Full Text
UDC 517.934
CORRECTNESS OF BOUNDARY VALUE PROBLEMS
FOR A SECOND-ORDER DIFFERENTIAL EQUATION
IN A BANACH SPACE. II*)
S. G. Krein, G. I. Laptev
In article [1] we considered a boundary value problem of the following form:
\[ \frac{d^2u}{dt^2}=Au-f(t)\qquad (0\leq t\leq T), \tag{0.1} \]
\[ L_1(u)\equiv \alpha_{11}u_0+\alpha_{12}u'_0+\beta_{11}u_T+\beta_{12}u'_T=f_1, \]
\[ L_2(u)\equiv \alpha_{21}u_0+\alpha_{22}u'_0+\beta_{21}u_T+\beta_{22}u'_T=f_2, \tag{0.2} \]
where \(u(t)\) is the unknown function with values in the complex Banach space \(E\); \(A\) is a closed linear operator with domain of definition \(D(A)\) everywhere dense in the space \(E\); \(f(t)\) is a given function, continuous on the interval \([0,T]\), with values in \(E\); \(\alpha_{ij}, \beta_{ij}\) \((i,j=1,2)\) are given complex numbers; \(f_1\) and \(f_2\) are given elements of the Banach space.
Concerning the operator \(A\), it was assumed that for all \(\lambda\geq 0\) the operator \((A+\lambda I)^{-1}\) exists, is defined on the whole space \(E\), and
\[ \left\|(A+\lambda I)^{-1}\right\|\leq \frac{c}{1+\lambda}\qquad (\lambda\geq 0). \tag{0.3} \]
An essential role is played by the analytic semigroup \(U(t)\), for which the operator \(-A^{1/2}\) is the infinitesimal generator.
The general solution of equation (0.1) is then written in the form
\[ u(t)=U_1(t)g_1+U_2(t)g_2+\int_0^T U_0(t,\tau)f(\tau)\,d\tau, \tag{0.4} \]
where
\[ U_1(t)=U(t)A^{-1/2},\qquad U_2(t)=U(T-t)A^{-1/2},\qquad U_0(t,\tau)= \]
\[ =\frac{1}{2}U(|t-\tau|)A^{-1/2}, \tag{0.5} \]
and \(g_1\) and \(g_2\) are arbitrary elements of \(E\).
In considering the boundary value problem, the operator determinant
\[ D= \begin{vmatrix} L_1(U_1) & L_1(U_2)\\ L_2(U_1) & L_2(U_2) \end{vmatrix} \tag{0.6} \]
is of great importance.
*) For Communication I, see the journal Differential Equations, No. 3, 1966.
For the solution of the boundary value problem the formula
\[ D u(t)=S_1(t)f_1+S_2(t)f_2+\int_0^T G_0(t,\tau)f(\tau)\,d\tau \tag{0.7} \]
is valid.
The operators entering into this formula can be written in the form of determinants
\[ S_1(t)= \begin{vmatrix} U_1(t) & U_2(t)\\ L_2(U_1) & L_2(U_2) \end{vmatrix}, \qquad S_2(t)=- \begin{vmatrix} U_1(t) & U_2(t)\\ L_1(U_1) & L_1(U_2) \end{vmatrix}, \]
\[ G_0(t,\tau)= \begin{vmatrix} U_1(t) & U_2(t) & U_0(t,\tau)\\ L_1(U_1) & L_1(U_2) & L_1(U_0)_t\\ L_2(U_1) & L_2(U_2) & L_2(U_0)_t \end{vmatrix}. \tag{0.8} \]
Every continuous function \(u(t)\) satisfying equation (0.7) is called a generalized solution of problem (0.1), (0.2).
Denote by \(C(E)\) the space of all functions \(u(t)\) continuous on \([0,T]\) with values in \(E\), endowed with the usual norm
\[
\|u\|_{C(E)}=\max_{0\le t\le T}\|u(t)\|_E.
\]
Definition. The boundary value problem (0.1), (0.2) is called uniformly correct on the interval \([0,T]\) if for all \(f_1,f_2\in E\) and \(f(t)\in C(E)\) there exists a unique generalized solution of this problem, continuously depending in the norm of the space \(C(E)\) on \(f_1,f_2\in E\) and \(f(t)\in C(E)\).
It follows immediately from formula (0.7) that for the uniform correctness of problem (0.1), (0.2) it is necessary and sufficient that the operator \(D\) have an inverse \(D^{-1}\) and that the operators \(D^{-1}S_1(t)\), \(D^{-1}S_2(t)\) be bounded uniformly in \(t\), while the integral operator
\[ D^{-1}\int_0^T G_0(t,\tau)f(\tau)\,d\tau \]
be bounded as an operator from \(C(E)\) into \(C(E)\). The investigation of the indicated operators is the aim of the present paper.
§ 1. REGULAR BOUNDARY CONDITIONS. GREEN’S FUNCTION
In the question of correctness of the boundary value problem (0.1), (0.2), regular boundary conditions play a special role (see [2]).
Denote by \(d_{ij}\) \((i<j)\) the minor of the matrix
\[ \begin{pmatrix} \alpha_{11} & \alpha_{12} & \beta_{11} & \beta_{12}\\ \alpha_{21} & \alpha_{22} & \beta_{21} & \beta_{22} \end{pmatrix} \tag{1.1} \]
of the coefficients of the boundary conditions (0.2), formed from its \(i\)-th and \(j\)-th columns.
The boundary conditions (0.2) will be regular if one of the following conditions is satisfied:
- \(d_{24}\ne 0\);
- \(d_{24}=0\), but \(|\alpha_{12}|+|\beta_{12}|>0\) and \(d_{23}-d_{14}\ne 0\);
- \(\alpha_{12}=\beta_{12}=\alpha_{22}=\beta_{22}=0\), but \(d_{13}\ne 0\).
Let us dwell on these cases in more detail. Write the determinant \(D\) in expanded form:
\[ D=-d_{24}I+(d_{14}-d_{23})A^{-\frac12}+d_{13}A^{-1} +2(d_{12}+d_{34})A^{-\frac12}U(T)+ \]
\[ +\bigl[d_{24}I+(d_{14}-d_{23})A^{-\frac12}-d_{13}A^{-1}\bigr]U(2T). \tag{1.2} \]
\(1^\circ.\) \(d_{24}\ne0\). Put \(d_{24}=-1\); then the characteristic determinant \(D\) takes the form
\[ D=I-R_1, \]
where \(R_1\) is a bounded operator (a linear combination of the bounded operators \(A^{-\frac12}U(T)\) and their products).
If we assume that unity belongs to the resolvent set of the operator \(R_1\), then the determinant \(D\) will have a bounded everywhere-defined inverse. It follows from this that the operators \(D^{-1}S_1(t)\), \(D^{-1}S_2(t)\) will be uniformly bounded on \([0,T]\), and the operator \(D^{-1}G_0(t,\tau)\) likewise for \(0\le t,\tau\le T\).
\(2^\circ.\) \(d_{24}=0\), \(|\alpha_{12}|+|\beta_{12}|>0\), and \(d_{23}-d_{14}\ne0\).
Let \(d_{23}-d_{14}=-1\). Under these conditions the matrix (1.1) can be transformed to the form
\[ \begin{pmatrix} \alpha_{11} & \alpha_{12} & \beta_{11} & \beta_{12}\\ \widetilde{\alpha}_{21} & 0 & \widetilde{\beta}_{21} & 0 \end{pmatrix}, \tag{1.3} \]
and the determinant \(D\), according to formula (1.2), assumes the form
\[ D=A^{\frac12}(I-R_2), \]
where \(R_2\) is a bounded operator.
Thus, even if the operator \((I-R_2)^{-1}\) is bounded, the operator
\[ D^{-1}=(I-R_2)^{-1}A^{-\frac12} \]
will nevertheless be unbounded. But let us look more closely at the matrix (1.3). The boundary condition determined by it, \(L_2(u)\equiv \widetilde{\alpha}_{21}u_0+\widetilde{\beta}_{21}u_T\), does not contain differentiation of the function \(u(t)\), and therefore the operators \(L_2(U_1)\), \(L_2(U_2)\), and \(L_2(U_0)_t\) retain as a factor the operator \(A^{-\frac12}\), which the functions \(U_i(t)\) \((i=1,2,0)\) have in accordance with formulas (0.5). But it is precisely these operators \(L_2(U_i)\) that stand in the last row of the third of the determinants (0.8), which gives the function \(G_0(t,\tau)\). This means, finally, that the operator \(D^{-1}G_0(t,\tau)\) will be a bounded everywhere-defined operator if \((I-R_2)^{-1}\) is such.
Analogous arguments are valid for the determinant \(S_1(t)\) from (0.8), in whose second row stand the operators \(L_2(U_i)\) for \(i=1,2\). In the determinant for \(S_2(t)\) there is also a factor \(A^{-\frac12}\), but not in the second row, where the operators \(L_1(U_i)\) stand, but in the first, since by definition
\[ U_1(t)=A^{-\frac12}U(t),\qquad U_2(t)=A^{-\frac12}U(T-t). \]
Thus, for the case under consideration the operator \(D^{-1}\) is unbounded; nevertheless the operators \(D^{-1}S_1(t)\), \(D^{-1}S_2(t)\), and \(D^{-1}G_0(t,\tau)\) will be bounded uniformly in \(t\) and \((t,\tau)\), if \((I-R_2)^{-1}\) is bounded.
\(3^\circ.\ a_{12}=\beta_{12}=\alpha_{22}=\beta_{22}=0\), but \(d_{13}\ne0\). These conditions are equivalent to the first boundary value problem \(u_0=f_1,\ u_T=f_2\) with matrix
\[ \begin{pmatrix} 1&0&0&0\\ 0&0&1&0 \end{pmatrix}. \tag{1.4} \]
For the determinant \(D\) one obtains the formula:
\[ D=A^{-1}[I-U(2T)], \]
i.e., the operator \(D^{-1}=[I-U(2T)]^{-1}\), if it exists, is unbounded because of the factor \(A\).
Here neither of the boundary conditions contains differentiation, and therefore all the operators \(L_i(U_j)\) for \(i=1,2;\ j=0,1,2\) retain the factor \(A^{-1/2}\). Taking it out of the second and third rows of the determinant \(G_0(t,\tau)\), we find that \(G_0(t,\tau)=A^{-1}H_0(t,\tau)\), where \(H_0(t,\tau)\) is a certain uniformly bounded operator-function. In order to obtain the operator \(A^{-1}\) in the functions \(S_1(t)\) and \(S_2(t)\), one has to use the factor \(A^{-1/2}\) in the second rows and in the elements \(U_1(t), U_2(t)\) of the first rows of these determinants (which we did not do in the determinant for \(G_0(t,\tau)\)). Thus we have succeeded in extracting the factor \(A^{-1}\) in the operator-functions \(S_1(t), S_2(t)\), and \(G_0(t,\tau)\), which will eliminate the unboundedness of the operator \(D^{-1}\).
Thus, for the case of regular boundary conditions the following can be stated.
Theorem 1. Let the boundary conditions of problem (0.1), (0.2) be regular. Then the operator \(D\) can be represented in one of the following three forms: a) \(c(I-R)\), b) \(cA^{-1/2}(I-R)\), c) \(cA^{-1}(I-R)\), where \(R\) are certain bounded operators and \(c\) are constants. If it is assumed that \(1\) is not a point of the spectrum of the operator \(R\), then the boundary value problem (0.1), (0.2) is uniformly well-posed on the interval \([0,T]\).
Let us dwell on the function \(G(t,\tau)=D^{-1}G_0(t,\tau)\), which it is natural to call the Green’s function, since it gives the solution of the inhomogeneous differential equation \(u''=Au-f(t)\) under homogeneous boundary conditions \(L_1(u)=L_2(u)=0\) in the form
\[ u(t)=\int_0^T G_0(t,\tau)f(\tau)\,d\tau . \]
The notation of the Green’s function is entirely analogous to the scalar case:
\[ G(t,\tau)=D^{-1} \begin{vmatrix} U_1(t)&U_2(t)&U_0(t,\tau)\\ L_1(U_1)&L_1(U_2)&L_1(U_0)_t\\ L_2(U_1)&L_2(U_2)&L_2(U_0)_t \end{vmatrix}, \tag{1.5} \]
and in its properties it also resembles the Green’s function for the scalar boundary value problem:
I) \(G(t,\tau)\) is a fundamental solution of equation (0.1);
II) for each fixed \(0\le \tau\le T\), as a function of \(t\), it satisfies the homogeneous boundary conditions
\[ L_2(G)_t=L_2(G)_t=0. \]
Both properties follow immediately from representation (1.5). Indeed, it follows from it, first, that \(G(t,\tau)\) is a linear combination (with bounded operator coefficients) of the particular solutions \(U_1(t)\), \(U_2(t)\), and \(U_0(t,\tau)\) of the homogeneous differential equation, i.e., is itself a fundamental solution, and, second, after applying the operator \(L_i\) \((i=1,2)\), one obtains:
\[ L_i(G)_t = D^{-1} \begin{vmatrix} L_i(U_1) & L_i(U_2) & L_i(U_0)_t\\ L_1(U_1) & L_1(U_2) & L_1(U_0)_t\\ L_2(U_1) & L_2(U_2) & L_2(U_0)_t \end{vmatrix}. \]
The first row of this determinant coincides either with the second (for \(i=1\)) or with the third (for \(i=2\)); that is, the determinant vanishes, and this means precisely that \(G(t,\tau)\) satisfies the homogeneous boundary conditions.
Let us also clarify when, under regular conditions, the generalized solutions of the uniformly correct problem will be true solutions. As was shown in [1], for this it is sufficient that the function \(f(t)\) satisfy the Hölder condition and that the elements \(g_1\) and \(g_2\) in (0.4) can be determined from the equations
\[ \begin{aligned} Dg_1&=L_2(U_2)f_1-L_1(U_2)f_2-L_2(U_2)L_1(g)+L_1(U_2)L_2(g),\\ Dg_2&=L_1(U_1)f_2-L_2(U_1)f_1-L_1(U_1)L_2(g)+L_2(U_1)L_2(g), \end{aligned} \tag{1.6} \]
where
\[ g(t)=\int_0^T G_0(t,\tau)f(\tau)\,d\tau. \]
Consider equations (1.6) in all three cases of regularity of the boundary conditions. In case \(1^\circ\), the operator \(D^{-1}\) is bounded; therefore \(g_1\) and \(g_2\) are found for arbitrary \(f_1,f_2\in E\) and \(f(t)\in C(E)\).
In case \(2^\circ\), we shall assume that matrix (1.1) has the form (1.3). Then the operators \(D^{-1}L_2(U_j)\) are bounded, while the operators \(D^{-1}L_1(U_j)\) are unbounded and have the form \(CA^{1/2}\), where \(C\) is a bounded operator; therefore \(g_1\) and \(g_2\) cannot be determined for an arbitrary element \(f_2\). They can be found if \(f_2\in D(A^{1/2})\).
Finally, in case \(3^\circ\) with matrix (1.4), all operators \(D^{-1}L_i(U_j)\) have the form \(CA^{1/2}\); therefore equations (1.5) are solvable for \(f_1,f_2\in D(A^{1/2})\).
Theorem 2. Suppose that the conditions of Theorem 1 for the uniform correctness of problem (0.1), (0.2) are satisfied. Then the generalized solutions will be true solutions if \(f(t)\) satisfies the Hölder condition, while \(f_1\) and \(f_2\) are arbitrary elements of \(E\) in case \(a)\), and \(f_1,f_2\in D(A^{1/2})\) in the remaining cases.
§ 2. NONREGULAR CONDITIONS
For nonregular conditions (0.2) only the following possibility remains ([2], p. 54):
\[ d_{24}=0,\quad |\alpha_{12}|+|\beta_{12}|>0,\quad d_{23}-d_{14}=0. \tag{2.1} \]
The determinant \(D\) has the form:
\[ D=A^{-1}\left[d_{13}+2(d_{12}+d_{34})\,A^{\frac12}U(T)-d_{13}U(2T)\right]. \]
For analytic semigroups the operator \(A^{\frac12}U(T)\) is bounded; hence the operator in square brackets is also bounded.
Let \(d_{13}\ne0\). Then \(D=A^{-1}(d_{13}-R_4)\), where \(R_4\) is a bounded operator. If \(d_{13}\) does not belong to the spectrum of the operator \(R_4\), then \(D^{-1}\) is proportional to \(A\) (with a bounded operator coefficient). At the same time, by virtue of the condition \(|\alpha_{12}|+|\beta_{12}|>0\), the operator \(L_1(u)\) contains differentiation, which “cancels” the factor \(A^{-\frac12}\) occurring in the functions \(U_i(t)\) \((i=0,1,2)\). As a result, the operator
\[ S_2(t)= \left| \begin{array}{cc} A^{-\frac12}U(t) & A^{-\frac12}U(T-t)\\ L_1(U_1) & L_1(U_2) \end{array} \right| \]
contains only \(A^{-\frac12}\) as a factor, and the product \(D^{-1}S_2(t)\) turns out to be an unbounded operator of the form \(RA^{\frac12}\). This means that the boundary value problem (0.1), (0.2) for the boundary conditions under consideration
\[ \left( \begin{array}{cccc} \alpha_{11} & \alpha_{12} & \beta_{11} & \beta_{12}\\ \tilde{\alpha}_{21} & 0 & \tilde{\beta}_{21} & 0 \end{array} \right), \qquad |\alpha_{12}|+|\beta_{12}|>0,\qquad d_{24}=d_{23}-d_{14}=0 \tag{2.2} \]
is, in general, ill-posed. But if one assumes that \(f_2\in D(A^{\frac12})\), then the element \(D^{-1}S_2(t)f_2\) will be well defined. Further, the determinants for \(S_1(t)\) and \(G_0(t,\tau)\) contain the factor \(A^{-1}\) because of their first rows, which give \(A^{-\frac12}\), and those rows which contain the operators \(L_2(U_i)\), which also give \(A^{-\frac12}\). Consequently, the operators \(D^{-1}S_1(t)\) and \(D^{-1}G_0(t,\tau)\) will be bounded, and from formula (0.7) one can obtain a generalized solution.
In this connection, boundary conditions of the form (2.2) will be called semi-correct.
Let us note that for any \(f_2\in E\) the function \(D^{-1}S_2(t)f_2\) will be defined for every \(t\in(0,T)\), since for such \(t\) the operators \(A^{\frac12}U(t)\) and \(A^{\frac12}U(T-t)\) are bounded. However, this function, and hence also the function \(u(t)\), may grow without bound as \(t\to0\) and \(t\to T\). If nevertheless one regards the function \(u(t)\) in some sense as a solution of the problem, then this solution will not depend continuously on \(f_2\) in the norm of \(C(E)\), but will depend continuously on it for each fixed \(t\in(0,T)\).
Finally, suppose that one more condition is added to (2.1): \(d_{13}=0\). Then
\[ D=2(d_{12}+d_{34})A^{-\frac12}U(T), \]
whence
\[ D^{-1}=\frac{1}{2(d_{12}+d_{34})}\,A^{\frac12}U^{-1}(T). \]
The operator \(D^{-1}\), generally speaking, is not applicable to any of the terms appearing on the right in (0.7). The boundary value problem (0.1), (0.2) turns out to be “strongly ill-posed.”
The general form of the corresponding boundary conditions will be as follows:
\[ \alpha u'(0)-\beta u'(T)=f_1,\qquad \alpha u(0)+\beta u(T)=f_2. \]
For \(\beta=0\) we obtain the conditions of the Cauchy problem.
Theorem 3. The Cauchy problem is ill-posed for equation (0.1).
§ 3. The case of a completely continuous \(A^{-1}\)
For regular boundary conditions, Theorem 1 reduces the question of correctness to the question of the existence of a bounded inverse to the operator \(I-R\). Such an operator certainly does not exist if \(D\) vanishes at some nonzero element. As shown in [1], this will occur if and only if the corresponding homogeneous boundary value problem has nontrivial solutions. However, the operator \((I-R)^{-1}\) may be unbounded in other cases as well.
Suppose, for example, that the operator \(A\) has a finite point \(\alpha\) in its continuous spectrum. Consider the regular boundary conditions determined by the matrix
\[ \begin{pmatrix} \sqrt{\alpha} & 1 & 0 & 0\\ 0 & 0 & \sqrt{\alpha} & 1 \end{pmatrix}. \]
Then
\[ D=(\alpha-A)A^{-1}[I-U(2T)] \quad\text{and}\quad I-R=\left(I-\frac{1}{\alpha}A\right)[I-U(2T)]. \]
If the factor \(I-U(2T)\) has a bounded inverse, then \(I-R\) does not have one because of the factor
\[ \left(I-\frac{1}{\alpha}A\right). \]
The indicated difficulties are removed when one considers a special but important class of operators \(A\) that have a completely continuous inverse. The theory obtained for this case is analogous to the theory of scalar boundary value problems. An example is given by the following.
Theorem 4. Let the boundary conditions (0.2) be regular and let the operator \(A^{-1}\) be completely continuous. In order that the boundary value problem (0.1), (0.2) be uniformly correct, it is necessary and sufficient that the corresponding homogeneous boundary value problem have only the zero solution. When this condition is satisfied, the Green’s function \(G(t,\tau)\) exists and the solution is given by the formula
\[ u(t)=D^{-1}S_1(t)f_1+D^{-1}S_2(t)f_2+\int_0^T G(t,\tau)f(\tau)\,d\tau. \]
If the domain of definition of the adjoint operator \(A'\) is dense in \(E'\), then this same condition is necessary and sufficient for the uniform correctness of the nonhomogeneous adjoint problem
\[ \frac{d^2y}{dt^2}=A'y-\varphi(t),\qquad L_1'(y)=\varphi_1,\qquad L_2'(y)=\varphi_2. \tag{3.1} \]
Proof. The proof is based on the following remark: if an operator \(T\) in a Banach space \(E\) has a completely ne-
has a continuous inverse and generates an analytic semigroup \(Z(t)\), then for all \(t>0\) the operators \(Z(t)\) are also completely continuous. This follows from the representation
\[ Z(t)=[T Z(t)]T^{-1}. \]
For analytic semigroups the operators \(T Z(t)\) are bounded for all \(t>0\), and the operator \(Z(t)\) is completely continuous as the product of a bounded operator and a completely continuous operator.
By assumption, the operator \(A^{-1}\) is completely continuous. All the operators \(A^{-\alpha}\) with \(\alpha>0\) are likewise completely continuous ([3], item 4); in particular, the operator
\[ A^{-\frac12} \]
is completely continuous. Then the semigroup \(U(t)\), generated by the operator \(-A^{\frac12}\), consists of completely continuous operators for all \(t>0\). By Theorem 1, under regular boundary conditions the equalities
\[ D=cA^{-\alpha}(I-R), \]
hold, where \(\alpha\) takes one of the values \(0,\frac12,1\), and \(R\) is a certain polynomial in the completely continuous operators \(A^{-\frac12}\) and \(U(T)\), and, consequently, is itself completely continuous. But then the point \(1\) of the complex plane is either an eigenvalue of the operator \(R\), or belongs to its resolvent set.
In the first case the determinant \(D\) has no inverse, and the homogeneous problem has a nontrivial solution. In the second case the operator \((I-R)^{-1}\) exists, is bounded, and is defined on the whole space; consequently, by Theorem 1 the boundary value problem (0.1), (0.2) is uniformly well posed.
By Theorem 4 of [1], the determinant \(D(A')\) of the adjoint boundary value problem, as an operator in \(E'\), is adjoint to the determinant \(D(A)\). The spectra of the adjoint operators \(D(A)\) and \(D(A')\) coincide; moreover, from the complete continuity of the operator \(R\) follows the complete continuity of the operator \(R'\). Therefore everything just said about problem (0.1), (0.2) remains valid for the inhomogeneous adjoint problem (3.1). Thus the proof of the theorem is complete.
References
- Krein S. G., Laptev G. I. Differential Equations, 2, No. 3, 353–360, 1966.
- Naimark M. A. Linear Differential Operators. Moscow, 1954.
- Balakrishnan A. V. Pacific Journ. Math., 10, No. 2, 1960.
Received by the editors
September 21, 1965
Voronezh State University