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UDC 517.941.1
ON BOUNDEDNESS OF SOLUTIONS OF A LINEAR SYSTEM OF DIFFERENTIAL EQUATIONS WITH VARIABLE COEFFICIENTS
V. V. KOROLEV
In this note some sufficient criteria are established for the boundedness of solutions of the system
\[ \dot{x}_i=\sum_{k=1}^{n} a_{ik}(t)x_k \quad (i=1,\ldots,n). \tag{1} \]
These criteria are not connected with the estimation of characteristic numbers, and conclusions on the boundedness of solutions are derived from direct consideration of the coefficients \(a_{ik}(t)\). The coefficients \(a_{ik}(t)\) are assumed to be real continuous functions on the half-axis \([t_0,+\infty)\).
Theorem 1. If the following conditions are satisfied:
\[ \text{a) }\quad \int_{t_0}^{+\infty} a_{ii}^{+}(t)\,dt<\infty, \quad \text{where} \quad a_{ii}^{+}(t)=\frac{a_{ii}(t)+|a_{ii}(t)|}{2}; \]
\[ \text{b) }\quad \sum_{\substack{k=1\\(k\ne i)}}^{n}|a_{ik}(t)|\le |a_{ii}(t)|+\varepsilon_i(t), \quad \varepsilon_i(t)\ge 0,\quad \int_{t_0}^{+\infty}\varepsilon_i(t)\,dt<\infty \quad (i=1,\ldots,n), \]
then all solutions of system (1) are bounded on the half-axis \([t_0,+\infty)\).
Proof. Let \(x(t)=\{x_1(t),\ldots,x_n(t)\}\) be an arbitrary (nonzero) solution of system (1).
Define the norm \(\|x(t)\|\) of the solution \(x(t)\) by the relation
\[ \|x(t)\|=\max_{1\le j\le n}\{|x_j(t)|\}. \]
Suppose that on some interval \([t',t'']\) the equality \(\|x(t)\|=|x_i(t)|\) holds. From (1) we find
\[ \frac{dx_i}{x_i} = \left( a_{i1}(t)\frac{x_1}{x_i} +\cdots +a_{ii}(t) +\cdots +a_{in}(t)\frac{x_n}{x_i} \right)dt \]
and
\[ \frac{dx_i}{x_i} \le \left(|a_{i1}(t)|+\cdots+a_{ii}(t)+\cdots+|a_{in}(t)|\right)dt. \]
Integrating over the interval \([t',t'']\), we obtain
\[ \|x(t'')\|\le \|x(t')\| \exp\left\{ \int_{t'}^{t''} \left[ a_{ii}(t)+ \sum_{\substack{k=1\\(k\ne i)}}^{n}|a_{ik}(t)| \right]dt \right\}. \tag{2} \]
Divide the interval \([t',t'']\) into two sets: \(E^{(1)}\{t:\ a_{ii}(t)<0\}\) and \(E^{(2)}\{t:\ a_{ii}(t)\geqslant 0\}\), and represent the integral on the right-hand side of (2) in the form
\[ \int_{t'}^{t''}\left[a_{ii}(t)+\sum_{\substack{k=1\\(k\ne i)}}^{n}|a_{ik}(t)|\right]dt = \int_{E^{(1)}}\left[a_{ii}(t)+\sum_{\substack{k=1\\(k\ne i)}}^{n}|a_{ik}(t)|\right]dt + \]
\[ + \int_{E^{(2)}}\left[a_{ii}(t)+\sum_{\substack{k=1\\(k\ne i)}}^{n}|a_{ik}(t)|\right]dt . \]
By virtue of b) we have
\[ \int_{E^{(1)}}\left[a_{ii}(t)+\sum_{\substack{k=1\\(k\ne i)}}^{n}|a_{ik}(t)|\right]dt \leqslant \int_{E^{(1)}}\varepsilon_i(t)\,dt, \]
\[ \int_{E^{(2)}}\left[a_{ii}(t)+\sum_{\substack{k=1\\(k\ne i)}}^{n}|a_{ik}(t)|\right]dt \leqslant \int_{E^{(2)}}\left[2a_{ii}(t)+\varepsilon_i(t)\right]dt = \]
\[ = \int_{t'}^{t''}2a_{ii}^{+}(t)\,dt+\int_{E^{(2)}}\varepsilon_i(t)\,dt . \]
Then from (2) we find
\[ \|x(t'')\|\leqslant \|x(t')\|\exp\left\{\int_{t'}^{t''}\left[2a_{ii}^{+}(t)+\varepsilon_i(t)\right]dt\right\} \leqslant \|x(t')\|\exp\left\{\int_{t'}^{t''}a(t)\,dt\right\}, \]
where
\[ a(t)=\sum_{i=1}^{n}\left[2a_{ii}^{+}(t)+\varepsilon_i(t)\right]. \]
It is clear that
\[ \int_{t_0}^{+\infty}a(t)\,dt<\infty . \]
Thus, it has been established that
\[ \|x(t'')\|\leqslant \|x(t')\|\exp\left\{\int_{t'}^{t''}a(t)\,dt\right\}. \tag{3} \]
Decompose the interval \([t_0,+\infty)\) into subintervals \([t_0,t_1]\), \([t_1,t_2]\), ..., \([t_m,t_{m+1}]\), ... so that on each of these intervals the equality \(\|x(t)\|=|x_j(t)|\) holds for some \(j\) \((j=1,\ldots,n)\), fixed for the entire subinterval.
On the basis of (3) we have the system of inequalities:
\[ \|x(t_1)\|\leqslant \|x(t_0)\|\exp\left\{\int_{t_0}^{t_1}a(t)\,dt\right\}, \]
\[ \|x(t_2)\|\leqslant \|x(t_1)\|\exp\left\{\int_{t_1}^{t_2}a(t)\,dt\right\}, \]
\[ \cdot\quad \cdot\quad \cdot\quad \cdot\quad \cdot\quad \cdot\quad \cdot\quad \cdot\quad \cdot\quad \cdot \]
\[ \|x(t_{m+1})\|\leqslant \|x(t_m)\|\exp\left\{\int_{t_{m+1}}^{t_m}a(t)\,dt\right\}, \]
\[ \cdot\quad \cdot\quad \cdot\quad \cdot\quad \cdot\quad \cdot\quad \cdot\quad \cdot\quad \cdot\quad \cdot \]
From these inequalities we find, for any \(t\in [t_0,+\infty)\),
\[ \|x(t)\|\leqslant \|x(t_0)\|\exp\left\{\int_{t_0}^{t}a(\tau)\,d\tau\right\} \leqslant \|x(t_0)\|\exp\left\{\int_{t_0}^{+\infty}a(t)\,dt\right\}. \]
The theorem is proved.
Remark. If condition a) of Theorem 1 is satisfied and there exist numbers \(\alpha_i>0\) \((i=1,\ldots,n)\) such that the inequalities
\[ \sum_{\substack{k=1\\(k\ne i)}}^{n} \alpha_k |a_{ik}(t)| \leq \alpha_i |a_{ii}(t)|+\varepsilon_i(t),\qquad \varepsilon_i(t)\geq 0, \]
\[ \int_{t_0}^{+\infty} \varepsilon_i(t)\,dt<\infty \qquad (i=1,\ldots,n), \]
hold, then the conclusion of Theorem 1 is valid. Here the norm of the solution is defined by the relation
\[ \|x(t)\|=\max_{1\leq j\leq n}\left\{\frac{|x_j(t)|}{\alpha_j}\right\}. \]
Theorem 2. If condition a) of Theorem 1 is satisfied and the inequalities
\[ \sum_{\substack{k=1\\(k\ne i)}}^{n} |a_{ki}(t)| \leq |a_{ii}(t)|+\varepsilon_i(t),\qquad \varepsilon_i(t)\geq 0, \]
\[ \int_{t_0}^{+\infty} \varepsilon_i(t)\,dt<\infty \qquad (i=1,\ldots,n), \]
hold, then the conclusion of Theorem 1 is valid.
Proof. Define the norm of the solution \(x(t)\) by the relation
\[ \|x(t)\|=\sum_{i=1}^{n} |x_i(t)|. \]
Let us note the following: the set of points of the interval \([t_0,+\infty)\) at which the derivative \(\dfrac{d|x_i|}{dt}\) does not exist (with \(i\) fixed) is at most countable. Indeed, the derivative \(\dfrac{d|x_i|}{dt}\) fails to exist only at such a point \(t^*\) where \(x_i(t^*)=0\) and, at the same time, \(\dot x_i(t^*)\ne 0\). By the continuity of \(\dot x_i(t)\), the relation \(\dot x_i(t)\ne 0\) holds in some neighborhood of the point \(t^*\), in which there are already no zeros of the function \(x_i(t)\), and, consequently, no points at which the derivative \(\dfrac{d|x_i|}{dt}\) fails to exist. Taking as the neighborhood mentioned above an interval with rational endpoints, we are convinced of the validity of the assertion made. Clearly, the set \(\sigma\) of points of the interval \([t_0,+\infty)\) at which at least one of the derivatives \(\dfrac{d|x_k|}{dt}\) \((k=1,\ldots,n)\) does not exist is also at most countable. Excluding such points from consideration, let us estimate the derivative \(\dfrac{d|x_i|}{dt}\).
If \(x_i(t)>0\), then we have
\[ \frac{d|x_i|}{dt}=\frac{dx_i}{dt}\leq |a_{i1}(t)|\,|x_1|+\cdots+a_{ii}(t)|x_i|+\cdots+|a_{in}(t)|\,|x_n|. \]
Similarly, for \(x_i(t)<0\) we find
\[ \frac{d|x_i|}{dt}=-\frac{dx_i}{dt}\leq |a_{i1}(t)|\,|x_1|+\cdots+a_{ii}(t)|x_i|+\cdots+|a_{in}(t)|\,|x_n|. \]
Summing similar inequalities over \(i\), we find
\[ \frac{d\left(\sum_{i=1}^{n}|x_i|\right)}{dt} \leq \left(a_{11}(t)+\sum_{j=2}^{n}|a_{j1}(t)|\right)|x_1|+\cdots+ \]
\[ +\left(a_{ii}(t)+\sum_{j=1\,(j\ne i)}^{n}|a_{ji}(t)|\right)|x_i| +\cdots+ \left(a_{nn}(t)+\sum_{j=1}^{n-1}|a_{jn}(t)|\right)|x_n|. \]
Multiplying both sides of the last inequality by
\[ \frac{dt}{\sum_{i=1}^{n}|x_i|} \]
and integrating over the interval \([t_0,t]\), we obtain:
\[ \int_{t_0}^{t}\frac{d\|x\|}{\|x\|} \leq \int_{t_0}^{t}\left[a_{11}(\tau)+\sum_{j=2}^{n}|a_{j1}(\tau)|\right]\frac{|x_1|}{\|x\|}\,d\tau+\cdots+ \int_{t_0}^{t}\left[a_{ii}(\tau)+\right. \]
\[ \left. +\sum_{j=1\,(j\ne i)}^{n}|a_{ji}(\tau)|\right]\frac{|x_i|}{\|x\|}\,d\tau+\cdots+ \int_{t_0}^{t}\left[a_{nn}(\tau)+\sum_{j=1}^{n-1}|a_{jn}(\tau)|\right]\frac{|x_n|}{\|x\|}\,d\tau. \]
Estimating the \(i\)-th term on the right-hand side, divide the interval \([t_0,t]\) into two sets:
\[ E_i^{(1)}\{\tau:\ a_{ii}(\tau)<0\} \quad\text{and}\quad E_i^{(2)}\{\tau:\ a_{ii}(\tau)\geq 0\}. \]
Then we obtain
\[ \int_{t_0}^{t}\left[a_{ii}(\tau)+ \sum_{j=1\,(j\ne i)}^{n}|a_{ji}(\tau)|\right]\frac{|x_i|}{\|x\|}\,d\tau \leq \int_{E_i^{(1)}}\varepsilon_i(\tau)\frac{|x_i|}{\|x\|}\,d\tau+ \]
\[ +\int_{t_0}^{t}2a_{ii}^{+}(\tau)\frac{|x_i|}{\|x\|}\,d\tau+ \int_{E_i^{(2)}}\varepsilon_i(\tau)\frac{|x_i|}{\|x\|}\,d\tau = \]
\[ = \int_{t_0}^{t}\left[2a_{ii}^{+}(\tau)+\varepsilon_i(\tau)\right]\frac{|x_i|}{\|x\|}\,d\tau \leq \int_{t_0}^{t}a(\tau)\frac{|x_i|}{\|x\|}\,d\tau, \]
where the function \(a(t)\) is defined as in Theorem 1.
As a result we obtain
\[ \int_{t_0}^{t}\frac{d\|x\|}{\|x\|} \leq \int_{t_0}^{t}a(\tau)\frac{|x_1|}{\|x\|}\,d\tau+\cdots+ \int_{t_0}^{t}a(\tau)\frac{|x_i|}{\|x\|}\,d\tau+\cdots+ \]
\[ +\int_{t_0}^{t}a(\tau)\frac{|x_n|}{\|x\|}\,d\tau = \int_{t_0}^{t}a(\tau)\,d\tau \quad\text{and}\quad \|x(t)\|\leq \|x(t_0)\|\exp\left\{\int_{t_0}^{+\infty}a(t)\,dt\right\}. \]
The theorem is proved.
Remark. If condition a) of Theorem 1 is satisfied and there exist numbers \(\beta_i>0\) \((i=1,\ldots,n)\) such that the inequalities
\[ \sum_{\substack{k=1\\(k\ne i)}}^{n}\beta_k|a_{ki}(t)| \le \beta_i|a_{ii}(t)|+\varepsilon_i(t), \qquad \varepsilon_i(t)\ge 0, \qquad \int_{t_0}^{+\infty}\varepsilon_i(t)\,dt<\infty \]
\[ (i=1,\ldots,n), \]
hold, then the conclusion of Theorem 1 is valid. Here the norm of a solution is defined by the relation
\[ \|x(t)\|=\sum_{i=1}^{n}\beta_i|x_i(t)|. \]
In the theorems presented, unlike in [1, 2], there is no requirement of boundedness or integrability on the half-axis \([t_0,+\infty)\) of the coefficients \(a_{ik}(t)\).
References
- Azbelev N. V., Tsalyuk Z. B. PMM, vol. XXVIII, no. 1, 1964.
- Moiseenko A. P. UMN, vol. XVIII, no. 1, 1962.
Received by the editors
June 28, 1965
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