MATHEMATICS

P. E. SOBOLEVSKII

ON APPROXIMATE METHODS FOR SOLVING DIFFERENTIAL EQUATIONS IN BANACH SPACE

(Presented by Academician A. N. Kolmogorov on 16 II 1957)

Consider the equation

[
\frac{dx}{dt}+A(t)x=f(t,x)\qquad (0\leq t\leq T),
\tag{1}
]

where (x(t)) is the unknown function with values in a Banach space (E); (A(t)) and (f(t,x)), for each (t\in[0,T]), are operators acting in (E). We shall assume that (A(t)) is a linear unbounded closed operator with domain (D(A)), independent of (t), and that (f(t,x)) is a bounded nonlinear operator.

In papers ((^1,^2)), under certain assumptions, it was established that there exists a solution (x(t)) of equation (1), defined on some segment ([0,h]), and satisfying the initial condition

[
x(0)=x^0\qquad (x^0\in D(A)).
\tag{2}
]

Let (A_n(t)) be bounded operators approximating uniformly in (t) the operator (A(t)) on its domain of definition:

[
\lim_{n\to\infty}\ \sup_{0\leq t\leq T}|[A_n(t)-A(t)]x|=0
\qquad (x\in D(A)).
\tag{3}
]

Let the bounded operators (f_n(t,x)) converge uniformly in (t) to the operator (f(t,x)) for each (x) from some ball (S) with center at the point (x^0). Finally, let (x_n^0) converge to (x^0). In the present paper we consider the question of under what conditions the solutions (x_n(t)) of the equations

[
\frac{dx}{dt}+A_n(t)x=f_n(t,x),
\tag{4}
]

satisfying the initial conditions

[
x_n(0)=x_n^0,
\tag{5}
]

converge to the solution (x(t)) of problem (1)—(2).

1. The homogeneous equation with a constant operator

[
\frac{dx}{dt}+Ax=0
\tag{6}
]

has the solution (x(t)=e^{-tA}x^0), satisfying the initial condition (2), if the operator (-A) is the infinitesimal generator of a strongly continuous semigroup ((^3)).

In the case of a constant operator (A), it is natural also to choose the approximating operators (A_n) to be constant, and to write the equation for the approximate solutions in the form

[
\frac{dx}{dt}+A_nx=0.
\tag{7}
]

The solutions of these equations under the initial conditions (5) are given by the formula:

[
x_n(t)=e^{-tA_n}x_n^0 .
]

Theorem 1. In order that the sequence of operators (e^{-tA_n}) converge strongly to the operator (e^{-tA}), uniformly in (t\in[0,T]), it is necessary and sufficient that the inequality

[
\sup_{0\le t\le T}|e^{-tA_n}|\le M,
]

hold, where (M) does not depend on (n).

The assertion of Theorem 1 is proved in the book (4) under the additional assumption that the operators (A_n) commute. This assumption is not satisfied for the operators (A_n) encountered in most approximate methods.

It follows from Theorem 1 that (x_n(t)) converge to (x(t)) uniformly in (t\in[0,T]). If (|A_nx_n^0-Ax_0|\to0), then, moreover, (dx_n/dt) also converge to (dx/dt) uniformly in (t\in[0,T]).

1. Let us now consider the homogeneous equation

[
\frac{dx}{dt}+A(t)x=0 .
\tag{8}
]

We shall assume that for every (t\in[0,T])

[
|[\lambda I+A(t)]^{-1}|\le \frac{1}{1+\lambda}\quad(\lambda>-1)
\tag{9}
]

and that the operator (C(t)=A(t)\,dA^{-1}(t)/dt) is bounded and strongly continuous with respect to (t).

It follows from the results of (5) that under these conditions there exists an operator (U(t,s)) ((t\ge s)), strongly continuous jointly in (t) and (s), and such that the function (x(t)=U(t,s)x^0), for (x^0\in D(A)), satisfies, for (t\ge s), equation (8) and the initial condition (x(s)=x^0). Thus, under the initial condition (2), the solution of equation (8) has the form (x(t)=U(t,s)x^0).

Consider the equation for approximate solutions

[
\frac{dx}{dt}+A_n(t)x=0 .
\tag{10}
]

If the bounded operator (A_n(t)) is strongly continuous in (t), then equation (10) has a solution for any initial condition (x_n^0). This solution, as in the case of the operator (A(t)), can be written in the form (x_n(t)=U_n(t,0)x_n^0), where the operator (U_n(t,s)) has properties analogous to those of the operator (U(t,s)).

Theorem 2. Suppose that all the operators (A_n(t)) satisfy condition (9). Then the operators (U_n(t,s)) converge strongly to the operator (U(t,s)), uniformly in (t) and (s) for (0\le s\le t\le T).

The proof is based on the formula

[
[U(t,s)-U_n(t,s)]x
=
\int_s^t U_n(t,\tau)[A(\tau)-A_n(\tau)]U(\tau,0)x\,d\tau,
\tag{11}
]

valid for every (x\in D(A)).

Let us note that from Theorem 2 there follow assertions analogous to those given at the end of the preceding section.

1. In (1), the continuous solutions of the integral equation

[
x(t)=U(t,0)x^0+\int_0^t U(t,s)f(s,x(s))\,ds
\tag{12}
]

were called generalized solutions of equation (1) under the initial condition (2). This name is justified by the fact that generalized solutions are ordinary solutions if certain smoothness assumptions on (f(t,x)) are fulfilled. In the case of a bounded operator (A(t)), equation (12) and problem (1)—(2) are equivalent.

Theorem 3. Let the operators (A(t)) and (A_n(t)) satisfy the conditions of items 1 or 2, according as whether or not they depend on (t). Let the operators (f_n(t,x)) be continuous in (t \in [0,T]) and, in the ball (S), satisfy a Lipschitz condition in (x) with a constant independent of (t) and (n). Then, beginning with some (n), problem (4)—(5) has solutions defined on some segment ([0,h]). These solutions converge uniformly in (t \in [0,h]) to the generalized solution of equation (1) under the initial condition (2).

Theorem 3 may be regarded as a justification of various approximate methods for solving equation (1).

4. In what follows equation (1) is considered in a separable Hilbert space (H).

The Bubnov—Galerkin method for the approximate solution of problem (1)—(2) consists in replacing equation (1) by the equations

[
\frac{dx}{dt}+P_n A(t)x=P_n f(t,x),
\tag{13}
]

where (P_n) is the operator of orthogonal projection onto the linear span (R_n) of the first (n) elements of the basis ({e_n}), while the initial condition (2) is replaced by the initial condition (5), with (x_n^0) chosen from (R_n) so that (|x_n^0-x^0|\to 0). Suppose that the basis ({e_n}) is composed of eigenvectors of a certain self-adjoint operator (C), for which (D(C)=D(A)).

Theorem 4. Let the operator (A(t)) satisfy the conditions of item 2, and let the operator (f(t,x)) be continuous in (t) on ([0,T]) and, in the ball (S), satisfy a Lipschitz condition in (x) with a constant independent of (t). Then, beginning with some (n), problem (13)—(5) has solutions defined on some segment ([0,h]). These solutions converge uniformly in (t \in [0,h]) to the generalized solution of equation (1) under the initial condition (2).

In proving that the operators (P_n A(t)) satisfy condition (9) on (R_n), we used the following lemma.

Lemma. For (9) to hold it is necessary and sufficient that

[
\operatorname{Re}(Ax,x)\ge (x,x)\quad (x\in D(A));\qquad
\operatorname{Re}(A^x,x)\ge (x,x)\quad (x\in D(A^)).
]

The sufficiency of these conditions was proved by V. E. Lyantse ({}^{(6)}); the proof of their necessity was communicated to us by S. G. Krein.

5. Consider the equation:

[
\frac{dx}{dt}+Ax=f(t,x,Bx),
\tag{14}
]

where (A) is a self-adjoint, positive-definite operator; (B) is a closed operator with (D(B)\supset D(A^{1/2})). Let a continuous function (y(t)) satisfy the equation

[
y(t)=f\left(t,e^{-tA}x^0+\int_0^t e^{-(t-s)A}y(s)\,ds,\,
Be^{-tA}x^0+B\int_0^t e^{-(t-s)A}y(s)\,ds\right).
\tag{15}
]

Then the function

[
x(t)=e^{-tA}x^0+\int_0^t e^{-(t-s)A}y(s)\,ds
]

is naturally called a generalized solution of equation (14)

under the initial condition (2), since it is its ordinary solution, provided certain smoothness assumptions on (f(t,x,y)) are satisfied ((^2)).

Application of the Bubnov—Galerkin method to equation (14) leads to the solution of the equations

[
\frac{dx}{dt}+P_nAx=P_nf(t,x,Bx).
\tag{16}
]

Theorem 5. Suppose that the operator (f(t,x,y)) is continuous in (t\in[0,T]) and, in the ball (S) containing (x_0) and (Bx_0) in its interior, satisfies a Lipschitz condition in the aggregate of the variables (x) and (y), with a constant independent of (t); suppose that (|A^{1/2}[x_n^0-x^0]|\to0). Then, starting from some (n), problem (16)—(5) has solutions (x_n(t)) defined on some segment ([0,h]). These solutions converge uniformly in (t) on ([0,h]) to the generalized solution of equation (14) under the initial condition (2).

If the basis ({e_n}) consists of eigenvectors of the operator (A), then Theorem 5 admits a substantial strengthening. In this case it is sufficient to require of the operator (B) that (D(B)) contain (D(A^{1-\alpha})), where (\alpha) is an arbitrarily small positive number. The assertion of Theorem 5 then holds if (|A^{1-\alpha}[x_n^0-x^0]|\to0).

We note that another approach to the study of the convergence of the Bubnov—Galerkin method for equations of the form (1) may be found in the works ((^{7-9})).

REFERENCES

(^1) M. A. Krasnosel’skii, S. G. Krein, P. E. Sobolevskii, DAN, 111, No. 1 (1956).
(^2) M. A. Krasnosel’skii, S. G. Krein, P. E. Sobolevskii, DAN, 112, No. 6 (1957).
(^3) E. Hille, Functional Analysis and Semigroups, IL, 1951.
(^4) F. Riesz, B. Sz.-Nagy, Lectures on Functional Analysis, IL, 1954.
(^5) T. Kato, J. Math. Soc. Japan, 5, No. 2 (1953).
(^6) V. E. Lyantse, Matem. sborn., 35 (77), 62 (1954).
(^7) M. I. Vishik, Matem. sborn., 39 (81), No. 1 (1956).
(^8) E. Hopf, Math. Nachr., No. 2 (1950—1951).
(^9) I. I. Vorovich, Tr. 3rd All-Union Mathematical Congress, 1, 1956.