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UDC 517.919.2+517.948
MATHEMATICS
A. E. GEL’MAN
ON PROPERTIES OF BOUNDED SOLUTIONS OF LINEAR DIFFERENTIAL EQUATIONS IN \(B\)-SPACES
(Presented by Academician V. I. Smirnov on 27 III 1967)
- Let \(Z\) be a complex \(B\)-space, and let \(A\) be some endomorphism* of it with spectrum not intersecting the imaginary axis. We shall say that a set \(\Omega\), whose elements are functions defined on \((-\infty,+\infty)\) with values in \(Z\), has property \(C\) if the equation
\[ \dot{x}+Ax=\varphi(t) \tag{1} \]
has in \(\Omega\) a unique solution for every \(\varphi(t)\in\Omega\).**
It is known \((^1)\) that the property \(C\) is possessed by the set of functions continuous and bounded on \((-\infty,+\infty)\), the set of continuous periodic functions, and the set of almost periodic functions. It seems important to indicate a criterion allowing one to judge whether a given set \(\Omega\) possesses property \(C\).
In the present paper this problem is solved for one family of sets \(\Omega\): conditions are indicated that are necessary and sufficient in order that a set \(\Omega\) from this family possess property \(C\). Some sufficient conditions are also indicated. The results of the monograph \((^1)\) listed above follow from the theorems of this paper.
- We introduce notation.
\(\Phi\) is the set of all strongly measurable and almost everywhere bounded functions with values in \(Z\), defined on \((-\infty,+\infty)\).
\(\nu\) is a functional defined on \(\Phi\) (the value \(+\infty\) is also allowed among its values) such that for any \(\varphi\) and \(\psi\) from \(\Phi\) and any complex \(\xi\) the following hold: 1) \(\nu(\varphi)\ge 0\); 2) \(\nu(\varphi+\psi)\le \nu(\varphi)+\nu(\psi)\); 3) \(\nu(\xi\varphi)=|\xi|\nu(\varphi)\); 4***) \(\nu[\varphi(t+\tau)]=\nu[\varphi(t)]\) for any \(\tau\in(-\infty,+\infty)\); 5) from the inequality \(\|\varphi(t)\|\le \|\psi(t)\|\), holding on \((-\infty,+\infty)\), follows the inequality \(\nu(\varphi)\le \nu(\psi)\).
\(\Phi_\nu\) is the subset of \(\Phi\) formed by all elements of \(\Phi\) satisfying the inequality \(\nu(\varphi)<+\infty\). It is obvious that \(\Phi_\nu\) is a complex linear system.
We shall call \(\varphi\) and \(\psi\) from \(\Phi\) \(\nu\)-equivalent if \(\nu(\varphi-\psi)=0\). It is obvious that \(\Phi_\nu\) can be represented as a sum of nonintersecting classes of elements pairwise \(\nu\)-equivalent; the functional \(\nu\) is constant on each such class.
\(C_\nu\) is the normed linear space whose elements are the above-mentioned classes of elements of \(\Phi_\nu\), and the norm of an element is taken to be the value of the functional \(\nu\) on such a class. It is not difficult to see that all spaces \(L_p(-\infty,+\infty)\), \(1\le p\le +\infty\), are spaces of type \(C_\nu\).
* By an endomorphism of a normed linear space this paper means a linear bounded operator mapping this space into itself.
** A solution of the differential equation (1) is understood to be a continuous function satisfying this equation almost everywhere on \((-\infty,+\infty)\).
*** In connection with the use of the shift operation, the paper allows a double notation: sometimes we write \(\varphi\in\Phi\), and sometimes \(\varphi(t)\in\Phi\).
\(E(X)\) is the set of endomorphisms of the linear normed space \(X\). It is obvious that \(E(Z) \subset E(C_\nu)\).
\(P_\tau\) is the shift operator, i.e., the operator defined on \(\Phi\) by the equality \(P_\tau[\varphi(t)] = \varphi(t+\tau)\); \(P\) is the set of all \(P_\tau\), \(\tau \in (-\infty,+\infty)\).
\(S_h\) is the operator defined on \(\Phi\) by the equality*
\[
S_h[\varphi(t)] = \frac1h \int_t^{t+h}\varphi(\tau)\,d\tau;
\]
\(S\) is the set of all \(S_h\), \(h \ne 0\), \(h \in (-\infty,+\infty)\). It is easy to see that \(P \subset E(C_\nu)\) and \(S \subset E(C_\nu)\).
- For the problem studied in this paper, the space \(C_\nu\) turns out to be a rather natural object, as is seen from the following theorem.
Theorem 1. If \(\varphi(t) \in C_\nu\), then equation (1) has a unique solution in \(C_\nu\).
In Theorem 1, the spaces \(L_p(-\infty,+\infty)\), \(1 \le p \le +\infty\), may be taken as \(C_\nu\). For \(p=+\infty\) one obtains the previously known result on the existence of a unique bounded solution of equation (1).
Despite its generality, Theorem 1 gives only one-sided information about the bounded solution of equation (1). To study more subtle properties of this solution, a new type of space is introduced.
Definition. Let \(\widetilde{\Phi}\) be a complete subspace of the space \(C_\nu\). We shall call \(\widetilde{\Phi}\) a \(B\)-space if \(E(Z) \subset E(\widetilde{\Phi})\).
Theorem 2. Let \(\widetilde{\Phi}\) be a \(B_\nu\)-space. In order that, for every \(\varphi(t) \in \widetilde{\Phi}\) and every \(A\), equation (1) have a unique solution in \(\widetilde{\Phi}\), it is necessary and sufficient that \(S \subset E(\widetilde{\Phi})\).
Example. Let \(\widetilde{\Phi}\) be the set of all continuous bounded functions defined on \((-\infty,+\infty)\) and having the limit \(\lim_{t\to+\infty}\varphi(t)\). If \(\nu\) is introduced by the equality
\[
\nu(\varphi)=\sup_{-\infty<t<+\infty}\|\varphi(t)\|,
\]
then \(\widetilde{\Phi}\) will be a \(B_\nu\)-space. For every \(\varphi(t)\in\widetilde{\Phi}\) and \(h\ne0\), the function
\[
\widetilde{\varphi}(h,t)=\frac1h\int_t^{t+h}\varphi(\tau)\,d\tau
\]
will be continuous; the limit will also exist:
\[
\lim_{t\to+\infty}\widetilde{\varphi}(h,t)=\lim_{t\to+\infty}\varphi(t).
\]
Consequently, \(\widetilde{\varphi}(t,h)\in\widetilde{\Phi}\), and the space \(\widetilde{\Phi}\) introduced by us satisfies all the conditions of Theorem 2.
The conditions of Theorem 2 are checked just as simply for a broad class of spaces. In particular, these conditions are satisfied by the \(B_\nu\)-spaces of \(T\)-periodic functions
\[
\left(\nu(\varphi)=\lim_{M\to\infty}\left(\frac1{2M}\int_{-M}^{+M}\|\varphi(t)\|^p\right)^{1/p},\ 1\le p\le+\infty\right);
\]
uniformly almost periodic functions
\[
\left(\nu(\varphi)=\sup_{-\infty\le t<+\infty}\|\varphi(t)\|\right);
\]
continuous functions, for each of which there exist two limits
\[
\lim_{t\to-\infty}\varphi(t)\quad\text{and}\quad \lim_{t\to+\infty}\varphi(t)
\]
\[
\left(\nu(\varphi)=\sup_{\infty<t<+\infty}\|\varphi(t)\|\right),
\]
and many others.
Despite the convenience of using Theorem 2 in many concrete cases, it seems expedient to indicate some sufficient conditions that would imply the fulfillment of the condition \(S \subset E(\widetilde{\Phi})\), but not contain
* Here and below the integral is understood in the sense of Bochner.
** In accordance with the notation introduced earlier, this should be understood as follows: \(A\) is any element of the set of all endomorphisms of \(Z\) whose spectra do not intersect the imaginary axis.
would allow passage to the limit in the sense of the topology of the space \(Z\). To obtain such results, however, one has to narrow the class of spaces under consideration.
Let us introduce the notation used in Theorems 3 and \(3'\).
Let \(\widetilde{\Phi}\) be a \(B_\nu\)-space. We shall call \(\widetilde{\Phi}\) a \(\widetilde{B}_\nu\)-space if the following conditions are satisfied: a) every function that differs from zero only on a set of measure \(0\) is \(\nu\)-equivalent to \(0\); b) convergence of a sequence \(\{\varphi_n(t)\}\) in the space \(\widetilde{\Phi}\) implies its convergence in measure on \((-\infty,+\infty)\).
Let \(\widetilde{\Phi}\) be such a \(B_\nu\)-space that \(P \subset E(\widetilde{\Phi})\). We shall denote by \(\widetilde{\Phi}_\tau=\{\varphi_\tau\}\) the set of all functions of the argument \(\tau\), with values in \(\widetilde{\Phi}\), of the form \(\varphi_\tau=\varphi(t+\tau)\), where \(\varphi(t)\in\widetilde{\Phi}\).
Theorem 3. Let \(\widetilde{\Phi}\) be a \(\widetilde{B}_\nu\)-space such that \(P \subset E(\widetilde{\Phi})\). If there exists an \(h>0\) such that all elements of the set \(\widetilde{\Phi}_\tau\) are strongly measurable on \([-h,+h]\), then \(S \subset E(\widetilde{\Phi})\).
Theorem 3 contains a requirement of measurability of functions from \(\widetilde{\Phi}_\tau\) that is difficult to verify; more convenient for applications is its corollary,
Theorem \(3'\). If \(\widetilde{\Phi}\) is such a \(\widetilde{B}_\nu\)-space that \(P \subset E(\widetilde{\Phi})\) and every function \(\varphi_\tau\) from \(\widetilde{\Phi}_\tau\) is continuous at the point \(\tau=0\), then \(S \subset E(\widetilde{\Phi})\).
We note that there exist \(\widetilde{B}_\nu\)-spaces satisfying the conditions of Theorem 2 but not satisfying the conditions of Theorems 3 and \(3'\). Such a space, in particular, is the space \(\widetilde{\Phi}\) from the example to Theorem 2; the \(B_\nu\)-spaces of periodic and almost periodic functions also satisfy the conditions of Theorem \(3'\).
In conclusion, we note that the requirement of boundedness of the operator \(A\) can be weakened; it is required only that it belong to a certain class of closed operators.
Leningrad Electrotechnical Institute
named after V. I. Ulyanov (Lenin)
Received
22 III 1967
REFERENCES
- M. G. Krein, Lectures on the Theory of Stability of Solutions of Differential Equations in Banach Space, Kiev, 1964.