§§ 2, 3) and of the local case (§ 4).

Theorem 1. For the existence of the Green’s matrix of problem (1):

\[ \frac{dy}{dt}+A(t)y=f(t), \tag{1.1} \]

\[ My(\alpha)+Ny(\beta)=0 \tag{1.2} \]

it is necessary and sufficient that the condition

\[ \det |M+NK(\beta,\alpha)|\ne 0 \tag{1.3} \]

be fulfilled.

Proof. As is known, the existence of the Green’s matrix of problem (1.1), (1.2) is equivalent to the unique solvability of the corresponding homogeneous boundary-value problem \((f(t)\equiv 0)\). Moreover, any solution of the homogeneous system

\[ \frac{dy}{dt}+A(t)y=0 \]

can be represented in the form \(y(t)=Y(t)c\), where \(c\) is an arbitrary constant \(n\)-dimensional vector. Taking into account the boundary condition (1.2), we obtain that \(y(t)\equiv 0\) if and only if \(\det |MY(\alpha)+NY(\beta)|\ne 0\), or \(\det |M+NY(\beta)Y^{-1}(\alpha)|\ne 0\). The theorem is proved.

When condition (1.3) is fulfilled, problem (1.1), (1.2) has a unique solution, which can be expressed through the Green’s matrix of this problem by the formula [2]:

\[ y(t)=\int_{\alpha}^{\beta}\Gamma(t,s)f(s)\,ds. \tag{1.4} \]

In turn, let us find the dependence of \(\Gamma(t,s)\) on the Cauchy matrix \(K(t,s)\) and the boundary condition (1.2), using the fact that any solution of system (1.1) can be written in the form

\[ y(t)=\int_{\alpha}^{t}K(t,s)f(s)\,ds+K(t,\alpha)c. \tag{1.5} \]

For the constant vector \(c\), by virtue of (1.2), we have the expression

\[ c=-\int_{\alpha}^{\beta}[M+NK(\beta,\alpha)]^{-1}NK(\beta,s)f(s)\,ds. \]

Comparing now (1.4) and (1.5), and taking into account the arbitrariness of \(f(t)\), we obtain

\[ \Gamma(t,s)= \begin{cases} K(t,s)-K(t,\alpha)[M+NK(\beta,\alpha)]^{-1}NK(\beta,s), & (\alpha\le s\le t\le \beta),\\ -K(t,\alpha)[M+NK(\beta,\alpha)]^{-1}NK(\beta,s), & (\alpha\le t<s\le \beta). \end{cases} \tag{1.6} \]

Similarly we find the second formula:

\[ \Gamma(t,s)= \begin{cases} -K(t,s)+K(t,\beta)[MK(\alpha,\beta)+N]^{-1}MK(\alpha,s), & (\alpha\le t\le s\le \beta),\\ K(t,\beta)[MK(\alpha,\beta)+N]^{-1}MK(\alpha,s), & (\alpha\le s<t\le \beta). \end{cases} \tag{1.7} \]

In particular, when \(A(t)=A\) is a constant \(n\times n\) matrix, we have

\[ K(t,s)=\exp[-A(t-s)], \tag{1.8} \]

since in this case one may take \(Y(t)=\exp(-At)\).

Lemma 1. Let \(A=|a_{ik}|_1^n\), \(a_{ik}\leqslant 0\) \((i\ne k)\). Then, for every \(\tau>0\),
\(\exp(-A\tau)\gg 0\).

Proof. For any fixed \(\tau>0\), choose \(\omega\gg |a_{ii}|\tau\) \((i=1,\ldots,n)\). Then \(-A\tau+\omega E=B\gg 0\) and \(\exp(-A\tau)\cdot \exp(\omega E)=\exp B\gg 0\), whence \(\exp(-A\tau)=\exp(-\omega)\cdot \exp B\gg 0\).

From the expressions for the Green’s matrix and Lemma 1 there follows directly

Theorem 2. Let \(A=|a_{ik}|_1^n\), \(a_{ik}\leqslant 0\) \((\geqslant 0)\) for \(i\ne k\),

\[ [M+N\exp\{-A(\beta-\alpha)\}]^{-1}N\leqslant 0 \quad \left( [M\exp\{A(\beta-\alpha)\}+N]^{-1}M\leqslant 0 \right). \]

Then the Green’s matrix \(\Gamma(t,s)\) of problem (1.1), (1.2), for \(A(t)\equiv A\), is nonnegative (nonpositive) in the square \(t,s\in[\alpha,\beta]\).

Remark. The Green’s matrix \(\Gamma_0(t,s)\) of problem (1.1), (1.2), for \(A(t)\equiv 0\) and \(\det |M+N|\ne 0\), has the form

\[ \Gamma_0(t,s)= \begin{cases} E-(M+N)^{-1}N, & t\geqslant s,\\ -(M+N)^{-1}N, & t<s, \end{cases} \tag{1.9} \]

so that \(\Gamma_0(t,s)\gg 0\) \((\leqslant 0)\) on the whole square \(t,s\in[\alpha,\beta]\) if and only if

\[ (M+N)^{-1}N\leqslant 0 \quad (\geqslant 0 \ \text{and}\ [(M+N)^{-1}N]_{ii}\geqslant 1,\ i=1,\ldots,n). \]

An analogous criterion of sign-constancy can be obtained starting from (1.7).

§ 2. Consider the boundary value problem

\[ \frac{dy}{dt}+Ay=f(t),\quad My(\alpha)+Ny(\beta)=0, \tag{2.1} \]

where \(A=|a_{ik}|_1^n\), \(M=|m_{ik}|_1^n\), \(N=|n_{ik}|_1^n\) are constant upper*) triangular matrices.

In this case \(M+N\exp[-A\cdot(\beta-\alpha)]\) is also an upper triangular matrix, and its determinant is equal to

\[ \prod_{i=1}^{n}\bigl[m_{ii}+n_{ii}\exp\{-a_{ii}(\beta-\alpha)\}\bigr]. \]

Consequently, for the existence of the Green’s matrix of problem (2.1) it is necessary and sufficient that

\[ \prod_{i=1}^{n}\bigl[m_{ii}+n_{ii}\exp\{-a_{ii}(\beta-\alpha)\}\bigr]\ne 0. \tag{2.2} \]

From (2.2) we see that \([\alpha,+\infty)\) is the maximal interval of unique solvability of problem (2.1) if, for each \(i\) \((i=1,\ldots,n)\), one of the following conditions is satisfied:

1) \(m_{ii}n_{ii}\geqslant 0,\quad m_{ii}^{2}+n_{ii}^{2}>0;\)

2) \(a_{ii}=0\) \((>0),\quad m_{ii}n_{ii}<0,\quad |m_{ii}|\geqslant |n_{ii}|\ (|m_{ii}|>|n_{ii}|);\)

3) \(a_{ii}<0,\quad m_{ii}n_{ii}<0,\quad |m_{ii}|\leqslant |n_{ii}|.\)

*) We note that all the results of §§ 2, 3 remain valid also for the “lower triangular” problem.

Before considering sufficient criteria for the sign constancy of the Green’s matrix of problem (2.1), we shall state two lemmas.

Lemma 2. Suppose that in the determinant of order \(m\) \((m>1)\)

\[ \Delta_m= \left| \begin{array}{cccccc} b_{11} & b_{12} & \cdot & \cdot & b_{1m-1} & b_{1m}\\ \lambda_1 & b_{22} & \cdot & \cdot & b_{2m-1} & b_{2m}\\ 0 & \lambda_2 & \cdot & \cdot & b_{3m-1} & b_{3m}\\ \cdot & \cdot & \cdot & \cdot & \cdot & \cdot\\ 0 & 0 & \cdot & \cdot & \lambda_{m-1} & b_{mm} \end{array} \right| \]

we have \(b_{ik}\leq 0\) \((i\leq k;\ i,k=1,\ldots,m)\), \(\lambda_i>0\) \((i=1,\ldots,m-1)\). Then \(\Delta_m=(-1)^m p_m\), where \(p_m\geq 0\).

We carry out the proof by induction on \(m\). For \(m=2\) and \(m=3\) the assertion is obvious. Suppose it is true for \(m=k\):

\[ \Delta_k=(-1)^k p_k,\qquad p_k\geq 0. \]

Then

\[ \Delta_{k+1}= \left| \begin{array}{cccccc} b_{11} & b_{12} & \cdot & \cdot & b_{1k} & b_{1k+1}\\ \lambda_1 & b_{22} & \cdot & \cdot & b_{2k} & b_{2k+1}\\ 0 & \lambda_2 & \cdot & \cdot & b_{3k} & b_{3k+1}\\ \cdot & \cdot & \cdot & \cdot & \cdot & \cdot\\ 0 & 0 & \cdot & \cdot & \lambda_k & b_{k+1\,k+1} \end{array} \right| = \]

\[ = b_{11}\Delta_k^{(1)}-\lambda_1\Delta_k^{(2)} = b_{11}(-1)^k p_k^{(1)}-\lambda_1(-1)^k p_k^{(2)} = \]

\[ = (-1)^{k+1}\left[\lambda_1p_k^{(2)}-b_{11}p_k^{(1)}\right], \qquad \text{where } p_k^{(1)}\geq 0,\ p_k^{(2)}\geq 0. \]

Thus, \(\Delta_{k+1}=(-1)^{k+1}p_{k+1}\), where
\[ p_{k+1}=\lambda_1p_k^{(2)}-b_{11}p_k^{(1)}\geq 0. \]
This proves Lemma 2 for every natural \(m>1\).

Remark. If \(b_{ik}\leq 0\) \((i\geq k;\ i,k=1,\ldots,m)\), \(\lambda_i>0\) \((i=1,\ldots,m-1)\), then, evidently,

\[ \Delta_m= \left| \begin{array}{cccccc} b_{11} & \lambda_1 & 0 & \cdot & \cdot & 0\\ b_{21} & b_{22} & \lambda_2 & \cdot & \cdot & 0\\ \cdot & \cdot & \cdot & \cdot & \cdot & \cdot\\ b_{m-1\,1} & b_{m-1\,2} & b_{m-1\,3} & \cdot & \cdot & \lambda_{m-1}\\ b_{m1} & b_{m2} & b_{m3} & \cdot & \cdot & b_{mm} \end{array} \right| =(-1)^m p_m,\qquad p_m\geq 0. \]

Lemma 3. Let \(C=\|c_{ik}\|_1^n\) be an upper triangular matrix, with \(c_{ii}>0\) \((i=1,\ldots,n)\), \(c_{ik}\leq 0\) \((i<k)\). Then the matrix \(C\) is positively invertible, i.e. \(C^{-1}\geq 0\).

Proof. As is known,

\[ (C^{-1})_{ik}=\frac{A_{ki}}{\det|C|}, \]

where \(A_{ki}\) is the cofactor of the element \(c_{ki}\) of the matrix \(C\), and

\[ \det|C|=\prod_{i=1}^{n}c_{ii}>0. \]

The lemma is obvious for \(n=2,\ n=3\).

Let \(n>3\). Since \(A_{ii}=\det|C|/c_{ii}>0\) and \(A_{ki}=0\) \((k<i)\), it remains to show that \(A_{ki}\geq 0\) for \(k>i\).

Take \(k=i+1\), where \(i=1,n-1\) and \(1<i<n-1\). Then

\[ A_{21}=(-c_{12})\prod_{i=3}^{n} c_{ii}\geqslant 0;\qquad A_{nn-1}=(-c_{n-1\,n})\prod_{i=1}^{n-2} c_{ii}\geqslant 0, \]

\[ A_{i+1\,i}=(-c_{i\,i+1})\det |C|/c_{ii}\geqslant 0. \]

Let now \(k>i+1\). Setting \(i=1,\ n-2\) and \(1<i<n-2\), taking into account the conditions imposed on \(c_{ik}\), and applying Lemma 2, we successively obtain:

\[ A_{k1}=(-1)^{k+1}(-1)^{k-1}p_{k-1}\prod_{i=k+1}^{n} c_{ii}\geqslant 0; \]

\[ A_{nn-2}=\prod_{i=1}^{n-3} c_{ii}\,(c_{n-2\,n-1}c_{n-1\,n}-c_{n-1\,n-1}c_{n-2\,n})\geqslant 0; \]

\[ A_{ki}=(-1)^{k+i}\prod_{m=1}^{i-1}c_{mm}(-1)^{k-i}p_{k-i}\prod_{m=k+1}^{n}c_{mm}\geqslant 0. \]

Corollary. If \(C=|c_{ik}|_1^n\) is an upper triangular matrix, \(c_{ii}<0\) \((i=1,\ldots,n)\), \(c_{ik}\geqslant 0\) \((i<k)\), then \(C^{-1}\leqslant 0\).

Remark. The conditions \(c_{ii}>0\) \((<0)\), \(i=1,\ldots,n\), for any \(n>1\), are also necessary for positive (negative) invertibility of the upper triangular matrix \(C=|c_{ik}|_1^n\). The conditions \(c_{ik}\leqslant 0\) \((\geqslant 0)\), \(i<k\), for \(n>2\), are not necessary for this purpose, although they are essential for the validity of Lemma 3 (the corollary).

From Theorem 2, by virtue of Lemma 3, it follows that

Theorem 3. \(1''.\) Let \(a_{ik}\leqslant 0,\ m_{ik}\leqslant 0\) \((\geqslant 0)\) for \(i<k,\ N\leqslant 0\) \((\geqslant 0)\), and
\[ m_{ii}+n_{ii}\exp[-a_{ii}(\beta-\alpha)]>0\quad (<0),\quad i=1,\ldots,n. \tag{2.3} \]
Then the Green’s matrix \(\Gamma(t,s)\) of the boundary-value problem (2.1) is nonnegative in the square \(t,s\in[\alpha,\beta]\).

\(2''.\) If \(a_{ik}\geqslant 0,\ n_{ik}\leqslant 0\) \((\geqslant 0)\), \(i<k;\ M\leqslant 0\) \((\geqslant 0)\), and the conditions (2.3) are fulfilled, then \(\Gamma(t,s)\leqslant 0\) for \(t,s\in[\alpha,\beta]\).

We note two corollaries of Theorem 3.

Corollary 1. The Green’s matrix of problem (2.1) is nonnegative in any square \(t,s\in[\alpha,\beta]\), if \(a_{ik}\leqslant 0,\ m_{ik}\leqslant 0\) \((\geqslant 0)\) for \(i<k,\ N\leqslant 0\) \((\geqslant 0)\), and for each \(i\) \((i=1,\ldots,n)\) one of the following conditions is satisfied \((1)\) and \(2)\) for \(m_{ik}\leqslant 0,\ N\leqslant 0;\ 1')\) and \(2')\) for \(m_{ik}\geqslant 0,\ N\geqslant 0\):

\[ \begin{aligned} &1)\quad n_{ii}=0,\quad m_{ii}>0;\\ &2)\quad n_{ii}<0,\quad a_{ii}=0\quad (>0),\quad m_{ii}>|n_{ii}|\quad (m_{ii}\geqslant |n_{ii}|); \end{aligned} \tag{2.4} \]

\[ \begin{aligned} &1')\quad n_{ii}=0,\quad m_{ii}<0;\\ &2')\quad n_{ii}>0,\quad a_{ii}=0\quad (>0),\quad m_{ii}<-n_{ii}\quad (m_{ii}\leqslant -n_{ii}). \end{aligned} \tag{2.5} \]

In particular, for the periodic boundary-value problem \((M=-N=E)\) with a constant upper triangular matrix

\[ A=|a_{ik}|_1^n,\qquad a_{ii}\ne 0\quad (i=1,\ldots,n) \]

the Green’s matrix \(\Gamma(t,s)\) exists and is nonnegative for any \(\beta>\alpha\), if \(a_{ii}>0\) \((i=1,\ldots,n)\), \(a_{ik}\leqslant 0\) \((i<k)\).

Corollary 2. Let \(a_{ik}\leqslant 0,\ m_{ik}\leqslant 0\) \((\geqslant 0)\) for \(i<k,\ N\leqslant 0\) \((\geqslant 0)\), and among the diagonal elements of the matrix \(A\) there are negative,

where \(a_{i_p i_p}\) \((p=1,\ldots,r;\ r\le n)\) are all those of such elements for which \(n_{i_p i_p}a_{i_p i_p}\ne0\), and for each \(i\ne i_p\) one of the conditions (2.4) (respectively (2.5)) is satisfied. Then the Green matrix of problem (2.1) for fixed \(\alpha\) is nonnegative in all squares

\[ t,s\in [\alpha,\beta]\subset[\alpha,T_{\max}), \]

where

\[ T_{\max}=\alpha+\min_p\left\{\frac{1}{|a_{i_p i_p}|}\ln\left|\frac{m_{i_p i_p}}{n_{i_p i_p}}\right|\right\}. \]

Analogous corollaries (with the result \(\Gamma(t,s)\le0\)) are also generated by assertion \(2''\) of Theorem 3 (in their formulation one need only interchange the roles of \(m_{ii}\) and \(n_{ii}\), and change the signs of \(a_{ii}\) \((a_{i_p i_p})\) to the opposite ones).

§ 3. The results of § 2 are generalized to the case of a boundary value problem with variable coefficients:

\[ \frac{dy}{dt}+A(t)y=f(t), \tag{3.1} \]

\[ My(\alpha)+Ny(\beta)=0, \tag{3.2} \]

where \(A(t)=|a_{ik}(t)|_1^n,\ M=|m_{ik}|_1^n,\ N=|n_{ik}|_1^n\) are upper triangular matrices, \(A(t)\) and \(f(t)\) are continuous on \([\alpha,\beta]\).

Theorem 4. For the Green matrix of problem (3.1), (3.2) to exist, it is necessary and sufficient that

\[ m_{ii}+n_{ii}\exp\left[-\int_\alpha^\beta a_{ii}(t)\,dt\right]\ne0 \quad (i=1,\ldots,n). \tag{3.3} \]

Proof. It is easy to construct a fundamental solution \(Y(t)\) of the matrix equation

\[ \frac{dY}{dt}+A(t)Y=0, \]

which is an upper triangular matrix, and such that

\[ Y(\alpha)=E,\quad [Y(t)]_{ii}=\exp\left[-\int_\alpha^t a_{ii}(t_1)\,dt_1\right]\quad (i=1,\ldots,n). \]

Now the assertion of the theorem follows from the fact that

\[ \det|M+NK(\beta,\alpha)| = \prod_{i=1}^n \left\{ m_{ii}+n_{ii}\exp\left[-\int_\alpha^\beta a_{ii}(t)\,dt\right] \right\}. \]

Theorem 5. \(1_1.\) The Green matrix \(\Gamma(t,s)\) of problem (3.1), (3.2) is nonnegative in the square \(t,s\in[\alpha,\beta]\), if

\[ 1)\quad a_{ik}(t)\le0 \text{ on } [\alpha,\beta],\quad m_{ik}\le0\ (\ge0) \text{ for } i<k;\ N\le0\ (\ge0) \tag{3.4} \]

and

\[ 2)\quad m_{ii}+n_{ii}\exp\left[-\int_\alpha^\beta a_{ii}(t)\,dt\right]>0\ (<0) \quad \text{for } i=1,\ldots,n. \tag{3.5} \]

\(2_1.\) If \(a_{ik}(t)\ge0\) on \([\alpha,\beta]\), \(n_{ik}\le0\ (\ge0)\) for \(i<k\), \(M\le0\ (\ge0)\), and conditions (3.5) are fulfilled, then \(\Gamma(t,s)\le0\) for \(t,s\in[\alpha,\beta]\).

Proof. \(1_1.\) First of all, by (3.5) the Green matrix

\(\Gamma(t,s)\) of problem (3.1), (3.2) exists. Moreover, for the matrices \(M, N\) and
\(A_1=\{\max\limits_{t\in[\alpha,\beta]} a_{ik}(t)\}_1^n=\{a_{ik}^{(1)}\}_1^n\) all the conditions of Theorem 3, \(1^\circ\), are satisfied; consequently, the Green matrix \(\Gamma_1(t,s)\) of the problem

\[ \frac{dy}{dt}+A_1 y=f(t),\qquad My(\alpha)+Ny(\beta)=0 \tag{3.6} \]

also exists and is nonnegative for \(t,s\in[\alpha,\beta]\).

Let \(f(t)\) be an arbitrary continuous vector function, nonnegative on \([\alpha,\beta]\), \(f(t)\geq 0\) \((\not\equiv 0)\). Then the solution

\[ y_1(t)=\int_\alpha^\beta \Gamma_1(t,s)f(s)\,ds \]

of problem (3.6) is nonnegative on \([\alpha,\beta]\), \(y_1(t)\geq 0\) \((\not\equiv 0)\).

Next, problem (3.1), (3.2) is equivalent to the integral equation

\[ y(t)=\int_\alpha^\beta \Gamma_1(t,s)[A_1-A(s)]\,y(s)\,ds +\int_\alpha^\beta \Gamma_1(t,s)f(s)\,ds, \]

i.e.

\[ y(t)=\int_\alpha^\beta R(t,s)y(s)\,ds+y_1(t). \]

Here \(R(t,s)=\Gamma_1(t,s)[A_1-A(s)]\geq 0\) and \(y_1(t)\geq 0\) \((\not\equiv 0)\), therefore \(y(t)\geq 0\) and \(y(t)\geq y_1(t)\), i.e.

\[ \int_\alpha^\beta \Gamma(t,s)f(s)\,ds \geq \int_\alpha^\beta \Gamma_1(t,s)f(s)\,ds, \]

whence

\[ \Gamma(t,s)\geq \Gamma_1(t,s)\geq 0,\quad t,s\in[\alpha,\beta]. \]

\(2_1\). To prove assertion \(2_1\), it suffices, following the scheme given above, to start from
\(A_2=\{\min\limits_{t\in[\alpha,\beta]} a_{ik}(t)\}_1^n\) and \(f(t)\leq 0\) \((\not\equiv 0)\).

Corollary. a). Let

\[ A_1=\{\max_{t\in[\alpha,\beta]} a_{ik}(t)\}_1^n=\{a_{ik}^{(1)}\}_1^n \quad\text{and}\quad A_2=\{\min_{t\in[\alpha,\beta]} a_{ik}(t)\}_1^n=\{a_{ik}^{(2)}\}_1^n. \]

For the Green matrix \(\Gamma(t,s)\) and the solution \(y(t)\) of problem (3.1), (3.2), under the fulfillment of conditions (3.4) and

\[ m_{ii}+n_{ii}\exp[-a_{ii}^{(2)}(\beta-\alpha)]>0\quad(<0), \qquad i=1,\ldots,n, \tag{3.7} \]

the estimates

\[ 0\leq \Gamma_1(t,s)\leq \Gamma(t,s)\leq \Gamma_2(t,s), \quad t,s\in[\alpha,\beta] \tag{3.8} \]

and

\[ 0\leq y_1(t)\leq y(t)\leq y_2(t),\quad \text{if } f(t)\geq 0,\quad t\in[\alpha,\beta]. \tag{3.9} \]

hold.
\[ \begin{gathered} (\geq)\qquad(\geq)\qquad(\geq)\qquad(\leq) \end{gathered} \]

Here \(\Gamma_i(t,s)\) and \(y_i(t)\) are, respectively, the Green matrix and the solution of the problem

\[ \frac{dy}{dt}+A_i y=f(t),\qquad My(\alpha)+Ny(\beta)=0\quad (i=1,2). \]

b). If conditions (3.7) are satisfied and \(a_{ik}(t)\geq 0\) on \([\alpha,\beta]\), \(n_{ik}\leq 0\) \((\geq 0)\) for \(i<k\), \(M\leq 0\) \((\geq 0)\), the following estimates hold (in the notation of item a):

\[ \Gamma_1(t,s)\leq \Gamma(t,s)\leq \Gamma_2(t,s)\leq 0,\qquad t,s\in[\alpha,\beta] \]

and

\[ \underset{(\geq)}{y_1(t)}\leq \underset{(\geq)}{y(t)}\leq \underset{(\geq)}{y_2(t)}\leq 0, \quad \text{if } f(t)\geq 0,\quad t\in[\alpha,\beta]. \]

Example. For problem (3.1), (3.2), with \(\alpha=0,\ \beta=1\),

\[ A(t)= \left\| \begin{array}{cc} t+1 & t^2-1\\ 0 & -t^2+3 \end{array} \right\|, \qquad M= \left\| \begin{array}{cc} 2 & -1\\ 0 & 1 \end{array} \right\|, \qquad N= \left\| \begin{array}{cc} -1 & -2\\ 0 & -1 \end{array} \right\| \]

we have

\[ A_1= \left\| \begin{array}{cc} 2 & 0\\ 0 & 3 \end{array} \right\|, \qquad A_2= \left\| \begin{array}{cc} 1 & -1\\ 0 & 2 \end{array} \right\| \]

and, therefore,

\[ 0\leq \Gamma_1(t,s)\leq \Gamma(t,s)\leq \Gamma_2(t,s),\qquad t,s\in[0,1]. \]

Here

\[ \Gamma_i(t,s)= \begin{cases} K_i(t,s)+\Gamma_i^*(t,s), & s\leq t,\\ \Gamma_i^*(t,s), & t<s, \end{cases} \qquad (i=1,2) \]

\[ K_1(t,s)=\exp[-3(t-s)] \left\| \begin{array}{cc} \exp(t-s) & 0\\ 0 & 1 \end{array} \right\|, \]

\[ K_2(t,s)=\exp[-(t-s)] \left\| \begin{array}{cc} 1 & 1-\exp[-(t-s)]\\ 0 & \exp[-(t-s)] \end{array} \right\|, \]

\[ \Gamma_1^*(t,s)= \]

\[ = \frac{\exp[-3-2(t-s)]}{[2-\exp(-2)][1-\exp(-3)]} \left\| \begin{array}{cc} \exp 1-\exp(-2) & 3\exp s\\ 0 & [2-\exp(-2)]\exp[-(t-s)] \end{array} \right\|, \]

\[ \Gamma_2^*(t,s)= \]

\[ = \frac{\exp[-1-(t-s)]}{[2-\exp(-1)][1-\exp(-2)]} \left\| \begin{array}{cc} 1-\exp(-2) & b(t,s)\\ 0 & [2-\exp(-1)]\exp[-1-(t-s)] \end{array} \right\|, \]

\[ b(t,s)=1-\exp(-2)+2[2-\exp(-t)]\exp[-(1-s)]+\exp[-2-(t-s)]. \]

Remark 1. There are also consequences analogous to consequences 1 and 2 from Theorem 3.

Remark 2. The results of §3 extend to boundary value problems

\[ \frac{dy}{dt}+A(t)y=f(t),\qquad My(\alpha)+Ny(\beta)=o, \]

where \(M=\{M_1,\ldots,M_r\}\), \(N=\{N_1,\ldots,N_r\}\), \(A(t)=\{A_1(t),\ldots,A_r(t)\}\) are quasi-diagonal matrices whose corresponding diagonal blocks \(M_i, N_i, A_i(t)\) are upper (lower) triangular matrices of the same order satisfying the conditions of Theorem 5.

§ 4. Returning again to the “general” problem (1.1), (1.2), we note the following further results of a local character.

Lemma 4. If \(\det|M+N|\ne 0\), then, whatever the continuous \(n\times n\) matrix \(A(t)\), there exists an interval of unique solvability of problem (1.1), (1.2).

The validity of the lemma follows from the fact that the determinant
\(\det |M+NK(\beta,\alpha)|=\varphi(\tau)\), being a continuous function of \(\tau=\beta-\alpha\), is nonzero at \(\tau=0\).

Theorem 6. Let \(\det |M+N|\ne 0\), \((M+N)^{-1}N\leqslant 0\) \(((M+N)^{-1}M\leqslant 0)\), and \(A(t)\leqslant 0\) \((\geqslant 0)\). Then there exists \(T>\alpha\) such that the Green’s matrix \(\Gamma(t,s)\) of problem (1.1), (1.2) is nonnegative (nonpositive) in every square \(t,s\in[\alpha,\beta]\subset[\alpha,T)\).

Proof. Let, for example, \((M+N)^{-1}N\leqslant 0\), \(A(t)\leqslant 0\). By Lemma 4 the solution \(y(t)\) of problem (1.1), (1, 2) exists for sufficiently small \(\beta-\alpha\). This solution satisfies the integral equation

\[ y(t)=\int_{\alpha}^{\beta}\Gamma_0(t,s)[-A(s)y(s)+f(s)]\,ds, \]

or

\[ y(t)=\int_{\alpha}^{\beta}R(t,s)y(s)\,ds+\varphi(t), \]

where \(\Gamma_0(t,s)\) is the Green’s matrix of problem (1.1), (1.2) for \(A(t)\equiv 0\), \(R(t,s)=-\Gamma_0(t,s)A(s)\), \(\varphi(t)=\int_{\alpha}^{\beta}\Gamma_0(t,s)f(s)\,ds\). Since in this case \(R(t,s)\geqslant 0\) and \(\varphi(t)\geqslant 0(\ne 0)\) for every continuous vector-function \(f(t)\geqslant 0(\ne 0)\), it follows that \(y(t)\geqslant 0\), or

\[ y(t)=\int_{\alpha}^{\beta}\Gamma(t,s)f(s)\,ds\geqslant 0, \]

whence \(\Gamma(t,s)\geqslant 0\) in the square \(t,s\in[\alpha,\beta]\).

In the case \((M+N)^{-1}M\leqslant 0\) and \(A(t)\geqslant 0\) the proof is analogous. Just as simply one proves also

Theorem 7. Let \(A_1(t)\leqslant A_2(t)\leqslant 0\) and \((M+N)^{-1}N\leqslant 0\). Then there exists \(T>\alpha\) such that \(\Gamma_1(t,s)\geqslant \Gamma_2(t,s)\geqslant 0\) for \(t,s\in[\alpha,\beta]\subset[\alpha,T)\), where \(\Gamma_i(t,s)\) is the Green’s matrix of problem (1.1), (1.2) for \(A(t)=A_i(t)\) \((i=1,2)\).

Corollary. If \((M+N)^{-1}N\leqslant 0\), \(A(t)=|a_{ik}(t)|_1^n\leqslant 0\), then for some \(T>\alpha\) the following estimates of the Green’s matrix \(\Gamma(t,s)\) and the solution \(y(t)\) of problem (1.1), (1, 2) are valid:

\[ \Gamma_1(t,s)\geqslant \Gamma(t,s)\geqslant \Gamma_2(t,s)\geqslant 0,\quad \text{if } t,s\in[\alpha,\beta]\subset[\alpha,T) \]

and

\[ y_1(t)\geqslant y(t)\geqslant y_2(t)\geqslant 0,\quad \text{if } f(t)\geqslant 0 \text{ on }[\alpha,\beta], \]

\[ (\leqslant)\qquad(\leqslant)\qquad(\leqslant) \]

where \(\Gamma_m(t,s)\) and \(y_m(t)\) \((m=1,2)\) are respectively the Green’s matrix and the solution of problem (1.1), (1.2) for \(A(t)=A_m(t)\),

\[ A_1=\left|\min_{t\in[\alpha,\beta]}a_{ik}(t)\right|_1^n,\qquad A_2=\left|\max_{t\in[\alpha,\beta]}a_{ik}(t)\right|_1^n. \]

References

1. Khokhryakov A. Ya. Differential Equations, 2, No. 3, 371—381, 1966.
2. Kamke E. Handbook of Ordinary Differential Equations. Moscow, 1965.

Received by the editors
November 1, 1965

Mogilev Pedagogical Institute