Abstract
Full Text
A. I. PEROV
ON MULTIDIMENSIONAL LINEAR DIFFERENTIAL EQUATIONS WITH CONSTANT COEFFICIENTS
(Presented by Academician I. G. Petrovskii, 15 X 1963)
MATHEMATICS
1. Let \(E_x\) be an \(m\)-dimensional space over the field \(P_x\), and let \(E_y\) be an \(n\)-dimensional space over the field \(P_y\). Suppose that \(P_{xy}=P_x\cap P_y\) is also a field. An operator \(a\) acting from \(E_x\) into \(E_y\) will be called linear if
\(a(x_1+x_2)=ax_1+ax_2\) \((x_1,x_2\in E_x)\) and \(a(\alpha x)=\alpha ax\) for \(\alpha\in P_{xy}\). The set of linear operators from \(E_x\) into \(E_y\), under the usual definitions of the operations of addition and multiplication by scalars from \(P_y\), forms a linear space over the field \(P_y\), which we shall denote by \(E_{xy}\). In what follows we shall encounter the spaces \(E_{y(xy)}\) and \(E_{x(yy)}\). The fields that occur will be either the field of complex numbers or the field of real numbers.
Let \(A\in E_{y(xy)}\). The operator \(\widetilde A\in E_{x(yy)}\), uniquely determined by the equation
\((Ay)x=(\widetilde A x)y\) \((x\in E_x,\ y\in E_y)\), will be called the adjoint. The adjoint operator to an operator \(B\in E_{x(yy)}\) is defined analogously. We note that in the case when the spaces \(E_x\) and \(E_y\) are endowed with norms, the equality \(\|A\|=\|\widetilde A\|\) holds.
Let \(E'_y=E_\varphi\) be the space conjugate to \(E_y\). An operator \(A'\in E_{\varphi(x\varphi)}\) will be called conjugate to the operator \(A\in E_{y(xy)}\) if
\(((Ay)x,\varphi)=(y,(A'\varphi)x)\) for all \(x\in E_x,\ y\in E_y\), and \(\varphi\in E_\varphi\).
2. Let \(E_x\) be an \(m\)-dimensional space and \(E_y\) an \(n\)-dimensional space (each of them may be either real or complex). Consider the Cauchy problem for the linear equation \((A\in E_{y(xy)})\)
\[ \frac{dy}{dx}=Ay, \tag{1} \]
\[ y(\xi)=\eta \qquad (\xi\in E_x,\ \eta\in E_y). \tag{2} \]
Theorem 1. In order that the Cauchy problem (1)—(2) be solvable for every \(\eta\), it is necessary and sufficient that the condition
\[ \widetilde A h\,\widetilde A k=\widetilde A k\,\widetilde A h,\qquad h,k\in E_x \tag{3} \]
be satisfied.
If condition (3) is satisfied, the solution is written in the form
\[ y(x)=e^{\widetilde A(x-\xi)}\eta \tag{4} \]
and can be obtained by the usual method of successive approximations:
\[ y_0(x)\equiv \eta;\qquad \frac{dy_{k+1}}{dx}=Ay_k,\quad y_{k+1}(\xi)=\eta \quad (k=0,1,2,\ldots). \tag{5} \]
We note that condition (3) is superfluous if \(m=1\). In the case when \(E_x\) and \(E_y\) are real, Theorem 1 follows from the general theorems of the work \((^1)\).
Alongside problem (1)—(2), it is useful simultaneously to study the Cauchy problem for the conjugate equation
\[ \frac{d\varphi}{dx}=-A'\varphi, \tag{6} \]
\[ \varphi(\xi)=\theta \qquad (\xi\in E_x,\ \theta\in E_\varphi). \tag{7} \]
Theorem 2. If problem (1)—(2) is solvable for arbitrary \(\eta\), then problem (6)—(7) is solvable for arbitrary \(\theta\), and conversely,—if problem (6)—(7) is solvable for arbitrary \(\theta\), then problem (1)—(2) is solvable for arbitrary \(\eta\).
When the solvability condition is fulfilled, the solution of problem (6)—(7) is written in the form
\[ \varphi(x)=e^{-\widetilde{A}'(x-\xi)}\theta = e^{-[\widetilde{A}(x-\xi)]'\theta} \tag{8} \]
and can be obtained by the method of successive approximations:
\[ \varphi_0(x)\equiv\theta;\qquad \frac{d\varphi_{k+1}}{dx}=-A'\varphi_k,\qquad \varphi_{k+1}(\xi)=\theta \quad (k=0,1,2,\ldots). \tag{9} \]
The characteristic property of adjoint equations \((y(x),\varphi(x))\equiv \mathrm{const}\) also holds in our case.
- Let us now consider the Cauchy problem for operator equations
\[ \frac{dY}{dx}=\widetilde{A}Y,\qquad Y(\xi)=H, \tag{10} \]
\[ \frac{d\Phi}{dx}=-\widetilde{A}'\Phi,\qquad \Phi(\xi)=\theta. \tag{11} \]
It is easy to show that from the solvability of problem (1)—(2) for arbitrary \(\eta\) there follows the solvability of problem (10) for arbitrary \(H\), and conversely. Further, from the solvability of problem (10) for arbitrary \(H\) there follows the solvability of problem (11) for arbitrary \(\theta\); the converse is true. The solutions of problems (10) and (11) can be represented in the form
\[ Y(x)=e^{\widetilde{A}(x-\xi)}H,\qquad \Phi(x)=e^{-[\widetilde{A}(x-\xi)]'\theta}. \tag{12} \]
The operator function \(Y(x;\xi)=e^{\widetilde{A}(x-\xi)}\) will be called the fundamental operator-function (of equation (1)).
The solution \(Y(x)=e^{\widetilde{A}x}\) of problem (10) for \(\xi=0\), \(H=1\), as is easy to see, satisfies the following conditions
1) \(Y(0)=1\);
2) \(Y(x)\) is continuous at least for one value of \(x\);
3) \(Y(x+\xi)=Y(x)Y(\xi)\) for \(x,\xi\in E_x\).
For \(m=1\), Pólya’s theorem \(\left({}^{2}\right)\) is known, according to which an operator function satisfying conditions 1)—3) can be written in the form \(Y(x)=e^{xa}\). A generalization of this theorem to the case \(m>1\) is
Theorem 3. Let an operator function \(Y(x)\), defined for all \(x\in E_x\), satisfy conditions 1)—3). Then it can be represented in the form
\[ Y(x)=e^{\widetilde{A}x}, \tag{13} \]
where
\[ \widetilde{A}x=\lim_{\varepsilon\to 0}\frac{Y(\varepsilon x)-1}{\varepsilon} \tag{14} \]
and \(\widetilde{A}\) satisfies condition (3).
This theorem is proved according to the scheme proposed in \(\left({}^{3}\right)\).
- In all that follows we shall assume the space \(E_x\) to be real, and the space \(E_y\) to be complex.
Let \(A\in E_{y(xy)}\). A vector \(f\in E_y\) is called an eigenvector of the operator \(A\) \((\widetilde{A})\), if one can indicate a linear functional \(\lambda(x)\) for which, for all \(x\in E_x\), the equality holds
\[ (Af)x=\lambda(x)f \qquad \text{or} \qquad (\widetilde{A}x)f=\lambda(x)f. \tag{15} \]
The complex-valued functional \(\lambda(x)\) is called an eigenfunctional of the operator \(A\) \((\widetilde{A})\).
Theorem 4. Let the operator \(\widetilde{A}\) satisfy condition (3). Then one can specify subspaces \(E_1,\ldots,E_s\) such that
\[ E_y = E_1 \oplus \cdots \oplus E_s . \tag{16} \]
Each of the subspaces \(E_j\) is invariant with respect to the operators \(\widetilde{A}x\) \((j=1,\ldots,s)\); denote the induced operator by \(\widetilde{A}_j x\) \((j=1,\ldots,s)\). Then
\[ \widetilde{A}x=\widetilde{A}_1x\oplus \cdots \oplus \widetilde{A}_sx . \tag{17} \]
Each of the operators \(\widetilde{A}_j x\) is represented in the form
\[ \widetilde{A}_j x=\lambda_j(x)\mathbf{1}_j+\widetilde{T}_j x, \tag{18} \]
where all the eigenfunctionals \(\lambda_j(x)\) \((j=1,\ldots,s)\) are pairwise distinct, \(\mathbf{1}_j\) is the identity operator in the space \(E_j\) \((j=1,\ldots,s)\), and \(\widetilde{T}_j x\) is a nilpotent-valued operator \((j=1,\ldots,s)\).
It follows from this theorem that \(e^{\widetilde{A}x}\) can be represented in the form
\[ e^{\widetilde{A}x}=e^{\widetilde{A}_1x}\oplus \cdots \oplus e^{\widetilde{A}_sx}, \tag{19} \]
\[ e^{\widetilde{A}_j x} = e^{\lambda_j(x)} \left\{ \mathbf{1}_j+\widetilde{T}_j x+\cdots+ \frac{(\widetilde{T}_j x)^{n_j-1}}{(n_j-1)!} \right\}, \tag{20} \]
where \(n_j\) is the dimension of the space \(E_j\) \((j=1,\ldots,s)\).
Our subsequent results are connected with the decompositions (19) and (20).
- In this item we shall determine when all solutions of the operator equation
\[ \frac{dY}{dx}=\widetilde{A}Y \tag{21} \]
are bounded, almost periodic, or periodic. Here an operator function is called periodic if the linear span of its group of periods coincides with the entire space on which it is defined. (For the definition of almost periodic functions defined on a group, see the monograph \((^4)\).)
From formulas (19) and (20) it follows that \(\exp \widetilde{A}x\) is bounded if and only if the equalities
\[ \operatorname{Re}\lambda_j(x)=0,\qquad \widetilde{T}_j x=0,\qquad j=1,\ldots,s;\ x\in E_x \tag{22} \]
hold.
When these restrictions are satisfied, the decomposition (19) takes the form
\[ \exp \widetilde{A}x = \exp(i\,\operatorname{Im}\lambda_1(x))\,\mathbf{1}_1 \oplus \cdots \oplus \exp(i\,\operatorname{Im}\lambda_s(x))\,\mathbf{1}_s, \tag{23} \]
from which it follows immediately that a bounded solution of equation (21) is always almost periodic.
It is of interest to find conditions under which the operator function \(\exp \widetilde{A}x\) will be periodic. Let \(\nu_j(x)\) \((j=1,\ldots,s)\) be the imaginary parts of the eigenfunctionals \(\lambda_1,\ldots,\lambda_s\). Let \(L\) be the linear span of the functionals \(\nu_1,\ldots,\nu_s\) in the conjugate space \(E'_x\). It is easy to show that the annihilator \(L^0\) is a maximal constant manifold of the function \(\exp \widetilde{A}x\) \((^5)\) (for the terminology and notation of the present article, see \((^6)\)).
Theorem 5. The operator function \(\exp \widetilde{A}x\) is periodic if and only if in \(L\) one can specify a basis \(e_1,\ldots,e_k\) for which
\[ v_j=r_j^1 e_1+\cdots+r_j^k e_k\qquad (j=1,\ldots,s), \tag{24} \]
where the coefficients \(r_j^i\) are rational numbers.
- Suppose that the space \(E_y\) is normed. A direction \(h\in E_x\) will be called a direction of boundedness if, with \(c_h\) a constant, \(\|\exp(t\widetilde{A}h)\|\leq c_h\) for \(0\leq t<+\infty\). The totality of such \(h\) forms a convex pointed cone—the cone of boundedness.
A direction \(h\in E_x\) will be called a direction of tending to zero if \(\|\exp(t\widetilde{A}h)\|\to 0\) as \(t\to+\infty\). The totality of such \(h\) forms a convex blunt cone—the cone of tending to zero. It is not difficult to see that the cone is determined by the inequalities
\[ \operatorname{Re}\lambda_j(x)<0,\qquad j=1,\ldots,s, \tag{25} \]
where \(\{\lambda_j\}\) are all the eigenfunctionals of the operator \(A\). According to Carver’s theorem \((^7)\), the system of inequalities (25) is consistent if and only if the zero functional is not contained in the convex hull of the functionals \(\operatorname{Re}\lambda_j(x)\) \((j=1,\ldots,s)\).
In conclusion we formulate one more theorem.
Theorem 6. The following equality holds:
\[ \lim_{t\to+\infty}\frac{\ln\|\exp(t\widetilde{A}x)\|}{t} = \max_{1\leq j\leq s}\operatorname{Re}\lambda_j(x). \tag{26} \]
The author takes this opportunity to express his gratitude to Academician I. G. Petrovsky.
Voronezh State
University
Received
9 X 1963
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