UDC 517.91+517.948

MATHEMATICS

N. V. MEDVEDEV

ON CONDITIONS FOR THE EXISTENCE OF BOUNDED SOLUTIONS OF DIFFERENTIAL EQUATIONS IN A BANACH SPACE

(Presented by Academician I. G. Petrovskii, 29 IV 1968)

Consider in a Banach space \(E\) the differential equation

\[ dx/dt=A(t)x+f(t), \tag{1} \]

where \(x=x(t)\) is the sought function with values in \(E\); \(A(t)\) is, generally speaking, for each fixed \(t\in(-\infty,\infty)\) an unbounded linear operator with domain \(D\), dense in \(E\) and independent of \(t\); \(f(t)\) is a bounded strongly continuous function on \((-\infty,\infty)\),

\[ \sup_t \|f(t)\| \leq N. \]

There are various conditions for the existence of a bounded solution of this equation. For example, in [1] the case is indicated when equation (1) has at least one generalized solution bounded on \([0,\infty)\), and in [2] the Galerkin method is used to establish the existence of a bounded solution on the entire number axis.

In the present paper, by a somewhat different route, a method is indicated for finding a bounded solution of equation (1). Here the method of approximating a bounded solution of equation (1) by a bounded solution of the corresponding difference equation is essentially used:

\[ x(t)=(I+hA(t))^{-1}(x(t+h)-hf(t)). \tag{2} \]

In what follows it is assumed that the operator \(A(t)\) has a bounded inverse operator, and also that all the operators \(A(t)A^{-1}(s)\) are bounded and satisfy the condition

\[ \|I-A(t)A^{-1}(s)\|\leq M|t-s|,\qquad t,s\in(-\infty,\infty), \tag{3} \]

where \(I\) is the identity operator; \(M\) is some positive number. In addition, it is assumed that for all \(h\in(0,H)\) there exist bounded linear operators \((I+hA(t))^{-1}\) with the estimate

\[ \|(I+hA(t))^{-1}\|\leq 1-ph,\qquad t\in(-\infty,\infty), \tag{4} \]

where \(p\) is some positive number, and \(H\) is a sufficiently small positive number.

Lemma 1. Let conditions (3) and (4) be fulfilled. Then equation (2), for all \(h\in(0,H)\), has one and only one continuous bounded solution \(x(t,h)\) on \((-\infty,\infty)\); moreover, the estimate

\[ \sup_{t,h}\|x(t,h)\|\leq N/p \]

holds.

The following proposition establishes the relation between the bounded solutions of equations (1) and (2).

Lemma 2. Let \(A(t)\) be a bounded operator, defined for all \(t\) on all of \(E\) and satisfying conditions (3) and (4). Then, if from the set of solutions \(x(t,h_m)\) of equation (2), where \(h_m\to0\) as \(m\to\infty\), one can select a subsequence converging uniformly on each finite interval to some continuous function \(x(t)\), then \(x(t)\) is

is called a bounded solution of equation (1), and if \(x(t)\) is a bounded solution of equation (1), then

\[ \lim_{h\to 0} x(t,h)=x(t). \]

Let

\[ A_n(t)=A(t)\left(I+\frac{1}{n}A(t)\right)^{-1} \]

and let \(n\) take integer values greater than \(2H^{-1}\). Then the operators \(A_n(t)\) for all \(h\in(0,H/2)\) satisfy conditions (3) and (4). Moreover, the constants \(p\) and \(M\) are the same and do not depend on \(n\). Bounded solutions of equation (1) can be approximated by bounded solutions of the equation

\[ dx/dt=A_n(t)x+f(t). \tag{5} \]

Lemma 3. Let the operator \(A^2(t)\) have a domain of definition \(D(A^2(t))\) independent of \(t\). Let \(f(t)\), for each \(t\), take values in \(D(A^2(t))\) and satisfy the estimate \(\sup \|A^2(t)f(t)\|<+\infty\). In addition, let all operators \(A^2(t)A^{-2}(s)\) be bounded and let the inequality

\[ \|I-A^2(t)A^{-2}(s)\|\le M_1|t-s|,\qquad t,s\in(-\infty,\infty), \tag{6} \]

hold, where \(M_1\) is some constant number. If conditions (3) and (4) hold and the inequality \(\max\{M,M_1\}<p\) holds, then the sequence of bounded solutions \(x_n(t)\) of the difference equation corresponding to equation (5) converges uniformly in \(h\) to the function \(x(t,h)\).

In what follows, a continuously differentiable function \(x(t)\), bounded on \((-\infty,\infty)\) and satisfying equation (1), will be called a classical solution of this equation. The uniqueness of the classical bounded solution follows from the next lemma.

Lemma 4. Let the equation

\[ dx/dt=A_n(t)x+g_n(t), \]

where \(g_n(t)\) is some continuous, generally speaking unbounded on \((-\infty,\infty)\), function, have a continuous solution \(x_n(t)\) uniformly bounded in \(n\) on the whole number axis. If the function \(g_n(t)\) tends to zero as \(n\to\infty\), uniformly in \(t\) on every finite interval, then for any fixed \(t\) the function \(x_n(t)\) tends to zero as \(n\to\infty\).

For the proof, for \(\varepsilon>0\) on some interval one constructs a set of points \(\tau_{nh}\) such that

\[ \|x_n(t)\|\le \|x_n(\tau_{nh})\|,\qquad \|x_n(\tau_{nh}+h)\|-\|x_n(\tau_{nh})\|\le \varepsilon h. \]

Then from the identity

\[ x_n(t+h)=x_n(t)+hA_n(t)x_n(t)+hg_n(t)+\alpha(t,x_n(t),h), \tag{7} \]

where the quantity \(\alpha(t,x_n(t),h)\), for fixed \(n\), tends to zero together with \(h\) uniformly in \(t\) on every finite interval, one can obtain the estimate

\[ \|x_n(\tau_{nh})\|\le \frac{1}{p}\left(\varepsilon+\|g_n(\tau_{nh})\|+\|\alpha(\tau_{nh},x_n,h)\|\right), \]

from which the lemma follows. Incidentally, the same idea is used in the proof of Lemmas 2 and 3.

Lemma 5. Let \(x(t)\) be a classical bounded solution of equation (1), and let \(x_n(t)\) be a continuous solution of equation (5), uniformly bounded in \(n\) on \((-\infty,\infty)\). Then

\[ \lim_{n\to\infty} x_n(t)=x(t). \]

The proof of this lemma is based on Lemma 4.

Definition 1. If from the set of functions \(x_n(t,h_m)\), where \(h_m\to 0\) as \(m\to\infty\), one can select a subsequence converging to some function \(x_n(t)\), and from the obtained set of functions \(x_n(t)\), in turn, a subsequence converging to the function \(x^*(t)\), then

the function \(x^*(t)\) is called a generalized bounded solution of equation (1).

This definition is justified by the fact that, if equation (1) has a classical bounded solution, then, by Lemma 5, the generalized solution coincides with the classical one.

Theorem 1. Suppose that conditions (3) and (4) are satisfied. Suppose that \(f(t)\), for every \(t\), takes values in \(D\) and satisfies the condition
\[ \sup \|A(t)f(t)\|<+\infty . \]
Then, if \(f(t)\) is a uniformly continuous function on \((-\infty,\infty)\), \(A^{-1}(t)\) is a completely continuous operator for each fixed \(t\), and \(p>M\), equation (1) has at least one bounded generalized solution. If, in addition, the operator \(A^2(t)\) has a domain of definition independent of \(t\), satisfies condition (6), and \(p>M_1\), then equation (1) has a unique bounded generalized solution.

Proof. Since the functions \(A(t)x(t,h)\) and \(A(t)x_n(t,h)\) exist and are bounded uniformly in \(t\), \(n\), and \(h\), the values of the functions \(x(t,h)\) and \(x_n(t,h)\) for each \(t\) form compact sets in \(E\). Then, from their equicontinuity on the entire real axis, by Lemmas 1 and 2, the existence of a generalized solution follows. Further, by Lemmas 2 and 3, the equation
\[ \frac{dx}{dt}=A(t)x+\left(I+\frac{1}{m}A(t)\right)^{-1}f(t) \]
for any integer \(m>2H^{-1}\) has a unique generalized bounded solution, whence the uniqueness of the generalized solution of equation (1) follows.

Theorem 2. Suppose there exists a strongly continuous derivative \((A^{-1}(t))'\), which for each \(t\) is a linear bounded operator. If the generalized solution \(x^*(t)\) takes values in \(D\) and is a continuously differentiable function, then this solution is a classical bounded solution of equation (1).

If \(A(t)=A\) is a constant operator, then the requirement of complete continuity of the operator \(A^{-1}\) may be omitted. Suppose that the resolvent of the operator \(-A\) satisfies the condition
\[ \|(\lambda I+A)^{-1}\|\leq \frac{1}{\operatorname{Re}\lambda-\omega} \qquad (\operatorname{Re}\lambda>\omega), \tag{8} \]
where \(\omega\) is some negative number. Then the existence of a bounded solution of equation (5) follows from the results of [2], with
\[ x_n(t)=\int_t^{t+1} e^{-(t-s)A_n}F_n(s)\,ds, \]
where
\[ F_n(t)=-f(t)-e^{-A_n}f(t+1)-e^{-2A_n}f(t+2)-\cdots . \]

Theorem 3. Suppose that \(f(t)\) takes values in the domain of definition of the operator \(A^2\). Suppose that the function \(A^2 f(t)\) is continuous on \((-\infty,\infty)\) and that
\[ \sup \|A^2 f(t)\|<+\infty . \]
If condition (8) holds for the operator \(A\), then equation (1) has one and only one generalized bounded solution.

Proof. Since the operator \(A\) satisfies condition (4), using the identity of the form (7) for the functions \(x_n(t)\) and the method of proof of Lemma 4, we establish that the functions \(x_n(t)\) form a fundamental sequence.

The concept of a generalized solution bounded on \((-\infty,\infty)\) can also be introduced for a system of the form
\[ dx/dt=A(t)x+f(t,x,y), \qquad dy/dt=B(t)y+g(t,x,y), \tag{9} \]

where \(A(t)\) and \(B(t)\) are, for each \(t\), unbounded linear operators with dense domains of definition in the Banach spaces \(E_1\) and \(E_2\), respectively.

It is assumed that the operators \(A(t)\), \(B(t)\) and the functions \(f(t,x(t),y(t))\), \(g(-t,x(-t),y(-t))\), for any uniformly continuous and bounded functions \(x(t)\) and \(y(t)\) on \((-\infty,\infty)\), ensure the existence and uniqueness of generalized bounded solutions of the following equations:

\[ \begin{aligned} du/dt &= A(t)u + f(t,x(t),y(t)),\\ dv/dt &= -B(-t)v - g(-t,x(-t),y(-t)). \end{aligned} \tag{10} \]

In the space of ordered pairs of the form \((x(t),y(t))\), where \(x(t)\) and \(y(t)\) are uniformly continuous and bounded on \((-\infty,\infty)\) functions with values respectively in \(E_1\) and \(E_2\), introduce an operator \(W\), which assigns to the pair \((x(t),y(t))\) the pair \((u(t),v_1(t))\), where \(u(t)\) and \(v(t)=v_1(-t)\) are generalized bounded solutions, respectively, of equations (10).

Definition 2. A fixed point of the operator \(W\) is called a generalized bounded solution of system (9).

Using various conditions for the existence of a fixed point of the operator \(W\), one can obtain conditions for the existence of a generalized bounded solution of system (9), as well as conditions under which this solution is a true bounded solution.

In conclusion, the author expresses gratitude to M. M. Vainberg and R. S. Gusarova for discussion of the results of the work.

Vladimir State
Pedagogical Institute
named after P. I. Lebedev-Polyansky

Received
28 IV 1968

REFERENCES

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