UDC 517.911.

ON THE GREEN'S MATRIX OF A PERIODIC BOUNDARY VALUE PROBLEM FOR A SYSTEM OF LINEAR DIFFERENTIAL EQUATIONS

A. Ya. Khokhryakov

In the present article we study the existence and behavior of the Green's matrix of the periodic boundary value problem

\[ \frac{dx}{dt}+A(t)x=f(t),\qquad x(\alpha)=x(\beta), \tag{1} \]

where \(x=\{x_1,\ldots,x_n\}\); \(f(t)=\{f_1(t),\ldots,f_n(t)\}\) are \(n\)-dimensional vectors; \(A(t)\) is a continuous \(n\times n\) matrix. On the basis of information obtained about the Green's matrix, a number of consequences are considered. In particular, an estimate and a sufficient condition are given for the existence and uniqueness of an \(\omega\)-periodic solution of system (1).

In the paper the following definitions and notation are used.

1. By the characteristic numbers of a constant \(n\times n\) matrix \(Q\) we shall mean the roots of the equation \(\det|\lambda E-Q|=0\), where \(E\) is the identity \(n\times n\) matrix.

2. Let \(x=\{x_1,\ldots,x_n\}\); \(y=\{y_1,\ldots,y_n\}\); \(A=|a_{ij}|_1^n\), \(B=|b_{ij}|_1^n\). The inequality \(x\geqslant y\) means that \(x_1\geqslant y_1,\ldots,x_n\geqslant y_n\), and the inequality of matrices \(A\geqslant B\) is understood in the sense \(a_{ij}\geqslant b_{ij}\), \(i,j=1,\ldots,n\). The zero vector will be denoted by \(0\), and the zero matrix by \(O\).

3. Following A. G. Teterev [1], we shall say that a matrix \(Q\) is positively (negatively) invertible if \(Q^{-1}\geqslant O\) \((Q^{-1}\leqslant O)\), \((\ne O)\).

4. A diagonal matrix \(Q\) all of whose diagonal elements are positive (negative) will be denoted by \(Q^+\) \((Q^-)\).

5. By \(K_A(t,s)\) we shall denote the Cauchy matrix of the system \(\dfrac{dx}{dt}+A(t)x=0\), and by \(\Gamma_A(t,s)=|\gamma_{ij}(t,s)|_1^n\) the Green's matrix of problem (1).

6. The interval \([\alpha,T)\) \((\alpha<T)\) will be called an interval of unique solvability of problem (1) if problem (1) has a unique solution for any \(\beta\in[\alpha,T)\), whatever the continuous vector-function \(f(t)\) may be. The maximal interval of unique solvability of problem (1) for fixed \(\alpha\) will be denoted by \([\alpha,T_{\max})\).

1°. Let the constant matrix \(Q\) be nonsingular\(^*\). Then among the characteristic numbers \(\lambda_1,\ldots,\lambda_n\) of the matrix \(Q\) there are none equal to zero.

\(^*\) In this item the matrix \(Q\) is considered over the field of complex numbers.

By the corresponding linear transformation \(y=Px\), the system

\[ \frac{dx}{dt}+Qx=0,\qquad x(\alpha)=x(\beta) \tag{1.1} \]

is transformed into the system

\[ \frac{dy}{dt}+PQP^{-1}y=0,\qquad y(\alpha)=y(\beta), \tag{2.1} \]

where

\[ B=PQP^{-1}= \left\| \begin{array}{cccc} L_{k_1}(\lambda_1) & 0 & \cdots & 0\\ 0 & L_{k_2}(\lambda_2) & \cdots & 0\\ \cdots & \cdots & \cdots & \cdots\\ 0 & 0 & \cdots & L_{k_r}(\lambda_r) \end{array} \right\|, \]

and \(L_{k_j}(\lambda_j)\) is the known Jordan block, \(k_1+k_2+\cdots+k_r=n\).

Let \(Y(t)\) be the fundamental matrix of the system \(\dfrac{dy}{dt}+By=0\). According to the general theory of linear systems of differential equations with constant coefficients, \(Y(t)\) can be represented in the following form:

\[ Y(t)= \left\| \begin{array}{cccc} \exp L_{k_1}(\lambda_1)t & 0 & \cdots & 0\\ 0 & \exp L_{k_2}(\lambda_2)t & \cdots & 0\\ \cdots & \cdots & \cdots & \cdots\\ 0 & 0 & \cdots & \exp L_{k_r}(\lambda_r)t \end{array} \right\|, \tag{3.1} \]

\[ \exp L_{k_j}(\lambda_j)t = \exp \lambda_j t \left\| \begin{array}{ccccc} 1 & t & \dfrac{t^2}{2!} & \cdots & \dfrac{t^{k_j-1}}{(k_j-1)!}\\ 0 & 1 & t & \cdots & \dfrac{t^{k_j-2}}{(k_j-2)!}\\ \cdots & \cdots & \cdots & \cdots & \cdots\\ 0 & 0 & 0 & \cdots & 1 \end{array} \right\|. \]

From formula (3.1) it is clear that \([\alpha,T)\) will be an interval of unique solvability of problem (2.1) if \(\exp \lambda_j(t-\alpha)\ne 1\) \((j=1,\ldots,r)\) for any \(t\), \(\alpha<t<T\). Indeed, under the indicated conditions

\[ \det\|Y(\alpha)-Y(\beta)\|= \]

\[ =\prod_{j=1}^{r}\left\{[\exp \lambda_j\alpha]^{k_j}\,[1-\exp \lambda_j(\beta-\alpha)]^{k_j}\right\}\ne 0, \]

and therefore, for any \(\beta\in(\alpha,T)\), problem (2.1) has the unique solution \(y=0\). Hence, also from the nonsingularity of the linear transformation \(y=Px\), the unique solvability of problem (1.1) on the interval \([\alpha,T)\) follows.

Thus, the following has been established.

Theorem 1. Let the matrix \(Q\) be nonsingular, and let \(\lambda_1,\ldots,\lambda_n\) be the characteristic numbers of the matrix \(Q\). In order that \([\alpha,T)\) be an interval of unique solvability of problem (1.1), it is necessary and sufficient that

\[ \exp \lambda_j(t-\alpha)\ne 1 \quad (j=1,\ldots,n) \]

for any \(t:\alpha<t<T\).

Remark. If the real parts of all characteristic numbers \(\lambda_j\) of the matrix \(Q\) are different from zero, then the maximal interval of unique solvability of the problem will be \([\alpha,+\infty)\). If \(\operatorname{Re}(\lambda_j)=0\) for some \(j\), then, denoting \(\lambda_{j_1}=\sqrt{-1}\,\beta_1,\ldots,\lambda_{j_m}=\sqrt{-1}\,\beta_m\), we shall have

\[ T_{\max}-\alpha=\min\left\{\frac{2\pi}{\beta_1},\ldots,\frac{2\pi}{\beta_m}\right\}. \]

Corollary 1. Let \([\alpha,\beta]\subset[\alpha,T)\). Then in the square \(t,s\in[\alpha,\beta]\) there exists the Green’s matrix of problem (1.1) and, consequently, the solution \(x(t)\) of the problem

\[ \frac{dx}{dt}+Qx=f(t),\quad x(\alpha)=x(\beta), \tag{4.1} \]

where \(f(t)\) is a continuous vector-function. Moreover,

\[ x=\int_{\alpha}^{\beta}\Gamma(t,s)f(s)\,ds. \tag{4'.1} \]

Corollary 2. Let \(f(t)\) be a continuous \(\omega(=\beta-\alpha)\)-periodic vector-function and let \([\alpha,\beta]\subset[\alpha,T)\). Then there exists a unique \(\omega\)-periodic solution of system (4.1).

For the formulation of the next corollary, consider the problem

\[ L[y]\equiv y^{(n)}+a_{n-1}y^{(n-1)}+\cdots+a_0y=0, \tag{5.1} \]

\[ y^{(i)}(\alpha)=y^{(i)}(\beta)\quad (i=0,1,\ldots,n-1), \]

where \(a_i=\mathrm{const}\), \(a_0\ne0\). Hence it is obvious that among \(\lambda_1,\ldots,\lambda_n\), the roots of the equation

\[ \lambda^n+a_{n-1}\lambda^{n-1}+\cdots+a_0=0 \]

there are none equal to zero.

Corollary 3. Let \(a_0\ne0\). 1) The interval \([\alpha,T)\) will be an interval of unique solvability of problem (5.1) if and only if, for \(t:\alpha<t<T\), the inequality

\[ \exp \lambda_i(t-\alpha)\ne 1 \quad (i=1,\ldots,n) \]

holds.

2) In the square \(t,s\in[\alpha,\beta]\) there exists the Green’s function \(\Gamma(t,s)\) of problem (5.1). Further, whatever continuous vector-function \(f(t)\), the problem

\[ L[y]=f(t),\qquad y^{(i)}(\alpha)=y^{(i)}(\beta)\quad (i=0,1,\ldots,n-1) \]

has a unique solution

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

The assertion of Corollary 3 is easy to verify if one notes that (5.1) is equivalent to problem (1.1) with coefficient matrix

\[ Q_1= \left\| \begin{array}{cccccc} 0&-1&0&\ldots&0\\ 0&0&-1&\ldots&0\\ \ldots&\ldots&\ldots&\ldots&\ldots\\ 0&0&0&\ldots&-1\\ a_0&a_1&a_2&\ldots&a_{n-1} \end{array} \right\|,\qquad \det Q_1=a_0. \]

The characteristic numbers of the matrix \(Q_1\) are different from zero. Therefore, for the system with coefficient matrix \(Q_1\), all the conditions of Theorem 1 are fulfilled and, consequently, the assertions not only of Corollary 3 but also of Corollary 2 will hold.

\(2^\circ\). In this subsection we consider conditions ensuring preservation of the sign of the Green’s matrix \(\Gamma(t,s)\) of the homogeneous boundary-value problem (1.1).

We first give the dependence of \(\Gamma(t,s)\) on the Cauchy matrix \(K(t,s)\). Consider the solution \(x(t)\) of the problem

\[ \frac{dx}{dt}+Qx=f(t),\qquad x(\alpha)=x(\beta), \]

which can be written, by means of the Cauchy matrix \(K(t,s)\), in the following form:

\[ x(t)=\int_{\alpha}^{\beta}\Gamma(t,s)f(s)\,ds =\int_{\alpha}^{t}K(t,s)f(s)\,ds+K(t,\alpha)c. \]

Choosing the coefficient \(c\) in such a way that this solution satisfies the boundary condition, we obtain the expression for the Green’s function:

\[ \Gamma(t,s)= \begin{cases} K(t,s)+K(t,\alpha)[E-K(\beta,\alpha)]^{-1}K(\beta,s), & \text{for } t\ge s,\\ K(t,\alpha)[E-K(\beta,\alpha)]^{-1}K(\beta,s), & \text{for } t<s. \end{cases} \tag{1.2} \]

Analogously, we find the second formula

\[ \Gamma(t,s)= \begin{cases} -K(t,s)-K(t,\beta)[E-K(\alpha,\beta)]^{-1}K(\alpha,s), & \text{for } t\le s,\\ -K(t,\beta)[E-K(\alpha,\beta)]^{-1}K(\alpha,s), & \text{for } t>s. \end{cases} \tag{2.2} \]

Theorem 2. Let \(Q\) be a nonsingular constant \(n\times n\) matrix. In order that the Green’s matrix \(\Gamma(t,s)\) be nonnegative (nonpositive) in the square \(t,s\in[\alpha,\beta]\subset[\alpha,T)\), it is necessary that the matrix \(Q\) be

positive (negative) inverse. In order that the element \(\gamma_{ij}(t,s)\) of the matrix \(\Gamma(t,s)\) be identically equal to zero, it is necessary that the algebraic complement \(Q_{ij}\) of the element \(q_{ij}\) of the matrix \(Q\) be equal to zero.

Proof. Consider the boundary-value problem, written in matrix form,

\[ \frac{dX}{dt}+QX=E,\qquad X(\alpha)=X(\beta). \tag{3.2} \]

Obviously, problem (3.2) is equivalent to the system of \(n\) vector problems

\[ \frac{dx^i}{dt}+Qx^i=e^i,\qquad x^i(\alpha)=x^i(\beta)\quad (i=1,\ldots,n), \tag{4.2} \]

where \(x^i\) is the vector—the \(i\)-th column of the matrix \(X\), and \(e^i\) is the vector—the \(i\)-th column of the identity matrix \(E\).

It is easy to verify the formula

\[ x^i=Q^{-1}e^i\qquad (i=1,\ldots,n) \]

for the solution \(x^i\) of problem (4.2). The last formula shows that

\[ X=Q^{-1}E=Q^{-1}=\int_\alpha^\beta \Gamma(t,s)\,ds. \]

The theorem is proved.

Remark. As the example shows, the requirement \(Q^{-1}\geq 0\) \((\leq 0)\) is not a sufficient condition for preservation of the sign of the Green’s matrix.

We proceed to the study of sufficient conditions for preservation of the sign of the Green’s matrix for systems with constant coefficients.

Let the constant matrix \(Q=Q^+\). Since the characteristic numbers of the matrix \(Q^+\) coincide with its diagonal elements, the characteristic numbers of the Cauchy matrix \(K^+(\beta,\alpha)\) will be the numbers \(\exp[-q_{ii}(\beta-\alpha)]<1\) \((i=1,\ldots,n)\). Therefore, in formula (1.2) the matrix \(E-K^+(\beta,\alpha)\) is positively invertible [2], i.e., \([E-K^+(\beta,\alpha)]^{-1}\geq 0\). We note further that, in the case under consideration, \(K^+(t,s)\) is diagonal with positive elements. Hence from formula (1.2) it follows that the Green’s matrix \(\Gamma^+(t,s)\) is diagonal and the diagonal elements are positive.

In a completely analogous way, using formula (2.2), we verify the negativity of the diagonal elements of the matrix \(\Gamma^-(t,s)\), if \(Q=Q^-\).

One could arrive at this conclusion directly, solving the boundary-value problem separately for each equation of the system with the coefficient matrix \(Q^+\) \((Q^-)\), without resorting to formulas (1.2), (2.2).

3°. The further study of the Green’s matrix is based on one criterion for the existence and nonnegativity (nonpositivity) of such a matrix for the general two-point boundary-value problem

\[ L[x]\equiv \frac{dx}{dt}+A(t)x=f(t), \tag{1.3} \]

\[ Mx(\alpha)+Nx(\beta)=0, \tag{2.3} \]

where \(A(t)\) is a continuous \(n\times n\) matrix; \(M\) and \(N\) are constant \(n\times n\) matrices; \(x\) and \(f\) are \(n\)-dimensional vectors. Problem (1.3), (2.3), in particular, passes into problem (1) if \(M=-N=E\).

We shall say that \(W(t,s)\) is a matrix of the type of the Green’s matrix of problem (1.3), (2.3), if the following hold for \(W(t,s)\):

1) for fixed \(s\in(\alpha,\beta)\),
\[ MW(\alpha,s)+NW(\beta,s)=0; \]

2)
\[ W(s+0,s)-W(s-0,s)=E. \]

Theorem 3*). In order that the Green’s matrix \(\Gamma(t,s)\) of problem (1.3), (2.3) exist and be nonnegative (nonpositive), it is necessary and sufficient that, in the square \(t,s\in[\alpha,\beta]\), there exist a matrix \(W(t,s)\) of the type of the Green’s matrix satisfying the conditions:

a) for \(t,s\in[\alpha,\beta]\) the inequalities \(W(t,s)\geqslant0\) \((\leqslant0)\), \(L[W(t,s)]\leqslant0\) hold (the latter for \(t\ne s\));

b) the matrix \(-R(t,s)=L[W(t,s)]\) is such that unity is less than the modulus of any eigenvalue of the integral equation
\[ u=\lambda\int_{\alpha}^{\beta}R(t,s)u(s)\,ds+f(t). \tag{3.3} \]

Proof. Let \(v(t)\) be the solution of equation (3.3) for \(\lambda=1\). If \(f(t)\geqslant0\) \((\ne0)\), then \(v(t)\geqslant0\) \((\ne0)\). Indeed, as is known [3], the system (3.3) \((\lambda=1)\) can be replaced by one scalar integral equation
\[ U(t)=\int_{\alpha}^{\,n\beta-(n-1)\alpha} R_0(t,s)U(s)\,ds+\Phi(t), \tag{4.3} \]
where \(U,\Phi\) are scalar functions, defined in a known way on the interval \((\alpha,n\beta-(n-1)\alpha)\) through \(v=\{v_1,\ldots,v_n\}\), \(f(t)=\{f_1,\ldots,f_n\}\), respectively; \(R_0(t,s)\) is also a scalar function, defined in the square \(t,s\in[\alpha,n\beta-(n-1)\alpha]\) through the elements of the matrix \(R(t,s)\). Since \(f(t)\geqslant0\), \(R(t,s)\geqslant0\), it follows, in accordance with the definition, that \(\Phi(t)\geqslant0\), \(R_0(t,s)\geqslant0\) for \(t,s\in[\alpha,n\beta-(n-1)\alpha]\). Moreover, unity is less than the modulus of any eigenvalue of the integral equation
\[ y(t)=\lambda\int_{\alpha}^{\,n\beta-(n-1)\alpha}R_0(t,s)y(s)\,ds+\Phi(t). \]

Thus it has been established that, for the integral equation (4.3), the hypotheses of Uryson’s theorem are satisfied. According to the assertion of this theorem, the solution \(U(t)\) of equation (4.3) preserves its sign on the interval \([\alpha,n\beta-(n-1)\alpha]\), more precisely \(U(t)\geqslant0\).

Returning from the variables \(U,\Phi,R_0\) to \(v(t),R(t,s),f(t)\), we obtain the inequality \(v(t)\geqslant0\).

The vector
\[ z(t)=\int_{\alpha}^{\beta}W(t,s)v(s)\,ds, \]
obviously satisfies the boundary

*) Theorem 3 is close in idea to an unpublished report delivered by V. V. Ostroumov at the Izhevsk seminar.

conditions (2.3). Moreover, \(L[z(t)] = f(t)\):

\[ L[z(t)] = \int_{\alpha}^{\beta} L[W(t,s)]v(s)\,ds + v(t) = \]

\[ = -\int_{\alpha}^{\beta} R(t,s)v(s)\,ds + v(t) = f(t). \tag{5.3} \]

Thus, for any continuous \(f(t)\), the solution of problem (1.3), (2.3) exists \((x(t)=z(t))\), and this solution preserves the sign coinciding with the sign of \(W(t,s)\) for any \(f(t) \geqslant 0\). Sufficiency is proved.

Necessity will follow from the assumption \(W(t,s)=\Gamma(t,s)\), where \(\Gamma(t,s)\) is the Green’s matrix of problem (1.3), (2.3).

Remark. Condition b) of Theorem 3, for example, will be satisfied if there exists a positive vector-function \(v(t)\) such that

\[ -\int_{\alpha}^{\beta} R(t,s)v(s)\,ds > 0 \]

(where equality to zero is possible only on a set nowhere dense on the interval \([\alpha,\beta]\)).

Returning to the question of the behavior of the sign of the Green’s matrix \(\Gamma(t,s)\) of the periodic boundary value problem, we note that Theorem 2 gives a necessary condition for preservation of the sign of \(\Gamma(t,s)\). This condition consists in the positive (negative) invertibility of the matrix \(Q\) of coefficients of a linear system of differential equations with constant coefficients.

A. G. Tegerev gave a necessary and sufficient condition for positive (nonnegative) invertibility of the matrix \(Q\) [1]. Moreover, he proved the following. If the matrices \(A_1 < A_2\) are positively invertible, then any matrix \(A\) satisfying the inequalities \(A_1 < A < A_2\) is also positively invertible, and in this case \(A_2^{-1} < A^{-1} < A_1^{-1}\). From Theorem 4 considered below and from A. G. Tegerev’s theorem it follows that a nonsingular matrix \(Q\) satisfying the condition \(Q \leq Q^+\) \((Q^- \leq Q)\) will be nonnegatively (nonpositively) invertible if and only if there does not exist such a nonsingular matrix \(T\) that \(Q \leq T \leq Q^+\) \((Q^- \leq T \leq Q)\). Such matrices \(Q, Q^+\) \((Q^-)\) will be called \(d^+\)-\((d^-)\)-ordered.

Theorem 4. Let the matrices \(Q\) and \(Q^+\) \((Q^-)\) be \(d^+\)-ordered \((d^-\)-ordered). Then the Green’s matrix \(\Gamma(t,s)\) of problem (1.1) is nonnegative (nonpositive) in the square \(t,s \in [\alpha,\beta] \subset [\alpha,T]\).

Proof. Let \(Q\) and \(Q^+\) be \(d^+\)-ordered. In this case \(\Gamma(t,s)\) is a diagonal matrix with positive diagonal elements.

Introduce the sequence of matrices

\[ Q^+ > Q_1 > \ldots > Q_{k-1} > Q_k = Q \tag{6.3} \]

so that the norm \(\|Q_i - Q_{i-1}\| \leq \mu\) \((i=1,\ldots,k)\) is small, for example smaller than the modulus of any eigenvalue of the equation

\[ w(t) = \lambda \int_{\alpha}^{\beta} \Gamma(t,s)w(s)\,ds. \]

In this case, without loss of generality, one may assume that the interval of unique solvability of the problem

\[ \frac{dz}{dt}+Q_i z=0,\qquad z(\alpha)=z(\beta)\qquad (i=1,2,\ldots,k-1) \]

is no smaller than the interval \([\alpha,T)\) of unique solvability of problem (1.1). Consider the problem

\[ L_1[v]\equiv \frac{dv}{dt}+Q_1v=f(t),\qquad v(\alpha)=v(\beta)\quad (f(t)\geqslant 0), \tag{7.3} \]

for which the matrix \(\Gamma^+(t,s)\) will be of the type of a Green’s matrix. Moreover,

\[ L_1[\Gamma^+(t,s)] = [Q_1-Q^+]\Gamma^+(t,s)\leqslant 0\qquad (t\ne s). \]

Further, by virtue of the choice of \(Q_1\) (see (6.3)), one will be less than the modulus of each eigenvalue of the equation

\[ u=\lambda\int_\alpha^\beta [Q^+-Q_1]\Gamma^+(t,s)u(s)\,ds+f(t). \]

Then, in accordance with the assertion of Theorem 3, the nonnegativity of the Green’s matrix \(\Gamma_1(t,s)\) of problem (7.3) follows.

Assume by induction that \(\Gamma_{k-1}(t,s)\geqslant 0\) in the square \(t,s\in[\alpha,\beta]\subset[\alpha,T)\), and consider the problem

\[ L_k[x]\equiv \frac{dx}{dt}+Q_kx=\frac{dx}{dt}+Qx=f(t),\qquad x(\alpha)=x(\beta). \]

We have

\[ L_k[\Gamma_{k-1}(t,s)] = [Q_k-Q_{k-1}]\Gamma_{k-1}(t,s)\leqslant 0\qquad (t\ne s). \]

Since one is less than the modulus of any eigenvalue of the equation

\[ u=\lambda\int_\alpha^\beta [Q_{k-1}-Q_k]\Gamma_{k-1}(t,s)u(s)\,ds+f(t), \]

then, on the basis of the assertion of Theorem 3, we again ascertain the inequality \(\Gamma_k(t,s)=\Gamma(t,s)\geqslant 0\) for \(t,s\in[\alpha,\beta]\subset[\alpha,T)\).

The case \(Q^-\leqslant Q\) is proved analogously.

Remark. From Theorem 4, by virtue of Theorem 3, there follows the positive (negative) invertibility of the matrix \(Q\).

Example. The problem

\[ \frac{dx_1}{dt}+2x_1-x_2=0,\qquad x_1(0)=x_2(1), \]

\[ \frac{dx_2}{dt}-x_1+x_2=0,\qquad x_2(0)=x_2(1) \]

has a nonnegative Green's matrix, since the matrix

\[ Q= \begin{Vmatrix} 2 & -1\\ -1 & 1 \end{Vmatrix} \]

of coefficients satisfies Theorem 4 (for the matrix \(Q^+\), for example, one may take

\[ \begin{Vmatrix} 2 & 0\\ 0 & 1 \end{Vmatrix} \]
).

4°. The results of the preceding subsections can easily be generalized to system (1). Using the idea of the proof of Theorem 4, it can easily be proved

Theorem 5. Let, for the continuous matrix \(A(t)\), there exist such \(d^+\)-ordered (\(d^-\)-ordered) constant matrices \(Q\) and \(Q^+\) (\(Q^-\)) that

\[ Q \leqslant A(t) \leqslant Q^+ \quad (Q \geqslant A(t) \geqslant Q^-). \]

Then: 1) the boundary value problem (1) is uniquely solvable on the interval \([\alpha,T)\) of unique solvability of problem (1.1); 2) the matrix \(\Gamma_A(t,s)\) of problem (1) preserves its sign in the square \(t,s \in [\alpha,\beta]\), and moreover \(\operatorname{sgn}\Gamma_A(t,s)=\operatorname{sgn}\Gamma(t,s)\).

Corollary. If the matrix \(A(t)\), satisfying the conditions of Theorem 5, and the continuous vector-function \(f(t)\) are \(\omega(=\beta-\alpha)\)-periodic, then the system

\[ \frac{dx}{dt}+A(t)x=f(t) \]

has a unique \(\omega\)-periodic solution \(x(t)\). If, moreover, \(A\leqslant Q^+\), \(f\geq 0\), then \(x\geq 0\) \((\ne 0)\).

Example. The system

\[ \frac{dx_1}{dt}+\left(1+\sin^2 t\right)x_1-\frac14 \exp(\sin t)x_2=1, \]

\[ \frac{dx_2}{dt}-x_1+(3+\cos t)x_2=\sin^2 t, \tag{1.4} \]

has a unique \(2\pi\)-periodic solution \(x=\{x_1,x_2\}\geq 0\).

5°. Let us consider estimates of the solution of problem (1). First we prove a comparison theorem for the boundary value problems

\[ \frac{du}{dt}+Q_1u=f(t),\qquad u(\alpha)=u(\beta), \tag{1.5} \]

\[ \frac{dv}{dt}+Q_2v=f(t),\qquad v(\alpha)=v(\beta). \tag{2.5} \]

Lemma. Let \([\alpha,T)\) be an interval of unique solvability of problems (1.5), (2.5), and let \(Q_1(t)\geqslant Q_2(t)\), and let \(f(t)\geq 0\) \((\ne 0)\). If, moreover, the Green's matrices of problems (1.5), (2.5) are simultaneously nonnegative (nonpositive) in the square \(t,s\in[\alpha,\beta]\subset[\alpha,T)\), then the solutions of problems (1.5), (2.5) satisfy the condition \(u(t)\leqslant v(t)\).

Proof. The difference \(\eta=u-v\), according to (1.5), (2.5), satisfies the boundary value problem

\[ \frac{d\eta}{dt}+Q_1\eta=(Q_2(t)-Q_1(t))v,\qquad \eta(\alpha)=\eta(\beta). \]

Hence, by formula (4.1), we have

\[ \eta=u-v=\int_\alpha^\beta \Gamma_1(t,s)(Q_2(s)-Q_1(s))v(s)\,ds\leq 0, \]

where \(\Gamma_1(t,s)\) is the Green's matrix of problem (1.5).

Corollary. If the conditions of the lemma are fulfilled, the Green matrices \(\Gamma_1(t,s)\), \(\Gamma_2(t,s)\) of the problems (1.5), (2.5) satisfy the inequality
\[ \Gamma_2(t,s)\geqslant \Gamma_1(t,s). \]

Passing to the estimate of the solution of problem (1), suppose that for \(A(t)=\|a_{ij}(t)\|_1^n\) one can specify two \(d^+\)-ordered constant matrices \(Q\) and \(Q^+\) such that
\[ Q\leqslant A(t)\leqslant Q^+. \]
Then, in accordance with Theorem 5, the matrix \(\Gamma_A(t,s)\geqslant 0\). Denoting by
\[ Q_1=\left\|\max_t a_{ij}(t)\right\|_1^n,\qquad Q_2=\left\|\min_t a_{ij}(t)\right\|_1^n, \]
we write the obvious inequalities
\[ Q\leqslant Q_2\leqslant A(t)\leqslant Q_1\leqslant Q^+. \]

Let \(z(t)\), \(x(t)\), \(y(t)\) be the solutions of the boundary value problems with coefficient matrices \(Q_2\), \(A(t)\), \(Q_1\), respectively; then, taking the lemma into account, we shall have the inequalities
\[ z(t)\geqslant x(t)\geqslant y(t) \]
or
\[ \int_\alpha^\beta \Gamma_2(t,s)f(s)\,ds \geqslant x(t)\geqslant \int_\alpha^\beta \Gamma_1(t,s)f(s)\,ds, \tag{3.5} \]
\(f(t)\geqslant 0\). Let
\[ f^2=\{\max_t f_1,\ldots,\max_t f_n\},\qquad f^1=\{\min_t f_1,\ldots,\min_t f_n\}; \]
then from inequalities (3.5) we shall have
\[ \left(\int_\alpha^\beta \Gamma_2(t,s)\,ds\right)f^2 \geqslant x(t)\geqslant \left(\int_\alpha^\beta \Gamma_1(t,s)\,ds\right)f^1, \]
or, in accordance with Theorem 2,
\[ Q_2^{-1}f^2\geqslant x(t)\geqslant Q_1^{-1}f^1. \tag{4.5} \]

With respect to the last estimates it should be noted that in the case when \(A(t)\) is a constant matrix, the solution \(x(t)\) of problem (1) can attain both the lower and the upper estimate. In this sense estimate (4.5) cannot be improved.

Let us give estimates of the solution \(x=\{x_1,x_2\}\) of system (1.4). Here
\[ A(t)= \begin{Vmatrix} 1+\sin^2 t-\dfrac{1}{4}\exp\sin t & \\ -1 & 3+\cos t \end{Vmatrix}; \]
\[ Q_1= \begin{Vmatrix} 2 & -\dfrac{1}{4e}\\ -1 & 4 \end{Vmatrix}, \qquad Q_2= \begin{Vmatrix} 1 & -\dfrac{e}{4}\\ -1 & 2 \end{Vmatrix}; \]
\[ f^1=\{1,0\},\qquad f^2=\{1,1\}. \]

Inequalities (4.5) lead to the following estimates for the components of the solution \(x(t)\):
\[ \frac{8+e}{8-e}\geqslant x_1(t)\geqslant \frac{16e}{32e-1}; \qquad \frac{8}{8-e}\geqslant x_2(t)\geqslant \frac{4e}{32e-1}. \]

\(6^\circ\). In conclusion we note that all the results of the present article concerning problem (1) carry over to boundary periodic problems of the form
\[ \frac{d^2x}{dt^2}+A(t)x=f(t),\qquad x(\alpha)=x(\beta),\qquad x'(\alpha)=x'(\beta), \]

\[ \frac{d^3 x}{dt^3}+A(t)x=f(t), \qquad x(\alpha)=x(\beta), \quad x'(\alpha)=x'(\beta), \quad x''(\alpha)=x''(\beta), \]

where \(A(t)\) is a continuous \(n\times n\) matrix for which there exist such \(d^+\)-ordered (\(d^-\)-ordered) matrices \(Q^-\) and \(Q^+\) (\(Q^-\)) that
\[ Q^- \leq A(t) \leq Q^+ \quad (Q^+ \geq A(t) \geq Q^-); \]
\(x\) and \(f\) are \(n\)-dimensional vectors.

The main results of the work were reported at the Izhevsk seminar. The author is grateful to the participants of the seminar for their attention to and interest in boundary value problems with periodic conditions.

References

1. Teterev A. T. Izv. vuzov, Matematika, No. 3, 143–150, 1964.
2. Gantmacher F. R. Theory of Matrices. Tekhizdat, Moscow, 1953.
3. Mikhlin S. G. Lectures on Linear Integral Equations. Fizmatgiz, 1959.

Received by the editors
November 12, 1965

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