Full Text
UDC 517.948 : 513.88
MATHEMATICS
G. M. VAINIKKO
COMPACT APPROXIMATION OF OPERATORS AND APPROXIMATE SOLUTION OF OPERATOR EQUATIONS
(Presented by Academician V. I. Smirnov on 25 III 1969)
The concept of compact approximation of operators was introduced in \((^{1,2})\). The situation considered below is more general in several respects.
- Let \(E, F, E_n, F_n\) \((n=1,2,\ldots)\) be Banach spaces. Suppose there exist operators \(*\) \(p_n \in \mathcal L(E,E_n)\), \(q_n \in \mathcal L(F,F_n)\) with the following properties:
\[ p_n E = E_n,\qquad q_n F = F_n \qquad (n=1,2,\ldots); \tag{1} \]
\[ \|p_n\|\le c,\qquad \|q_n\|\le c \qquad (n=1,2,\ldots;\ c=\mathrm{const}); \tag{2} \]
\[ \|\xi_n\|_{E_n}\ge \gamma \inf_{\substack{x\in E,\; p_nx=\xi_n}}\|x\|_E, \qquad \|\eta_n\|_{F_n}\ge \gamma \inf_{\substack{y\in F,\; q_ny=\eta_n}}\|y\|_F \tag{3} \]
for any \(**\) \(\xi_n\in E_n,\ \eta_n\in F_n\) \((n=1,2,\ldots;\ \gamma=\mathrm{const}>0)\);
\[ \lim_{n\to\infty}\|p_nx\|>0,\qquad \lim_{n\to\infty}\|q_ny\|>0 \quad \text{for } \forall x\in E,\ y\in F\ (x\ne0,\ y\ne0). \tag{4} \]
Definition 1. A sequence of operators \(T_n\in \mathcal L(E_n,F_n)\) compactly approximates an operator \(T\in \mathcal L(E,F)\) with respect to \(\{p_n\}\) and \(\{q_n\}\), if the following conditions are satisfied:
a) \(\|q_nTx-T_np_nx\|\to0\) as \(n\to\infty\) for every \(x\in E\);
b) for any sequence \(\{\xi_n\}\) \((\xi_n\in E_n,\ \|\xi_n\|\le1,\ n=1,2,\ldots)\) there exist such \(x_n\in E,\ y_n\in F\) and \(c'=\mathrm{const}\) that \(p_nx_n=\xi_n,\ q_ny_n=T_n\xi_n,\ \|y_n\|\le c'\|T_n\xi_n\|\) \((n=1,2,\ldots)\), and the sequence \(\{y_n-Tx_n\}\) is compact in \(F\).
Remark 1. If \(T\in \mathcal L(E,F)\) is completely continuous, then condition b) is equivalent to the following condition (cf. \((^1)\)): for any sequence \(\{\xi_n\}\) \((\xi_n\in E_n,\ \|\xi_n\|\le1,\ n=1,2,\ldots)\) there exist such \(y_n\in F\) that \(q_ny_n=T_n\xi_n\) \((n=1,2,\ldots)\), and the sequence \(\{y_n\}\) is compact in \(F\).
Consider the equations
\[ Tx=y \qquad (y\in F), \tag{5} \]
\[ T_n\xi_n=q_ny. \tag{6} \]
Theorem 1. Suppose the following conditions are satisfied:
1) the sequence \(T_n\in \mathcal L(E_n,F_n)\) compactly approximates \(T\in \mathcal L(E,F)\);
2) \(T\) has an inverse \(T^{-1}\in \mathcal L(F,E)\);
3) the operators \(T_n\) are such that from
\[ \inf_{\xi_n\in E_n,\ \|\xi_n\|=1}\|T_n\xi_n\|>0 \]
it follows that there exists an inverse \(T_n^{-1}\in \mathcal L(F_n,E_n)\).
Then equation (6) has, for all sufficiently large \(n\), a unique solution \(\xi_n^*\in E_n\), and \(\|\xi_n^*-p_nx^*\|\to0\) as \(n\to\infty\), where \(x^*=T^{-1}y\) is the solu-
* By \(\mathcal L(X,Y)\) is denoted the space of linear continuous operators from \(X\) to \(Y\).
** The formula \(\varphi_n\xi_n=p_n^{-1}(\xi_n)\) defines an isomorphism \(\varphi_n\in \mathcal L(E_n,E/p_n^{-1}(0))\), for which \(\|\varphi_n\|\le 1/\gamma,\ \|\varphi_n^{-1}\|\le c\) \((n=1,2,\ldots)\); here \(p_n^{-1}(\xi_n)\) is the complete preimage of the element \(\xi_n\in E_n\). Similarly, \(F_n\) is isomorphic to the quotient space \(F/q_n^{-1}(0)\).
solution of equation (5). The estimate is valid
\[ c_1\|q_nTx^* - T_np_nx^*\| \le \|\xi_n^* - p_nx^*\| \le c_2\|q_nTx^* - T_np_nx^*\| \qquad (c_1,c_2=\mathrm{const}>0). \tag{7} \]
Proof. From 1) it follows that
\[ \|T_n\| \le 1/c_1=\mathrm{const}\qquad (n=1,2,\ldots). \tag{8} \]
Let us show that, for sufficiently large \(n\), there exist \(T_n^{-1}\in \mathcal L(F_n,E_n)\) and
\[ \|T_n^{-1}\|\le c_2=\mathrm{const}\qquad (n=n_0,n_0+1,\ldots). \tag{9} \]
Suppose that for some \(\xi_n\in E_n\) \((\|\xi_n\|=1,\ n=1,2,\ldots)\) we have \(\|T_n\xi_n\|\to0\) as \(n\to\infty\). Choose \(x_n\in E,\ y_n\in F\) \((n=1,2,\ldots)\) such that \(p_nx_n=\xi_n,\ q_ny_n=T_n\xi_n,\ \|y_n\|\le c'\|T_n\xi_n\|\to0\), and \(\{y_n-Tx_n\}\) is compact in \(F\). Then \(\{x_n\}\) is compact in \(E\), and from condition a) of Definition 1 we conclude that \(\|q_nTx_n-T_np_nx_n\|\to0\) as \(n\to\infty\). Since \(\|T_np_nx_n\|=\|T_n\xi_n\|\to0\), it follows that \(\|q_nTx_n\|\to0\), and also \(\|q_nTx'\|\to0\) as \(n\to\infty\) for every limit point \(x'\) of the sequence \(\{x_n\}\); by (4), \(Tx'=0\) and \(x'=0\). Hence \(x_n\to0\) as \(n\to\infty\), which contradicts (2) and the equalities \(\|p_nx_n\|=\|\xi_n\|=1\). Together with 3), this contradiction proves (9).
From inequalities (8), (9) and the equality
\[ T_n(\xi_n^*-p_nx^*)=q_nTx^*-T_np_nx^* \]
the estimate (7) follows. The convergence \(\|\xi_n^*-p_nx^*\|\to0\) follows from (7) and condition a) of Definition 1. Theorem 1 is proved.
Inequalities (9), together with condition a) of Definition 1, mean, in the terminology of \((^3,^4)\), that the sequence \(T_n\in\mathcal L(E_n,F_n)\) stably approximates \(T\in\mathcal L(E,F)\). In \((^3,^4)\) it is proved that stability of the approximation is not only a sufficient but also a necessary condition for the convergence \(\|\xi_n^*-p_nx^*\|\to0\) for any right-hand side \(y\in F\) of equation (5). Theorem 1 can be reformulated as follows: if the sequence \(T_n\in\mathcal L(E_n,F_n)\) compactly approximates \(T\in\mathcal L(E,F)\) and if conditions 2) and 3) of Theorem 1 are fulfilled, then the approximation is stable. The converse assertion is true only in the following weakened form.
Remark 2. Let the sequence \(\widetilde T_n\in\mathcal L(E_n,F_n)\) stably approximate \(T\in\mathcal L(E,F)\). Let \(TE=F\), and let \(T\) admit a compact approximation by some sequence of operators \(T_n\in\mathcal L(E_n,F_n)\), for which condition 3) of Theorem 1 is fulfilled. Then there exist \(\widetilde q_n\in\mathcal L(F,F_n)\) with properties (1)—(4) such that, with respect to \(\{p_n\}\) and \(\{\widetilde q_n\}\), the sequence \(\widetilde T_n\) compactly approximates \(T\).
2. Consider the case where the operators \(T:E\to F\) and \(T_n:E_n\to F_n\) in equations (5) and (6) are nonlinear.
Lemma 1. Let the operator \(T_n\) be Fréchet differentiable in the ball \(\{\xi_n\in E_n:\|\xi_n-\xi_n^0\|\le\delta_0\}\), where \(\delta_0>0,\ \xi_n^0\in E_n\). Let \(T_n'(\xi_n^0)\in\mathcal L(E_n,F_n)\) have an inverse \([T_n'(\xi_n^0)]^{-1}\in\mathcal L(F_n,E_n)\), and suppose that for some \(\delta\) and \(\theta\) \((0<\delta\le\delta_0,\ 0\le\theta<1)\) the inequalities
\[ \sup_{\xi_n\in E_n,\ \|\xi_n-\xi_n^0\|\le\delta} \|[T_n'(\xi_n^0)]^{-1}[T_n'(\xi_n)-T_n'(\xi_n^0)]\|\le\theta, \tag{10} \]
\[ \alpha_n\equiv \|[T_n'(\xi_n^0)]^{-1}[T_n\xi_n^0-q_ny]\|\le \delta(1-\theta). \tag{11} \]
Then equation (6) has, in the ball \(\|\xi_n-\xi_n^0\|\le\delta\), a unique solution \(\xi_n^*\), and the estimate is valid
\[ \alpha_n/(1+\theta)\le \|\xi_n^*-\xi_n^0\|\le \alpha_n/(1-\theta). \tag{12} \]
The proof is analogous to the proof of Theorem 2 from \((^5)\).
Theorem 2. Suppose that the following conditions are fulfilled:
1) equation (5) has a solution \(x^*\);
2) the operator \(T\) is Fréchet differentiable at the point \(x^*\), and \(T'(x^*)\in\mathcal L(E,F)\) has an inverse \([T'(x^*)]^{-1}\in\mathcal L(F,E)\);
3) the operators \(T_n\) \((n=1,2,\ldots)\) are Fréchet differentiable in the corresponding balls \(\|\xi_n-p_nx^*\|\leqslant \delta_0\) \((\delta_0=\mathrm{const}>0)\), and for any \(\varepsilon>0\) there exist \(n_\varepsilon\) and \(\eta_\varepsilon\) \((0<\eta_\varepsilon\leqslant\delta_0)\) such that \(\|T_n'(\xi_n)-T_n'(p_nx^*)\|\leqslant\varepsilon\) for \(n\geqslant n_\varepsilon\), \(\|\xi_n-p_nx^*\|\leqslant\eta_\varepsilon\);
4) \(\|q_nTx^*-T_np_nx^*\|\to0\) as \(n\to\infty\) \((q_nTx^*=q_ny)\);
5) the sequence \(T_n'(p_nx^*)\in\mathcal L(E_n,F_n)\) compactly approximates the operator \(T'(x^*)\in\mathcal L(E,F)\);
6) the operators \(T_n'(p_nx^*)\) are such that from
\[ \inf_{\xi_n\in E_n,\ \|\xi_n\|=1}\|T_n'(p_nx^*)\xi_n\|>0 \]
there follows the existence of an inverse \([T_n'(p_nx^*)]^{-1}\in\mathcal L(F_n,E_n)\).
Then there exist \(N\) and \(\delta>0\) such that, for \(n\geqslant N\), equation (6) has in the ball \(\|\xi_n-p_nx^*\|\leqslant\delta\) a unique solution \(\xi_n^*\). As \(n\to\infty\), the convergence \(\|\xi_n^*-p_nx^*\|\to0\) holds with the estimate (7).
Proof is based on Lemma 1, in which we put \(\xi_n^0=p_nx^*\). From 2), 5), and 6) we conclude that, for sufficiently large \(n\), the inverses \([T_n'(p_nx^*)]^{-1}\in\mathcal L(F_n,E_n)\) exist, and their norms are bounded in the aggregate. Fix \(\theta\) \((0<\theta<1)\). Using condition 3), we find \(\delta>0\) such that, for sufficiently large \(n\), (10) will be satisfied. By virtue of 4), for sufficiently large \(n\) (11) is also fulfilled. From (12) we obtain the estimate (7); from (7) and 4) follows the convergence \(\|\xi_n^*-p_nx^*\|\to0\) as \(n\to\infty\). Theorem 2 is proved.
- Let now \(E=F,\ E_n=F_n,\ p_n=q_n\) \((n=1,2,\ldots)\). In studying the closeness of solutions of the operator equations
\[ x=Tx, \tag{13} \]
\[ \xi_n=T_n\xi_n \tag{14} \]
with completely continuous operators \(T\) and \(T_n\), it is natural to use the concept of the rotation of vector fields (⁶). We shall assume that the Banach spaces \(E\) and \(E_n\) \((n=1,2,\ldots)\) are real. Let \(\Omega\) be a bounded domain (a connected open set) in \(E\); denote \(\Omega_n=p_n\Omega\). Then \(\Omega_n\) is a bounded domain in \(E_n\). By \(\overline\Omega,\ \dot\Omega,\ \overline\Omega_n,\ \dot\Omega_n\) we denote the closures and boundaries of the domains \(\Omega\) and \(\Omega_n\). It is clear that \(p_n\overline\Omega\subset\overline\Omega_n\).
Definition 2. A sequence of operators \(T_n:\overline\Omega_n\to E_n\) compactly approximates the completely continuous operator \(T:\overline\Omega\to E\) with respect to \(\{p_n\}\), if the following conditions are fulfilled:
a) \(\|p_nTx-T_np_nx\|\to0\) as \(n\to\infty\) for every \(x\in\overline\Omega\);
b) for any sequence \(\{\xi_n\}\) \((\xi_n\in\overline\Omega_n,\ n=1,2,\ldots)\) there exist such \(y_n\in E\) that \(p_ny_n=T_n\xi_n\) \((n=1,2,\ldots)\) and the sequence \(\{y_n\}\) is compact in \(E\).
Below we assume that (4) is fulfilled in the following strengthened form:
\[ \lim_{n\to\infty}\|p_nx\|\geqslant c_0\|x\|\quad\text{for all }x\in E\ (c_0=\mathrm{const}>0). \tag{4′} \]
Lemma 2. Suppose the following conditions are fulfilled:
1) the sequence of completely continuous operators \(T_n:\overline\Omega_n\to E_n\) compactly approximates the completely continuous operator \(T:\overline\Omega\to E\);
2) the operators \(T_n\) \((n=1,2,\ldots)\) are such that from \(\|\xi_n-p_nx\|\to0\) \((\xi_n\to\overline\Omega_n,\ x\in\overline\Omega)\) it follows that \(\|T_n\xi_n-T_np_nx\|\to0\) as \(n\to\infty\);
3)
\[ \lim_{n\to\infty}\inf_{\xi_n\in\dot\Omega_n}\|p_nx-\xi_n\|>0 \quad\text{for every }x\notin\dot\Omega\ (x\in E); \]
4) the operator \(T\) has no fixed points on the boundary \(\dot\Omega\).
Then, for all sufficiently large \(n\), the operator \(T_n\) has no fixed points on the boundary \(\dot\Omega_n\), and the equality of rotations holds:
\[ \gamma(\xi_n-T_n\xi_n;\dot\Omega_n)=\gamma(x-Tx;\dot\Omega) \quad (n=n_0,n_0+1,\ldots). \]
Remark 3. Condition 3) of Lemma 2 is fulfilled if (2) and (4′) hold with \(c=c_0=1\), and \(\Omega\) is a ball in \(E\).
Theorem 3. Suppose conditions 1)—3) of Lemma 2 are satisfied, and suppose equation (13) has an isolated solution \(x^* \in \Omega\) of nonzero index (6), unique in \(\overline{\Omega}\). Then equation (14), for sufficiently large \(n\), has at least one solution \(\xi_n^* \in \overline{\Omega}_n\), and \(\|\xi_n^* - p_n x^*\|\to 0\) as \(n\to\infty\) for all solutions \(\xi_n^* \in \overline{\Omega}_n\).
- As an application, we consider the method of mechanical quadratures for solving integral equations. Let \(D\) be a metric compact set, \(\rho\) a metric in \(D\), and \(\nu\) a measure defined on some algebra of subsets of \(D\). Assume that \(|\nu|(D)<\infty\) (where \(|\nu|\) is the total variation of \(\nu\)) and that the ball \(S(t_0,r)=\{t\in D:\rho(t,t_0)<r\}\) is \(\nu\)-measurable for any \(t_0\in D\), \(r>0\), with \(|\nu|(S(t_0,r))>0\). Clearly, every function continuous on \(D\) is \(\nu\)-integrable. Consider the integral equation
\[ x(t)=\int_D K(t,s,x(s))\,\nu(ds), \tag{15} \]
a convergent quadrature process
\[ \int_D z(s)\,\nu(ds)=\sum_{j=1}^n \alpha_{jn}z(s_{jn})+\Phi_n(z)\qquad (n=1,2,\ldots) \]
\[ (\Phi_n(z)\to 0 \text{ for continuous functions } z(s)) \]
and the system of equations
\[ \xi_{in}=\sum_{j=1}^n \alpha_{jn}K(s_{in},s_{jn},\xi_{jn})\qquad (i=1,\ldots,n). \tag{16} \]
Theorem 4. Suppose equation (15) has a solution \(x^*(t)\) (continuous on \(D\)), and suppose the kernel \(K(t,s,x)\) is continuous and has a continuous derivative \(\partial K(t,s,x)/\partial x\) for \(t,s\in D\), \(|x-x^*(s)|\le \delta_0\) \((\delta_0=\mathrm{const}>0)\). Suppose the equation
\[ y(t)=\int_D H(t,s)y(s)\nu(ds)\qquad (H(t,s)=\partial K(t,s,x^*(s))/\partial x) \]
has only the zero solution. Then there exist \(N\) and \(\delta>0\) such that, for \(n\ge N\), the system of equations (16) has a unique solution \((\xi_{1n}^*,\ldots,\xi_{nn}^*)\) such that \(|\xi_{jn}^*-x^*(s_{jn})|\le \delta\) \((j=1,\ldots,n)\). The convergence
\[ \max_{1\le j\le n}|\xi_{jn}^*-x^*(s_{jn})|\to 0\quad \text{as } n\to\infty \tag{17} \]
holds, with the estimate
\[ c_1\varepsilon_n\le \max_{1\le j\le n}|\xi_{jn}^*-x^*(s_{jn})|\le c_2\varepsilon_n\qquad (c_1,c_2=\mathrm{const}>0), \]
where
\[ \varepsilon_n=\max_{1\le i\le n}|\Phi_n(z_{in})|,\qquad z_{in}(s)=K(s_{in},s,x^*(s)). \]
The proof is based on Theorem 2, which is applied with \(E=F=C(D)\), \(E_n=F_n=m_n\), \(p_nx=q_nx=(x(s_{1n}),\ldots,x(s_{nn}))\).
Theorem 3 makes it possible to establish the convergence (17), assuming only the continuity of the kernel \(K(t,s,x)\) and the existence of a solution \(x^*(t)\) of nonzero index (6) of equation (15).
Similar results in the case of Lebesgue measure on \(D=[a,b]\subset R^1\) were obtained by another method in \({}^{5}\), and for linear equations in a case similar to the one considered, in \({}^{1}\).
Tartu State University
Received
26 II 1969
References
\({}^{1}\) G. M. Vainikko, Uch. zap. Tartu State Univ., 220 (1968).
\({}^{2}\) G. M. Vainikko, Zh. Vychisl. Mat. i Mat. Fiz., 9, No. 4, 739 (1969).
\({}^{3}\) N. N. Gudovich, Zh. Vychisl. Mat. i Mat. Fiz., 6, No. 5 (1966).
\({}^{4}\) S. G. Krein, Linear Differential Equations in Banach Space, “Nauka,” 1967.
\({}^{5}\) G. M. Vainikko, Zh. Vychisl. Mat. i Mat. Fiz., 7, No. 4 (1967).
\({}^{6}\) M. A. Krasnosel’skii, Topological Methods in the Theory of Nonlinear Integral Equations, 1956.