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UDC 517.914.2
ON STABILITY WITH RESPECT TO THE FIRST APPROXIMATION
N. A. IZOBOV
We consider \(n\)-dimensional systems
\[ \frac{dx}{dt}=P(t)x, \tag{1} \]
\[ \frac{dy}{dt}=P(t)y+f(t,y) \tag{2} \]
with a matrix \(P(t)\) bounded and continuous for \(t \geqslant 0\), such that the characteristic exponents \(\lambda_i\) of system (1) are negative:
\[ \lambda_1 \leqslant \ldots \leqslant \lambda_n, \tag{3} \]
and the vector function \(f(t,y)\) satisfies the condition
\[ \|f(t,y)\| \leqslant g(t)\|y\|. \tag{4} \]
In [1], criteria are given for the asymptotic stability of the trivial solution of system (2)
\[ \left(\int_0^\infty g(t)\exp[(\sigma_\Gamma+\alpha)t]\,dt<+\infty,\ \alpha>0,\ \sigma_\Gamma \text{ is the Grobman irregularity coefficient of system (1)}\right) \]
for arbitrary \(f(t,y)\) satisfying (4).
In the present note, for systems (1) we introduce the notion of a quantity \(\gamma\), smaller than \(\sigma_\Gamma\) for a certain class of systems (and coinciding with \(\sigma_\Gamma\) for the remaining systems), but possessing a property analogous to the property of \(\sigma_\Gamma\) indicated above, and we establish one property of such systems.
\(1^\circ\). Let \(X(t)\) be a binormal matrix of solutions [2] of system (1); \(\lambda_i\) the characteristic exponent of the \(i\)-th column-solution of the matrix \(X(t)\); \(\delta_j\) the characteristic exponent of the \(j\)-th row of the matrix \(X^{-1}(t)\), \(\sigma_p \equiv \lambda_p+\delta_p\), \(p=1,\ldots,n\). In what follows we consider systems (1) for which
\[ \sigma_k=\max_p\{\sigma_p\}>\max_{p\ne k}\{\sigma_p\}=\sigma_l. \tag{5} \]
We note that \(\sigma_\Gamma=\sigma_k\) [2].
Let
\[ L_{ji}(U)\equiv \begin{cases} \displaystyle \delta_j+\lambda_i+\sum_{p=j}^{i-1}u_p, & j<i,\\[1.2em] \displaystyle \delta_j+\lambda_i-\sum_{p=i}^{j-1}u_p, & j>i. \end{cases} \]
where the point \(U(u_1,\ldots,u_{n-1})\) is a point of the Euclidean space \(E_{n-1}(u_n=0)\). By definition,
\[ \gamma(U)=\max_{j,i}\{L_{ji}(U)\}\qquad (j,i=1,\ldots,n) \]
(the maximum is taken over all pairs \((j,i)\) for which the corresponding \(L_{ji}(U)\) are defined, i.e., for which \(j\ne i\)). Let us specify a bounded domain \(G\subset E_{n-1}\). The point \(U(u_1,\ldots,u_{n-1})\in G\) if
\[ \lambda_j \leq \sum_{p=j}^{i-1} u_p \leq -\lambda_i,\qquad j<i. \tag{6} \]
Obviously, \(G\) is convex; \(G\) is nonempty, since the point \(U_\Gamma(\lambda_1-\lambda_2,\ldots,\lambda_{n-1}-\lambda_n)\in G\), and moreover is an interior point of it (not lying on the boundary). By definition,
\[ \gamma=\inf_{U\in G}\gamma(U). \]
Lemma.
\[ \gamma=\max\left\{\frac12(\sigma_k+\sigma_l),\ \sigma_k+\max_{p\ne k}\{\lambda_p\}\right\}. \]
Proof. We agree that
\[ \sum_{p=s}^{r}u_p=0,\qquad \text{if } r<s. \]
Compute \(\gamma(U_0)\):
\[ U_0(u_1^0,\ldots,u_{n-1}^0), \]
\[ u_i^0=\lambda_i-\lambda_{i+1},\qquad 1\leq i\leq k-2,\quad k+1\leq i\leq n-1, \tag{7} \]
\[ u_{k-1}^0=\lambda_{k-1}-\lambda_l+\frac12(\delta_k-\delta_l+\lambda_l-\lambda_k), \tag{8} \]
\[ u_k^0=\lambda_l-\lambda_{k+1}-\frac12(\delta_k-\delta_l+\lambda_l-\lambda_k). \tag{9} \]
Remark. If \(k=1\), then the coordinates of the point \(U_0\) are computed by formulas (7), (9), while if \(k=n\), by formulas (7), (8).
First compute \(L_{ki}(U_0)\); for this consider the case \(i>k\) (the case \(i<k\) is analogous). Since, according to (7),
\[ \sum_{p=k+1}^{i-1}u_p^0=\lambda_{k+1}-\lambda_i, \]
we have
\[ L_{ki}(U_0)=\delta_k+\lambda_i+u_k^0+\sum_{p=k+1}^{i-1}u_p^0 =\frac12(\sigma_k+\sigma_l),\qquad i>k. \]
In the case \(i<k\),
\[ L_{ki}(U_0)=\delta_k+\lambda_i-\sum_{p=i}^{k-2}u_p^0-u_{k-1}^0, \]
and therefore the relation
\[ L_{ki}(U_0)=\frac12(\sigma_k+\sigma_l),\qquad i\ne k, \]
is always valid.
Now estimate \(L_{jk}(U_0)\), \(j\ne k\):
\[ L_{jk}(U_0)=\delta_j+\lambda_k+\sum_{p=j}^{k-2}u_p^0-u_{k-1}^0 =\frac12(\sigma_j+\sigma_k)+\frac12(\sigma_j-\sigma_l),\qquad j<k, \]
\[ L_{jk}(U_0)=\delta_j+\lambda_k-u_k^0-\sum_{p=k+1}^{j-1}u_p^0 =\frac12(\sigma_j+\sigma_k)+\frac12(\sigma_j-\sigma_l),\qquad j>k. \]
Therefore, for any admissible \(j\),
\[
L_{jk}(U_0)\leq \frac12(\sigma_k+\sigma_l).
\]
Obviously, it now suffices to estimate only those \(L_{ji}(U_0)\) for which \(j\ne k,\ i\ne k\). Compute
\[
\sum_{p=j}^{i-1}u_p^0
\]
(and similarly \(\sum_{p=i}^{j-1}u_p^0\)). Since \(j\ne k,\ i\ne k\), the sum \(\sum_{p=j}^{i-1}u_p^0\) either contains none of the terms \(u_{k-1}^0,\ u_k^0\), or contains the entire sum \(u_{k-1}^0+u_k^0\). Hence
\[
\sum_{p=j}^{i-1}u_p^0=\lambda_j-\lambda_i,\qquad
\sum_{p=i}^{j-1}u_p^0=\lambda_i-\lambda_j,
\]
and then
\[
L_{ji}(U_0)=\delta_j+\lambda_i+\lambda_j-\lambda_i=\sigma_j,\qquad j\ne k,\ i\ne k.
\]
Finally,
\[
\gamma(U_0)=\frac12(\sigma_k+\sigma_l).
\]
Let us show that \(\gamma=\gamma(U_0)\), if \(U_0\in G\). Suppose the contrary: there is a point
\[
U'(u_1',\ldots,u_{n-1}')\in G
\]
for which \(\gamma(U')<\gamma(U_0)\).
Let \(l<k\) (the case \(k<l\) is analogous). Obviously,
\[
\sum_{p=l}^{k-1}u_p'\ne \frac12(\delta_k-\delta_l+\lambda_l-\lambda_k),
\]
since otherwise we would have
\[
\gamma(U')\geq L_{lk}(U')=\frac12(\sigma_k+\sigma_l).
\]
Denote
\[
\sum_{p=l}^{k-1}u_p'-\frac12(\delta_k-\delta_l+\lambda_l-\lambda_k)\equiv \alpha.
\]
Then
\[
\gamma(U')\geq \max\{L_{kl}(U'),\,L_{lk}(U')\}
=\max\{\gamma(U_0)-\alpha,\,\gamma(U_0)+\alpha\}.
\]
We have obtained a contradiction. Finally,
\[
\gamma=\frac12(\sigma_k+\sigma_l),\qquad U_0\in G.
\]
Let us now consider the case when \(U\in {}_0G\). Let the coordinates of the point
\[
U_1(u_1^{(1)},\ldots,u_{n-1}^{(1)})
\]
be determined as follows:
\[
u_i^{(1)}=\lambda_i-\lambda_{i+1},\qquad 1\leq i\leq k-2,\quad k+1\leq i\leq n-1, \tag{7'}
\]
\[
u_{k-1}^{(1)}=\lambda_{k-1}-\lambda_k-\max_{p\ne k}\{\lambda_p\}, \tag{8'}
\]
\[
u_k^{(1)}=\lambda_k-\lambda_{k+1}+\max_{p\ne k}\{\lambda_p\}. \tag{9'}
\]
Here the preceding remark should be taken into account. Let us verify that \(U_1\in G\).
First of all, \(\lambda_i \leqslant u_i^{(1)} \leqslant -\lambda_{i+1}\); this follows from the negativity of \(\lambda_i\) and from the inequalities
\[ -\lambda_{k+1}+\max_{p\ne k}\{\lambda_p\} \geqslant 0,\quad k<n, \]
\[ \lambda_{k-1}-\max_{p\ne k}\{\lambda_p\}\leqslant 0,\quad k>1. \]
Therefore it now suffices to consider all possible pairs \(\{j,i\}\) for which \(j<i-1\). Obviously, for \(\{j,k\}\), \(\{k,i\}\) we have, respectively,
\[ \sum_{p=j}^{k-1} u_p^{(1)} =\lambda_j-\lambda_k-\max_{p\ne k}\{\lambda_p\}, \tag{10} \]
\[ \sum_{p=k}^{i-1} u_p^{(1)} =\lambda_k-\lambda_i+\max_{p\ne k}\{\lambda_p\} \tag{11} \]
(the relations (10) and (11) are also valid for \(j=k-1\), \(i=k+1\), respectively). But
\(\lambda_j-\max_{p\ne k}\{\lambda_p\}\leqslant 0\) for \(j<k\),
\(\max_{p\ne k}\{\lambda_p\}-\lambda_i\geqslant 0\) for \(i>k\). Therefore (10) and (11) satisfy conditions (6). For all remaining \(\{j,i\}\), \(j<i-1\),
\[ \sum_{p=j}^{i-1}u_p^{(1)}=\lambda_j-\lambda_i, \]
and the fulfillment of (6) is obvious.
Thus, \(U_1\in G\). Let us show that \(U_1\) is a boundary point of \(G\). Let \(\lambda_s=\max_{p\ne k}\{\lambda_p\}\). If \(s<k\), then from (10) we have
\[ \sum_{p=s}^{k-1}u_p^{(1)} =\lambda_s-\lambda_k-\max_{p\ne k}\{\lambda_p\} =-\lambda_k, \tag{12} \]
whereas if \(s>k\), then from (11)
\[ \sum_{p=k}^{s-1}u_p^{(1)} =\lambda_k-\lambda_s+\max_{p\ne k}\{\lambda_p\} =\lambda_k. \tag{13} \]
Comparing (12) and (13) with (6), we obtain what is required.
Let us compute \(\gamma(U_1)\). It is easy to see that
\[ L_{ki}(U_1)=\sigma_k+\max_{p\ne k}\{\lambda_p\}. \]
Let \(\Pi\) be the set of points of the line passing through the points \(U_\Gamma\) and \(U_0\). It is not hard to verify that the point \(U_1\in\Pi\). Obviously, \(u_n=L_{ji}(U)\), \(U\in\Pi\), is the equation of a line in the \(n\)-dimensional space \(E_n\), and since
\[ L_{ki}(U_\Gamma)>L_{ki}(U_0),\quad L_{ki}(U_\Gamma)>L_{ki}(U_1), \]
and the point \(U_1\in G\), the point \(U_1\) lies between the points \(U_\Gamma\) and \(U_0\) on the line \(\Pi\) (otherwise \(L_{ki}(U_1)>L_{ki}(U_\Gamma)\)) and divides the “segment” \(U_\Gamma U_0\) in some positive ratio \(\tau\). Using the inequalities obtained earlier and the obvious inequalities
\[ L_{ji}(U_0)\leqslant \frac12(\sigma_k+\sigma_l),\quad L_{ji}(U_\Gamma)\leqslant \sigma_i,\quad j\ne k, \]
we have
\[ L_{ji}(U_1)=\frac{L_{ji}(U_\Gamma)+\tau L_{ji}(U_0)}{1+\tau} \leq \frac{\frac{1}{2}(\sigma_k+\sigma_l)+\tau\cdot\frac{1}{2}(\sigma_k+\sigma_l)}{1+\tau} = \frac{1}{2}(\sigma_k+\sigma_l). \tag{14} \]
Since \(U_1\) is an interior point of the segment \(U_\Gamma U_0\), from \(L_{ki}(U_\Gamma)>L_{ki}(U_0)\) it follows that
\[ L_{ki}(U_1)>L_{ki}(U_0) \tag{15} \]
(recall that the point \(\overline U_1\bigl(u_1^{(1)},\ldots,u_{n-1}^{(1)},L_{ki}(U_1)\bigr)\) lies on the line passing through the points \(\overline U_\Gamma\bigl(\lambda_1-\lambda_2,\ldots,\lambda_{n-1}-\lambda_n,L_{ki}(U_\Gamma)\bigr)\) and \(\overline U_0\bigl(u_1^0,\ldots,u_{n-1}^0,L_{ki}(U_0)\bigr)\)), and hence, taking (14) into account,
\[ L_{ji}(U_1)<\sigma_k+\max_{p\ne k}\{\lambda_p\},\qquad j\ne k. \]
Thus, \(\gamma(U_1)=\sigma_k+\max_{p\ne k}\{\lambda_p\}\). By virtue of (12) or (13) (the point \(U_1\) is a boundary point) and (6), \(\gamma=\gamma(U_1)\), for for any point \(U\in G\)
\[ L_{ks}(U)=\gamma(U_1)+\varepsilon(U)\geq \gamma(U_1), \]
since
\[ \varepsilon(U)= \begin{cases} \displaystyle \sum_{p=k}^{s-1} u_p-\lambda_k, & k<s,\\[1.2em] \displaystyle -\sum_{p=s}^{k-1} u_p-\lambda_k, & k>s, \end{cases} \]
is a nonnegative quantity.
Finally,
\[ \gamma= \begin{cases} \displaystyle \frac{1}{2}(\sigma_k+\sigma_l), & \text{if } U_0\in G,\\[1.2em] \displaystyle \sigma_k+\max_{p\ne k}\{\lambda_p\}, & \text{if } U_0\notin G. \end{cases} \tag{*} \]
If \(U_0\notin G\), then from inequality (15) it follows that
\[ \sigma_k+\max_{p\ne k}\{\lambda_p\}>\frac{1}{2}(\sigma_k+\sigma_l). \]
Therefore, to prove the lemma it is sufficient to show that in the case \(U_0\in G\) the inequality
\[ \frac{1}{2}(\sigma_k+\sigma_l)\geq \sigma_k+\max_{p\ne k}\{\lambda_p\} \tag{16} \]
holds. From the inequalities
\[ L_{ki}(U_\Gamma)>L_{ki}(U_1),\qquad L_{ki}(U_\Gamma)>L_{ki}(U_0) \]
and from the fact that the point \(U_1\) is a boundary point of the domain \(G\), it follows that the point \(U_0\) is a point of the segment \(U_\Gamma U_1\). Hence, as above, \(L_{ki}(U_0)\geq L_{ki}(U_1)\), i.e. (16) holds. On the basis of (*) we have the assertion of the lemma.
Corollary 1. The quantity \(\gamma\) does not depend on the choice of the binormal matrix \(X(t)\).
The proof follows from Theorem 2 of [2] and the expression for \(\gamma\) established by the lemma.
Corollary 2. \(\sigma_l < \gamma < \sigma_\Gamma\).
Corollary 3. If system (1) is of second order, then \(\gamma\) does not exceed the Perron number \(\sigma_\Pi\) [2].
Let us note that in the case \(\sigma_l=\sigma_k\), \(\gamma\) coincides with Grobman’s irregularity coefficient.
\(2^\circ\). Together with systems (1), (2) we consider the system
\[ \frac{dy}{dt}=[P(t)+Q(t)]y . \tag{2'} \]
Denote
\[ \overline{\omega}[r]\equiv \overline{\lim}_{t\to+\infty}\frac{1}{t}\ln\|r(t)\|, \]
where \(r(t)\) is a scalar, a vector, or a matrix. The following is true.
Theorem 1. If \(\overline{\omega}[Q]<-\gamma\), then the characteristic exponents of system \((2')\) are negative.
Proof. Denote
\[ H(t)=((h_{ji}(t)))\equiv X^{-1}(t)Q(t)X(t), \]
where \(j\) is the row number; \(\overline{\omega}[Q]+\gamma=-\varepsilon,\ \varepsilon>0\). Obviously,
\[ \overline{\omega}[h_{ji}]\leq \delta_j+\lambda_i-\gamma-\varepsilon \quad (j,i=1,\ldots,n). \tag{17} \]
We shall establish for \(\overline{\omega}[h_{kk}]\) a sharper estimate; for this purpose compute \(h_{kk}\):
\[ h_{kk}=\sum_{s=1}^{n}\left(\sum_{i=1}^{n}x'_{ki}q_{is}\right)x_{sk} =\sum_{s=1}^{n}\left(\sum_{i=1}^{n}x'_{ki}x_{sk}q_{is}\right), \tag{18} \]
where \(X=((x_{ji}))\), \(X^{-1}=((x'_{ji}))\), \(Q=((q_{ji}))\), and \(j\) is the row number. The estimate for \(\overline{\omega}[h_{kk}]\) needed by us will be obtained if an estimate of the quantity \(\overline{\omega}[x'_{ki}x_{sk}]\) is given in a suitable way. Multiplying the \(s\)-th row of the matrix \(X(t)\) by the \(i\)-th column of its inverse matrix \(X^{-1}(t)\), we have
\[ \sum_{p=1}^{n}x_{sp}x'_{pi}=\delta_{si}, \]
where \(\delta_{si}\) is the Kronecker symbol; whence
\[ x_{sk}x'_{ki}=\delta_{si}-\sum_{p\ne k}x_{sp}x'_{pi}. \tag{19} \]
But
\[ \overline{\omega}[x_{sp}x'_{pi}]\leq \lambda_p+\delta_p=\sigma_p \]
for arbitrary \(s\) and \(i\). Therefore from (19) we have \((\sigma_p\geq 0)\)
\[ \overline{\omega}[x_{sk}x'_{ki}]\leq \max_{p\ne k}\{\sigma_p\}=\sigma_l, \]
and from (18), finally,
\[ \overline{\omega}[h_{kk}]\leq \sigma_l-\gamma-\varepsilon. \tag{20} \]
By the definition of \(\gamma\), there is a point \(U_m\in G\) such that \(\gamma=\gamma(U_m)\). In view of the convexity of the domain \(G\) and the existence of its interior point \(U_\Gamma\), in any neighborhood of the point \(U_m\) there are interior points of the domain \(G\). The function \(\gamma(U)\) is continuous; therefore, for the chosen \(\varepsilon\) there is an interior point \(U_\varepsilon(u_1^\varepsilon,\ldots,u_{n-1}^\varepsilon)\) of the domain \(G\) such that
\[ |\gamma(U_\varepsilon)-\gamma|<\frac{\varepsilon}{2}. \tag{21} \]
(if \(U_m\) is an interior point, i.e., \(U_m\) coincides with \(U_0\), then the point \(U_0\) may be taken as \(U_\varepsilon\)).
In equation \((2')\), written in matrix form, we make the change of variables
\[ Y(t)=X(t) \begin{pmatrix} e^{-\alpha_1 t} & 0\\ \cdot & \\ & \cdot & \\ 0 & & e^{-\alpha_n t} \end{pmatrix} Z(t), \tag{22} \]
where \(Z(t)\) is the fundamental matrix of the system
\[ \frac{dZ}{dt} = \begin{pmatrix} \alpha_1 & 0\\ \cdot & \\ & \cdot & \\ 0 & & \alpha_n \end{pmatrix} Z+B(t)Z, \tag{23} \]
\[ B(t)\equiv \begin{pmatrix} e^{\alpha_1 t} & 0\\ \cdot & \\ & \cdot & \\ 0 & & e^{\alpha_n t} \end{pmatrix} X^{-1}(t)Q(t)X(t) \begin{pmatrix} e^{-\alpha_1 t} & 0\\ \cdot & \\ & \cdot & \\ 0 & & e^{-\alpha_n t} \end{pmatrix}, \]
and \(\alpha_i\), \((i=1,\ldots,n)\), are determined from the following linear system of equations:
\[ \alpha_n=0, \]
\[ \alpha_i-\alpha_{i+1}=u_i^\varepsilon \quad (i=1,\ldots,n-1). \]
Let us now estimate \(\bar\omega[B]\). From the definition of \(B(t)\) it follows that
\[ b_{ji}(t)=h_{ji}(t)\exp[(\alpha_j-\alpha_i)t]. \]
Hence
\[ \bar\omega[b_{ji}]\leqslant (\delta_j+\lambda_i+\alpha_j-\alpha_i)-\gamma-\varepsilon, \qquad j\ne i. \]
Let \(j<i\); then
\[ \alpha_j-\alpha_i = \sum_{p=j}^{i-1}(\alpha_p-\alpha_{p+1}) = \sum_{p=j}^{i-1}u_p^\varepsilon, \]
and therefore
\[ \delta_j+\lambda_i+(\alpha_j-\alpha_i)=L_{ji}(U_\varepsilon). \]
An analogous result is obtained also in the case \(j>i\). Thus,
\[ \bar\omega[b_{ji}]\leqslant L_{ji}(U_\varepsilon)-\gamma-\varepsilon, \qquad j\ne i, \]
i.e., on the basis of (21) and the definition of \(\gamma(U)\), we have
\[ \bar\omega[b_{ji}] \leqslant \gamma(U_\varepsilon)-\gamma-\varepsilon \leqslant |\gamma(U_\varepsilon)-\gamma|-\varepsilon < -\frac{\varepsilon}{2}, \qquad j\ne i. \]
According to Corollary 2 of the lemma and estimate (20),
\[ \bar\omega[b_{ii}]\leqslant \sigma_1-\gamma-\varepsilon \quad (i=1,\ldots,n). \]
Thus, always \(\bar\omega[B]<0\). Therefore, according to [3], the totality of characteristic exponents of system (23) is the set \(\{\alpha_1,\ldots,\alpha_n\}\).
Let \(Z(t)=((Z_{ji}(t)))\) be a normal matrix of solutions of system (23), such that the characteristic exponent of the \(i\)-th solution-column is equal to \(\alpha_i\). Denote
\[ S(t) \equiv \begin{pmatrix} e^{-\alpha_1 t} & 0\\ \cdot & \\ & \cdot & \\ 0 & & e^{-\alpha_n t} \end{pmatrix} Z(t). \]
Then \(s_{ji}(t)=z_{ji}(t)e^{-\alpha_j t}\), and therefore \(\bar\omega[s_{ji}]\leqslant \alpha_i-\alpha_j\). From our substitution
\[ y_{ji}(t)=\sum_{p=1}^{n} x_{jp}s_{pi}, \qquad Y(t)=\bigl((y_{ji}(t))\bigr) \]
it follows, taking into account the preceding estimate, that
\[ \bar\omega[x_{jp}s_{pi}]\leqslant \lambda_p+\alpha_i-\alpha_p. \]
If \(i<p\), then
\[ \alpha_i-\alpha_p=\sum_{r=i}^{p-1}(\alpha_r-\alpha_{r+1}) =\sum_{r=i}^{p-1}u_r^\varepsilon, \]
and therefore
\[ \bar\omega[x_{jp}s_{pi}]\leqslant \lambda_p+\sum_{r=i}^{p-1}u_r^\varepsilon . \]
But the point \(U_\varepsilon\) is an interior point of the domain \(G\); therefore from (6) we have
\[ \beta_{ip}\equiv \lambda_p+\sum_{r=i}^{p-1}u_r^\varepsilon<0,\qquad i<p. \]
If, however, \(i>p\), then
\[ \lambda_p+\alpha_i-\alpha_p =\lambda_p-(\alpha_p-\alpha_i) =\lambda_p-\sum_{r=p}^{i-1}u_r^\varepsilon . \]
And again from (6) (the point \(U_\varepsilon\) is interior) we have
\[ \beta_{ip}\equiv \lambda_p-\sum_{r=p}^{i-1}u_r^\varepsilon<0,\qquad i>p. \]
In the case \(p=i\), \(\bar\omega[x_{ji}s_{ii}]\leqslant \lambda_i\). Denote
\[ \beta_i=\max\{\max_{p\ne i}\{\beta_{ip}\},\lambda_i\}. \]
Then \(\lambda_i'\leqslant\beta_i\), where \(\lambda_i'\) is the characteristic exponent of the \(i\)-th column-solution of the fundamental matrix of solutions \(Y(t)\). Obviously, \(\beta_i<0\) \((i=1,\ldots,n)\). The theorem is proved.
Let
\[ \beta(\varepsilon)=\max_i\{\beta_i\}, \]
and let \(y(t)\) be an arbitrary solution of system (2′).
Corollary 1. For any \(Q(t)\), \(\bar\omega[Q]+\gamma=-\varepsilon\), \(\varepsilon>0\), the inequality \(\bar\omega[y]\leqslant\beta(\varepsilon)<0\) holds.
Corollary 2. If there exists an \(\varepsilon>0\) such that
\[ \int_{0}^{\infty}\|Q(t)\|e^{(\gamma+\varepsilon)t}\,dt<+\infty, \tag{24} \]
then \(\bar\omega[y]\leqslant\beta(\varepsilon)\) for all \(Q(t)\) satisfying (24).
The proof follows from [3].
Theorem 2. If there exists a positive \(\varepsilon\) such that
\[ \int_{0}^{\infty} g(t)e^{(\gamma+\varepsilon)t}\,dt<+\infty, \]
then the trivial solution of system (2) is asymptotically stable.
Proof. According to the Bylov–Grobman principle of linear inclusion [4], every nontrivial solution \(y(t)\) of system (2) is a solution of a linear system
\[ \frac{dv}{dt}=P(t)v+Q_y(t)v \]
of the form \((2')\), where the matrix \(Q_y(t)\), depending on \(y\), has the property
\[ \|Q_y(t)\|\leq g(t) \tag{25} \]
for all \(y\). Then, according to the substitution (22),
\[ y(t)=X(t) \begin{pmatrix} e^{-a_1t} & 0\\ \cdot & \cdot\\ 0 & e^{-a_nt} \end{pmatrix} z_y(t), \tag{22'} \]
where \(z_y(t)\) is a solution of system (23) with the corresponding matrix \(B_y(t)\). By the well-known Gronwall lemma,
\[ \|z_y(t)\|\leq \|z_y(0)\| \exp\left\{\int_0^t \|B_y(\tau)\|\,d\tau\right\} e^{\left[\max_p\{a_p\}\right]t},\qquad t\geq 0. \]
Let
\[ a_s=\max_p\{a_p\}. \]
From the definition of the matrix \(B_y(t)\), taking (21) into account, it follows that, for a given \(\varepsilon\), there is a positive constant \(b=b(\varepsilon)\), depending only on \(\varepsilon\), such that for every \(y(t)\), by virtue of (25),
\[ \|B_y(t)\|\leq b g(t)e^{(\gamma+\varepsilon)t}. \]
Therefore
\[ \|z_y(t)\|\leq N\|z_y(0)\|e^{a_s t},\qquad N=\exp\left[b\int_0^\infty g(t)e^{(\gamma+\varepsilon)t}\,dt\right]. \]
From the substitution \((22')\), for any \(\varepsilon_1>0\) there is an \(M(\varepsilon_1)>0\) such that
\[ \|y(t)\|\leq M\|z_y(t)\| \exp\left[\left(\max_p\{\lambda_p-a_p\}+\varepsilon_1\right)t\right]. \]
But
\[ \max_p\{\lambda_p-a_p\}+a_s\leq \beta(\varepsilon),\qquad \|z_y(0)\|\leq \|X^{-1}(0)\|\,\|y(0)\|. \]
Finally,
\[ \|y(t)\|\leq MN\|X^{-1}(0)\|\,\|y(0)\|e^{[\beta(\varepsilon)+\varepsilon_1]t}. \]
From this estimate, choosing \(\varepsilon_1\) so that \(\beta(\varepsilon)+\varepsilon_1<0\), the validity of the theorem follows.
On the basis of Corollary 3 of the lemma, in the case \(n=2\) the following is true.
Corollary. If \(\bar\omega[Q]<-\sigma_{\Pi}\), then for every solution \(y(t)\) of system \((2')\), \(\omega[y]<0\).
An even stronger result is true, namely the following.
Remark. Under the conditions of the corollary to Theorem 2,
\[ \bar\omega[y]\leq \max_p\{\lambda_p\}. \]
Proof. Obviously, consideration is required only in the case
\[ \sigma_{\Pi}<\sigma_{\Gamma}. \]
But in this case
\[ \sigma_{\Pi}=\max_p\{\lambda_p+\delta_{3-p}\}\qquad (p=1,2), \]
and the validity of the remark is obvious if one makes the substitution
\[ y(t)=X(t)z(t) \]
in equation \((2')\). Using estimate (20), we see that \(z(t)\)—
solution of a system having a linear asymptotic equilibrium ([5], p. 74).
Let us note, in connection with the last remark, that by changing the domain \(G\) in a suitable way, one can obtain certain modifications of the quantity \(\gamma\) possessing definite properties.
\(3^\circ\). In conclusion, let us clarify one property of the quantity \(\gamma_0=\dfrac{1}{2}(\sigma_k+\sigma_l)\).
The following assertion is valid.
If \(\bar\omega[Q]<-\gamma_0\), then the set of characteristic exponents \(\{\lambda'_1,\ldots,\lambda'_n\}\) of system \((2')\), written in a certain order, satisfies the condition
\[ \lambda'_i \leq \lambda_i+\frac{1}{2}(\sigma_k-\sigma_l). \]
Proof. In equation \((2')\) we make the substitution (22), where the \(\alpha_i\) are now defined as follows:
\[ \alpha_i=\frac{1}{2}(\delta_n-\delta_i+\lambda_i-\lambda_n)\quad (i=1,\ldots,n). \]
On the basis of (17), (20), and the definition of \(\alpha_i\), we have \(\bar\omega[B]<0\). Therefore the set of characteristic exponents of system (22), as before, is \(\{\alpha_1,\ldots,\alpha_n\}\). Using the estimate for \(s_{pi}\) proved in Theorem 1, the inequality
\[
\bar\omega[x_{jp}s_{pi}]\leq \lambda_p+\alpha_i-\alpha_p
=\lambda_i+\frac{1}{2}(\sigma_p-\sigma_i)
\]
is valid for every \(j\) and fixed \(i\). Hence from the substitution \(y_{ji}(t)=\sum_{p=1}^{n} x_{jp}s_{pi}\) it follows that
\[ \bar\omega[y_{ji}]\leq \lambda_i+\frac{1}{2}(\sigma_k-\sigma_i), \]
i.e., there exists a fundamental matrix of solutions \(Y(t)\) such that the characteristic exponent of its \(i\)-th column does not exceed the quantity
\[
\lambda_i+\frac{1}{2}(\sigma_k-\sigma_i).
\]
This implies the validity of the assertion.
Corollary. If \(\bar\omega[Q]<-\gamma_0\), then system \((2')\) has at least one nontrivial solution whose characteristic exponent does not exceed \(\lambda_k\).
Everything stated above can be reformulated in the corresponding way also for system (2).
References
- D. M. Grobman, Mat. sb., 30 (72), 1, 1952, pp. 121–166.
- R. E. Vinograd, DAN SSSR, 119, No. 3, 1958.
- D. M. Grobman, DAN SSSR, 86, No. 1, 19–22, 1952.
- B. F. Bylov, D. M. Grobman, UMN, 17, issue 3 (105), 1962.
- L. Cesari, Asymptotic Behavior and Stability of Solutions of Ordinary Differential Equations. Mir Publishers, Moscow, 1964.
Received by the editors
November 17, 1965
Belorussian State University
named after V. I. Lenin