Abstract
Full Text
UDC 517.917
MATHEMATICS
A. I. PEROV
NEW CONDITIONS FOR COMPLETE SOLVABILITY
AND QUESTIONS OF REDUCIBILITY
(Presented by Academician S. L. Sobolev on 18 X 1968)
Let \(E_x\) and \(E_y\) be finite-dimensional Banach spaces, the first of them real and the second complex. Denote by \(L(E_x;E_y)\) the Banach space of all linear operators acting from \(E_x\) into \(E_y\), and let \(L(E_x;E'_y)=L(E_x;L(E_y;E_y))\).
In this paper we study linear multidimensional differential equations of the form
\[ Y'(x)h=A(x)hY(x)\qquad(\forall h\in E_x), \tag{1} \]
\[ y'(x)h=A(x)hy(x)+f(x)h\qquad(\forall h\in E_x), \tag{2} \]
where \(A(x)\) and \(f(x)\) are operator functions defined on a convex domain \(G_x\subset E_x\) and taking values in the spaces \(L(E_x;E'_y)\) and \(L(E_x;E_y)\), respectively. In the first equation the unknown is an operator function \(Y(x)\) with values in \(L(E_y;E_y)\), while in the second equation it is a vector function \(y(x)\) with values in the space \(E_y\). Recall that a multidimensional differential equation is called completely solvable if every Cauchy problem is uniquely solvable.
The paper gives new conditions (necessary and sufficient) for complete solvability of multidimensional differential equations and studies the method of successive approximations; for the case of an almost periodic right-hand side, conditions for complete solvability are formulated in terms of Fourier coefficients and frequencies; with the aid of an estimate for the resolvent of a permutable operator, conditions are indicated for the reducibility of an equation with almost periodic coefficients.
Let \(\varphi(x)\) be a continuous operator function defined on the domain \(G_x\) and taking values in \(L(E_x;E_y)\), and let \(\partial s_\varepsilon\) be the boundary of the triangle \(s_\varepsilon=s(x;\varepsilon h,\varepsilon k)\), whose positive orientation is specified by the order of the vertices \(x, x+\varepsilon h, x+\varepsilon k\). Put
\[ \operatorname{rot}\varphi(x;h,k)=\lim_{\varepsilon\to+0}\frac{1}{\varepsilon^2} \int_{\partial s_\varepsilon}\varphi(\xi)\,d\xi, \tag{3} \]
if the limit exists. In the case where \(\operatorname{rot}\varphi(x;h,k)\) is a bilinear operator in \(h,k\), we write \(\operatorname{rot}\varphi(x)hk\). Note that if \(\varphi(x)\) is differentiable, then \(\operatorname{rot}\varphi(x)hk=\Lambda_{hk}\varphi'(x)hk\) \((\Lambda_{hk}Bhk=(Bhk-Bkh)/2)\).
1. Of great interest in the study of multidimensional differential equations is the question of conditions for complete solvability. If the right-hand side is continuously differentiable, then the answer is given by Frobenius’ theorem (see, for example, \((^1)\), p. 356). In the monograph \((^2)\), R. Nevanlinna gives the condition for complete solvability of the operator equation (1) in the form
\[ U(x;h,k)=1+o(\delta^2), \tag{4} \]
where \(U(x;h,k)\) is the multiplicative integral of \(A(x)\), computed over \(\partial s=\partial s(x;h,k)\), and \(\delta\) is the diameter of the triangle \(s\). Developing this assertion, we arrive at the following theorem.
Theorem 1. Let \(A(x)\) and \(f(x)\) be continuous. Then complete solvability of equation (2) holds if and only if the conditions \((\forall h,k\in E_x)\)
\[ \operatorname{rot} A(x)hk=\Lambda_{hk}A(x)hA(x)k, \tag{5} \]
\[ \operatorname{rot} f(x)hk=\Lambda_{hk}A(x)hf(x)k. \tag{6} \]
The proof is based on the formulas
\[ U(x;h,k)=1+\int A(\xi)\,d\xi-\Lambda_{hk}A(x)hA(x)k+o(\delta^2), \tag{7} \]
\[ \int_{\partial s} f(\xi)\,d\xi=\Lambda_{hk}A(x)hf(x)k+o(\delta^2), \tag{8} \]
which hold for any compact set \(K\subset G_x\), the above-cited assertion of R. Nevanlinna and one result of V. Shapiro \((^3)\).
We shall seek a solution of equation (2) satisfying the initial condition \(y(\xi)=\eta\), by the method of successive approximations
\[ y_0(x)\equiv \eta \]
\[ y_p'(x)h=A(x)hy_{p-1}(x)+f(x)h,\qquad y_p(\xi)=\eta \tag{9} \]
\[ (p=1,2,\ldots). \]
Theorem 2. The method of successive approximations (9) is realizable for arbitrary initial conditions if and only if \((\forall h,k\in E_x)\)
\[ \operatorname{rot} A(x)=0,\qquad \Lambda_{hk}A(x)hA(x)k=0; \tag{10} \]
\[ \operatorname{rot} f(x)=0,\qquad \Lambda_{hk}A(x)hf(x)k=0. \tag{11} \]
Let \(U(A;f)\) denote the totality of all continuous vector functions \(u(x)\) for which the curvilinear integral of \(\varphi(x)h=A(x)hu(x)+f(x)h\) is path-independent. It can be shown that \(U(A;f)\) contains all constants and is mapped into itself by the integral operator
\[ I_\eta u(x)=\eta+\int_{\xi}^{x} A(\zeta)\,d\zeta\,u(\zeta)+\int_{\xi}^{x} f(\zeta)\,d\zeta \tag{12} \]
for any \(\eta\in E_y\), if and only if conditions (10) and (11) are satisfied.
- Let us now consider equation (2) with almost periodic operator functions \(A(x)\) and \(f(x)\), and let
\[ A(x)\sim \sum A_\lambda e^{i\lambda x},\qquad f(x)\sim \sum f_\lambda e^{i\lambda x}; \tag{13} \]
in this case \(G_x=E_x,\ \lambda\in E_x^*\) (see (7)).
Theorem 3. Complete solvability for equation (2) with almost periodic operator functions \(A(x)\) and \(f(x)\) holds if and only if \((\forall h,k\in E_x)\)
\[ \Lambda_{hk}i\lambda h\,A_\lambda k = \Lambda_{hk}\left(\sum_{\mu+\nu=\lambda} A_\mu hA_\nu k\right), \tag{14} \]
\[ \Lambda_{hk}i\lambda h\,f_\lambda k = \Lambda_{hk}\left(\sum_{\mu+\nu=\lambda} A_\mu hf_\nu k\right). \tag{15} \]
We emphasize that no smoothness of \(A(x)\) and \(f(x)\) is assumed in the theorem. The proof of Theorem 3 is based on the use of Theorem 1 and the following assertion (the theorem on the rotor).
Theorem 4. Let \(f(x)\) be an almost periodic function, and suppose that one can specify an almost periodic function \(R(x)\) with values
in \(L_2(E_x;E_y)\), such that for any \(h,k\in E_x\)
\[ R(x)hR\sim\sum \Lambda_{hk}(i\lambda hf,\lambda k)e^{i\lambda x}. \tag{16} \]
Then
\[ \int_{\partial s} f(\xi)\,d\xi=\iint_s R(\xi)\,d\xi_1\,d\xi_2 . \tag{17} \]
In formula (17), on the right there is the affine integral (see \((^2)\)); \(L_2(E_x;E_y)\) is the space of bilinear operators defined on \(E_x\oplus E_x\) and with values in \(E_y\). Theorem 4 is most simply proved with the aid of Bochner–Fejér polynomials (see, for example, \((^4)\)), whose theory is not difficult to carry over to vector-valued (and operator-valued) functions.
- An operator \(A\in L[E_x;E_y]\) is called permutable if \(\Lambda_{hk}AhAk=0\). A linear functional \(\lambda\) is called an eigenfunctional of the operator \(A\) if \(Ah\xi=(\lambda h)\xi\) \((0\ne \xi\in E_y)\) (see \((^5)\)). The totality \(\sigma(A)\) of all eigenfunctionals is called the spectrum of the operator \(A\). The least natural number \(p\) for which \((\lambda h-Ah)^{p+1}y=0\) implies \((\lambda h-Ah)^p y=0\) is called the index \(n(\lambda)\) of the functional \(\lambda\).
It can be shown that \(\lambda\in\sigma(A)\) if and only if \(\det(\lambda h-Ah)=0\). In expanded form this equation has the form
\[ (\lambda h)^n-p_1(h)(\lambda h)^{n-1}+\cdots+(-1)^n p_n(h)=0, \tag{18} \]
where \(p_j(h)\) is a homogeneous functional of degree \(j\). In order to eliminate \(h\) from the written equation, call the product \(f\varphi\) of symmetric functionals \(f\) and \(\varphi\) the symmetric functional uniquely determined by the formula \((f\varphi)h^{p+q}=(fh^p)(\varphi h^q)\) (\(f\) and \(\varphi\) are assumed to be \(p\)- and, respectively, \(q\)-linear functionals). Let \(p_j\) be the symmetric \(j\)-linear functional uniquely determined from the relation \(p_jh^j=p_j(h)\) (as above, we use here the result from \((^6)\), p. 769). Then equation (18) takes the form
\[ \mathfrak D(\lambda)\stackrel{\mathrm{def}}{=}\lambda^n-p_1\lambda^{n-1}+\cdots+(-1)^n p_n=0 . \tag{19} \]
Let us note that for a permutable operator the formula
\[ \mathfrak D(\lambda)=\prod_{\mu\in\sigma(A)}(\lambda-\mu)^{n(\mu)} \tag{20} \]
is valid.
If \(\lambda\notin\sigma(A)\), then the equation \((\lambda h-Ah)y=fh\) is uniquely solvable if and only if \(\Lambda_{hk}(\lambda h-Ah)fk=0\) \((f\in L(E_x;E_y))\). The inverse operator \(R(\lambda;A)\) will be called the resolvent of the permutable operator \(A\).
Theorem 5. Let \(A\) be a permutable operator and \(\lambda\notin\sigma(A)\). Then
\[ \|R(\lambda;A)\|\le c(A)\left(\sum_{j=1}^{n(A)}\frac{1}{[\rho(\lambda)]^j}\right), \tag{21} \]
where \(\rho(\lambda)\) is the distance from \(\lambda\) to \(\sigma(A)\); \(n(A)\) is the maximum of the indices of the eigenfunctionals of the operator \(A\).
Let us now consider the operator \(\mathfrak C\in L\{E_x;E_y\}\), given by the formula
\[ \mathfrak C hY=AhY+YBh, \tag{22} \]
where \(L\{E_x;E_y\}=L[E_x;L(E_y;E_y)]\) and \(A,B\in L[E_x;E_y]\), \(Y\in L(E_y;E_y)\).
Theorem 6. The operator \(\mathfrak C\) is permutable if and only if the operators \(A\) and \(B\) are permutable; \(\sigma(\mathfrak C)=\sigma(A)+\sigma(B)\), and
\[ n(\nu)\le \max_{\lambda+\mu=\nu}[n(\lambda)+n(\mu)]-1, \]
where \(\lambda\in\sigma(A)\), \(\mu\in\sigma(B)\), and \(\nu\in\sigma(\mathfrak C)\).
- Consider the operator equation, completely solvable for all \(\varepsilon\),
\[ Y'(x)h=(A+\varepsilon B(x))hY(x), \tag{23} \]
where \(A\in L[E_x;E_y]\), and the operator function \(B(x)\) is an almost periodic function, \(B(x)\sim\sum B_\mu e^{i\mu x}\), with \(B_0=0\). The operator \(A\) is permutable, and suppose that its spectrum consists of the functionals \(\lambda^1,\ldots,\lambda^p\). Denote by \(\sigma_B\) the spectrum of the almost periodic function \(B(x)\), and let \(j\sigma_B\) be the set of all sums of the form \(\mu^1+\cdots+\mu^j\), where \(\mu^i\in\sigma_B\) \((i=1,\ldots,j)\). Suppose that \(\|B()\|_*=\sum\|B_\mu\|<\infty\) and
\[ \delta_j=\inf \|i\nu-(\lambda^t-\lambda^s)\|, \tag{24} \]
where \(\nu\in j\sigma_B\) and \(t,s=1,2,\ldots,p\). Put
\[ \Delta_j=\sum_{k=1}^{2n(A)-1}\frac{1}{\delta_j^k}, \tag{25} \]
\[ \frac{1}{\varkappa}=c(\mathfrak{C})\|B()\|_*\lim_{p\to\infty}(\Delta_1\cdots\Delta_p)^{1/p}, \tag{26} \]
where \(c(\mathfrak{C})\) is the constant in an estimate of type (21) for the resolvent of the operator \(\mathfrak{C}\), where \(\mathfrak{C}hY=AhY+YB h\).
Theorem 7. Under the assumptions stated above, the solution of equation (23) with \(Y(0,\varepsilon)=1\) is representable in the form
\[ Y(x,\varepsilon)=Z(x,\varepsilon)e^{Ax}, \tag{27} \]
where the series
\[ Z(x,\varepsilon)=\sum_{j=0}^{\infty}\varepsilon^j Z_j(x) \tag{28} \]
converges uniformly and absolutely for \(|\varepsilon|<\varkappa\), and each \(Z_j(x)\) is an almost periodic operator function whose spectrum lies in \(j\sigma_B\).
Voronezh State
University
Received
2 X 1968
REFERENCES
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