UDC 517.925

ON A FAMILY OF EXTREMALS

A. N. ERUGIN

Definition 1. If \(s_1(t)=s(t)s_2(t)\), where \(|s|\leqslant \mathrm{const}\) for \(t\geqslant t_0\), and if as \(t\to\infty\) \(s\) does not tend to zero, then \(s_1, s_2\) are called functions of the same order; while if as \(t\to\infty\) \(s\to 0\), then \(s_1\) is called a function of higher order than \(s_2\).

This definition makes it possible to compare quantities oscillating with arbitrary amplitude.

Definition 2. If among the solutions of a differential equation there are solutions of higher order, in the sense of Definition 1, in comparison with other solutions, then such solutions are called extremal, or simply extremals.

If a system of differential equations is given, then the order of solutions must be compared by norm.

These definitions are quoted from [1], where a theorem on an extremal of a nonlinear equation is proved, and one theorem concerning extremals of a system of \(n\) equations is formulated; it is also indicated that the concept of an extremal makes it possible to give a classification of the solutions of the system.

The purpose of this article is to study the extremals of a nonlinear system of \(n\) differential equations with a purely diagonal matrix of the linear terms, and to classify the solutions of this system.

The system is given by

\[ u_l' = a_l(t)u_l+s_l(t,u)\qquad (l=1,2,\ldots,n), \tag{1} \]

where

\[ \text{1) }\quad |s_l(t,\bar u)-s_l(t,\bar{\bar u})|\leqslant c_l(t)\sum_{l=1}^{n}|\bar u_l-\bar{\bar u}_l| \qquad (l\leqslant n); \]

2) \(a_l(t)\), \(c_l(t)\), \(s_l(t,u(t))\) are integrable functions; for this it is sufficient, for example, to take continuous \(a_l(t), c_l(t), s_l(t,u)\) \((l\leqslant n)\);

3) \(a_l(t)\), \(s_l(t,u)\) are such that

\[ \lim_{t\to\infty} I_l(t)=\mathrm{const}, \]

\[ I_l(t)=\int_{t_0}^{t}s_l(t,u(t))S_l^{-1}(t)\,dt, \qquad S_l(t)=\exp\int_{t_0}^{t}a_l(t)\,dt, \]

\[ \lim_{t\to\infty}\int_{t_0}^{t}c_l(t)\,dt=\mathrm{const} \qquad (l\leqslant n). \]

Thus, we have

\[ \tilde I_l(t)=\int_{t}^{\infty}s_l(t,u(t))S_l^{-1}(t)\,dt. \]

The comparison of the orders of solutions is made in the norm

\[ \|u\|=\left(\sum_{l=1}^{n} u_l^s\right)^{\frac1s} \qquad (s \geqslant 1)\ \text{integer}. \]

It is further clear that there are the following types of behavior of \(S_l(t)\) as \(t \to \infty\): \(S_l \to 0\); \(|S_l| \leqslant \mathrm{const}\), \(S_l\) does not tend to zero; \(S_l\) is unbounded.

We shall assume that the \(u_l\) are numbered so that the order of \(S_{l_1}\) is greater than, or equal to, the order of \(S_{l_2}\), if \(l_1 > l_2\).

Accordingly, one can formulate three series of theorems.

1st series.

Theorem 1. If \(S_l \to 0\) \((l \leqslant n)\) and if the order of \(S_2\) is greater than the order of \(S_1\), then among the solutions of system (1) there is an \((n-1)\)-parameter family of extremals \(U_{n-1}\), having order equal to the smaller of the orders of \(S_1 \tilde I_1, S_2\). If the order of \(S_1 \tilde I_1, S_3\) is greater than the order of \(S_2\), then among the extremals of the family \(U_{n-1}\) there is an \((n-2)\)-parameter family of extremals \(U_{n-2}\), whose order is greater than the order of the extremals of the family \((U_{n-1}-U_{n-2})\), and so on; and if the order of \(S_l \tilde I_l\) \((l \leqslant s)\), \(S_{s+2}\) is greater than the order of \(S_{s+1}\), then among the extremals of the \((n-s)\)-parameter family of extremals \(U_{n-s}\) there is an \((n-(s+1))\)-parameter family of extremals \(U_{n-(s+1)}\), whose order is greater than the order of the extremals of the family \((U_{n-1}-U_{n-(s+1)})\), and so on.

Thus, if the order of \(S_l \tilde I_l, S_{s+2}\) is greater than the order of \(S_{s+1}\) \((0 \leqslant l \leqslant s;\ 0 \leqslant s \leqslant n-1)\), this condition will henceforth be called the order condition, and the family of extremals of system (1) obtained is as follows:

\[ U_{n-1}\supset U_{n-2}\supset \cdots \supset U_2\supset U_1\supset U_0, \]

where \(U_{n-l}\) is an \((n-l)\)-parameter family of extremals whose order is greater than the order of the extremals of the family \(U_{n-(l+1)}\), and \(U_0\) is a certain extremal whose order is greater than the order of the extremals of the family \((U_1-U_0)\).

Let us note that \(\tilde u=u(t,t_0,\tilde u_1^0,\tilde u_2^0,\ldots,\tilde u_l^0,C_{n-(l+1)},\ldots,C_n)\in U_{n-l}\), where \(\tilde u_l^0=-I_l(\infty)\), \(C_s(s>l)\) are arbitrary constants.

Remark. If \(a_l, s_l\) are such that, for some \(l\), the order of one of the functions \(S_s\tilde I_s\) \((s \leqslant l)\) does not exceed the order of \(S_{l+1}\), or if the order of \(S_{l+2}\) is equal to the order of \(S_{l+1}\), then among the extremals of the family \(U_{n-l}\) there is no \((n-(l+1))\)-parameter family of extremals \(U_{n-(l+1)}\) whose order is greater than the order of the extremals of the family \((U_{n-l}-U_{n-(l+1)})\). However, among the extremals of the family \(U_{n-l}\) there is an \((n-(l+1))\)-parameter family of extremals \(\tilde U_{n-(l+1)}\), such that the order of \(\tilde u_{l+1}\in \tilde u\in \tilde U_{n-(l+1)}\) is greater than the order of \(\tilde u_{l+1}\in \tilde u\in (U_{n-l}-\tilde U_{n-(l+1)})\), while the order of \(\tilde u_s\in \tilde u\in \tilde U_{n-(l+1)}\) \((s\ne l+1)\) is equal to the order of \(\tilde u_s\in \tilde u\in U_{n-l}\) \((s\ne l+1)\). Further, among the extremals of the family \(\tilde U_{n-(l+1)}\) there is an \((n-(l+2))\)-parameter family of extremals \(\tilde U_{n-(l+2)}\), such that the order of \(\tilde u_{l+2}\in \tilde u\in \tilde U_{n-(l+2)}\) is greater than the order of \(\tilde u_{l+2}\in \tilde u\in (\tilde U_{n-(l+1)}-\tilde U_{n-(l+2)})\), while the order of \(\tilde u_s\in \tilde u\in \tilde U_{n-(l+2)}\) \((s\ne l+2)\) is equal to the order of \(\tilde u_s\in \tilde u\in (\tilde U_{n-(l+1)}-\tilde U_{n-(l+2)})\) \((s\ne l+2)\), and so on.*

These extremals will henceforth be called conditional extremals, and these families of such extremals will be called conditionally extremal and denoted by \(\tilde U_{n-l}\).

We note further that from this the meaning of the order condition is clear, and henceforth, for simplicity of formulation, we shall assume that this condition is satisfied and shall not stipulate this specially, except in certain cases where the application of this condition has special features.

We shall henceforth denote the family of extremals of system (1) by \(\tilde U\).

Theorem 2. If \(|S_l| \leq \mathrm{const}\), \(S_l\) does not tend to zero \((l \leq n)\), then \(\tilde U\) is obtained as follows:

\[ \tilde U_{n-1} \supset \tilde U_{n-2} \supset \cdots \supset \tilde U_2 \supset \tilde U_1 \supset U_0, \]

i.e., there is one extremal and families of conditional extremals.

Theorem 3. If \(S_l\) are unbounded \((l \leq n)\), then \(\tilde U\) is obtained as follows:

\[ U_{n-1} \supset U_{n-2} \supset \cdots \supset U_0. \]

2nd series.

Theorem 1. If \(|S_l| \leq \mathrm{const}\), \(S_l\) does not tend to zero \((l \leq s < n)\); \(S_l \to 0\) \((l > s)\), then \(\tilde U\) is obtained as follows:

\[ \tilde U_{n-1} \supset \tilde U_{n-2} \supset \cdots \supset \tilde U_{n-s} \supset U_{n-(s+1)} \supset \cdots \supset U_1 \supset U_0. \]

Remark. It is not difficult to see that the order condition is satisfied for \(S_l\) \((l > s)\), while for \(S_l\) \((l \leq s)\) this condition has no meaning.

Theorem 2. If \(S_l\) are unbounded \((l \leq s < n)\), \(S_l\) are bounded \((l > s)\), then \(\tilde U\) is obtained as follows:

\[ U_{n-1} \supset \cdots \supset U_{n-s} \supset \tilde U_{n-(s+1)} \supset \cdots \supset \tilde U_1 \supset \tilde U_0. \]

Remark. If \(S_l \to 0\) \((l > s)\) and \(a_l, S_l\) are such that, as \(t \to \infty\), \(S_l \tilde I_l \to 0\) \((l \leq s)\), then

\[ U_{n-s} \supset U_{n-(s+1)} \supset \cdots \supset U_1 \supset U_0. \]

If \(|S_l| \leq \mathrm{const}\), \(S_l\) does not tend to zero \((l > s)\) and \(a_l, S_l\) are such that, as \(t \to \infty\), \(S_l \tilde I_l \to 0\) \((l \leq s)\), then \(\tilde U_1 \supset U_0\).

3rd series.

Theorem. If \(S_l\) are unbounded \((l \leq s_1 < n)\), \(|S_l| \leq \mathrm{const}\), \(S_l\) does not tend to zero \((s_1 < l \leq s_2 < n)\), \(S_l \to 0\) \((l > s_2)\), then \(\tilde U\) is obtained as follows:

\[ U_{n-1} \supset \cdots \supset U_{n-s} \supset \tilde U_{n-(s+1)} \supset \cdots \supset \tilde U_1 \supset \tilde U_0. \]

Remark. If \(a_l, S_l\) are such that, as \(t \to \infty\), \(S_l \tilde I_l \to 0\) \((l \leq s_1)\), then \(\tilde U_{n-s_2} \supset U_{n-(s_2+1)} \supset \cdots \supset U_1 \supset U_0\).

References

1. Erugin A. N. Differential Equations, 2, No. 8, 1027—1030, 1966.

Received by the editors
October 12, 1966

Leningrad Branch
of the V. A. Steklov
Mathematical Institute