Full Text
Yu. I. LYUBICH
ON CONDITIONS FOR SOLVABILITY OF THE ABSTRACT CAUCHY PROBLEM
(Presented by Academician S. N. Bernstein on 20 VII 1963)
Let us consider* the differential equation
\[ \frac{dx(t)}{dt}=Ax(t)\qquad (0\leq t<T;\; T\leq \infty) \tag{1} \]
with a linear operator \(A\) in a Banach space, and pose the question: for which vectors \(x_0\) does there exist a solution of equation (1) satisfying the initial condition
\[ x(0)=x_0. \tag{2} \]
In order to answer this question, we shall generalize the classical Laplace transform in such a way that the generalized (or, as we shall call it, local) Laplace transform is applicable on a finite time interval \([0,T)\), as well as on the infinite interval \([0,\infty)\) without restrictions on growth.
Let \(f(\lambda)\) be a locally integrable** vector-valued function defined for all sufficiently large \(\lambda>0\), say for \(\lambda\geq \lambda_f\). We shall say that the function \(f(\lambda)\) belongs to the class \(\Lambda_T\) \((0<T\leq \infty)\) if there exists a vector-valued function \(\widetilde f(\tau,t)\) \((0\leq \tau\leq t,\;0<t<T)\), integrable** with respect to \(\tau\), such that for every \(t\in(0,T)\) the representation
\[ f(\lambda)=\int_0^t \widetilde f(\tau,t)e^{-\lambda\tau}\,d\tau+\varepsilon(\lambda,t) \]
holds, with a remainder term \(\varepsilon(\lambda,t)\) satisfying the condition
\[ \overline{\lim_{\lambda\to+\infty}}\lambda^{-1}\ln\|\varepsilon(\lambda,t)\|\leq -t. \]
We shall call the function \(\widetilde f(\tau,t)\) the local Laplace original for \(f(\lambda)\), and \(f(\lambda)\) the local Laplace transform (image) for \(\widetilde f(\tau,t)\).
Lemma 1. For a given function \(f(\lambda)\in\Lambda_T\), the local Laplace original \(\widetilde f(\tau,t)\) is uniquely determined*** and does not depend on the parameter \(t\).
Therefore, in what follows one may write \(\widetilde f(\tau)\) instead of \(\widetilde f(\tau,t)\). The proof of this lemma is based on the fact (see, for example, (²), pp. 412–413) that if a scalar function \(\psi(t)\) \((0\leq t\leq a,\; a<\infty)\), summable on \([0,a]\), is such that
\[ \overline{\lim_{\lambda\to+\infty}}\lambda^{-1}\ln \left|\int_0^a \psi(t)e^{\lambda t}\,dt\right| \leq b, \]
where \(0\leq b<a\), then \(\psi(t)=0\) almost everywhere on \((b,a)\).
* For definiteness we consider the so-called ACP ((¹), pp. 631–632), but the method we use is general.
** In the strong sense (see, for example, (¹), pp. 92–103).
*** This cannot be said of the local Laplace transform.
The following proposition describes an analytic procedure for inverting the local Laplace transform*.
Lemma 2. Let \(f(\lambda)\in \Lambda_T\), and let \(\tilde f(\tau)\) be the local Laplace original for \(f(\lambda)\). Put
\[ F(t)=\frac{1}{2\pi i}\int_a^\infty \frac{f(\lambda)e^{\lambda t}}{\lambda}\,d\lambda \qquad (a\geq \lambda_f,\ a>0;\ \operatorname{Re} t<0). \]
The function \(F(t)\) is analytically continued to the domain
\[
\Pi_T=\{t\mid \operatorname{Re} t<T,\ t\notin [0,T)\}
\]
and
\[ \lim_{\sigma\downarrow 0}\,[F(t+i\sigma)-F(t-i\sigma)] = \int_0^t \tilde f(\tau)\,d\tau \qquad (0<t<T). \tag{3} \]
Another method of inverting the local Laplace transform, not requiring complex variables, follows from a formula of Phragmén \((^3);\ (^{4}),\) pp. 21–23), but we shall not dwell on this.
Denote by \(\Lambda_T^{(1)}\) the subclass of the class \(\Lambda_T\) singled out by the requirement of smoothness (in the strong sense) of the original \(\tilde f(\tau)\) \((0\leq \lambda<T)\). In this subclass the inversion formula is simplified. Namely, instead of \(F(t)\) one should consider the function
\[ \Phi(t)=\frac{1}{2\pi i}\int_{\lambda_f}^{\infty} f(\lambda)e^{\lambda t}\,d\lambda, \]
and instead of (3) one may write
\[ \Phi(t+i0)-\Phi(t-i0)=\tilde f(t) \qquad (0<t<T). \tag{4} \]
We now turn to the abstract Cauchy problem (1)—(2). Suppose that there exists a real \(\lambda=\lambda_A\) such that the values \(\lambda\geq \lambda_A\) do not belong to the spectrum of the operator \(A\).
Theorem 1. If the resolvent \(R_\lambda\) of the operator \(A\) satisfies the condition**
\[ h_A\equiv \overline{\lim_{\lambda\to+\infty}}\ \lambda^{-1}\ln \|R_\lambda\|<\infty, \tag{5} \]
then \(R_\lambda x_0\in \Lambda_{T-h_A}^{(2)}\) for all those vectors \(x_0\) for which the problem (1)—(2) \((T>h_A)\) has a smooth solution.
Proof. Apply the operator \(R_\lambda\) to both sides of equation (1):
\[ \frac{dR_\lambda x(t)}{dt}=x(t)+\lambda R_\lambda x(t) \qquad (0\leq t<T). \]
Hence
\[ R_\lambda x_0 = -\int_0^t x(\tau)e^{-\lambda\tau}\,d\tau + e^{-\lambda t}R_\lambda x(t) \qquad (0\leq t<T). \tag{6} \]
In view of (5), \(R_\lambda x_0\in \Lambda_{T-h_A}^{(1)}\), and, analogously to the classical situation, the local Laplace transform for \(R_\lambda x_0\) is equal to \(-x(\tau)\).
* We note that the local Laplace original coincides with the classical one if the latter exists.
** This condition ensures, according to the results of \((^5,^6)\), the uniqueness of the solution of problem (1)—(2).
Corollary. Let condition (5) be satisfied. Put
\[ K_t=\frac{1}{2\pi i}\int_{\lambda_A}^{\infty} R_\lambda e^{\lambda t}\,d\lambda \qquad (\operatorname{Re} t<-h_A). \]
If \(x(t)\) is a smooth solution of problem (1)—(2), then the vector-function \(K_t x_0\) is analytically continued into the domain \(\Pi_{T-h_A}\), and there
\[ K_t x_0\big|_{t-i0}^{t+i0}=x(t)\qquad (0<t<T-h_A). \]
This proposition substantially supplements the uniqueness theorems obtained by the author in \((^{5,6})\).
Theorem 1 can be inverted in the following sense.
Theorem 2. If the vector \(x_0\) is such that \(R_\lambda x_0\in \Lambda_T^{(1)}\), then problem (1)—(2) has a smooth solution.
Here nothing is assumed about the behavior of \(\|R_\lambda\|\) as \(\lambda\to+\infty\).
Proof. Denote the local Laplace original of the function \(R_\lambda x_0\) by \(-x(\tau)\) \((0\leq \tau<T)\):
\[ R_\lambda x_0=-\int_0^t x(\tau)e^{-\lambda\tau}\,d\tau+e^{-\lambda t}\eta(\lambda,t) \qquad (0\leq t<T). \tag{7} \]
Here
\[ \overline{\lim_{\lambda\to+\infty}}\lambda^{-1}\ln \|\eta(\lambda,t)\|\leq 0. \tag{8} \]
We shall show that the function \(x(t)\) satisfies equation (6) (i.e., that \(\eta(\lambda,t)=R_\lambda x(t)\)).
Apply to both sides of relation (7) the operator \(R_\mu\) \((\mu\geq \lambda_A)\) and use Hilbert’s identity
\[ R_\mu R_\lambda=\frac{R_\lambda-R_\mu}{\lambda-\mu}. \]
Putting \(\lambda-\mu=\omega\), we obtain:
\[ R_\lambda x_0-R_\mu x_0 = -\omega\int_0^t e^{-\lambda\tau}R_\mu x(\tau)\,d\tau +\omega e^{-\lambda t}R_\mu\eta(\lambda,t). \]
Hence, by virtue of (7),
\[ \omega\int_0^t e^{-\lambda\tau}R_\mu x(\tau)\,d\tau -\int_0^t x(\tau)e^{-\lambda\tau}\,d\tau = R_\mu x_0+ [\omega R_\mu\eta(\lambda,t)-\eta(\lambda,t)]e^{-\lambda t}. \tag{9} \]
But
\[ \int_0^t x(\tau)e^{-\lambda\tau}\,d\tau = e^{-\omega t}\int_0^t x(\tau)e^{-\mu\tau}\,d\tau + \omega\int_0^t e^{-\omega\tau}\,d\tau \int_0^\tau x(\sigma)e^{-\mu\sigma}\,d\sigma. \tag{10} \]
Putting
\[ \rho(\mu,\tau)=e^{-\mu\tau}R_\mu x(\tau)-\int_0^\tau x(\sigma)e^{-\mu\sigma}\,d\sigma, \]
we obtain from (9) and (10)
\[ \omega\int_0^t e^{-\omega\tau}\rho(\mu,\tau)\,d\tau = R_\mu x_0+ e^{-\omega t}\int_0^t x(\tau)e^{-\mu\tau}\,d\tau + [\omega R_\mu\eta(\lambda,t)-\eta(\lambda,t)]e^{-\lambda t}, \]
whence
\[ \int_0^t e^{\omega \tau}\,[\rho(\mu,t-\tau)-R_\mu x_0]\,d\tau = \]
\[ = \frac{1}{\omega}R_\mu x_0+\frac{1}{\omega}\int_0^t x(\tau)e^{-\mu \tau}\,d\tau+ \left[R_\mu\eta(\mu+\omega,t)-\frac{1}{\omega}\eta(\mu+\omega,t)\right]e^{-\mu t}. \]
Fixing \(\mu\) and letting \(\omega\) tend to \(+\infty\), we obtain, by virtue of (8),
\[ \varlimsup_{\omega\to+\infty} \frac{\ln\left\|\int_0^t e^{\omega \tau}[\rho(\mu,t-\tau)-R_\mu x_0]\,d\tau\right\|}{\omega} \leq 0. \]
It follows from this that
\[ \rho(\mu,t-\tau)=R_\mu x_0 \qquad (0\leq \tau\leq t,\ \mu\geq \lambda_A), \]
i.e., that \(x(t)\) does indeed satisfy equation (6). Differentiating (6) with respect to \(t\), we obtain
\[ R_\lambda\left[\frac{dx(t)}{dt}-\lambda x(t)\right]=x(t), \]
whence it follows that \(x(t)\in D_A\) and equation (1) holds. Setting \(t=0\) in (6), we arrive at equality (2). Thus the theorem is completely proved.
Kharkov State University
named after A. M. Gorky
Received
17 VII 1963
REFERENCES
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