UDC 517.941

ON THE GEOMETRIC LOCATION AND ESTIMATION OF THE GROWTH OF SOLUTIONS OF PERTURBED SYSTEMS

B. F. BYLOV

§ 1. PRELIMINARY REMARKS

In the present article we give a generalization of the well-known Perron theorem [1], as well as of the results of a number of works [2—4] devoted to the study of the behavior of solutions of a perturbed system of the form

\[ \frac{dx}{dt}=P(t)x+f(t,x), \tag{1.1} \]

where \(x\) is an \(n\)-dimensional vector, \(P(t)\) is a diagonal matrix, and \(f(t,x)\) is a vector function satisfying the conditions

\[ f(t,0)=0, \]

\[ \|f(t,x')-f(t,x'')\|\leq \delta(t)\|x'-x''\|. \tag{1.2} \]

Before specifying the conditions to which we subject the system under consideration and formulating the main result, we shall make several preliminary remarks.

1. We shall regard the system of indices \((1,2,3,\ldots,n)\) as divided, in the order of succession of the indices, into \(q\) nonempty groups \(n_1,n_2,\ldots,n_q\).

In order not to introduce new notation, we shall assume that \(n_k\) also denotes the number of indices entering into the \(k\)-th group, so that \(n_1+n_2+\cdots+n_q=n\). We denote by \(n_0\) and \(n_{q+1}\) empty systems of indices.

1. By \(L^n\) we denote the \(n\)-dimensional complex Euclidean space and fix in it some orthonormal basis \(e_1,\ldots,e_n\). Every vector \(x\) with coordinates \(\{x_1,\ldots,x_n\}\) will be regarded as the radius vector of the point \((x_1,\ldots,x_n)\) in \(L^n\), and subsequently, as desired, \(x\) may be understood either as a vector or as a point.

By \(L^{n_{\alpha_1} n_{\alpha_2}\cdots n_{\alpha_p}}\) we denote the linear subspace in \(L^n\) whose basis consists of the vectors \(e_i\) taken under the condition \(i\subset n_{\alpha_1},\ldots,n_{\alpha_p}\).

1. Let \(h>0\) be a certain number. We shall call cones \(K^{n_1\cdots n_l}(h)\) and \(K^{n_m\cdots n_q}(h)\) \((l=1,\ldots,q;\; m=1,\ldots,q)\) the open sets in \(L^n\) defined respectively by the inequalities

\[ K^{n_1\cdots n_l}(h):\quad h\sum_{i\subset n_1,\ldots,n_l}|x_i|^2> \sum_{i\subset n_{l+1},\ldots,n_q}|x_i|^2, \tag{1.3} \]

\[ K^{n_m\cdots n_q}(h):\quad h\sum_{i\subset n_m,\ldots,n_q}|x_i|^2> \sum_{i\subset n_1,\ldots,n_{m-1}}|x_i|^2. \tag{1.4} \]

If the coordinates of the vector \(x=\{x_1,\ldots,x_n\}\) satisfy (1.3) or (1.4), then we shall agree to say that the vector \(x\) belongs to the corresponding cone. If one performs the decomposition \(x=a+b\), where \(a\subset L^{n_1,\ldots,n_l}\), \(b\subset L^{n_{l+1},\ldots,n_q}\) (possible in a unique way by virtue of the orthogonality of the corresponding subspaces), then, obviously, condition (1.3) for the belonging of the vector \(x\) to the cone \(K^{n_1\cdots n_l}(h)\) can be written in the form \(h\|a\|^2>\|b\|^2\), and condition (1.4) in the form \(h\|b\|^2>\|a\|^2\).

It follows from the definition, in particular, that \(K^{n_1\cdots n_q}(h)\) coincides with \(L^n\) except for the point \(O\).

We denote by \(H^{n_1\cdots n_l}(h)\) and \(H^{nm\cdots n_q}(h)\) the sets of boundary points of the corresponding cones, assuming that the vertex of each cone, i.e. the point \(O\), is not included in these sets.

The closure of a set \(A\subset L^n\) will be denoted by \(\overline A\), and, in order not to confuse the closure sign with the sign of complex conjugation (which will occur later), we shall agree to denote the latter by the symbol \(\sim\). From the definitions it easily follows that

\[ H^{n_1\cdots n_k}(h)=H^{n_{k+1}\cdots n_q}\left(\frac1h\right), \tag{1.5} \]

\[ K^{n_1\cdots n_k}(h)\cup H^{n_1\cdots n_k}(h)\cup K^{n_{k+1}\cdots n_q}\left(\frac1h\right)\cup 0=L^n, \tag{1.6} \]

\[ \overline K^{\,n_1\cdots n_k}(h)\cup K^{n_{k+1}\cdots n_q}\left(\frac1h\right)=L^n \quad (k=1,2,\ldots,q-1). \tag{1.7} \]

Put

\[ K^{n_1\cdots n_l}(h)\cap K^{nm\cdots n_q}(h)=\Pi^{nm\cdots n_l}(h). \]

For \(m\le l\), each of these intersections is nonempty, since \(\Pi^{nm\cdots n_l}(h)\) certainly contains those points for which
\[ \sum_{i\subset nm,\ldots,n_l}|x_i|^2\ne 0 \]
and \(x_i=0\) for \(i\subset n_m,\ldots,n_l\). In particular, as is not difficult to see,

\[ \Pi^{n_1}(h)=K^{n_1}(h) \quad\text{and}\quad \Pi^{n_q}(h)=K^{n_q}(h). \tag{1.8} \]

Lemma 1.1. If \(0<h<1\) and \(m>l\), then the cones \(K^{n_1\cdots n_l}(h)\) and \(K^{nm\cdots n_q}(h)\) have no common points.

Proof. Suppose, contrary to the assertion, that there is a point \(x\) belonging to the intersection of these cones. Then for the coordinates of this point, with the aid of (1.3) and (1.4), and taking into account that \(l<m\), we have

\[ h^2\sum_{i\subset n_1,\ldots,n_{m-1}} |x_i|^2 \ge h^2\sum_{i\subset n_1,\ldots,n_l}|x_i|^2 > h\sum_{i\subset n_{l+1},\ldots,n_q}|x_i|^2 \ge \]

\[ \ge h\sum_{i\subset nm,\ldots,n_q}|x_i|^2 > \sum_{i\subset n_1,\ldots,n_{m-1}} |x_i|^2 . \]

Comparing the left- and right-hand sides of the resulting strict inequality, we notice that it is contradictory, since \(h^2<1\). The contradiction arose from the assumption of the existence of a point of intersection of the cones considered in the lemma. The lemma is proved.

Corollary. The sets \(\Pi(h)^{nm_1\cdots n_{l_1}}\) and \(\Pi(h)^{nm_2\cdots n_{l_2}}\) do not intersect if \(l_1<m_2\).

1. Let \(L^n\) be decomposed into a direct sum \(L^n=L_1\oplus L_2\) of orthogonal subspaces. A manifold \(E\subset L^n\) is called a regular image of class \(\Lambda(h)\) of the subspace \(L_1\) if \(E\) is the set of points of the form \(x=a+\varphi(a)\), where \(a\subset L_1\) and \(\varphi(a)\) is a single-valued operator defined on \(L_1\) and such that \(\varphi(a)\subset L_2\), \(\varphi(0)=0\), and

\[ \|\varphi(a')-\varphi(a'')\|<h\|a'-a''\| \tag{1.9} \]

for any \(a'\ne a''\subset L_1\).

It follows readily from this definition that the equality \(x=a+\varphi(a)=\Phi(a)\) establishes a homeomorphic mapping of the subspace \(L_1\) onto \(E\), satisfying a Lipschitz condition with constant \(1+h\). Moreover, the inverse mapping coincides on \(E\) with the operator \(P_1\) of projection onto the subspace \(L_1\) and, consequently, the inverse operator \(\Phi^{-1}\) satisfies a Lipschitz condition with constant equal to one.

1. To study the location of the solutions of system (1.1) in the space \(L^n\), we shall use the topological principle of Ważewski [5]; in this connection we introduce the following definitions*).

Let \(B\subset A\) be certain subsets of \(L^n\). If there exists a continuous mapping of \(A\) onto \(B\) leaving the points of \(B\) fixed, then \(B\) is called a retract of \(A\), and the mapping itself a retraction.

Suppose that system (1.1) possesses the properties of existence, uniqueness, and continuability of solutions (and consequently also continuous dependence on initial data), and that there is some open domain \(G\subset L^n\) with boundary \(\Gamma\). A point \(x_0\subset \Gamma\) is called an exit point with respect to the domain \(G\) if, for an arbitrarily prescribed instant \(t_0\), the solution \(x(t)\) satisfying the initial condition \(x(t_0)=x_0\), for all \(t<t_0\), belongs to \(G\); and \(x_0\subset \Gamma\) is called a strict exit point if, in addition, there exists an \(\varepsilon>0\) such that for all \(t\) in the interval \(t_0<t<t_0+\varepsilon\) the solution \(x(t)\) lies outside \(G\cup \Gamma\). Denote by \(A\) the set of exit points and by \(A^*\) the set of strict exit points. Let, moreover, some set \(D\subset G\cup \Gamma\) be given and put \(S=A\cap D\). Ważewski’s principle consists in the following assertion.

If \(A=A^*\) and \(S\) is a retract of \(A\), but is not a retract of \(D\), then on \(D-S\), at an arbitrarily prescribed instant \(t_0\), there is a point \(x_0\) through which passes a solution \(x(t)\) \((x(t_0)=x_0)\) that remains in the domain \(G\) for all \(t\).

If one wishes to use Ważewski’s principle in its usual formulation, one may pass to the space \(L^n\times I\) (\(I\) is the \(t\)-axis), putting \(G_1=G\times I\), \(\Gamma_1=\Gamma\times I\), \(A_1=A\times I\), \(D_1=(D,t_0)\). Here the last set must be understood as the totality of points \((x_0,t_0)\) of the space \(L^n\times I\) such that \(x_0\subset D\) and \(t=t_0\). It is obvious that, as before, \(A_1=A_1^*\), and the set \(S_1=A_1\cap D_1\) is a retract of \(A_1\), but not of \(D_1\). (The retraction of \(A_1\) onto \(S_1\) can be established by first mapping a point \((x_0,t)\subset A_1\) to the point \((x_0,t_0)\)—projection along the \(t\)-axis—and then, with the aid of the given retraction of \(A\) onto \(S\), to a point of the set \(S_1=(S,t_0)\).)

§ 2. FORMULATION AND PROOF OF THE MAIN THEOREM

We shall assume that the diagonal matrix \(P\) in system (1.1) is real, continuous on the whole axis, and decomposes into blocks

\[ P=\operatorname{diag}\{P_1,\ldots,P_q\}, \]

* In Ważewski’s work the solutions of the system are considered as trajectories in the \((n+1)\)-dimensional space \((t,x)\). For us, however, it will be more convenient to remain in the space \(L^n\), and therefore we somewhat modify the definitions given by Ważewski.

where each block \(P_k\) is a diagonal matrix of order \(n_k\), and for \(i \subset n_k\) the elements \(p_i=p_i(t)\) are given the general estimates

\[ r_k(t)\leq p_i(t)\leq R_k(t)\qquad (i\subset n_k,\ k=1,2,\ldots,q), \tag{2.1} \]

where \(r_k(t)\) and \(R_k(t)\) are certain continuous functions satisfying, in the case \(q>1\), the separation condition

\[ R_k(t)\leq r_{k+1}(t)-c(t) \tag{2.2} \]

with some continuous function \(c(t)>0\) having divergent integrals

\[ \int_{-\infty}^{0} c(\tau)\,d\tau=\int_{0}^{\infty} c(\tau)\,d\tau=\infty . \tag{2.3} \]

In particular, \(c(t)\) may be constant. From (2.1) and (2.2) it follows that, if \(p_i\subset P_m\), \(p_j\subset P_l\), and \(m<l\), the inequality

\[ p_j\geq p_i+c(t)(l-m) \]

will hold. Therefore, for \(l>m\) we shall agree to call the block \(P_l\) senior in comparison with the block \(P_m\).

Cases in which the requirements listed above can be weakened will be given at the end of the paper.

We shall also write system (1.1) in coordinate form as

\[ \frac{dx_1}{dt}=p_1(t)x_1+f_1(t,x), \]

\[ \frac{dx_2}{dt}=p_2(t)x_2+f_2(t,x), \tag{2.4} \]

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

\[ \frac{dx_n}{dt}=p_n(t)x_n+f_n(t,x). \]

Theorem. If system (1.1) satisfies conditions (1.2), (2.1), (2.2), and (2.3), then for any \(h_0(0<h_0<1)\) and \(t_0(-\infty<t_0<+\infty)\) one can specify an \(\varepsilon>0\) such that, when the inequality

\[ \delta(t)<\varepsilon c(t)\qquad (-\infty<t<+\infty) \tag{2.5} \]

holds, there exist in \(L^n\) two pyramids composed of manifolds

\[ 0=E_{t_0}^{n_0}\subset E_{t_0}^{n_1}\subset E_{t_0}^{n_1n_2}\subset\ldots\subset E_{t_0}^{n_1\cdots n_q}=L^n, \tag{2.6} \]

\[ 0=E_{t_0}^{n_{q+1}}\subset E_{t_0}^{n_q}\subset E_{t_0}^{n_{q-1}n_q}\subset\ldots E_{t_0}^{n_1\cdots n_q}=L^n, \tag{2.7} \]

depending on \(t_0\) and such that

1) Every nontrivial solution \(x(t)\) which begins at the moment \(t_0\) at a point \(x_0\subset E_{t_0}^{n_1\cdots n_l}\), belongs for all \(t\) to the cone \(K^{n_1\cdots n_l}(h_0)\), and for any \(t\geq s\) the following estimate is valid for the solution under consideration:

\[ \exp\left(\int_s^t [r_1(\tau)-\delta(\tau)]\,d\tau\right) \leq \frac{\|x(t)\|}{\|x(s)\|} < \]

\[ <\sqrt{1+h_0}\, \exp\left(\int_s^t [R_l(\tau)+\delta(\tau)(1+h_0)]\,d\tau\right). \tag{2.8} \]

In this case the manifold \(E_{t_0}^{n_1\cdots n_l}\) \((l<q)\) is a regular image of class \(\Lambda(\sqrt{h_0})\) of the subspace \(L^{n_1\cdots n_l}\).

2) Every nontrivial solution \(x(t)\), starting at the moment \(t_0\) at a point \(x_0\subset E_{t_0}^{n_m\cdots n_q}\), belongs for all \(t\) to the cone \(K^{n_m\cdots n_q}(h_0)\), and for any \(t\ge s\) the estimate

\[ \frac{1}{\sqrt{1+h_0}}\exp\left(\int_s^t [r_m(\tau)-\delta(\tau)(1+h_0)]\,d\tau\right) < \frac{\|x(t)\|}{\|x(s)\|} \le \]

\[ \le \exp\left(\int_s^t [R_q(\tau)+\delta(\tau)]\,d\tau\right). \tag{2.9} \]

holds for the solution under consideration.

In this case the manifold \(E_{t_0}^{n_m\cdots n_q}\) \((m>1)\) is a regular image of class \(\Lambda(\sqrt{h_0})\) of the subspace \(L^{n_m\cdots n_q}\).

3) Each set

\[ E_{t_0}^{n_m\cdots n_l}=E_{t_0}^{n_1\cdots n_l}\cap E_{t_0}^{n_m\cdots n_q}\qquad (m\le l) \]

is a regular image of class

\[ \Lambda\left(\sqrt{\frac{2h_0}{1-h_0}}\right) \]

of the subspace \(L^{n_m\cdots n_l}\), and a nontrivial solution \(x(t)\), starting at the moment \(t_0\) at a point \(x_0\) of this manifold, belongs for all \(t\) to \(\Pi^{n_m\cdots n_l}(h_0)\) and for all \(t\ge s\) satisfies the inequality

\[ \frac{1}{\sqrt{1+h_0}}\exp\left(\int_s^t [r_m(\tau)-\delta(\tau)(1+h_0)]\,d\tau\right) < \frac{\|x(t)\|}{\|x(s)\|} < \]

\[ < \sqrt{1+h_0}\, \exp\left(\int_s^t [R_l(\tau)+\delta(\tau)(1+h_0)]\,d\tau\right). \tag{2.10} \]

4) For an arbitrarily chosen \(t_0\), the space \(L^n\), in accordance with (2.6) and (2.7), can be decomposed in two ways into set-theoretic sums

\[ L^n=\bigcup_{l=1}^{q}(E_{t_0}^{n_1\cdots n_l}-E_{t_0}^{n_1\cdots n_{l-1}})\cup O, \]

\[ L^n=\bigcup_{m=1}^{q}(E_{t_0}^{n_m\cdots n_q}-E_{t_0}^{n_{m+1}\cdots n_q})\cup O, \]

the summands in each of which (the stages of the pyramids (2.6) and (2.7)) have no common points, and therefore

\[ L^n= \bigcup_{(m,l)} \left[ (E_{t_0}^{n_1\cdots n_l}-E_{t_0}^{n_1\cdots n_{l-1}}) \cap (E_{t_0}^{n_m\cdots n_q}-E_{t_0}^{n_{m+1}\cdots n_q}) \right]\cup O, \tag{2.11} \]

where the summands enclosed in square brackets have no common points. For \(l<m\) the corresponding sets are empty, while for \(l\ge m\) each set

\[ (E_{t_0}^{n_1\cdots n_l}-E_{t_0}^{n_1\cdots n_{l-1}}) \cap (E_{t_0}^{n_m\cdots n_q}-E_{t_0}^{n_{m+1}\cdots n_q}) \]

has topological dimension \(n_m+n_{m+1}+\cdots+n_l\), and for any solution beginning at time \(t_0\) at a point \(x_0\) belonging to this set, one can indicate times \(t_2 \leq t_0 \leq t_1\) such that, for \(t \leq t_2\), the solution \(x(t)\) will belong to \(\Pi^{n_m}(h_0)\), and for all \(s \leq t \leq t_2\) the inequality

\[ \frac{1}{\sqrt{1+h_0}} \exp\left(\int_s^t [r_m(\tau)-\delta(\tau)(1+h_0)]\,d\tau\right) < \frac{\|x(t)\|}{\|x(s)\|} < \sqrt{1+h_0}\, \exp\left(\int_s^t [R_m(\tau)+\delta(\tau)(1+h_0)]\,d\tau\right), \tag{2.12} \]

will hold, while for \(t \geq t_1\) the solution \(x(t)\) will belong to \(\Pi^{n_l}(h_0)\), and for all \(t_1 \leq s \leq t\) the inequality

\[ \frac{1}{\sqrt{1+h_0}} \exp\left(\int_s^t [r_l(\tau)-\delta(\tau)(1+h_0)]\,d\tau\right) < \frac{\|x(t)\|}{\|x(s)\|} < \sqrt{1+h_0}\, \exp\left(\int_s^t [R_l(\tau)+\delta(\tau)(1+h_0)]\,d\tau\right). \tag{2.13} \]

will hold.

5) For arbitrary \(t_0\) there exists a common homeomorphism \(\Phi_{t_0}\) of the space \(L^n\) onto itself, under which

\[ \Phi_{t_0}\bigl(L^{n_1\cdots n_l}\bigr)=E_{t_0}^{n_1\cdots n_l}\quad (l=1,\ldots,q), \tag{2.14} \]

\[ \Phi_{t_0}\bigl(L^{n_m\cdots n_q}\bigr)=E_{t_0}^{n_m\cdots n_q}\quad (m=1,\ldots,q), \tag{2.15} \]

\[ \Phi_{t_0}\bigl(L^{n_m\cdots n_l}\bigr)=E_{t_0}^{n_m\cdots n_l}\quad (m\leq l), \tag{2.16} \]

\[ \Phi_{t_0}\bigl([L^{n_1\cdots n_l}-L^{n_1\cdots n_{l-1}}]\cap [L^{n_m\cdots n_q}-L^{n_{m+1}\cdots n_q}]\bigr) = \]

\[ = \bigl(E_{t_0}^{n_1\cdots n_l}-E_{t_0}^{n_1\cdots n_{l-1}}\bigr)\cap \bigl(E_{t_0}^{n_m\cdots n_q}-E_{t_0}^{n_{m+1}\cdots n_q}\bigr). \tag{2.17} \]

The homeomorphism \(\Phi_{t_0}\) satisfies a Lipschitz condition and has the structure

\[ \Phi_{t_0}(a)=a+\varphi_{t_0}(a),\qquad \varphi_{t_0}(a)=0, \]

where \(\varphi_{t_0}(a)\) satisfies a Lipschitz condition with constant \(H(h_0)\), depending on \(h_0\) in such a way that \(H(h_0)\to 0\) as \(h_0\to 0\). It follows from this that, for small \(h_0\), the homeomorphism \(\Phi_{t_0}\) is close to the identity mapping. The inverse homeomorphism \(\Phi_{t_0}^{-1}\) satisfies a Lipschitz condition and on each of the sets \(E_{t_0}^{n_k}\) \((k=1,\ldots,q)\) coincides with the operator of projection onto the corresponding subspace \(L^{n_k}\).

For convenience of exposition, we shall divide the proof of the theorem into a number of lemmas, indicating in each of them the smallness of \(\varepsilon\) needed for the corresponding assertion to hold. In doing so we shall use a simple device that makes it possible to pass from the study of the location of solutions of system (1.1) with respect to the “younger” cones \(K^{n_1\cdots n_l}(h)\) to the study of the location of solutions with respect to the “older” cones \(K^{n_m\cdots n_q}(h)\). The essence of this device is as follows.

Together with system (1.1), we shall consider the system

\[ \frac{dx}{d\theta}=-P(-\theta)-f(-\theta,x), \tag{2.18} \]

obtained from (1.1) by the change of time \(t=-\theta\). Here

\[ -P(-\theta)=\operatorname{diag}\{-P_1(-\theta),\ldots,-P_q(-\theta)\} \tag{2.19} \]

and, in accordance with (2.1), (2.2), and (2.3), we shall have, for \(i\subset n_k\),

\[ -R_k(-\theta)\leq -p_i(-\theta)\leq -r_k(-\theta) \tag{2.20} \]

and, for \(q>1\),

\[ -r_{k+1}(-\theta)\leq -R_k(-\theta)-c(-\theta), \tag{2.21} \]

where

\[ \int_0^\infty c(-\theta)\,d\theta = \int_{-\infty}^0 c(-\theta)\,d\theta = \infty . \tag{2.22} \]

Thus, for system (2.18), analogous conditions of separation of the blocks are satisfied with the function \(c(-\theta)\) and with the sole difference that in (2.19) the blocks are arranged in decreasing order of seniority. This difference is immaterial, since, if desired, one can change the numbering of the coordinates. At the same time, the ratio

\[ \frac{\delta(-\theta)}{c(-\theta)} \]

is estimated by the same constant as the ratio

\[ \frac{\delta(t)}{c(t)}. \]

Therefore every assertion established for solutions of system (1.1) and conditioned only by the smallness of the ratio \(\delta(t)/c(t)\) will also be valid for solutions of system (2.18) in the corresponding form, taking into account the reverse arrangement of the blocks in (2.19) by seniority and the modified estimates (2.20) and (2.21).

Under the inverse substitution \(\theta=-t\), this property of solutions of system (2.18) will reduce to a certain new property of solutions of system (1.1). Let us note, in addition to what has been said, that if \(x(t)\) is a solution of system (1.1), then \(x(-\theta)=x_1(\theta)\) is a solution of system (2.18), and conversely. Every solution \(x(t)\) determines in \(L^n\) a certain trajectory. It is obvious that the substitution \(t=-\theta\) leaves each trajectory, understood as the geometric locus of points, unchanged. We shall agree to call this method of investigation, outlined in general terms, the principle of duality.

Let an integer \(m\) \((2\leq m\leq q)\) and \(h_0\) \((0<h_0<1)\) be fixed numbers. Take an arbitrary \(h\) satisfying the inequality

\[ h_0\leq h\leq \frac{1}{h_0}, \tag{2.23} \]

and consider the function

\[ V(m,h,u)=h\sum_{i\subset n_m,\ldots,n_q}|u_i|^2-\sum_{i\subset n_1,\ldots,n_{m-1}}|u_i|^2. \tag{2.24} \]

Put \(u=x'-x''\), where \(x'=x'(t)\) and \(x''=x''(t)\) are any two noncoinciding solutions of system (1.1). In particular, one may take \(x'=x(t)\) to be an arbitrary nontrivial solution and \(x''\equiv0\); then \(u=x(t)\).

Lemma 2.1. If the functions \(c(t)\) and \(\delta(t)\) in (1.2) and (2.2) are such that

\[ \delta(t)<c(t)\frac{h_0^2}{1+h_0}, \tag{2.25} \]

and for some instant \(t_0\) the inclusion

\[ u(t_0)\subset H^{n_m\ldots n_q}(h)\qquad \left(m\geqslant 2,\ h_0\leqslant h\leqslant \frac1{h_0}\right), \]

holds, then

\[ \left.\frac{dV(m,h,u)}{dt}\right|_{t=t_0}>0. \tag{2.26} \]

Proof. Using the representation of system (1.1) in the form (2.4), we carry out the following calculations:

\[ \begin{aligned} \frac12\frac{dV}{dt} &=h\,\operatorname{Re}\sum_{i\subset n_m,\ldots,n_q} \bigl[p_i u_i+f_i(t,x')-f_i(t,x'')\bigr]\tilde u_i \\ &\quad-\operatorname{Re}\sum_{i\subset n_1,\ldots,n_{m-1}} \bigl[p_i u_i+f_i(t,x')-f_i(t,x'')\bigr]\tilde u_i \\ &=h\sum_{i\subset n_m,\ldots,n_q}p_i|u_i|^2 -\sum_{i\subset n_1,\ldots,n_{m-1}}p_i|u_i|^2 \\ &\quad+h\,\operatorname{Re}\sum_{i\subset n_m,\ldots,n_q} \bigl[f_i(t,x')-f_i(t,x'')\bigr]\tilde u_i \\ &\quad-\operatorname{Re}\sum_{i\subset n_1,\ldots,n_{m-1}} \bigl[f_i(t,x')-f_i(t,x'')\bigr]\tilde u_i . \end{aligned} \]

With the aid of conditions (2.1), (2.2), (2.23), (1.2), and the Bunyakovsky inequality, we obtain the estimate:

\[ \begin{aligned} \frac12\frac{dV}{dt} &\geqslant h r_m(t)\sum_{i\subset n_m,\ldots,n_q}|u_i|^2 -\bigl[r_m(t)-c(t)\bigr]\sum_{i\subset n_1,\ldots,n_{m-1}}|u_i|^2 \\ &\quad-\frac1{h_0}\sum_{i=1}^n |f_i(t,x')-f_i(t,x'')|\,|u_i| \geqslant \\ &\geqslant r_m(t)\left\{ h\sum_{i\subset n_m,\ldots,n_q}|u_i|^2 -\sum_{i\subset n_1,\ldots,n_{m-1}}|u_i|^2 \right\} \\ &\quad+c(t)\sum_{i\subset n_1,\ldots,n_{m-1}}|u_i|^2 -\frac1{h_0}\|f(t,x')-f(t,x'')\|\,\|u\| \geqslant \tag{2.27}\\ &\geqslant r_m(t)V +c(t)\sum_{i\subset n_1,\ldots,n_{m-1}}|u_i|^2 -\frac{\delta(t)}{h_0}\sum_{i=1}^n |u_i|^2 \geqslant \\ &\geqslant r_m(t)V +\left[c(t)-\frac{\delta(t)}{h_0}\right] \sum_{i\subset n_1,\ldots,n_{m-1}}|u_i|^2 -\frac{\delta(t)}{h_0^2}\,h \sum_{i\subset n_m,\ldots,n_q}|u_i|^2 \end{aligned} \]

(the introduction of the factor \(\frac{h}{h_0}\geqslant 1\) can only increase the subtrahend and strengthen the inequality).

Since from condition (2.25) it follows that \(c(t)-\dfrac{\delta(t)}{h_0}>\dfrac{\delta(t)}{h_0^2}\), continuing the preceding estimate, we obtain

\[ \frac{1}{2}\,\frac{dV}{dt}\geq r_m(t)V+ \frac{\delta(t)}{h_0^2} \left\{ \sum_{i\subset n_1,\ldots,n_{m-1}} |u_i|^2 -h\sum_{i\subset n_m,\ldots,n_q}|u_i|^2 \right\}, \]

whence, finally,

\[ \frac{1}{2}\,\frac{dV}{dt}\geq \left\{r_m(t)-\frac{\delta(t)}{h_0^2}\right\}V. \tag{2.28} \]

Let us note in particular that when

\[ \sum_{i\subset n_1,\ldots,n_{m-1}} |u_i|^2\ne 0 \]

the sign of a strict inequality holds in (2.28), since in this case replacing the factor \(c(t)-\dfrac{\delta(t)}{h_0}\) by \(\dfrac{\delta(t)}{h_0^2}\) leads to a strict decrease of the right-hand side of inequality (2.27). But precisely this circumstance holds under the conditions of the lemma at the instant \(t_0\), for the inclusion \(u(t_0)\subset H^{n_m\ldots n_q}(h)\) is equivalent to the equality

\[ h\sum_{i\subset n_m,\ldots,n_q}|u_i(t_0)|^2 = \sum_{i\subset n_1,\ldots,n_{m-1}}|u_i(t_0)|^2 \quad (\|u(t_0)\|\ne 0). \tag{2.29} \]

The right-hand side of this equality is nonzero, since otherwise we would have \(\|u(t_0)\|=0\). Thus, at the instant \(t_0\) under consideration, the sign of a strict inequality holds in (2.28). At the same time, from (2.29) it follows that \(V(m,h,u(t_0))=0\), and therefore (2.28) at the instant \(t_0\) turns into the required inequality (2.26).

Lemma 2.2. If the conditions of Lemma 2.1 are satisfied, then

\[ u(t)\subset K^{n_m\ldots n_q}(h)\quad \text{for all } t>t_0 \]

and

\[ u(t)\subset K^{n_1\ldots n_{m-1}}\left(\frac{1}{h}\right)\quad \text{for all } t<t_0 . \tag{2.30} \]

Proof. We have
\[ V(m,h,u(t))= V(m,h,u(t_0))+ \left.\frac{dV(m,h,u(t))}{dt}\right|_{t=t_0} (t-t_0)+\alpha(t)(t-t_0), \]
where \(\alpha(t)\to 0\) as \(t\to t_0\). Since from \(u(t_0)\subset H^{n_m\ldots n_q}(h)\) it follows that \(V(m,h,u(t_0))=0\), and by Lemma 2.1

\[ \left.\frac{dV(m,h,u(t))}{dt}\right|_{t=t_0}>0, \]

there exists an \(\varepsilon>0\) such that for all \(t\) in the interval \((t_0-\varepsilon,t_0)\) the inequality \(V(m,h,u(t))<0\) holds, while for all \(t\) in the interval \((t_0,t_0+\varepsilon)\) the inequality \(V(m,h,u(t))>0\) holds. According to the definitions of the cones (1.3) and (1.4), it follows that

\[ u(t)\subset K^{n_1\ldots n_{m-1}}\left(\frac{1}{h}\right) \quad \text{for } t\subset (t_0-\varepsilon,t_0) \]

and

\[ u(t)\subset K^{n_m\ldots n_q}(h) \quad \text{for } t\subset (t_0,t_0+\varepsilon). \]

It is easy to see that each of these intervals is in fact infinite. Suppose, for example, that at some instant \(t_1>t_0\) the vector \(u(t)\) leaves \(H^{n_m\ldots n_q}(h)\), while for \(t\subset (t_0,t_1)\) it belongs to \(K^{n_m\ldots n_q}(h)\). Repeating—

applying the preceding arguments to the instant \(t_1\), we find an \(\varepsilon_1\) such that

\[ u(t)\subset K^{n_1\ldots n_{m-1}}\left(\frac{1}{h}\right) \quad \text{for } t\subset (t_1-\varepsilon_1,t_1), \]

which contradicts the inclusion

\[ u(t)\subset K^{n_m\ldots n_q}(h) \]

on the interval \((t_0,t_1)\), since the cones \(K^{n_m\ldots n_q}(h)\) and \(K^{n_1\ldots n_{m-1}}\left(\frac{1}{h}\right)\) have no common points. Nor can \(u(t)\) pass through the vertex of the cone \(K^{n_m\ldots n_q}(h)\), since otherwise we would have \(u(t)\equiv 0\). The case \(t<t_0\) is considered analogously.

Let us note once more that in the preceding arguments the quantities \(h\) and \(m\) may be chosen arbitrarily from the conditions \(h_0\leq h\leq \frac{1}{h_0}\) and \(m=2,\ldots,q\), which is taken into account in the formulation of the following obvious assertions.

Corollary 1. Each point of the set \(H^{n_1\ldots n_l}(h)\) is a point of strict exit with respect to the domain \(K^{n_1\ldots n_l}(h)\) \((l=1,\ldots,q-1)\) (see Definition in No. 5, § 1).

Corollary 2. If \(u=x'-x''\) \((x'\ne x'')\) and it is known that

\[ u(t_0)\subset K^{n_1\ldots n_l}(h)\quad (l<q), \]

then

\[ u(t)\subset K^{n_1\ldots n_l}(h) \]

for all \(t\leq t_0\).

Corollary 3. If \(u=x'-x''\) \((x'\ne x'')\) and it is known that

\[ u(t_0)\subset K^{n_m\ldots n_q}(h)\quad (m>1), \]

then

\[ u(t)\subset K^{n_m\ldots n_q}(h) \]

for all \(t\geq t_0\).

The assertions of Corollaries 2 and 3 are also valid for a separately taken nontrivial solution \(x(t)\), as is seen by setting \(u=x(t)\).

We now introduce the following definition. Let \(a\subset L^{n_1\ldots n_l}\) \((a\ne 0)\) be an arbitrarily fixed vector. The set \(D(l,h,a)\) of points \(x\) of the space \(L^n\) admitting the representation \(x=a+b\), where \(b\subset L^{n_{l+1}\ldots n_q}\) and \(h\|a\|^2\geq \|b\|^2\), will be called a spacer of the cone \(\overline K^{\,n_1\ldots n_l}(h)\) \(\left(l=1,\ldots,q-1,\ h_0\leq h\leq \frac{1}{h_0}\right)\).

From the geometric point of view \(D(l,h,a)\) is an \((n_{l+1}+\cdots+n_q)\)-dimensional ball of radius \(\sqrt h\,\|a\|\). The boundary of this ball consists of points \(x=a+b\) satisfying the conditions

\[ h\|a\|^2=\|b\|^2,\qquad b\subset L^{n_{l+1}\ldots n_q}, \]

and, consequently, belongs to \(H^{n_1\ldots n_l}(h)\).

Lemma 2.3. On every spacer \(D(l,h_0,a)\) of the cone \(\overline K^{\,n_1\ldots n_l}(h_0)\), at an arbitrarily chosen instant \(t_0\), there is a point

\[ x_0\subset [D(l,h_0,a)-H^{n_1\ldots n_l}(h_0)] \]

such that the solution \(x(t)\), determined by the initial condition \(x(t_0)=x_0\), will belong for all \(t\) to the cone \(K^{n_1\ldots n_l}(h_0)\) \((l\leq q-1)\).

Proof. We shall use Wazewski’s principle (see No. 5, § 1), for which purpose we set

\[ G=K^{n_1\ldots n_l}(h_0),\qquad \Gamma=H^{n_1\ldots n_l}(h_0), \]

and as the set \(D\) we take any spacer \(D(l,h_0,a)\) of the cone \(\overline K^{\,n_1\ldots n_l}(h_0)\). According to Corollary 1 of Lemma 2.2, we have

\[ A=A^*=H^{n_1\ldots n_l}(h_0). \]

For clarity let us note that the point \(x=0\)—the vertex of the cone \(\overline K^{\,n_1\ldots n_l}(h_0)\)—is not

belongs to neither of the sets \(A\) and \(A^{*}\), since the solution beginning at an arbitrary time \(t_0\) at this point coincides with the trivial one and does not belong to the domain \(G=K^{n_1\ldots n_l}(h_0)\) for any \(t\). The set

\[ S=A\cap D=H^{n_1\ldots n_l}\cap D(l,h_0,a), \]

as was already noted earlier, is the sphere bounding the ball \(D\).

We now turn to the construction of a retraction of \(A\) onto \(S\). Let \(x_1\subset A\); then \(x_1\) admits a unique representation \(x_1=a_1+b_1\), where \(a_1\subset L^{n_1\ldots n_l}\), \(b_1\subset L^{n_{l+1}\ldots n_q}\), and \(h_0\|a_1\|^2=\|b_1\|^2\). Since \(x_1\ne 0\), it follows that \(\|a_1\|\ne0\), \(\|b_1\|\ne0\). Consider the transformation \(x=F(x_1)\), defined by the equality

\[ x=a+\frac{\sqrt{h_0}\,\|a\|}{\|b_1\|}\,b_1, \]

where \(a\) is the vector by means of which the spacer \(D(l,h_0,a)\) is defined.

It is obvious that the summand \(\dfrac{\sqrt{h_0}\,\|a\|}{\|b_1\|}b_1\) belongs to \(L^{n_{l+1}\ldots n_q}\), and since the norm of this vector is equal to \(\sqrt{h_0}\,\|a\|\), \(x\) belongs to the sphere bounding \(D(l,h_0,a)\), i.e. to the set \(S\). Thus, \(F(x_1)\) effects a mapping of \(A\) onto \(S\), and this mapping is continuous, since in the decomposition \(x_1=a_1+b_1\) the vector \(b_1\) depends continuously on \(x_1\) and does not vanish. Moreover, \(F(x_1)\) leaves the points \(x_1\subset S\) fixed. Indeed, let \(x_1\subset S\); then \(x_1=a+b_1\), where \(\|b_1\|^2=h_0\|a\|^2\), and, according to the definition of \(F(x_1)\), in this case we have

\[ F(x_1)=a+\frac{\sqrt{h_0}\,\|a\|}{\|b_1\|}\,b_1=a+b_1=x_1. \]

Thus, it has been established that \(F\) is a retraction of \(A\) onto \(S\). At the same time, \(S\), as the boundary of the ball \(D\), cannot be a retract of this ball, since there is no continuous mapping of a ball onto the sphere bounding it which leaves the points of the sphere fixed. Indeed, assuming the existence of such a mapping and then additionally mapping each point of the sphere into the diametrically opposite one, we obtain a continuous mapping of the closed ball into itself without a fixed point, which contradicts the well-known Brouwer theorem.

To complete the proof of the lemma it remains now to apply Ważewski’s principle.

Let \(T(m,h_0,a)\) denote the spacer of the cone \(\bar K^{nm\ldots n_q}(h_0)\) \((m\ge2)\), i.e. the set of points of the form \(a+b\), where \(a\subset L^{nm\ldots n_q}\) is fixed \((a\ne0)\), \(b\subset L^{n_1\ldots nm-1}\), and \(h_0\|a\|^2\ge\|b\|^2\).

With the aid of the duality principle described earlier, the following is established.

Lemma 2.3′. On each spacer \(T(m,h_0,a)\) of the cone \(K^{nm\ldots n_q}(h_0)\), at an arbitrarily chosen time \(t_0\), there is a point \(x_0\subset [T(m,h_0,a)-H^{nm\ldots n_q}(h_0)]\) such that the solution \(x(t)\), determined by the initial condition \(x(t_0)=x_0\), will for all \(t\) belong to the cone \(K^{nm\ldots n_q}(h_0)\) \((m\ge2)\).

Proof. We make the change of time \(t=-\theta\) and arrive at system (2.18), for which the assertion of Lemma 2.3 is valid, with the sole difference that, in view of the arrangement of the blocks in the reverse order of seniority, in (2.19) one will have to consider the cones \(K^{n_m\ldots n_q}(h_0)\) \((m\ge 2)\) and their spacers \(T(m,h_0,a)\).

Let \(x_1(\theta)\) be a solution of system (2.18) which begins at the instant \(\theta_0=-t_0\) at some point \(x_0\) of the spacer \(T(m,h_0,a)\) and remains in \(K^{n_m\ldots n_q}(h_0)\) for all \(t\). Then \(x(t)=x_1(-t)\) will be a solution of system (1.1) beginning at the instant \(t_0\) at the point \(x_0\) and remaining in \(K^{n_m\ldots n_q}(h_0)\), since the trajectories \(x_1(\theta)\) and \(x(t)\) coincide.

Lemma 2.4. For the norm of the vector \(u=x'-x''\), where \(x'\) and \(x''\) are any two distinct solutions of system (1.1), the estimate
\[ \exp\left(\int_s^t [r_1(\tau)-\delta(\tau)]\,d\tau\right) \le \frac{\|u(t)\|}{\|u(s)\|} \le \]
\[ \le \exp\left(\int_s^t [R_q(\tau)+\delta(\tau)]\,d\tau\right), \tag{2.31} \]
is valid for all \(t\ge s\). If, however, it is known that there exists an instant \(t_0\) such that \(u(t)\subset K^{n_1\ldots n_l}(h)\) \((l<q,\ h>0)\) for all \(t\le t_0\), then the upper estimate in (2.31) can be sharpened, so that for all \(s\le t\le t_0\)
\[ \exp\left(\int_s^t [r_1(\tau)-\delta(\tau)]\,d\tau\right) \le \frac{\|u(t)\|}{\|u(s)\|} < \]
\[ < \sqrt{1+h}\, \exp\left(\int_s^t [R_l(\tau)+\delta(\tau)(1+h)]\,d\tau\right). \tag{2.32} \]

In particular, if \(u(t)\subset K^{n_1\ldots n_l}(h)\) for all \(t\), then estimate (2.32) is valid for all \(s\le t\).

Proof. Carrying out calculations analogous to those made in deriving inequality (2.27), we have
\[ \frac12\frac{d}{dt} \sum_{i\subset n_1,\ldots,n_l}|u_i|^2 = \sum_{i\subset n_1,\ldots,n_l} p_i |u_i|^2+ \]
\[ +\operatorname{Re} \sum_{i\subset n_1,\ldots,n_l} [f_i(t,x')-f_i(t,x'')]\bar u_i, \tag{2.33} \]
whence, taking into account (2.1), (2.2), and (1.2), we obtain for \(l=q\)
\[ [r_1(t)-\delta(t)]\sum_{i=1}^n |u_i|^2 \le \frac12\frac{d}{dt}\sum_{i=1}^n |u_i|^2 \le [R_q(t)+\delta(t)]\sum_{i=1}^n |u_i|^2 . \]

Integrating this inequality, we arrive at the required estimate (2.31). If, however, it is known that \(u(t)\subset K^{n_1\ldots n_l}(h)\) \((l<q)\) for \(t\leqslant t_0\), then for all \(t\leqslant t_0\) the inequality

\[ h\sum_{i\subset n_1,\ldots,n_l}|u_i|^2> \sum_{i\subset n_{l+1},\ldots,n_q}|u_i|^2, \tag{2.34} \]

holds; applying this to the estimate of the right-hand side of (2.33), we obtain

\[ \begin{aligned} \frac12\frac{d}{dt}\sum_{i\subset n_1,\ldots,n_l}|u_i|^2 &\leqslant R_l(t)\sum_{i\subset n_1,\ldots,n_l}|u_i|^2 +\delta(t)\sum_{i=1}^{n}|u_i|^2 \leqslant \\ &\leqslant R_l(t)\sum_{i\subset n_1,\ldots,n_l}|u_i|^2 +\delta(t)\sum_{i\subset n_1,\ldots,n_l}|u_i|^2 +\delta(t)\sum_{i\subset n_{l+1},\ldots,n_q}|u_i|^2 < \\ &< [R_l(t)+\delta(t)(1+h)] \sum_{i\subset n_1,\ldots,n_l}|u_i|^2 . \end{aligned} \]

Dividing both sides of this inequality by the sum \(\sum\limits_{i\subset n_1,\ldots,n_l}|u_i|^2\), which does not vanish by virtue of (2.34), we obtain, by integration \((s\leqslant t\leqslant t_0)\),

\[ \frac{\displaystyle\sum_{i\subset n_1,\ldots,n_l}|u_i(t)|^2} {\displaystyle\sum_{i\subset n_1,\ldots,n_l}|u_i(s)|^2} \leqslant \exp\left(2\int_s^t [R_l(\tau)+\delta(\tau)(1+h)]\,d\tau\right). \tag{2.35} \]

From (2.34) there further follows the estimate of the numerator

\[ \begin{aligned} \sum_{i\subset n_1,\ldots,n_l}|u_i(t)|^2 &= \frac{1}{1+h}\sum_{i\subset n_1,\ldots,n_l}|u_i(t)|^2 +\frac{h}{1+h}\sum_{i\subset n_1,\ldots,n_l}|u_i(t)|^2 > \\ &> \frac{1}{1+h}\sum_{i\subset n_1,\ldots,n_l}|u_i(t)|^2 +\frac{1}{1+h}\sum_{i\subset n_{l+1},\ldots,n_q}|u_i(t)|^2 = \\ &= \frac{1}{1+h}\sum_{i=1}^{n}|u_i(t)|^2 . \end{aligned} \]

Using this estimate and extending the summation in the denominator of the left-hand side of (2.35) to all indices, which can only additionally strengthen the inequality, we obtain

\[ \frac{\displaystyle\sum_{i=1}^{n}|u_i(t)|^2} {\displaystyle\sum_{i=1}^{n}|u_i(s)|^2} < (1+h)\exp\left(2\int_s^t [R_l(\tau)+\delta(\tau)(1+h)]\,d\tau\right), \]

whence estimate (2.32) follows.

Lemma 2.4′. If \(u=x'-x'' \subset K^{n_m\ldots n_q}(h)\) \((m>1,\ h>0)\) for all \(t\geq t_0\), then for any \(t_0\leq s\leq t\) the inequality
\[ \frac{1}{\sqrt{1+h}}\exp\left(\int_s^t [r_m(\tau)-\delta(\tau)(1+h)]\,d\tau\right) <\frac{\|u(t)\|}{\|u(s)\|}\leq \]
\[ \leq \exp\left(\int_s^t [R_q(\tau)+\delta(\tau)]\,d\tau\right). \tag{2.36} \]

In particular, if \(u\subset K^{n_m\ldots n_q}(h)\) for all \(t\), then (2.36) is valid for all \(s\leq t\).

Proof. Put
\[ x_1'(\theta)=x'(-\theta),\qquad x_1''(\theta)=x''(-\theta). \]
Then \(x_1'(\theta)\) and \(x_1''(\theta)\) will be noncoinciding solutions of system (2.18), such that their difference
\[ u_1(\theta)=x_1'(\theta)-x_1''(\theta)=u(-\theta) \]
for all \(\theta<-t_0\) belongs to \(K^{n_m\ldots n_q}(h)\).

Since for system (2.18) Lemma 2.4 is valid with the corresponding replacement of the system of indices \(n_1\ldots n_l\) by \(n_m\ldots n_q\) and taking into account the separation conditions written in the form (2.20) and (2.21), for any \(\theta_1\leq \theta_2\leq -t_0\) we have
\[ \frac{\|u_1(\theta_2)\|}{\|u_1(\theta_1)\|} < \sqrt{1+h}\, \exp\left(\int_{\theta_1}^{\theta_2} [-r_m(-\theta)+\delta(-\theta)(1+h)]\,d\theta\right). \tag{2.37} \]

Put in this estimate \(\theta_1=-t,\ \theta_2=-s\), assuming that \(t\) and \(s\) are chosen arbitrarily from the condition \(-t\leq -s\leq -t_0\). Then, making the change of variable of integration \(\tau=-\theta\) in the exponent, we obtain
\[ \frac{\|u_1(-s)\|}{\|u_1(-t)\|} < \sqrt{1+h}\, \exp\left(\int_t^s [r_m(\tau)-\delta(\tau)(1+h)]\,d\tau\right). \]

Since from the way \(u_1(\theta)\) is defined it follows that \(u_1(-s)=u(s)\) and \(u_1(-t)=u(t)\), the preceding inequality implies the lower estimate in (2.36).

The upper estimate was established earlier by inequality (2.31), valid for all \(t\geq s\).

Lemma 2.5. Let \(c(t)\) and \(\delta(t)\) in (1.2) and (2.2) be subject to condition (2.25) and, moreover,
\[ 2\delta(t)\left(1+\frac{1}{h_0}\right)\leq \gamma c(t), \tag{2.38} \]
where \(\gamma\) is some number \((0<\gamma<1)\). Then if the solutions \(x'(t)\) and \(x''(t)\)
\[ (x'(t)\ne x''(t)) \]
belong, for some \(h\) \(\left(h_0\leq h\leq \frac{1}{h_0}\right)\), to the cone
\[ K^{n_1\ldots n_l}(h)\quad (l<q) \]
for all \(t\) (or \(x'(t)\subset K^{n_1\ldots n_l}(h)\) for all \(t\) and \(x''(t)\equiv 0\)), then the vector \(u(t)=x'(t)-x''(t)\) belongs to \(K^{n_1\ldots n_l}(h_0)\) for all \(t\).

Proof. Suppose, contrary to the assertion of the lemma, that there exists such a moment \(t_0\) that
\[ u(t_0)\subset K^{n_1\ldots n_l}(h_0). \]
Then
\[ u(t_0)\subset K^{n_{l+1}\ldots n_q}\left(\frac{1}{h_0}\right) \]
and

\(u(t_0)\ne 0\) (otherwise the solutions \(x'\) and \(x''\) would coincide). By Lemma 2.2, the point \(t_0\) may be assumed such that \(u(t_0)\subset K^{n_{l+1}\ldots n_q}\left(\dfrac{1}{h_0}\right)\). It also follows from Corollary 3 of Lemma 2.2 that \(u(t)\subset K^{n_{l+1}\ldots n_q}\left(\dfrac{1}{h_0}\right)\) for all \(t\ge t_0\). In this case, according to Lemma \(2.4'\), for all \(t\ge t_0\) the estimate

\[ \frac{\|u(t)\|}{\|u(t_0)\|} > \frac{1}{\sqrt{1+\dfrac{1}{h_0}}} \exp\left( \int_{t_0}^{t} \left[ r_{l+1}(\tau)-\delta(\tau)\left(1+\frac{1}{h_0}\right) \right]d\tau \right), \]

holds, whence, for all \(t\ge t_0\), we have

\[ \|x'(t)-x''(t)\|>B_1\exp\left( \int_{t_0}^{t} \left[ r_{l+1}(\tau)-\delta(\tau)\left(1+\frac{1}{h_0}\right) \right]d\tau \right), \tag{2.39} \]

where \(B_1>0\) is some constant.

Since, on the other hand, by the hypothesis of the lemma \(x'(t), x''(t)\subset K^{n_1\ldots n_l}(h)\) for all \(t\) (or \(x''\equiv 0\)), then by Lemma 2.4, putting successively in (2.32) \(u=x'(t)\), \(u=x''(t)\) and \(s=t_0\), we obtain the estimates

\[ \frac{\|x'(t)\|}{\|x'(t_0)\|} < \sqrt{1+h}\, \exp\left( \int_{t_0}^{t} \left[ R_l(\tau)+\delta(\tau)(1+h) \right]d\tau \right), \]

\[ \frac{\|x''(t)\|}{\|x''(t_0)\|} < \sqrt{1+h}\, \exp\left( \int_{t_0}^{t} \left[ R_l(\tau)+\delta(\tau)(1+h) \right]d\tau \right) \quad(\text{if }x''\ne 0), \]

valid for all \(t\ge t_0\). The last inequalities make it possible to give the following estimate, valid for \(t\ge t_0\):

\[ \|x'(t)-x''(t)\|<B_2\exp\left( \int_{t_0}^{t} \left[ R_l(\tau)+\delta(\tau)(1+h) \right]d\tau \right), \tag{2.40} \]

where \(B_2>0\) is some constant.

From (2.39) and (2.40) we have, in an obvious way,

\[ \exp\left( \int_{t_0}^{t} \left[ r_{l+1}(\tau)-R_l(\tau)-\delta(\tau)\left(2+h+\frac{1}{h_0}\right) \right]d\tau \right) < \frac{B_2}{B_1}. \tag{2.41} \]

Next using conditions (2.2), (2.38) and taking into account that \(h\le \dfrac{1}{h_0}\), we find

\[ r_{l+1}(t)-R_l(t)-\delta(t)\left(2+h+\frac{1}{h_0}\right) \ge \]

\[ \ge c(t)-2\delta(t)\left(1+\frac{1}{h_0}\right) \ge (1-\gamma)c(t). \]

Then from (2.3) it follows that the left-hand side in (2.41) tends to \(+\infty\) as \(t \to +\infty\), which contradicts estimate (2.41). The contradiction arose from the assumption that there exists a moment \(t_0\) such that \(u(t_0) \subset K^{n_1\cdots n_l}(h_0)\). The lemma is proved.

Corollary 1. If for some nontrivial solution \(x(t)\) it is known that, for all \(t\), it belongs to the cone

\[ K^{n_1\cdots n_l}(h) \left( l < q,\ h_0 \leq h \leq \frac{1}{h_0} \right), \]

then in fact this solution belongs to the cone \(K^{n_1\cdots n_l}(h_0)\).

For the proof it suffices to put \(x' = x(t)\) and \(x'' = 0\) in the lemma.

Corollary 2. If some nontrivial solution \(x(t)\) is such that

\[ x(t_0) \subset K^{n_1\cdots n_l}(h_0), \]

then there exists a moment \(t_1\) such that, for all \(t \geq t_1\), the solution \(x(t)\) will be immersed in the cone \(K^{n_{l+1}\cdots n_q}(h_0)\).

Proof. If \(x(t_0) \subset \overline{K}^{\,n_{l+1}\cdots n_q}(h_0)\), then by Lemma 2.2 and its Corollary 3,

\[ x(t) \subset K^{n_{l+1}\cdots n_q}(h_0) \]

for all \(t > t_0\), and as the moment \(t_1\) one may take any \(t_1 > t_0\). If, however, \(x(t_0) \subset \overline{K}^{\,n_{l+1}\cdots n_q}(h_0)\), then

\[ x(t_0) \subset K^{n_1\cdots n_l}\left(\frac{1}{h_0}\right), \]

and by Corollary 2 of Lemma 2.2,

\[ x(t) \subset K^{n_1\cdots n_l}\left(\frac{1}{h_0}\right) \]

for all \(t \leq t_0\). If, in addition, it turned out that \(x(t)\) belongs to

\[ K^{n_1\cdots n_l}\left(\frac{1}{h_0}\right) \]

for all \(t \geq t_0\), then by Corollary 1 of Lemma 2.5 we would have

\[ x(t) \subset K^{n_1\cdots n_l}(h_0) \]

for all \(t\), which contradicts the condition

\[ x(t_0) \subset K^{n_1\cdots n_l}(h_0). \]

Thus, for \(t \geq t_0\), the solution \(x(t)\) cannot remain indefinitely in

\[ K^{n_1\cdots n_l}\left(\frac{1}{h_0}\right), \]

and at some moment will reach the boundary of the cone

\[ K^{n_1\cdots n_l}\left(\frac{1}{h_0}\right), \]

and, as \(t\) increases further, will be immersed in \(K^{n_{l+1}\cdots n_q}(h_0)\).

Corollary 3. If \(a \subset L^{n_1\cdots n_q}\) is an arbitrary fixed vector \((a \ne 0)\), \(D(l,h_0,a)\) is a support of the cone \(K^{n_1\cdots n_l}(h_0)\) constructed with the aid of \(a\), and \(x_0\) is a point through which, at the moment \(t_0\), passes a solution \(x(t)\) immersed in \(K^{n_1\cdots n_l}(h_0)\) for all \(t\), then \(x_0\) is the unique point with this property among the set of points of the support \(D(l,h_0,a)\).

Proof. From the inclusion \(x_0 \subset D(l,h_0,a)\) it follows that

\[ x_0 = a + b', \]

where

\[ b' \subset L^{n_{l+1}\cdots n_q}. \]

Suppose, contrary to the assertion, that among the points of the support \(D(l,h_0,a)\) there is a point

\[ x_1 = a + b'' \quad (x_1 \ne x_0), \]

through which, at the moment \(t_0\), passes a solution that remains, for all \(t\), in \(K^{n_1\cdots n_l}(h_0)\). Then, by the lemma, the difference of these solutions must, for all \(t\), belong to \(K^{n_1\cdots n_l}(h_0)\).

In particular, at \(t=t_0\), it follows from this that

\[ x_1 - x_0 \subset K^{n_1\cdots n_l}(h_0), \]

but this is impossible, since

\[ x_1 - x_0 = b'' - b' \subset L^{n_{l+1}\cdots n_q}. \]

(Continued in the next issue.)

Received by the editors
December 30, 1965

Moscow Aviation
Technological Institute