Full Text
D. P. MILMAN
FORMULATION AND METHODS OF SOLVING THE GENERAL BOUNDARY-VALUE PROBLEM OF OPERATOR THEORY FROM THE VIEWPOINT OF FUNCTIONAL ANALYSIS.
PROBLEMS OF CAUCHY AND DIRICHLET TYPE
(Presented by Academician I. G. Petrovskii, 14 XI 1964)
MATHEMATICS
1. Let \(R\) and \(R_1\) be separable linear topological spaces (l.t.s.); let \(L\) be a linear operator mapping \(R\) onto \(R_1\); let \(N_L\) be the set of all elements annihilated by \(L\); assume \(N_L\) is closed and of dimension greater than zero.
The assertion “for the operator \(L\) a linear boundary-value problem is given” has a general logical aspect. Without entering here into a detailed discussion, in this section we indicate our conclusion concerning the appropriate form of the general formulation of this problem.
In such a problem there are given a certain class \(\tau\) of boundary conditions of the same type and a subspace \(R_\tau' \subseteq R_1\) of admissible right-hand sides of the equation \(Lx=y\). A quasi-complement \(M\) to \(N_L\) will be called a maximal of \(L\) relative to \(R'=LM\). It turns out that, in the general logical aspect, the class \(\tau\) is completely characterized by a certain maximal \(M_\tau\) relative to \(R_\tau'\). This assertion has the following circumstances in view.
For every \(x \in N_L+M_\tau\) there is a unique decomposition \(x=z+u\), \(z\in N_L\), \(u\in M_\tau\). The mapping \(x\to z\) defines a linear operator \(P_{M_\tau}\), namely: \(P_{M_\tau}x=z\). The operator \(P_{M_\tau}\) proves to be closed in \(N_L+M_\tau\). For every \(z\in N_L\) the set \(z+M_\tau\) is a closed linear manifold mapped by means of \(L\) one-to-one onto \(R_\tau'\).
In view of what has been said, the problem of finding a solution \(x\) of the system
\[ Lx=y \quad \text{with the boundary condition} \quad P_{M_\tau}x=z, \tag{1} \]
where \(z\in N_L\) and \(y\in R_\tau'\) are given, has a unique solution.
For a fixed maximal \(M_\tau\), the totality \(\tau\) of all boundary conditions is exhausted by running through all elements \(z\in N_L\). If \(z\in N_L\), but \(y\notin R_\tau'\), then problem (1) has no solution. For any fixed \(y\in R_\tau'\) and different \(z_1\) and \(z_2\) from \(N_L\), problem (1) has different solutions.
Let \(L_{M_\tau}\) denote the restriction of \(L\) to \(M_\tau\), and \(L_{M_\tau}^{-1}\) the inverse operator (it maps \(R_\tau'=LM_\tau\) one-to-one onto \(M_\tau\)). We have
\[ x=P_{M_\tau}x+L_{M_\tau}^{-1}Lx,\qquad x\in N_L+M_\tau. \tag{2} \]
It follows from (2) that the solution of (1) has the form \(x=z+L_{M_\tau}^{-1}y\).
Problem (1) is the simplest aspect of the general linear boundary-value problem. It is clear that the requirement of continuity of the operator \(L_{M_\tau}^{-1}\) is equivalent to the continuous dependence of the solution on the right-hand side \(y\) of the equation. We shall call a maximal \(M\) principal when the operator \(P_M\) is continuous; in this case the boundary datum \(z\) depends continuously on the solution \(x\) in problem (1).
In the elementary situation, when \(L\) is the operator of differentiation of functions of a real variable, \(M_\tau\) is the set of those of them,
which vanish at the point \(\tau\) of the real axis,
\[ P_{M_\tau}x=x(\tau),\qquad L_{\tau}^{-1}=\int_{\tau}^{t}, \]
and equality (2) is the classical equality of analysis
\[ x(t)=x(\tau)+\int_{\tau}^{t}x'(\xi)\,d\xi . \]
2. Let \(M_0\) be a principal maximal subspace and \(N_L\) a complete null l.t.s. (every Cauchy sequence has a limit). Then the general form of a maximal subspace \(M\) relative to \(LM_0\) is as follows:
\[ M=\{x-Sx\}_{x\in M_0}, \tag{3} \]
where \(S\) is a closed linear operator from \(M_0\) into \(N_L\); moreover \(R=N_L+M_0=N_L+M\).
When \(N_L\) and \(M_0\) are subspaces of type \((F)\), all maximal subspaces relative to \(LM_0\) are principal. When the maximal subspace \(M\) is given, one may choose \(S=P_M\).
3. Illustration for the infinitesimal operator \(I\) of a strongly continuous semigroup \(\{u(t)\}_{0\le t<\infty}\) of linear bounded operators in a Banach space \(B\). We assume \(u(0)=1\), \(D_I\) is dense in \(B\), \(\|u(t)\|\le C\), \(0\le t<\infty\) (\(C\) is a constant). Let \(B_\infty\) be the totality of those \(x\in B\) for which there exists \(u(+\infty)x=\lim_{t\to+\infty}u(t)x\); let \(B_\infty^{0}\) be the part of \(B_\infty\) where \(u(+\infty)x=0\). Then \(M_\infty=B_\infty^{0}\cap D_I\) is a principal maximal subspace for \(I\) on the space \(B_\infty\cap D_I\), \(D_\pi=IM_\infty\), \(I_{M_\infty}^{-1}=-\Pi\), where \(\Pi\) is the potential operator of the semigroup.
To \(M_\infty\) there corresponds the boundary-value problem: \(Ix=-y\) with \(\lim_{t\to+\infty}u(t)x=x_0\), where \(x_0\in N_I\), \([N_I=u(+\infty)M_\infty]\), and \(y\in D_\Pi\) are given. The solution, as is known, is
\[ x=x_0+(-I)_{M_\infty}^{-1}y=x_0+\Pi y=x_0+\int_{0}^{\infty}u(\tau)y\,d\tau . \]
4. Formal derivation of the annihilated space and of a maximal subspace of the difference of operators. Application. Let \(A\) and \(F\) be some linear operators with domains \(D_A\) and \(D_F\); let \(M_0\) be a maximal subspace for \(A\); \(FD_F\subseteq AM_0\), and let the operator \((1-A_{M_0}^{-1}F)^{-1}\) exist on \(D_A\) with values in \(D_A\cap D_F\). Formal transformations show that
\[ N_{A-F}=(1-A_{M_0}^{-1}F)^{-1}N_A \]
and the set
\[ M_1=(1-A_{M_0}^{-1}F)^{-1}M_0 \]
is, in the case of closedness, a maximal subspace for \(A-F\). Moreover, the following holds:
\[ z=(A-F)(1-A_{M_0}^{-1}F)^{-1}A_{M_0}^{-1}z,\qquad z\in AM_0; \tag{4} \]
\[ (A-F)_{M_1}^{-1}z=(1-A_{M_0}^{-1}F)^{-1}A_{M_0}^{-1}z,\qquad z\in AM_0\cap AM_1. \]
Retaining the notation of item 3, denote by \(B_{(0,\infty)}\) the l.t.s. whose elements are all functions \(\{y(t)\}_{0\le t<\infty}\) with values in \(B\), continuous, continuously differentiable, and satisfying the condition
\[ \lim_{\to\infty}\frac{\ln\|y(t)\|}{t}<\infty \]
(convergence is with respect to continuous variation of \(\|y(t)\|\) and \(\|y'(t)\|\) for each \(t\), \(0\le t<\infty\)).
Consider in \(B_{(0,\infty)}\) the operators \(\hat A=d/dt-I\) and \(F\), generated by a bounded linear operator \(F\) in \(B\). It can be shown that
\[ \hat A_{M_0}^{-1}\tilde z(t)=\int_{0}^{t}u(t-\tau)z(\tau)\,d\tau \]
for \(0\le t<\infty\) and \(\{z(t)\}=\tilde z\in B_{(0,\infty)}\). Hence one obtains convergence of the series
\[ \sum_{n=0}^{\infty}(A_{M_0}^{-1}F)^n\tilde z(t),\qquad 0\le t<\infty, \]
for \(\tilde z\in B_{(0,\infty)}\); for \(\tilde z\in D_A\) the sum is an element of \(D_A\). In view of this, the rules mentioned are applicable for obtaining \(N_{A-F}\) and \(M_1\), and formula (4). Hence it follows,
that the function \(y(t)=(1-A_{M_0}^{-1}F)^{-1}A_{M_0}^{-1}z(t)\), in the class of functions \(B_{(0,\infty)}\), is the unique solution of the problem
\[
(d/dt-I-F)y(t)=z(t),\qquad 0\le t<\infty,\qquad y(0)=0;
\]
\[
\bar z=\{z(t)\}\in B_{(0,\infty)}.
\]
It is known that \(I_1=I+F\) is infinitesimal for some semigroup of operators \(\{u_1(t)\}_{0\le t<\infty}\). From (4) it easily follows that
\[ u_1(t)z=(1-A_{M_0}^{-1}F)^{-1}u(t)z,\qquad 0\le t<\infty,\qquad z\in B. \]
This is a conveniently surveyable form of the perturbation formula for the semigroup (2).
5. Construction of the main maximal, when \(R\) is a subspace of the space \(C_Q\) of functions that are material and continuous on a bicompactum \(Q\); problems of Dirichlet type. We shall assume that \(L\) is a closed operator and \(N_L\) contains constants and separates the points of \(Q\). Let \(\Gamma\) be the \(T\)-boundary of \(Q\) with respect to \(N_L\).* We shall suppose that \(\Gamma\ne Q\) and that in \(R\) there is a dense subset \(R'\) of those functions \(y\in R\) for each of which there is an \(x\in N_L\) such that \(x(q)\equiv y(q)\) on \(\Gamma\).
It turns out that the subset \(M_0\) of all \(y\in R'\) annulled on \(\Gamma\) is the main maximal.
The boundary problem corresponding to this maximal will be called a problem of Dirichlet type (it naturally generalizes the classical Dirichlet problem, in which \(\Gamma\) is the ordinary boundary of the domain).
The boundary conditions for this item can also be given in the form of linear equations on the \(T\)-boundary; boundary problems of this kind should naturally be regarded as boundary-value problems.
6. Taylor-type formula for a cascade of boundary problems. By a cascade of linear boundary problems we shall understand a set \(\{L_k,N_{L_k},M_k\}_{1\le k\le n}\) for which the relation
\[ N_{L_k}+M_k\subseteq L_{k+1}M_{k+1},\qquad 1\le k\le n-1. \]
is satisfied. Denote by \(R^{I}\) the collection of elements \(x\) for which
\[ x\in N_{L_n}+M_n,\qquad L_kL_{k+1}\cdots L_nx\in N_{L_{k-1}}+M_{k-1},\qquad 2\le k\le n. \]
For \(x\in R^{I}\) the following Taylor-type formula is valid:
\[
x=P_{M_n}x+L_{nM_n}^{-1}P_{M_{n-1}}L_nx+\cdots+L_{nM_n}^{-1}\cdots L_{2M_2}^{-1}P_{M_1}L_2\cdots L_nx+
\]
\[
+L_{nM_n}^{-1}\cdots L_{1M_1}^{-1}L_1\cdots L_nx.
\tag{*}
\]
When \(L_n=L_0,\quad M_k=M_0,\quad 1\le k\le n,\quad L_0^{k}x\in N_{L_0}+M_0\subseteq L_0M_0,\quad 0\le k\le n-1,\) formula (*) gives us
\[ x=\sum_{k=0}^{n-1}(L_{0M_0}^{-1})^kP_{M_0}L_0^{k}x+(L_{0M_0}^{-1})^nL_0^{n}x. \tag{**} \]
Formula (**) is a generalization of the usual Taylor formula (when \(L_0\) is the differentiation operator).
7. To each \(x\in R^{I}\) assign the set \(\bar x=\{x,L_nx,\ldots,L_2L_3\cdots L_nx\}\). Let a function \(f\) be given that assigns to each \(x\in R^{I}\) an element \(f(\bar x)\in L_1M_1\); the operators \(\{L_k\}_{1\le k\le n}\) are assumed closed. Suppose also that, for known \(z\) and \(y\), for which \(L_kz\) and \(L_kM_k^{-1}y\) are defined, we know how to find the latter.
It is required to find a solution of the system
\[ \text{equations }\quad L_1L_2\cdots L_nx=f(\bar x) \tag{K_I} \]
under the Cauchy boundary conditions
\[ P_{M_n}x=z_0,\qquad P_{M_k}L_{k+1}\cdots L_nx=z_{n-k}, \]
where \(1\le k\le n-1\), the elements \(z_{n-k}\in N_{L_k}\), \(1\le k\le n\), are given.
\[
\text{* } q_0\in\Gamma \text{ means: for any neighborhood } U(q_0) \text{ there is } x_u\in N_L \text{ such that }
\]
\[
\max_{q\in Q/U(q_0)} x_u(q)<\max_{q\in Q} x_u(q)\ (^{3,4}).
\]
By \((\mathrm{K}_{\mathrm{I}}^y)\) we denote problem \((\mathrm{K}_{\mathrm{I}})\) in which the function \(f\) is replaced by the known element \(y\) from \(L_1M_1\). The solution of \((\mathrm{K}_{\mathrm{I}}^y)\) is obtained directly from formula \((*)\).
Denote
\[
\zeta_0=z_0+L_{nM_n}^{-1}z_1+\cdots+L_{nM_n}^{-1}\cdots L_{2M_2}^{-1}z_{n-1},\qquad F(y)=
\]
\[
=f\bigl(\zeta_0+L_{nM_n}^{-1}\cdots L_{1M_1}^{-1}y\bigr).
\]
It follows from formula \((*)\) that problem \((\mathrm{K}_{\mathrm{I}})\) is equivalent to the successive solution of the equation \(y=F(y)\) and of problem \((\mathrm{K}_{\mathrm{I}}^y)\).
Suppose that in problem \((\mathrm{K}_{\mathrm{I}})\) one has
\[
N_{L_1}+M_1\subseteq L_1M_1\subseteq L_2M_2
\]
and all spaces \(\{L_kM_k\}_{2\le k\le n}\) lie in a certain Banach structure \(R_1\)*. Suppose, moreover, that the operators \(\{L_{kM_k}^{-1}\}_{1\le k\le n}\) are structurally monotone and bounded, the operator \(L_{1M_1}^{-1}\) is Volterra, and the function \(f(x_0,x_1,\ldots,x_{n-1})\) satisfies the following structural Lipschitz condition:
\[
\bigl|f(x_0,x_1,\ldots,x_{n-1})-f(\widetilde x_0,\widetilde x_1,\ldots,\widetilde x_{n-1})\bigr|
\le
C_f\sum_{j=0}^{n-1}|x_j-\widetilde x_j|,
\]
where \(C_f\) is a linear, bounded, structurally monotone operator commuting with \(L_{1M_1}^{-1}\).
Under these conditions, problem \((\mathrm{K}_{\mathrm{I}})\) is reducible, by means of formula \((*)\), to an equation of the form \(y=\varphi(y)\), solvable by the contraction principle by the method of successive approximations.
Problem \((\mathrm{K}_{\mathrm{I}})\) is a certain variant of a Cauchy-type problem.
- Suppose that linear operators \(\{L_k\}_{0\le k\le m}\) are given; \(M_0\) is maximal for \(L_0\); \(R^{\mathrm{II}}\) is the space of elements \(x\) for which there exist sets
\[ \bar x=\{L_0^{n_0}L_1^{n_1}\cdots L_m^{n_m}x\}_{\,n_0<n,\; n_0+n_1+\cdots+n_m\le n} \]
of elements belonging to \(N_{L_0}+M_0\); \(L_0\) commutes with \(\{L_k\}_{1\le k\le n}\) in \(R^{\mathrm{II}}\), and
\[ R^{\mathrm{II}}\subseteq L_0(M_0\cap R^{\mathrm{II}}). \]
Suppose that a function \(f\) is given, assigning to the set \(\bar x\), for each \(x\in R^{\mathrm{II}}\), an element \(f(\bar x)\) from \(L_0(M_0\cap R^{\mathrm{II}})\).
The second variant of the Cauchy-type problem is the problem of solving the system
\[
L_0^n(x)=f(\bar x),
\qquad\qquad (\mathrm{K}_{\mathrm{II}})
\]
\[
P_{M_0}x=z_0,\qquad P_{M_0}L_0^k x=z_k,\qquad 1\le k\le n-1,
\]
where \(\{z_k\}_{0\le k\le n-1}\subseteq N_{L_0}\cap R^{\mathrm{II}}\) are given.
For problem \((\mathrm{K}_{\mathrm{II}})\), reductions and a method of solution analogous to those mentioned above for problem \((\mathrm{K}_{\mathrm{I}})\) are possible.
- Let \(L\), \(N_L\), \(M\), \(P_M\) have their former meaning; \(N_L+M\subseteq LM\); \(\{a_k\}_{0\le k\le n}\) are linear operators defined in \(LM\), \(a_n=1\), and let
\[ \Psi_k(L_M^{-1})=\sum_{m=0}^{n}a_m(L_M^{-1})^{k-m},\qquad 0\le k\le n. \]
It turns out that if the operator \(\Psi_n(L_M^{-1})\) has an inverse on \(LM\), then one can propose, in a convenient form, the solution \(x\) of the problem
\[ \sum_{k=0}^{n}a_kL^k x=y,\qquad P_ML^k x=z_k,\qquad 0\le k\le n-1, \]
where \(y\in LM\), \(\{z_k\}_{0\le k\le n-1}\subset N_L\) are given. For the case \(z_k=0\), \(0\le k\le n-1\), one has
\[ x=(L_M^{-1})^n[\Psi_n(L_M^{-1})]^{-1}y. \]
Odessa Electrotechnical
Institute of Communications
Received
11 XI 1964
CITED LITERATURE
- E. Hille, R. Phillips, Functional Analysis and Semi-Groups, IL, 1962.
- D. P. Milman, DAN, 57, No. 2, 449 (1947).
- D. P. Milman, DAN, 87, No. 1, 9 (1952).
* \(0<x<y\) implies \(\|x\|\le \|y\|\), and there is a constant \(a\) such that \(\|\!|x|\!\|\le a\|x\|\).