Full Text
UDC 517.916
ON WEAKLY IRREGULAR SYSTEMS
N. A. IZOBOV
In the work [1], R. E. Vinograd established that, for the regularity of the system
\[ \frac{dx}{dt}=P(t)x \tag{1} \]
with coefficient matrix \(P(t)\) bounded and continuous for \(t \geqslant 0\), it is necessary and sufficient that there exist exact characteristic exponents for the solutions \(\bar{x}_i\) of its normal system of solutions \(X(t)=\{\bar{x}_1,\ldots,\bar{x}_n\}\), and that the functions
\[ \frac{1}{t}\ln |\sin(\bar{x}_k,\bar{x}_k^{*})|\quad (k=2,\ldots,n) \]
have zero limits as \(t\to+\infty\). The first condition is equivalent to the following: for every \(\delta>0\) there exists \(T(\delta)>0\) such that, for all \(t,\tau\geqslant T(\delta)\), the inequality
\[ \left|\frac{1}{t}\ln \|\bar{x}_i(t)\|-\frac{1}{\tau}\ln \|\bar{x}_i(\tau)\|\right|<\delta \quad (i=1,\ldots,n) \tag{2} \]
holds (for definitions of the vector and matrix norms and of the quantities \(\bar{\omega}[x]\), \(\omega[x]\), see [6]). If one requires the inequality (2) to be fulfilled not for all \(\tau\geqslant T(\delta)\), but only for \(\tau\) “sufficiently close” to \(t\), while preserving the second condition, then one obtains a certain generalization of regular systems.
Definition A. We shall call the system (1) weakly irregular if, for its normal system of solutions \(X\), the following conditions hold:
1) \(\omega[\sin(\bar{x}_k,\bar{x}_k^{*})]=0\) \((k=2,\ldots,n)\);
2) for every \(\delta>0\) there is a number \(T(\delta)>0\) such that, for all \(t\geqslant T(\delta)\) and \(\tau\in[\delta t,t]\), the inequality (2) is satisfied. In the case when only condition 2) is present, we shall call the system (1) an \(M\)-system (slowly varying).
Together with the system (1), consider the perturbed system
\[ \frac{dy}{dt}=P(t)y+f(t,y) \tag{3} \]
with vector function \(f(t,y)\) continuous in \(t\) and \(y\), \(f(t,0)\equiv 0\). As is known, in the case of regularity of the system (1) the question of stability of the trivial solution of the system (3), under very broad assumptions on \(f(t,y)\), is completely resolved by the system of first approximation. An analogous result is also valid in the case when the system (1) is weakly irregular.
\(1^\circ\). Suppose that the system (1) is an \(M\)-system, and \(f(t,y)\) satisfies the Lipschitz condition.
\[ \|f(t,y^{(2)})-f(t,y^{(1)})\|\leq g(t)\|y^{(2)}-y^{(1)}\|. \]
Denote by \(\alpha_i\) the angle between the vector \(\bar x_i\) and the subspace spanned by the vectors \(\bar x_k\) \((k\ne i)\), \(\nu_i(t)=|\operatorname{cosec}\alpha_i|\), \(\nu(t)=\max_i\{\nu_i(t)\}\) [2]; \(\mu=\bar\omega[\nu(t)]\).
Theorem 1. If the characteristic exponents of the solutions of system (1) are negative and, for some \(\tilde\alpha>0\), the inequality
\[ \int_0^\infty e^{(\mu+\tilde\alpha)\tau} g(\tau)\,d\tau<+\infty, \tag{4} \]
holds, then the trivial solution of system (3) is asymptotically stable.
Proof. First of all, we note that it is sufficient to restrict ourselves to perturbations \(f(t,y)\) for which
\[ \bar\omega[g(t)]<-\mu \tag{5} \]
(see [3]). Applying the transformation of V. M. Millionshchikov [2]
\[ y=\sum_{i=1}^n \frac{\xi_i}{\|\bar x_i\|}\,\bar x_i \tag{6} \]
to system (3), we reduce it to the form
\[ \frac{d\xi}{dt}=L(t)\xi+\Psi(t,\xi), \tag{7} \]
where
\[ L(t)=\operatorname{diag}\left[\frac{d}{dt}\ln\|\bar x_1(t)\|,\ldots,\frac{d}{dt}\ln\|\bar x_n(t)\|\right], \]
and \(\Psi(t,\xi)\), \(\Psi(t,0)\equiv0\), satisfies the condition
\[ \|\Psi_i(t,\xi^{(2)})-\Psi_i(t,\xi^{(1)})\|\leq \nu(t)g(t)\|y^{(2)}-y^{(1)}\|. \]
From (6) it follows that
\[ \|y\|\leq \sum_{i=1}^n |\xi_i|\,\frac{\|\bar x_i\|}{\|\bar x_i\|}\leq n\|\xi\| \tag{8} \]
and
\[ \|\Psi(t,\xi)\|\leq n^2\nu(t)g(t)\|\xi\|. \tag{9} \]
System (7) for \(\varepsilon=1\) is equivalent to the system of equations
\[ \frac{du}{dt}=L(t)u+\varepsilon\Psi(t,u), \]
whose right-hand side contains the positive parameter \(\varepsilon\). We shall seek the solutions \(u=u(t,\varepsilon)\) of this system, with initial values from a sufficiently small neighborhood of the origin \(O_0\), in the form of formal series in integral nonnegative powers of \(\varepsilon\), using the method of N. P. Erugin [4, p. 56]. Thus, let
\[ u=\sum_{i=0}^\infty u_i(t)\varepsilon^i,\qquad t\geq T. \tag{10} \]
(the meaning of the constant \(T\) will be explained below). The vectors \(u_i(t)\) are defined successively by the formulas
\[ u_0(t)=R(t)x_0,\qquad \|x_0\|\leqslant 1, \]
\[ u_i(t)=R(t)\int_T^t R^{-1}(\tau)\Psi(\tau,u_{i-1}(\tau))\,d\tau, \tag{11} \]
where the matrix \(R(t)\) is a diagonal matrix of the form
\[ R(t)=\operatorname{diag}\bigl[\|\bar{x}_1(t)\|,\ldots,\|\bar{x}_n(t)\|\bigr]. \]
The series (10) is majorized by the series
\[ \sum_{i=0}^{\infty}\|u_i(t)\|\varepsilon^i . \tag{12} \]
We shall show that the series (12) converges for every \(\varepsilon\) when \(t\geqslant T(\varepsilon)\). To this end we estimate the functions \(\|u_i(t)\|\) \((i=0,1,2,\ldots)\).
Let \(\lambda=\max_k(\lambda_k)\) be the largest characteristic exponent of the system (1);
\[ \mu+\bar{\omega}[g(t)]\equiv-\beta,\qquad \beta>0. \tag{13} \]
Take an arbitrary \(\alpha>0\), satisfying only the conditions \(\alpha<|\lambda|\), \(2\alpha<\beta\). Then, evidently, for the chosen \(\alpha\) there exists \(T_\alpha>0\) such that, for \(t\geqslant T_\alpha\), on the basis of (11), the definition of the quantity \(\mu\), and (13), the following inequalities hold simultaneously:
\[ \|u_0(t)\|<e^{(\lambda+\alpha)t},\qquad n^2\nu(t)<e^{\left(\mu+\frac{\alpha}{2}\right)t},\qquad g(t)<e^{\left(-\mu-\beta+\frac{\alpha}{2}\right)t}. \tag{14} \]
We estimate \(\|u_1(t)\|\). On the basis of inequality (9), taking into account the positivity of the elements of the matrices \(R(t)\) and \(R^{-1}(\tau)\), from equalities (11) we obtain the estimate
\[ \|u_1(t)\|\leqslant \left\|R(t)\int_{T_\alpha}^{t} g_1(\tau)\|u_0(\tau)\|R^{-1}(\tau) \begin{pmatrix} 1\\ 1\\ \cdot\\ \cdot\\ 1 \end{pmatrix} \,d\tau\right\| \equiv \|v^{(1)}\|, \tag{15} \]
where \(g_1(t)=n^2\nu(t)g(t)\). The components \(v_k^{(1)}\) of the vector \(v^{(1)}\) are computed by the following formulas:
\[ v_k^{(1)}=\|\bar{x}_k(t)\|\int_{T_\alpha}^{t} g_1(\tau)\|u_0(\tau)\| \frac{d\tau}{\|\bar{x}_k(\tau)\|} \qquad (k=1,\ldots,n). \tag{16} \]
Estimating the right-hand side of (16) with the aid of inequalities (14), we have for \(v_k^{(1)}\) the following estimate:
\[ v_k^{(1)}\leqslant \|\bar{x}_k(t)\|\int_{T_\alpha}^{t} e^{(-\beta+2\alpha+\lambda)\tau}\frac{d\tau}{\|\bar{x}_k(\tau)\|}. \]
Denoting \(s(t)=\|\bar{x}_k(t)\|e^{-\lambda t}\) and replacing the function \(e^{(-\beta+2\alpha)\tau}\) by the number \(1\), which bounds this function from above, the preceding inequality may be written in the form
\[ v_k^{(1)} \leqslant e^{\lambda t} s(t) \int_{T_\alpha}^{t}\frac{d\tau}{s(\tau)} \equiv e^{\lambda t} m(t), \quad t \geqslant T_\alpha . \tag{17} \]
Let us compute \(\overline{\omega}[m(t)]\). To this end we represent the function \(m(t)\) as the sum of two positive functions
\[ m(t)=s(t)\int_{T_\alpha}^{\delta t}\frac{d\tau}{s(\tau)} +s(\tau)\int_{\delta t}^{t}\frac{d\tau}{s(\tau)} \equiv m_1(t)+m_2(t) \]
(when \(\delta t<T_\alpha\), there is no need for such a representation) and estimate each of the functions \(m_1(t)\), \(m_2(t)\) separately.
In view of the boundedness of the coefficients of system (1) (\(\|P(t)\|\leqslant M\)), \(1/s(\tau)\leqslant e^{2M\tau}\), and therefore
\[ m_1(t)\leqslant s(t)\int_{T_\alpha}^{\delta t} e^{2M\tau}\,d\tau < s(t)e^{2M\delta t}. \]
And since from the definition of \(s(t)\) and \(\lambda\) there follows the inequality
\[ \overline{\omega}[s(t)]\leqslant \lambda_k-\lambda\leqslant 0, \tag{18} \]
we have \(\overline{\omega}[m_1(t)]\leqslant 2M\delta\).
We write the function \(m_2(t)\) in the form
\[ m_2(t)=s(t)\int_{\delta t}^{t} e^{\left[\frac{1}{t}\ln s(t)-\frac{1}{\tau}\ln s(\tau)\right] -\frac{\tau}{t}\ln s(t)}\,d\tau . \]
Since the function \(s(t)\) satisfies condition 2) of the definition of \(A\), for \(t\geqslant T(\delta)\) the inequality
\[ m_2(t)\leqslant s(t)\int_{\delta t}^{t} e^{\delta t-\frac{\tau}{t}\ln s(t)}\,d\tau \]
holds.
Consider the auxiliary function \(\varphi(t)\):
\[ \varphi(t)= \begin{cases} s(t)\cdot e^{\delta t}\cdot t, & \text{if } \ln s(t)\geqslant 0,\\ e^{\delta t}\cdot t, & \text{if } \ln s(t)<0. \end{cases} \]
It is not difficult to see that, for the values of \(t\) under consideration, \(m_2(t)<\varphi(t)\). Indeed, for those \(t\) for which \(\ln s(t)\geqslant 0\), the inequalities
\[ \int_{\delta t}^{t} e^{\delta t-\frac{\tau}{t}\ln s(t)}\,d\tau \leqslant \int_{\delta t}^{t} e^{\delta t}\,d\tau < e^{\delta t}t \]
are valid.
In the case where \(\ln s(t)\) is negative, the following inequalities are valid:
\[ \int_{\delta t}^{t} e^{\delta t-\frac{\tau}{t}\ln s(t)}\,d\tau \leqslant \int_{\delta t}^{t} e^{\delta t-\frac{t}{t}\ln s(t)}\,d\tau < \frac{1}{s(t)}e^{\delta t}t . \]
And in both cases the required inequality \(m_2(t)<\varphi(t)\) is obvious. On the basis of inequality (18) we obtain that \(\overline{\omega}[\varphi(t)]=\delta\). Thus, for every \(\delta>0\), \(\overline{\omega}[m(t)]\leq \max\{\delta,2M\delta\}\), which also means that \(\overline{\omega}[m(t)]=0\). Therefore, for the \(\alpha>0\) chosen by us earlier and for positive \(\varkappa\), \(\varkappa<\dfrac{1}{\varepsilon}\), there is a \(T=T(\alpha,\varepsilon)\geq T_\alpha\) such that for \(t\geq T\)
\[ n e^{-\alpha t} m(t)<\varkappa . \tag{19} \]
The preceding arguments remain valid if, instead of the “initial” time \(T_\alpha\) considered up to now, one takes the just-defined time \(T\). Thus,
\[ v_k^{(1)}<\frac{\varkappa}{n}e^{(\lambda+\alpha)t},\qquad t\geq T\quad (k=1,\ldots,n), \]
and then also
\[ \|u_1(t)\|<\varkappa e^{(\lambda+\alpha)t},\qquad t\geq T . \]
Now using the method of induction, assuming that
\[ \|u_{i-1}(t)\|<\varkappa^{\,i-1}e^{(\lambda+\alpha)t}, \]
from equalities (11), using (9) and (14), we obtain the analogue of inequality (17)
\[ v_k^{(i)}\leq \varkappa^{\,i-1} e^{\lambda t}\cdot m(t), \]
which, taking account of inequality (19), gives the required estimate
\[ \|u_i(t)\|<\varkappa^{\,i}e^{(\lambda+\alpha)t},\qquad t\geq T . \tag{20} \]
Therefore the series (12), for \(\varepsilon=1\), is majorized by the series \((0<\varkappa<1)\)
\[ e^{(\lambda+\alpha)t}\sum_{i=0}^{\infty}\varkappa^i = e^{(\lambda+\alpha)t}\frac{1}{1-\varkappa}, \]
and every solution \(\xi(t)\) of system (7) which for \(t=T=T(\alpha,1)\) takes the value \(\xi(T)=R(T)x_0\), where \(x_0\) is an arbitrary vector of the unit ball \(\|x_0\|\leq 1\), satisfies the estimate
\[ \|\xi(t)\|<\frac{1}{1-\varkappa}e^{(\lambda+\alpha)t},\qquad t\geq T . \]
Obviously, in view of the positivity of the elements of the diagonal matrix \(R(T)\), there exists a spherical neighborhood \(\rho_1(O_0)\) of the origin \(O_0\), of radius \(r_1>0\), entirely contained in the set of vectors \(\{R(T)x_0,\ \|x_0\|\leq 1\}\). Therefore, for all solutions \(y(t)\) of system (3) belonging, for \(t=T\), to a sufficiently small neighborhood \(\widetilde{\rho}(O_0)\) of the origin (such that the corresponding \(\xi(T)\in\rho_1(O_0)\)), on the basis of inequality (8) the estimate
\[ \|y(t)\|<\frac{n}{1-\varkappa}e^{(\lambda+\alpha)t},\qquad t\geq T \]
is valid.
By virtue of the continuous dependence of solutions of system (3) on the initial data, there exists a neighborhood \(\rho(O_0)\subset \widetilde{\rho}(O_0)\) such that from \(y(0)\in\rho(O_0)\) it follows
there will be \(y(T)\in \tilde{\rho}(O_0)\). And then for all solutions \(y(t)\) of system (3), \(y(0)\in \rho(O_0)\), there is a positive constant \(C=C(a)\), depending only on \(a\), such that the estimate
\[ \|y(t)\|<\frac{Cn}{1-\chi}\,e^{(\lambda+a)t},\qquad t\ge 0. \tag{21} \]
holds.
The theorem is proved.
Corollary 1. For the solutions of system (3) the inequality
\[ \varlimsup_{r\to 0}\ \sup_{\|y_0\|\le r}\{\bar{\omega}[y(t,y_0)]\}\le \lambda . \tag{22} \]
holds.
The proof follows from the arbitrariness of \(a\) and inequality (21).
Corollary 2. The characteristic exponents of the solutions of the linear system
\[ \frac{dy}{dt}=[P(t)+Q(t)]y, \]
where the norm of the matrix \(Q(t)\) satisfies condition (4), do not exceed the greatest characteristic exponent of system (1).
The proof follows from inequality (22) and from the fact that the characteristic exponents of the solutions of a linear system starting at \(t=0\) on a line of the space of initial data passing through \(O_0\) coincide.
In the last two corollaries no negativity of the characteristic exponents is assumed.
\(2^\circ\). We show that fulfillment of condition 1) of the definition of \(A\) is sufficient for the equality to zero of the quantity \(\mu\).
Indeed, condition 1) is equivalent to the requirement of the existence of a zero limit for the function
\[ \frac{1}{t}\ln\left|\frac{1}{\Delta(t)}\prod_{i=1}^{n}\|\bar{x}_i(t)\|\right|,\qquad \Delta(t)=\det X(t), \]
as \(t\to +\infty\) (condition \(1'\)). But the absolute value of the determinant \(\Delta(t)\) of the matrix \(X=\{\bar{x}_1,\ldots,\bar{x}_n\}\) does not depend on the permutation of the columns \(\bar{x}_i\); therefore, interchanging \(\bar{x}_n\) with \(\bar{x}_j\), using the formula
\[ |\det[\bar{x}_1,\ldots,\bar{x}_n]|=\prod_{1}^{n}\|\bar{x}_k(t)\|\cdot \prod_{2}^{n}|\sin(\bar{x}_k,\bar{x}_k^{*})| \]
from [1], we obtain the following value of the determinant in the new numbering:
\[ |\Delta(t)|=\prod_{1}^{n}\|\bar{x}_k\|\prod_{2}^{n-1}|\sin(\bar{x}_{k'},\bar{x}_{k'}^{*})|\,\frac{1}{v_j(t)}. \]
And since \(|\sin(\bar{x}_{k'},\bar{x}_{k'}^{*})|\le 1\), \(v_j(t)\ge 1\), it follows from condition \(1'\) that we obtain the required result. Therefore, as a consequence of Theorem 1 the following is valid:
Theorem 2. If the characteristic exponents of the solutions of the weakly irregular system (1) are negative and for some \(\tilde{\alpha}>0\) the inequality
\[ \int_{0}^{\infty} e^{\tilde{\alpha}\tau}g(\tau)\,d\tau<+\infty, \]
holds, then the trivial solution of system (3) is asymptotically stable.
It is evident that Corollaries 1 and 2 can be reformulated in an analogous manner.
Remark. In the definition of \(A\), condition 1) can be replaced, generally speaking, by the weaker requirement \(\mu=0\).
\(3^\circ\). Let, for the perturbed system (3), the vector-function \(f(t,y)\) satisfy the condition
\[ \|f(t,y)\|\leq N\|y\|^{1+a},\qquad N=\operatorname{const}>0,\quad a>0. \]
Then the following is valid.
Theorem 3. If the characteristic exponents of the solutions of the \(M\)-system (1) are negative and \(a\lambda+\mu<0\), then the trivial solution of system (3) is asymptotically stable.
Proof. Applying, as before, transformation (6) to system (3), we obtain system (7), for which the vector-function \(\Psi(t,\xi)\) satisfies the condition
\[ \|\Psi(t,\xi)\|\leq n^{2+a}Nv(t)\|\xi\|^{1+a}, \]
i.e., the “Lipschitz condition” with constant \(g_1=n^{2+a}Nv(t)\|\xi\|^a\). Denoting \(a\lambda+\mu=-\beta,\ \beta>0\), choosing \(\alpha>0\) so that the inequalities \(\alpha<|\lambda|\), \(\alpha(2+a)<\beta\) are simultaneously satisfied, and changing in an obvious way one (the middle one) of inequalities (14), we reduce the proof of this theorem to the proof of Theorem 1.
The theorem is proved.
We reformulate Corollary 1 of Theorem 1 in the following way.
Let
\[ \|f(t,y)\|\leq g(t)\|y\|^{1+a},\qquad \overline{\omega}[g(t)]=\mu. \]
If the characteristic exponents of the weakly irregular system (1) are negative and \(a\lambda+\mu<0\), then for the solutions \(y(t)\) of system (3) inequality (22) is valid.
From Theorem 3 it follows that
Theorem 4. If the characteristic exponents of the solutions of the weakly irregular system (1) are negative, then the trivial solution of system (3) is asymptotically stable.
\(4^\circ\). The starting factor in defining a weakly irregular system was the existence of exact exponents for the solutions of a regular system. Now, seeking to generalize (for the case \(n=2\)) the concept of a weakly irregular system, we shall proceed from the existence, for a regular system, of a zero limit for the function
\[ p(t)=\frac{1}{t}\ln\left[\frac{\|\bar{x}_1(t)\|}{\|\bar{x}_2(t)\|}e^{(\lambda_2-\lambda_1)t}\right]\quad \text{as } t\to+\infty. \]
We shall call system (1) generalized weakly irregular if \(\mu=0\) and the function \(e^{t\cdot p(t)}\) satisfies condition 2) of definition \(A\).
The concept of a generalized \(M\)-system is introduced analogously. We shall show that Theorem 1 (and thereby Theorem 2) is valid for it. Introduce continuous auxiliary functions \(p_1(t)\) and \(p_2(t)\),
\[ p_1(t)= \begin{cases} t\cdot p(t), & \text{if } p(t)<0,\\ 0, & \text{if } p(t)>0, \end{cases} \]
and \(p_2(t)=p_1(t)-t\cdot p(t)\). Suppose first that \(f(t,y)=Q(t)y\). In equation (3) make the substitution
\[ y=X(t)R^{-1}(t)z,\qquad R(t)=\operatorname{diag}\,[e^{\lambda_1 t+p_1(t)},\ e^{\lambda_2 t+p_2(t)}]. \tag{23} \]
Then \(z(t)\) is a solution of the system
\[ \frac{dz}{dt}=L(t)z+B(t)z, \tag{24} \]
in which the bounded matrix \(L(t)\), having at most a countable set of points of discontinuity, and the matrix \(B(t)\) have the form
\[ L(t)=R^{-1}(t)\frac{dR(t)}{dt_+},\qquad B(t)=R(t)X^{-1}(t)Q(t)X(t)R^{-1}(t). \]
Denoting \(|P(t)|=((|p_{ij}(t)|))\), we have
\[ \|B(t)\|\le \|R(t)|X^{-1}(t)|Q(t)||X(t)|R^{-1}(t)\|. \]
Substituting further the obvious matrix estimates
\[ |X^{-1}|\le \frac{1}{|\Delta(t)|} \begin{pmatrix} \|\overline{x}_2\| & \|\overline{x}_2\|\\ \|\overline{x}_1\| & \|\overline{x}_1\| \end{pmatrix}, \qquad |Q|\le \|Q(t)\| \begin{pmatrix} 1 & 1\\ 1 & 1 \end{pmatrix}, \qquad |X|\le \begin{pmatrix} \|\overline{x}_1\| & \|\overline{x}_2\|\\ \|\overline{x}_1\| & \|\overline{x}_2\| \end{pmatrix}, \]
in the right-hand side of the preceding inequality, using the relation \(p_1(t)-p_2(t)=t\cdot p(t)\), valid for any \(t\), we obtain
\[ \|B(t)\|\le 4\|Q(t)\|\nu_1(t), \tag{25} \]
for, according to [1], \(|\Delta(t)|=\|\overline{x}_1\|\cdot\|\overline{x}_2\|\cdot|\sin(\overline{x}_1,\overline{x}_2)|\). Therefore, when the hypotheses of Theorem 1 are fulfilled, for solutions of system (24) consequence 2 of Theorem 2 is valid, asserting that \(\overline{\omega}[z]\le \lambda\). From the substitution (23) itself follows the inequality
\[ \|y\|\le e(t)\|z\|,\qquad e(t)\equiv \max_i\{\|\overline{x}_i(t)\|e^{-\lambda_i t}\}. \tag{26} \]
Hence the validity of Theorem 1 for the systems considered in \(4^\circ\) follows.
According to the principle of linear inclusion of Bylov—Grobman [5], every solution \(y(t)\) of a system for which \(\|f(t,y)\|\le g(t)\|y\|\) is a solution of the linear system
\[ \frac{du}{dt}=P(t)u+Q(t,y(t))u, \]
for which \(\|Q(t,y)\|\le g(t)\) for all \(y\). Therefore, on the basis of (25), the additions \(B(t)z\) of system (24) satisfy the condition
\[ \|B(t)z\|\le 4g(t)\nu_1(t)\|z\| \]
and, on the basis of consequence 1 of Theorem 2 (the hypotheses of Theorem 1 are fulfilled), taking (25) into account, we obtain the required result.
The question of the validity of Theorem 3 for generalized \(M\)-systems remains open.
Remark. Condition 2) of the definition of \(A\) may be replaced by a somewhat more general condition, requiring that \(\tau\) belong to the interval \([d(\delta)t,t]\), where \(d(\delta)\to 0\) as \(\delta\to 0\).
\(5^\circ\). Examples. The irregular diagonal system
\[ \dot{x}_1=0, \]
\[ \dot{x}_2= \left[ \sin\ln\ln(t+e)+\frac{1}{\ln(t+e)}\cos\ln\ln(t+e) \right]x_2,\qquad t\ge 0 \]
is weakly irregular, since, first, \(\sin(\bar x_1,\bar x_2)=1\), and, second, applying the mean value theorem to the function \(\sin\ln\ln u\) on the interval \([\tau,t]\), \(\tau\in[\delta t,t]\), we are convinced that this function satisfies condition 2) of Definition \(A\).
As a generalized weakly irregular system one may take the system
\[ \begin{aligned} \dot x_1&=p(t)x_1,\\ \dot x_2&=\left[p(t)+\sin\ln\ln(t+e)+\frac{1}{\ln(t+e)}\cos\ln\ln(t+e)\right]x_2,\qquad t\geqslant0, \end{aligned} \]
in which \(p(t)\) is an arbitrary continuous bounded function.
\(6^\circ\). Denote
\[ \mu(t,\tau)=\max_i\left\{\left|\frac{1}{t}\ln|\bar x_i(t)|-\frac{1}{\tau}\ln|\bar x_i(\tau)|+\frac{1}{\tau}\ln v_i(\tau)\right|\right\}. \]
The quantity \(\mu\) (we shall call it the coefficient of irregularity of system (1)) is defined as follows:
\[ \mu=\varlimsup_{\delta\to0}\ \varlimsup_{t\to+\infty}\ \sup_{\tau\in[\delta t,t]}\{\mu(t,\tau)\}. \]
From the proof of Theorem 1 it follows
Theorem 5. For the solutions of system (3), in which \(P(t)\) is a bounded continuous matrix, \(\|f(t,y)\|\leqslant g(t)\|y\|\), and for some \(\tilde\alpha\)
\[ \int_0^\infty e^{(\mu+\tilde\alpha)\tau}g(\tau)\,d\tau<+\infty, \]
inequality (22) is valid.
From the proof of Theorem 3 it follows
Theorem 6. If the characteristic exponents of the solutions of system (1) with bounded and continuous matrix \(P(t)\) are negative,
\[ \|f(t,y)\|\leqslant N\|y\|^{1+a},\qquad a>0,\qquad a\lambda+\mu<0, \]
then the trivial solution of system (3) is asymptotically stable.
References
- Vinograd R. E. UMN, IX, issue 2 (60), 1954.
- Millionshchikov V. M. DAN SSSR, 162, No. 2, 1965.
- Bylov B. F. Differential Equations, 1, No. 4, 1965.
- Erugin N. P. Linear Systems of Ordinary Differential Equations with Periodic and Quasiperiodic Coefficients. Publishing House of the Academy of Sciences of the BSSR, Minsk, 1963.
- Bylov B. F., Grobman D. M. UMN, 17, issue 3 (105), 1962.
- Izobov N. A. Differential Equations, 1, No. 4, 469—477, 1965.
Received by the editors
17 June 1966
Belorussian State University
named after V. I. Lenin