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ON BOUNDED SOLUTIONS OF A SYSTEM OF DIFFERENTIAL EQUATIONS
N. V. Medvedev
Many investigators have devoted their work to the study of bounded solutions of systems of ordinary differential equations (see, for example, [1—4]). In the present article we study questions of the existence of bounded solutions of a system of equations and their asymptotic behavior.
- In what follows we shall need the following linear difference equation:
\[ A(t)x(t+\omega)-x(t)=f(t), \tag{1} \]
where \(x\) and \(f\) are real \(n\)-dimensional vectors, \(A(t)\) is a real square matrix of order \(n\), \(\omega\) is some positive number, and \(f(t)\) and \(A(t)\) are given on the interval \((-\infty,+\infty)\).
Let us note some cases of existence of a solution of equation (1).
Theorem 1. If there exist positive numbers \(K\) and \(q\) such that \(\|f(t)\|\le K\) and \(\|A(t)\|\le q<1\) for all \(t\), where \(\|f(t)\|\) and \(\|A(t)\|\) are the norms of \(f(t)\) and \(A(t)\), respectively, then equation (1) has a bounded solution in the form of the series
\[ x(t)=-f(t)-\sum_{k=0}^{\infty} A(t)A(t+\omega)\ldots A(t+k\omega)f(t+(k+1)\omega). \]
Theorem 2. If the matrix \(A(t)\) is invertible and \(\|f(t)\|\le K\), \(\|A^{-1}(t)\|\le q<1\) for all \(t\), then equation (1) has the solution
\[ x(t)=\sum_{k=1}^{\infty} A^{-1}(t-\omega)\ldots A^{-1}(t-k\omega)f(t-k\omega). \]
In the special case the following is true.
Theorem 3. Let \(A-E\) be an invertible matrix (\(E\) is the identity matrix). If \(A(t)\) and \(f(t)\) have one and the same period \(\omega\), then \(x(t)=(A-E)^{-1}f(t)\) is a solution of equation (1).
- Consider the nonlinear system
\[ \frac{dx}{dt}=Px+f(t,x), \tag{2} \]
where \(P\) is a real constant matrix of order \(n\), and \(f(t,x)\) is a vector function, continuous in the domain \(\{T\le t<+\infty;\ \|x\|\le R\}\) (here \(T\) and \(R\) are some positive numbers).
Theorem 4. Let \(P\) have no eigenvalues of the form \(\dfrac{2\pi k}{\omega} i\) (\(k\) an integer or zero, \(i\) the imaginary unit). Let \(f(t,x)\) be an \(\omega\)-periodic function in \(t\) such that \(\|f(t,x)\|\leq N\) for \(\|x\|\leq R\) and \(t\geq T\). If
\[ e^{\|P\|\omega}\left\|(e^{-P\omega}-E)^{-1}\right\|N\omega\leq R, \tag{3} \]
then system (2) has at least one \(\omega\)-periodic solution.
Proof. Since the matrix \(e^{-P\omega}-E\) is invertible, the operator defined by the right-hand side of the equation
\[ x(t)=e^{Pt}(e^{-P\omega}-E)^{-1}\int_t^{t+\omega} e^{-P\tau} f(\tau,x(\tau))\,d\tau \]
is continuous and, by virtue of (3), maps the convex set of continuous \(\omega\)-periodic functions \(x(t)\), for which \(\|x(t)\|\leq R\), into its compact part. Therefore, by the well-known Schauder principle, this operator has a fixed point, which, by Theorem 3, is a solution of system (2). The theorem is proved.
We note that condition (3) holds, for example, if: a) \(f(t,x)\) is a bounded function in the domain \(\{T\leq t<+\infty,\ \|x\|\leq R\}\), or b) the estimate \(\|f(t,x)\|\leq L\|x\|\) holds for sufficiently large \(\|x\|\) with sufficiently small \(L\). The proof of this theorem for case b) is given in [3].
Theorem 5. Let the eigenvalues of the matrix \(P\) have negative real parts. Let \(\|f(t,x)\|\leq N\) for \(t\geq T\) and \(\|x\|\leq R\). Then, if \(\mu<R\) for some \(\omega>0\) such that \(\|e^{P\omega}\|<1\), where
\[ \mu=\frac{N\|e^{P\omega}\|}{1-\|e^{P\omega}\|} \max_{T\leq t<+\infty}\int_t^{t+\omega}\|e^{P(t-\tau)}\|\,d\tau, \]
then for sufficiently small \(\|x_0\|\) there exists at least one solution \(x(t)\) on \([T,+\infty)\), satisfying the conditions \(x(T)=x_0,\ \|x(t)\|<R\).
Proof. Introduce the function
\[ f^{*}(t,x(t))= \begin{cases} f(t,x(t)) & \text{for } t\geq T,\\[4pt] \dfrac{1}{\omega} e^{P(t-T)}x_0 & \text{for } T-\omega\leq t<T,\\[4pt] 0 & \text{for } t<T-\omega. \end{cases} \]
Let \(F(t,x(t))\) be a solution of the difference equation
\[ e^{P\omega}F(t+\omega,x(t+\omega))-F(t,x(t))=f^{*}(t,x(t)). \]
If \(T+k\omega\leq t<T+(k+1)\omega\), where \(k\) is a nonnegative integer, then by Theorem 2 we have
\[ F=e^{P\omega}f(t-\omega,x(t-\omega))+\cdots+ \]
\[ +e^{Pk\omega}f(t-k\omega,x(t-k\omega))+\frac{1}{\omega}e^{P(t-T)}x_0. \]
Therefore, the inequality
\[ \|F(t,x(t))\| \leqslant \frac{N\|e^{P\omega}\|}{1-\|e^{P\omega}\|}+\frac{M}{\omega}\|x_0\| \tag{4} \]
holds for \(t \geqslant T\) and \(\|x(t)\|\leqslant R\); here
\[ M=\max_{0\leqslant \tau<+\infty}\|e^{P\tau}\|. \]
Consider the operator defined by the right-hand side of the equality
\[ y(t)=e^{Pt}\int_t^{t+\omega} e^{-P\tau}F(\tau,x(\tau))\,d\tau . \tag{5} \]
Since for \(t\in [T+k\omega,\,T+(k+1)\omega]\) we have
\[ \begin{aligned} y(t) &= e^{Pt}\Biggl( \int_{t-\omega}^{t} e^{-P\tau} f\,d\tau+\cdots+ \int_{t-k\omega}^{t-(k-1)\omega} e^{-P\tau} f\,d\tau \\ &\quad + \int_{t-(k+1)\omega}^{T} e^{-P\tau}\frac{1}{\omega}e^{P(\tau-T)}x_0\,d\tau + \int_T^{t-k\omega} e^{-P\tau} f\,d\tau \\ &\quad + \int_{T-\omega}^{t-(k+1)\omega} e^{-P\tau}\frac{1}{\omega}e^{P(\tau-T)}x_0\,d\tau \Biggr) \\ &= e^{Pt}\int_T^t e^{-P\tau} f(\tau,x(\tau))\,d\tau + e^{P(t-T)}x_0, \end{aligned} \tag{6} \]
then \(y(t)\) is continuous if \(x(t)\) is continuous. Moreover, in view of (4) and \(\mu<R\), for sufficiently small \(\|x_0\|\) the inequality \(\|y(t)\|<R\) holds for \(t\geqslant T\) and \(\|x(t)\|\leqslant R\). Consequently, by Schauder’s principle the operator (5) has a fixed point \(x(t)\) on the interval \([T,T_1]\) for any \(T_1>T\). By the same principle one can show that the obtained solution \(x(t)\) can be continued to the right of \(T_1\). Since in doing so \(x(t)\) does not leave the domain \(\|x\|<R\), it exists on the entire interval \([T,+\infty)\). The theorem is proved.
Let us note that if \(\mu>R\), then, in general, Theorem 5 is not true. A confirming example is the equation \(x'=-x+(x+x^2)\), for which \(\mu=R+R^2\). The condition \(\mu<R\) holds, for example, in cases a) and b) indicated for Theorem 4.
Theorem 6. Let the eigenvalues of the matrix \(P\) have positive real parts. Let \(\|f(t,x)\|\leqslant N\) for \(t>T\) and \(\|x\|\leqslant R\). Then, if \(\nu<R\) for some \(\omega>0\) such that \(\|e^{-P\omega}\|<1\), where
\[ \nu=\frac{N}{1-\|e^{-P\omega}\|} \max_{T<t<+\infty}\int_t^{t+\omega}\|e^{P(t-\tau)}\|\,d\tau, \]
there exists at least one solution of system (2), bounded on \([T,+\infty)\).
For the proof, as in the preceding case, we consider operator (5), where in the present case \(F(t,x(t))\) is a solution of the equation
\[ e^{-P\omega}F(t+\omega,x(t+\omega))-F(t,x(t))=f(t,x(t)). \]
Operator (5), by virtue of \(\nu<R\), has a fixed point \(x(t)\) on the interval \([T,T_1]\), and \(x(t)\), by Theorem 1, is a solution of system (1) and can be continued indefinitely to the right of \(T_1\).
We note that, taking into account the conditions in Theorems 5 and 6 imposed on the function \(f(t,x)\), one can obtain cases when system (2) has at least one bounded solution if \(P\) has no eigenvalues with nonzero real part.
In some cases one can judge the asymptotic behavior of system (2).
Theorem 7. Let \(P\) have eigenvalues with negative real parts, and suppose there exists a nonnegative function \(c(t)\), \(t\ge T\), satisfying the condition \(\|f(t,x)\|\le c(t)\|x\|\) for \(\|x\|\le R\). Let
\[ \int_\tau^t c(t)\,dt+\|e^{P(t-\tau)}\|\le \psi(t,\tau) \]
and
\[ \inf_{\tau<t<+\infty}\psi(t,\tau)\le \psi(t_0,\tau) \]
for any \(\tau>T\), where \(t_0=\varphi(\tau)\), \(\tau<\varphi(\tau)<+\infty\). Then, if \(\psi(t_0,\tau)\le q\) for all \(\tau>T\), where \(q\) is a constant and \(0<q<1\), the solution \(x(t)\), \(\|x(t)\|\le R\), of system (2) tends to zero as \(t\to+\infty\).
Proof. By virtue of (6) and \(\psi(t_0,\tau)\le q\), we have
\[ \max_{\tau_k<t<\tau_{k+1}}\|x(t)\|\le q\max_{\tau_{k-1}<t<\tau_k}\|x(t)\|, \]
where \(\tau_{k+1}=\varphi(\tau_k)\). Hence it follows that \(x(t)\to0\) as \(t\to+\infty\).
Let us consider some known cases when Theorem 7 is valid (see [4]). Let
\[ \int_T^{+\infty} c(t)\,dt\le p,\qquad p<q<1. \]
If \(\|e^{P(t-\tau)}\|\le Ke^{-\sigma(t-\tau)}\), where \(K\) and \(\sigma\) are certain constants, then in this case
\[ \psi(t,\tau)=p+Ke^{-\sigma(t-\tau)},\quad \varphi(\tau)=-\frac{1}{\sigma}\ln\frac{q-p}{K}+\tau,\quad \psi(t_0,\tau)=q, \]
i.e., the conditions of Theorem 7 are satisfied. Now suppose \(\|f(t,x)\|\le L\|x\|\), where \(L\) is a sufficiently small constant. Then
\[ \psi(t,\tau)=L(t-\tau)+Ke^{-\sigma(t-\tau)}, \]
\[ \varphi(\tau)=-\frac{1}{\sigma}\ln\frac{L}{\sigma K}+\tau,\quad \psi(t_0,\tau)=\frac{L}{\sigma}\left(1-\ln\frac{L}{\sigma K}\right). \]
Here \(t_0=\varphi(\tau)\) is defined as the point of minimum of the function \(\psi(t,\tau)\). Thus Theorem 7 holds if
\[ \frac{L}{\sigma}\left(1-\ln\frac{L}{\sigma K}\right)<1,\qquad L<\sigma K. \tag{7} \]
On the basis of (7) one can obtain more precise estimates for \(L\) than the estimate \(L < \sigma K^{-1}\) (see [4]). For example, condition (7) is satisfied if
\[ L < \frac{2\sigma}{2+\ln K}. \]
References
- Demidovich B. P. Matem. sb., No. 1, 73–94, 1956.
- Krasnosel’skii M. A., Perov A. I. DAN SSSR, 123, No. 2, 235–238, 1958.
- Pliss V. A. Nonlocal Problems in the Theory of Oscillations. Moscow, Nauka Publishing House, 1964.
- Coddington E. A., Levinson N. Theory of Ordinary Differential Equations. IL, 1958.
Received by the editors
March 22, 1965
Omsk Pedagogical
Institute