UDC 517.941

ESTIMATES OF GENERALIZED CHARACTERISTIC NUMBERS OF DIFFERENTIAL SYSTEMS

Yu. S. Bogdanov

In an \(n\)-dimensional real Euclidean space, choose a bounded domain \(\Xi\) containing the origin \(O_0\). Construct the \((n+1)\)-dimensional cylinder \(\mathrm H \equiv \Xi \times (-\infty,+\infty)\). Consider on \(\mathrm H\) the differential system

\[ dx/d\tau=f(x,\tau), \tag{1} \]

where \(f(\xi,\tau)\) is continuous for all \(\tau\), has continuous first-order partial derivatives with respect to all components \(\xi\in\Xi\), and vanishes for \(\xi=O_0\). Suppose that all solutions of (1) can be continued indefinitely in both directions. The solution of (1) with initial condition \(x(\tau_0)=\xi_0\) will be denoted by \(x(\tau,\xi_0,\tau_0)\); if \(\tau_0=0\), then \(x(\tau,\xi_0,\tau_0)\) will be denoted briefly by \(x(\tau,\xi_0)\).

Assume that a continuous function \(v\) is defined on \(\Xi\), positive on \(\Xi=\Xi\setminus\{O_0\}\), vanishing for \(\xi=O_0\), and increasing without bound as one approaches \(\Xi_{\mathrm{fr}}=\overline{\Xi}\setminus\Xi\). In addition, consider a real function \(d(\gamma_1,\gamma_2)\), continuous for all positive values of its arguments, such that for all \(\gamma>0,\ \gamma_3>\gamma_2>\gamma_1>0\) the following hold:

\[ d(\gamma_2,\gamma_1)=-d(\gamma_1,\gamma_2),\qquad d(\gamma_2,\gamma)>d(\gamma_1,\gamma), \]

\[ d(\gamma_3,\gamma_2)+d(\gamma_2,\gamma_1)\ge d(\gamma_3,\gamma_1),\qquad \bigcup_\gamma \{d(\gamma,\gamma_1)\}=(-\infty,+\infty). \]

Denote \(d(\gamma)\equiv d(\gamma,1)\). For each solution \(x_\xi\equiv x(\tau,\xi)\), \(\xi\ne O_0\), of system (1), define the small \(vd\)-number \(\overline{\Omega}_{x_\xi}\) (cf. [1, 2]) by

\[ \overline{\Omega}_{x_\xi} \equiv \max\left\{ \overline{\lim_{\tau\to+\infty}} \frac{1}{\tau}d\bigl[v(x_\xi(\tau))\bigr], \right. \]

\[ \left. - \underline{\lim_{\tau\to+\infty}} \frac{1}{\tau}d\bigl[v(x_\xi(-\tau))\bigr] \right\}. \tag{2} \]

In the present note we derive estimates of the small \(vd\)-numbers of solutions of the perturbed system

\[ dy/d\tau=f(y,\tau)+g(y,\tau) \tag{3} \]

in terms of auxiliary quantities—the adjacent small \(vd\)-numbers of the original system (1).

§ 1. ADJACENT NUMBERS OF A PAIR OF SETS

In the closed domain \(\overline{\Xi}\), choose some closed subset \(M\). Take an arbitrary \(\varepsilon>0\). Denote by \(\varepsilon M\) the \(\varepsilon\)-neighborhood of \(M\) relative to \(\Xi\):

\[ \varepsilon M\equiv \{\xi\mid \xi\in\Xi,\ \inf \|\xi-\xi'\|\le \varepsilon\}. \tag{4} \]

We shall call the quantity

\[ (\overline{\Omega})(M_{-},M_{+}) \equiv \lim_{\varepsilon\to +0}\max \left\{ \overline{\lim_{\tau\to +\infty}}\frac{1}{\tau}\sup_{\xi\in M_{+}} d\bigl[v(x_{\xi}(\tau))\bigr], \right. \]

\[ \left. -\underline{\lim_{\tau\to +\infty}}\frac{1}{\tau}\inf_{\xi\in M_{-}} d\bigl[v(x_{\xi}(-\tau))\bigr]\right\}. \tag{5} \]

the adjacent small \(vd\)-number of the pair of sets \(M_{-}\) and \(M_{+}\) from \(\Xi\) relative to system (1).

We shall call the pair \(M_{-}, M_{+}\) regular if both sets \(M_{-}\) and \(M_{+}\) are subsets of \(\Xi\), and otherwise irregular. If \((M_{-},M_{+})\) is an irregular pair, then

\[ (\overline{\Omega})(M_{-},M_{+})=+\infty . \]

Let us take a positive function \(\varphi(\tau)\), defined for all \(\tau>0\). The \(\varphi\)-adjacent small \(vd\)-number of the pair \((M_{-},M_{+})\) relative to (1) is called (see [4])

\[ \overline{\Phi}(M_{-},M_{+}) \equiv \max \left\{ \overline{\lim_{\tau\to +\infty}}\frac{1}{\tau}\sup_{\xi\in \varphi(\tau)M_{+}} d\bigl[v(x_{\xi}(\tau))\bigr], \right. \]

\[ \left. -\underline{\lim_{\tau\to +\infty}}\frac{1}{\tau}\inf_{\xi\in \varphi(\tau)M_{-}} d\bigl[v(x_{\xi}(-\tau))\bigr]\right\}. \tag{6} \]

Let us note some properties of adjacent numbers:

1) if \(M'_{-}\subset M_{-}\), \(M'_{+}\subset M_{+}\), then

\[ (\overline{\Omega})(M'_{\pm})\preceq(\overline{\Omega})(M_{\pm}),\qquad \overline{\Phi}(M'_{\pm})\preceq\overline{\Phi}(M_{\pm}); \tag{7} \]

2) if \(\varphi(\tau)\succ \varphi'(\tau)\) for all sufficiently large \(\tau\), then

\[ \overline{\Phi}(M_{\pm})\succ \overline{\Phi}'(M_{\pm}); \]

3) if \(\varphi(\tau)\to 0\), then

\[ \overline{\Phi}(M_{\pm})\preceq(\overline{\Omega})(M_{\pm}). \tag{8} \]

Take two points \(\xi_{-}\) and \(\xi_{+}\) from \(\Xi\). Denote

\[ \overline{\Omega}(\xi_{-},\xi_{+}) \equiv \max \left\{ \overline{\lim_{\tau\to +\infty}}\frac{1}{\tau}d\bigl[v(x_{\xi_{+}}(\tau))\bigr], \right. \]

\[ \left. -\underline{\lim_{\tau\to +\infty}}\frac{1}{\tau}d\bigl[v(x_{\xi_{-}}(\tau))\bigr]\right\}. \tag{9} \]

From the continuous dependence of the solutions of (1) on the initial conditions and from the continuity of the functions \(v\) and \(d\), it follows that there exists a function \(\varphi(\tau)\) such that, for all \(\tau>0\) and all \(\xi'_{\pm}\) from the \(\varphi(\tau)\)-neighborhoods \(u_{\pm}\) of the points \(\xi_{\pm}\), the following hold:

\[ \left|d\bigl[v(x_{\xi_{+}}(\tau))\bigr]-d\bigl[v(x_{\xi'_{+}}(\tau))\bigr]\right|\leq 1, \]

\[ \left|d\bigl[v(x_{\xi_{-}}(-\tau))\bigr]-d\bigl[v(x_{\xi'_{-}}(-\tau))\bigr]\right|\leq 1, \]

and therefore

\[ \overline{\Phi}(\xi_{-},\xi_{+})=\overline{\Omega}(\xi_{-},\xi_{+}). \tag{10} \]

§ 2. The limit set of a point for one system relative to another

For fixed \(\tau\), the vector-function \(x(\tau,\xi)\) realizes a mutually differentiable mapping of \(\Xi\) onto itself

\[ \xi^* = x(\tau,\xi), \qquad \xi \in \Xi, \qquad \tau = \mathrm{fix}. \tag{11} \]

Construct the inverse mapping

\[ \xi = x^*(\tau,\xi^*), \qquad \xi^* \in \Xi, \qquad \tau = \mathrm{fix} \tag{12} \]

and the differential system defining the family of mappings (12),

\[ dx^*/d\tau = f^*(x^*,\tau). \tag{13} \]

We shall call system (13) the adjoint to (1). If the general solution \(x(\tau,\xi)\) of the given system (1) is available, the adjoint system (13) can be constructed by means of the operations of inversion and differentiation of vector-functions (for analogous constructions, see [3]).

Suppose that system (3) satisfies the conditions formulated above for (1). Denote the general solution of (3) by \(y(\tau,\xi)\), \(\xi \in \Xi\), \(\tau \in (-\infty,+\infty)\). Put

\[ F(\tau,\xi) \equiv x^*[\tau,y(\tau,\xi)]. \tag{14} \]

The vector-function \(F\) depends continuously on its arguments. In the particular case \(g(\xi,\tau) \equiv 0\), it is clear that

\[ F(\tau,\xi) \equiv \xi. \]

The limit sets of the point \(\xi\) for system (3) relative to system (1) will be the subsets of \(\Xi\):

\[ F_{-}(\xi) = \bigcap_{\tau_0<0}\overline{\bigcup_{\tau<\tau_0} F(\tau,\xi)}, \tag{15} \]

\[ F_{+}(\xi) = \bigcap_{\tau_0>0}\overline{\bigcup_{\tau>\tau_0} F(\tau,\xi)}. \tag{16} \]

By standard means (see [4], pp. 273–276) one can show that \(F_{\pm}(\xi)\) are nonempty closed connected subsets of \(\Xi\).

§ 3. Estimate of small \(vd\)-numbers

Theorem. Let \(M_{-} \supset F_{-}(\xi)\), \(M_{+} \supset F_{+}(\xi)\). Then

\[ \overline{\Omega}\, y_{\xi} \leqslant (\overline{\Omega})(M_{-},M_{+}). \tag{17} \]

Proof. From the definition of \(F_{+}(\xi)\) it follows that, for arbitrary \(\varepsilon>0\), one can indicate a \(\tau_{\varepsilon}\) such that for all \(\tau>\tau_{\varepsilon}\)

\[ x^*[\tau,y_{\xi}(\tau)] \in \varepsilon F_{+}(\xi) \]

and, a fortiori,

\[ x^*[\tau,y_{\xi}(\tau)] \in \varepsilon M_{+}. \]

For any fixed \(\tau=\tau_0\), the vector-function \(x^*(\tau,\xi^*)\) is inverse to \(x(\tau,\xi)\), and therefore

\[ y_{\xi}(\tau) = x\{\tau,x^*[\tau,y_{\xi}(\tau)]\} \in x(\tau,\varepsilon M_{+}) \qquad \text{for } \tau>\tau_{\varepsilon}. \tag{18} \]

In the same way we prove

\[ y_{\xi}(\tau) \in x(\tau,\varepsilon M_{-}) \qquad \text{for } \tau<-\tau'_{\varepsilon}. \tag{19} \]

From (18) and (19) it follows that, for sufficiently large \(\tau\),

\[ d\bigl[v(y_\xi(\tau))\bigr] \leq \sup_{\xi'\in M_+} d\bigl[v(x(\tau,\xi'))\bigr], \]

\[ d\bigl[v(y_\xi(-\tau))\bigr] \geq \inf_{\xi'\in M_-} d\bigl[v(x(-\tau,\xi'))\bigr], \]

i.e. (see [2, 5]) (17) holds. The theorem is proved.

The following proposition is proved analogously.
Let the positive function \(\varphi(\tau)\) be such that, for all sufficiently large \(\tau\),

\[ F(\tau,\xi)\in \varphi(\tau)M_+ \quad \text{and} \quad F(-\tau,\xi)\in \varphi(-\tau)M_-. \]

Then

\[ \overline{\Omega}_{y_\xi}\leq \overline{\Phi}(M_-,M_+). \]

§ 4. A SUFFICIENT CRITERION FOR THE DEGENERATION OF THE LIMIT SET TO A POINT

We shall denote by \(\partial w/\partial z\) the Jacobi matrix for the vector-function \(w\) with respect to the components of the vector \(z\). As the norm \(\|A\|\) of the matrix \(A\) we choose the third (spherical) norm ([5], pp. 122–129). Finally, the distance from a point \(\xi\in\mathfrak E\) to the boundary of \(\mathfrak E\) will be denoted by \(\delta(\xi)\).

Theorem. Suppose that for all \(\xi\in\mathfrak E\) and for any real \(\tau\) one has

\[ \|\partial f(\xi,\tau)/\partial \xi\|<\alpha(\tau)\lambda(\xi), \tag{20} \]

\[ \|\partial x^*(\tau,\xi)/\partial \xi\|<\beta(\tau)\mu(\xi). \tag{21} \]

Suppose further that

\[ \|g(\xi,\tau)\|<\delta(\xi)\{9\mu(\xi)[1+2\lambda(\xi)\alpha(\tau)]\beta(\tau)e^{3|\tau|}\}^{-1}. \tag{22} \]

Then both limit sets \(F_{\pm}(\xi)\) of any point \(\xi\in\mathfrak E\) for system (3) relative to (1) are points of \(\mathfrak E\).

Proof. Take an arbitrary point \(\eta=(\xi,\tau)\in H\). From (1), (3), and (20) it follows that, for \(\sigma\) sufficiently close to \(\tau\),

\[ \|y(\sigma,\eta)-x(\sigma,\eta)\|\leq \int_{\tau}^{\sigma}\alpha(\omega)\lambda(\xi)\|y(\omega,\eta)- \]

\[ -x(\omega,\eta)\|\,d\omega+\int_{\tau}^{\sigma}\|g(\xi,\omega)\|\,d\omega, \]

and therefore ([6], pp. 47–48)

\[ \|y(\sigma,\eta)-x(\sigma,\eta)\|< \]

\[ <\int_{\tau}^{\sigma}\|g(\xi,\omega)\|\,d\omega+ \int_{\tau}^{\sigma}\alpha(\omega)\lambda(\xi) \left(\int_{\tau}^{\omega}\|g(\xi,\rho)\|\,d\rho\right) \exp\left(\int_{\omega}^{\sigma}\alpha(\rho)\lambda(\xi)\,d\rho\right)d\omega \leq \]

\[ \leq \|g(\xi,\tau)\|\cdot[1+2\alpha(\tau)\lambda(\xi)]\cdot|\sigma-\tau|, \qquad \sigma\ne\tau. \]

On the basis of (21), for sufficiently small \(|\sigma-\tau|\),

\[ \|x^*[\sigma,y(\sigma,\eta)]-x^*[\sigma,x(\sigma,\eta)]\|\leq \]

\[ \leq \beta(\tau)\mu(\xi)\|g(\xi,\tau)\|\cdot[1+2\alpha(\tau)\lambda(\xi)]\,|\sigma-\tau| \]

and, by virtue of (22)

\[ \left\|x^*[\sigma, y(\sigma,\eta)]-x^*[\sigma, x(\sigma,\eta)]\right\| \leq \frac{\delta(\xi)}{9\cdot e^{3|\tau|}}\,|\sigma-\tau|, \]

but (see § 2) \(x^*[\sigma, x(\sigma,\eta)]\equiv \xi\), and therefore

\[ \left\|x^*[\sigma, y(\sigma,\eta)]-\xi\right\| \leq \frac{\delta(\xi)}{9e^{3|\tau|}}\,|\sigma-\tau|. \tag{23} \]

Fix \(\tau\). Put \(F(\tau,\xi)\equiv \tilde{\xi}\). Estimate the difference

\[ F(\sigma,\xi)-F(\tau,\xi) = x^*[\sigma,y(\sigma,\xi)]-\tilde{\xi} = x^*[\sigma,y(\sigma,\tilde{\xi},\tau)]-\tilde{\xi}. \tag{24} \]

On the basis of (23) and (24), for \(\sigma\) sufficiently close to \(\tau\), we have

\[ \|F(\sigma,\xi)-F(\tau,\xi)\| \leq \frac{1}{9}\,\delta(\tilde{\xi})\,|\sigma-\tau|e^{-3|\tau|}. \tag{25} \]

The boundedness of the domain \(\Xi\) entails the boundedness of the quantity \(\delta(\xi)\); therefore it follows from (25) that, for each fixed \(\xi\), the curve \(\xi'=\xi'=F(\tau,\xi)\), \(-\infty<\tau<+\infty\), is rectifiable and hence has endpoints \(F_{\pm}(\xi)\). It remains to show that both points \(F_{\pm}(\xi)\) lie inside \(\Xi\).

On the basis of (25), the half-line \([0,+\infty)\) is covered by a system of open intervals \(\Delta_\tau\equiv(\tau-\gamma_\tau,\tau+\gamma_\tau)\) such that, for \(\sigma\in\Delta_\tau\), (25) holds. From the system \(\{\Delta_\tau\}\) one can select a sequence of overlapping intervals and then construct a sequence

\[ 0=\tau_0<\tau_1<\tau_2<\cdots<\tau_k<\cdots\to+\infty \tag{26} \]

such that for \(\tau,\sigma\in[\tau_{k-1},\tau_k]\), in the case \(\tau,\sigma\geq m\), it is true that

\[ \|F(\sigma,\xi)-F(\tau,\xi)\| \leq \frac{1}{9}\,\delta(\xi_{k-1})\,|\sigma-\tau|e^{-3m}, \tag{27} \]

where \(\xi_k=F(\tau_k,\xi)\). Without loss of generality, one may assume that the sequence (26) contains all natural numbers \(m\). Denote \(\rho_k\equiv\|\xi_k\|\). Suppose that the interval \([m-1,m]\) is covered by the intervals

\[ [\tau_{k_{m-1}},\tau_{k_{m-1}+1}],\, [\tau_{k_{m-1}+1},\tau_{k_{m-1}+2}],\,\ldots,\, [\tau_{k_m-1},\tau_{k_m}]. \tag{28} \]

From (27) we derive (\(l=k_{m-1}\), \(r=k_m\))

\[ \|F(\tau,\xi)-F(\tau_l,\xi)\| \leq \frac{1}{9}\rho_l(\tau_{l+1}-\tau_l)e^{-3m} \qquad \text{for } \tau\in[\tau_l,\tau_{l+1}], \]

\[ |\rho_{l+1}-\rho_l| \leq \frac{1}{9}\rho_l(\tau_{l+1}-\tau_l)e^{-3m}, \]

\[ \rho_{l+1} \leq \rho_l\left[1+\frac{1}{9}(\tau_{l+1}-\tau_l)e^{-3m}\right], \]

\[ \cdots\cdots\cdots\cdots\cdots\cdots \]

\[ \|F(\tau,\xi)-F(\tau_{r-1},\xi)\| \leq \frac{1}{9}\rho_{r-1}(\tau_r-\tau_{r-1})e^{-3m} \qquad \text{for } \tau\in[\tau_{r-1},\tau_r], \]

\[ |\rho_r-\rho_{r-1}| \leq \frac{1}{9}\rho_{r-1}(\tau_r-\tau_{r-1})e^{-3m}, \]

\[ \rho_r \leq \rho_{r-1}\left[1+\frac{1}{9}(\tau_r-\tau_{r-1})e^{-3m}\right]. \]

Consequently,

\[ \rho_r \leqslant \rho_l \prod_{\mu=l+1}^{r} \left[1+\frac{1}{9}(\tau_\mu-\tau_{\mu-1})e^{-3m}\right]< \]

\[ <\rho_l \exp \sum_{\mu=l+1}^{r}\frac{1}{9}(\tau_\mu-\tau_{\mu-1})e^{-3m}, \]

\[ \rho_r < \rho_l \exp\left(\frac{1}{9}e^{-3m}\right), \tag{29} \]

\[ \|F(\tau,\xi)-F(\tau_l,\xi)\|\leqslant \frac{1}{9}\rho_l\exp\left(\frac{1}{9}e^{-3m}-3m\right) \quad \text{for } \tau\in[m-1,m]. \tag{30} \]

If formulas (29) and (30) are applied \(m\) times, we obtain

\[ \rho_r < \rho_0 \prod_{k=1}^{m}\exp\left(\frac{1}{9}e^{-3k}\right)< \]

\[ <\rho_0\exp\sum_{k=1}^{\infty}(e^{-3})^k < \rho_0 e^{\frac{e^3}{e^3-1}}<\frac{6}{5}\rho_0, \]

\[ \|F(\tau,\xi)-\xi\|=\|F(\tau,\xi)-F(\tau_0,\xi)\|\leqslant \]

\[ \leqslant \frac{1}{9}\cdot\frac{6}{5}\cdot\rho_0 \sum_{k=1}^{\infty}\exp\left(\frac{1}{9}e^{-3k}-3k\right)< \]

\[ <\frac{2}{15}\rho_0\sum_{k=1}^{\infty}(e^{-2})^k <\frac{2}{15}\rho_0\frac{e^2}{e^2-1} <\frac{\rho_0}{2}. \]

Thus,

\[ \|F(\tau,\xi)-\xi\|<\frac{\rho_0}{2} =\frac{1}{2}\|\xi\| \quad \text{for } \tau\geqslant 0 \]

or

\[ \|F(\tau,\xi)\|>\frac{1}{2}\|\xi\|>0 \quad \text{for } \tau\geqslant 0. \tag{31} \]

It follows from (31) that \(F_+(\xi)\ne O_0\). Similarly we show that \(F(\tau,\xi)\) for \(\tau\geqslant 0\) is at a distance greater than \(\frac{1}{2}\delta(\xi)\) from the boundary \(\Xi\), and therefore

\[ F_+(\xi)\cap\Xi_{\mathrm{gr}}=\varnothing. \]

Thus, \(F_+(\xi)\subset\Xi\). By similar arguments we show that also \(F_-(\xi)\subset\Xi\). The theorem is proved.

§ 5. LOCATION OF THE SMALL \(vd\)-NUMBERS OF THE PERTURBED SYSTEM

From the arguments of § 4 it follows that, for sufficiently large \(\tau\),

\[ \|F(\tau,\xi)-F_+(\xi)\|<2\delta[F_+(\xi)]\cdot\frac{1}{9} \int_{\tau}^{+\infty} e^{-3\sigma}\,d\sigma, \]

\[ \|F(-\tau,\xi)-F_-(\xi)\|<2\delta\,|F_-(\xi)|\cdot \frac{1}{9}\int_\tau^{+\infty} e^{-3\sigma}\,d\sigma \]

or

\[ \|F(\pm\tau,\xi)-F_\pm(\xi)\|<\frac{1}{13}\delta[F_\pm(\xi)]e^{-3\tau}. \tag{32} \]

The estimates carried out in the proof of Theorem § 4 show that if \(g(\xi,\tau)\) is multiplied by a positive function \(\psi(\xi,\tau)\), even and discontinuous in \(\tau\), continuously differentiable in \(\xi\), monotonically decreasing for positive \(\tau\), and not exceeding unity in absolute value, then the estimate corresponding to (32) will take the form

\[ \|F(\pm\tau,\xi)-F_\pm(\xi)\|<\frac{1}{13}\delta[F_\pm(\xi)]e^{-3\tau}\cdot\psi[F_\pm(\xi),\tau]. \tag{33} \]

Let us choose definite functions \(v(\xi)\) and \(d(y)\). For system (1), on the basis of the last paragraph of § 1, we construct a function \(\varphi(\xi,\tau)\) such that (10) is satisfied. We require that

\[ \|g(\xi,\tau)\|<\delta(\xi)\{9\mu(\xi)[1+2\lambda(\xi)\alpha(\tau)]\beta(\tau)e^{3|\tau|}\}^{-1}\psi(\xi,\tau), \]

\[ \frac{1}{13}\delta(\xi)e^{-3|\tau|}\psi(\xi,\tau)<\varphi(\xi,\tau). \]

Then, first, the limiting sets of points \(\Xi\) for system (3) relative to (1) are points of \(\Xi\), and, second, the aggregate of small \(vd\)-numbers of nonzero solutions of system (3) will constitute a subset of the aggregate \(\{\overline{\Omega}(\xi_-,\xi_+)\}\), constructed for all possible pairs \((\xi_-,\xi_+)\in\Xi\times\Xi\).

References

1. Bogdanov Yu. S. DAN SSSR, 158, No. 1, 9—12, 1964.
2. Bogdanov Yu. S. Differential Equations, 1, No. 1, 41—52, 1965.
3. Bogdanova M. P. DAN BSSR, 6, No. 5, 285—287, 1962.
4. Nemytskii V. V., Stepanov V. V. Qualitative Theory of Differential Equations. Gostekhizdat, 1947, p. 448.
5. Faddeev D. K., Faddeeva V. N. Computational Methods of Linear Algebra. Fizmatgiz, 1960, p. 656.
6. Coddington E. A., Levinson N. Theory of Ordinary Differential Equations. IL, 1958, p. 474.

Received by the editors
January 15, 1966

Belorussian State University
named after V. I. Lenin