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ON EXTREMALS
A. N. ERUGIN
UDC 517.92
1. Introduction. If among the solutions of a differential equation or a system of differential equations there are some that differ from the others according to some feature, then these solutions are called extremal with respect to this feature, or, if it is clear which feature is meant, simply extremals.
It is clear that an extremal is an extremal only with respect to a definite feature, and with respect to another feature it may fail to be an extremal. The concept of an extremal is therefore inseparable from the corresponding feature of extremality. Thus, this phrase does not so much give a concrete concept as indicate a rule for the use of this term. And it is clear that the degree of meaningfulness of the study of extremals of a system is determined by the choice of the feature of extremality.
Definition 1. If \(s_1(t)=s(t)s_2(t)\), where \(|s|\leqslant \mathrm{const}\) for \(t\geqslant t_0\), then we shall call \(s\) the relative change of the function \(s_1\) in comparison with \(s_2\). In this case, if for \(t\geqslant t_0\), \(|s|\leqslant \mathrm{const}\), and as \(t\to\infty\) \(s\) does not tend to 0, then \(s_1, s_2\) are called functions of the same order; if as \(t\to\infty\), \(s\to 0\), then \(s_1\) is called a function of higher order in comparison with \(s_2\).
Let us note that if \(s_1, s_2\) are infinitely small (i.s.) quantities, then the concept of order coincides with the concept of the order of smallness of i.s. quantities. This definition makes it possible to compare quantities oscillating with arbitrary amplitude.
The aim of the article is the study of extremals of a certain class of nonlinear equations. The feature of extremality is the rate of change of solutions as the argument increases. Certain extremals with such a feature are indicated by many authors [1—7].
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 others, then such solutions are called extremal, or simply extremals.
If a system of differential equations is given, then one must compare the order according to the norm of the solutions; and if an equation of order \(n\) is given (\(n>1\)), then the order is compared according to some norm. This is discussed in more detail below.
2. Extremal of a linear equation. If the equation
\[ y' + p(t)y = s(t), \]
is given, where \(p, s\) are integrable functions, and moreover such that
\[ S(t)=\exp\int_{t_0}^{t} p(t)\,dt, \]
\[ \tilde I(t)=\int_{t_0}^{t} s(t)S(t)\,dt,\qquad \lim_{t\to\infty}\tilde I(t)=\mathrm{const}, \]
then the solution \(y(t,y_0)\), \(y_0=-\tilde I(\infty)\), is extremal, since for \(y_0,\ y_0'\ne \tilde y_0\) we obtain
\[ \lim_{t\to\infty} \frac{y(t,y_0)}{y(t,y_0')}=\operatorname{const}\ne 0, \]
but
\[ \lim_{t\to\infty} \frac{y(t,\tilde y_0)}{y(t,y_0)}=0. \]
Thus, the order of the extremal solution differs from the order of the other solutions of the equation.
3. Extremal of a nonlinear equation. Given the equation
\[ u' + p(t)u = s(t,u), \]
where for \(t\in T=[t_0,\infty)\), \(u\in U\): 1) \(|s(t,u_1)-s(t,u_2)|\le c(t)|u_1-u_2|\); 2) \(p(t)\), \(c(t)\), \(s(t,u(t))\) are integrable functions, for which it suffices, for example, to take continuous \(p(t)\), \(c(t)\), \(s(t,u)\); 3) \(p\), \(s\) are such that
\[ \lim_{t\to\infty} I(t)=\operatorname{const},\qquad I(t)=\int_{t_0}^{t} s(t,u(t))S(t)\,dt. \]
Theorem. If \(\displaystyle \int_{t_0}^{\infty} c(t)\,dt<\ln 2\), then the solution \(u(t,\tilde u_0)\), \(\tilde u_0=-I(\infty)\), is extremal.
Proof. Denote
\[
\Delta(t)=u(t,t_0,u_0')-u(t,t_0,u_0''),\qquad
\Delta_0=u_0'-u_0'',\qquad
\tilde S(u_0)=\tilde S(u_0,u(t,u_0))=-I(\infty).
\]
We obtain, for \(\Delta_0>0\),
\[ |\Delta(t)|\le \overline{\Delta}(t),\qquad \overline{\Delta}' + (p-c)\overline{\Delta}=0,\qquad \overline{\Delta}=\Delta_0\exp\left(-\int_{t_0}^{t}(p-c)\,dt\right), \]
\[ |\tilde S(u_0')-\tilde S(u_0'')| \le \Delta_0\int_{t_0}^{\infty} c\exp\left(\int_{t_0}^{t} c\,dt\right)dt = \Delta_0\tilde c,\qquad \tilde c=\operatorname{const}<1. \]
Hence it is clear [8] that there exists such a \(\tilde u_0\) and such a \(u(t,\tilde u_0)\) that \(\tilde u_0=-I(\infty)\).
Example. The conditions of the theorem are satisfied if, for \(t\in T\), \(u\in U\),
\[ s(t,u)=s_1(t)s_2(t,u),\qquad |s_1|\le \tilde s(t)>0,\qquad \lim_{t\to\infty}\int_{t_0}^{t}\tilde s\,dt=\operatorname{const}, \]
\[ \left|\frac{\partial s_2}{\partial u}\right|\le s_0=\operatorname{const},\qquad \lim_{t\to\infty}\int_{t_0}^{t} s(t,u(t))S(t)\,dt=\operatorname{const}. \]
Indeed, we obtain, for \(t\ge t_0\), \(u\in U\),
\[ |s_2(t,u_1)-s_2(t,u_2)|\le s_0|u_1-u_2|,\qquad \overline{\Delta}=\Delta_0\exp\left(-\int_{t_0}^{t}(p-ss_0)\,dt\right). \]
Further, it is clear that, for \(t\ge t_0'\), \(s_0\le s_0'\), we obtain
\[
s_0\int_{t_0}^{\infty}\tilde s\,dt<\ln 2.
\]
4. On a family of extremals of a system and of an equation of arbitrary order. If a system is given
\[ u' + p(t)u = s(t,u), \]
ON EXTREMALS
where \(p\) is a matrix of order \(n\), \(u(t)\), \(s(t,u)\) are vectors of order \(n\) with elements \(u_l(t)\), \(s_l(t,u)\) \((l \leq n)\), then, using the concept of an extremal, it is easy to classify the solutions of the system according to their asymptotic behavior. In doing so one must compare the order with respect to the norm \(\|u\|\), where
\[ \|u\|=\left(\sum_{s=1}^{n}|u_s|^l\right)^{\frac1l}, \]
\(l \geq 1\), an integer, or for an equation of order \(n\)
\[ u^{(n)}=s(t,u',\ldots,u^{(n-1)}) \]
\[ \|u\|=\left(\sum_{s=1}^{n}|u^{(s-1)}|^l\right)^{\frac1l},\qquad l \geq 1, \]
an integer. We give the simplest theorem of this type.
Theorem. If \(p\) is a diagonal matrix and its elements \(p_{ss}(t)\) \((s\leq n)\) are of one sign,
\[ |p_{ss}(t)| \geq ct^{-1},\qquad c=\mathrm{const}, \]
and if for \(t\geq t_0\)
\[ |s_l(t,\bar u_1,\bar u_2,\ldots,\bar u_n)-s_l(t,\bar{\bar u}_1,\bar{\bar u}_2,\ldots,\bar{\bar u}_n)| \leq c_l(t)\sum_{s=1}^{n}|\bar u_s-\bar{\bar u}_s|\qquad (l\leq n); \]
\[ \lim_{t\to\infty}\int_{t_0}^{t} c_l(t)\,dt=\mathrm{const},\qquad \lim_{t\to\infty}\int_{t_0}^{t} S_l(t)s_l(t,u(t))\,dt=\mathrm{const}, \]
\[ S_l(t)=\exp\left(-\int_{t_0}^{t}p_{ll}(t)\,dt\right)\qquad (l\leq n); \]
\[ |s_l(t,u(t))|\leq \tilde s_l(t),\qquad \ln \tilde s_l(t)\leq \ln \tilde s(t)+\int_{t_0}^{t}\tilde c(t)\,dt, \]
\[ \tilde c(t)=\min(-p_{ll}(t))\qquad (l\leq n); \]
\[ c_l(t),\ \tilde s_l(t),\ \tilde s(t)>0;\qquad \lim_{t\to\infty}\tilde s(t)=0, \]
then among the solutions of the system there is an \((n-1)\)-parameter family of extremals \(U_{n-1}\) having higher order in comparison with the other solutions; among the extremals of the family \(U_{n-1}\) there is an \((n-2)\)-parameter family of extremals \(U_{n-2}\) having higher order in comparison with the other extremals of the family \(U_{n-1}\), and so on; and among the extremals of the one-parameter family of extremals \(U_1\) there is an extremal \(u(t,\tilde u_0)\) having higher order in comparison with the other extremals of the family \(U_1\).
Thus we obtain the following classification of the solutions of the system:
\[ U_n \supset U_{n-1}\supset U_{n-2}\supset \cdots \supset U_3\supset U_2\supset U_1\supset u(t,\tilde u_0). \]
The theorem is also valid for a system obtained from the indicated system by a linear transformation with a bounded matrix of coefficients having a bounded inverse matrix. We note that the assumptions made are not necessary, but only sufficient. For example, the diagonal form of the matrix of linear terms is not necessary; however, if this assumption is not made, then the corresponding theorem is formulated much more complicatedly. We further note that under other assumptions the scheme of the family of extremals is sometimes different; namely, generally speaking, one obtains the following scheme of the family of extremals:
\[ U_s \supset U_{s-1}\supset \cdots \supset U_{l+1}\supset U_l' \qquad (0\leq l\leq s\leq n-1), \]
where by \(U_{s'}\), \((l \leq s' \leq s)\), is denoted an \(s'\)-parameter family of extremals having higher order than the extremals of the family \(U_{s'+1}\), \(U_0 \equiv u(t,\tilde u_0)\). However, we shall not discuss this in detail in the present paper.
References
- Perron O. Journ. für die reine und ang. Math., 142, 1913.
- Perron O. Math. Z., 6, 1920.
- Perron O. Math. Z., 17, 1923.
- Späth H. Acta Math., 51, 1927.
- Späth H. Math. Z., 30, 1929.
- Ascoli G. Scritti Matematici offerti a Luigi Berzolari. Pavia, 1936.
- Peyovich T. Bulletin de la Societe des mathematiciens et physiciens de la R. P. de Serbie, vol. IX, 3—4. Beograd, Yougoslavie, 1957.
- Lyusternik L. A., Sobolev V. I. Elements of Functional Analysis. Moscow–Leningrad, GITTL, 1951.
Received by the editors
April 20, 1966
Steklov Mathematical Institute,
Leningrad Branch