MATHEMATICS

Academician of the Academy of Sciences of the Azerbaijan SSR Z. I. KHALILOV

ON THE STABILITY OF SOLUTIONS OF AN EQUATION IN A BANACH SPACE*

Let \(B\) be a Banach space. Consider the collection of functions \(x(t)\), defined on the half-line \(J=[0,+\infty)\), with values in \(B\). By \(L\) we denote the collection of functions \(x(t)\) that are locally Bochner integrable and have a countable family of compatible norms

\[ \|x\|_n=\int_0^n \|x(t)\|\,dt,\qquad n=1,2,\ldots . \]

Let \(U(t,s)\), for any fixed values of \(t\) and \(s\) \((0\le s\le t<+\infty)\), be a linear bounded operator acting in the space \(B\) and satisfying the conditions:

\(1^\circ.\) \(U(t,s)\) is strongly continuous in the aggregate of \(t\) and \(s\); \(U(t,t)=I\) (\(I\) is the identity operator).

\(2^\circ.\) \(U(t,s)U(s,0)=U(t,0)\), \(0\le s\le t\).

Consider the nonlinear integral equation

\[ x(t)=U(t,0)x_0+\int_0^t U(t,s)h(x(s),s)\,ds,\qquad x_0\in B. \tag{1} \]

Obviously, \(x_0=x(0)\). We shall establish conditions under which the solutions of equation (1) are stable.

By \(M_1\) we denote the set of functions \(x(t)\) having the representation

\[ x(t)=U(t,0)x_0,\qquad x_0\in B. \tag{2} \]

Obviously, the functions belonging to \(M_1\) are continuous on \(J\).

By \(M_2\) we denote the set of functions \(x(t)\) having the representation

\[ x(t)=U(t,0)x_0+\int_0^t U(t,s)f(s)\,ds,\qquad x_0\in B, \tag{3} \]

where \(f(t)\in \mathbf{B}\); \(\mathbf{B}\) is a Banach space stronger than \(L\), whose norm we denote by \(|f|_{\mathbf{B}}\). For example, \(C\) is the set of continuous functions with norm \(|x|=\sup_{t\in J}\|x(t)\|\). It is not difficult to verify that the functions of the set \(M_2\) are also continuous on \(J\).

Lemma 1. The set \(X\) of all bounded functions from the set \(M_1\) is a subspace of the space \(C\).

Denote by \(B_0\) the collection of all elements of \(B\) that are initial values of bounded functions in \(M_1\), \(|x|<+\infty\). As examples show, \(B_0\), generally speaking, is not closed.

* The contents of the present note were reported at the Fifth All-Union Conference on Functional Analysis in Baku in October 1959.

Lemma 2. If \(B_0\) is closed, then there exists a positive number \(S\) such that, for all \(x(t)\in X\),

\[ |x|\le S\|x(0)\|, \]

where \(|x|\), here and below, denotes the norm in \(C\).

Proof is carried out by applying the theorem on the continuity of the inverse of a one-to-one linear operator to the operator \(T\), which assigns to each element of \(X\) its initial value \(x_0\).

Let there exist a set \(B_1\), a complement of the closed \(B_0\). It is not always closed. Introduce the projection operators \(P_0, P_1\), \(x_0=P_0x\), \(x_1=P_1x\), \(x\in B\), \(x_0\in B_0\), \(x_1\in B_1\).

Lemma 3. Suppose \(B_0\) is closed and has a closed complement \(B_1\). Let there correspond to each \(f(t)\in B\) at least one bounded function from \(M_2\). Then there exists a constant \(K>0\) such that to each \(f(t)\in B\) there corresponds an \(x(t)\in M_2\) satisfying the inequality \(|x|\le K|f|_B\); this function can be chosen so that \(x(0)\in B_1\), and then it is determined uniquely.

Proof. Let \(Y\) be the set of all functions \(x(t)\) satisfying the conditions: \(x(0)\in B_1\), \(x(t)\) is continuous and bounded, and

\[ x(t)=U(t,0)x_0+\int_0^t U(t,s)f(s)\,ds,\qquad f(t)\in B. \]

In this linear manifold \(Y\) introduce the norm

\[ |x|_Y=|x|+|f|_B. \]

It is easy to prove that \(Y\), so normed, is a Banach space.

Let \(T\) be the linear mapping of \(Y\) into \(B\): \(Tx=f\). From the inequality

\[ |Tx|_B=|f|_B\le |x|+|f|_B=|x|_Y \]

it follows that \(T\) is bounded and \(\|T\|\le 1\). It is not difficult to show that there exists \(T^{-1}\), \(D(T^{-1})=B\), if one takes \(x_1\in B_1\) for \(x(0)\).

We now apply to \(T\) the theorem on the continuity of the inverse of a one-to-one linear transformation. Then \(\|T^{-1}\|\) exists and \(|x_1|\le k|f|_B\), where \(K=\|T^{-1}\|-1\).

In what follows we shall assume that the nonlinear operator \(h(x(t),t)\) is such that, for every \(x(t)\in C\), \(|x|<a\), \(h(x(t),t)\in B\). Let there also exist a constant \(\gamma>0\) such that

\[ |h(x',t)-h(x'',t)|_B\le \gamma |x'-x''| \]

for every pair \(x'\) and \(x''\in C\), \(|x'|<a\), \(|x''|<a\).

Theorem 1. Let \(\beta=|h(0,t)|_B\). Under the conditions of Lemma 3, if \(K\gamma<1\) and \(\beta<K^{-1}(1-K\gamma)a\), then for each \(\xi_0\in B_0\), \(\|\xi_0\|<b=S^{-1}((1-K\gamma)a-K\beta)\), there exists a unique solution \(x(t)\) of equation (1) such that \(|x|<a\) and \(P_0x(0)=\xi_0\); this solution satisfies the inequality

\[ |x|\le (1-K\gamma)^{-1}(K\beta+S\|\xi_0\|). \tag{4} \]

The theorem is proved by the method of successive approximations on the basis of Lemmas 2 and 3.

From (4) follows the boundedness of solutions of equation (1).

Corollary. If \(h(0,t)=0\), then the zero solution of equation (1) is stable in the sense of Lyapunov.

Indeed, for \(\beta=0\), from (4) we have:
\[ |x|\le (1-K\gamma)^{-1}S\|\xi_0\|. \]

Let us now consider the Cauchy problem
\[ \frac{dx(t)}{dt}=A(t)x(t)+h(x(t),t),\qquad x(0)=x_0, \tag{5} \]
where the operator \(A(t)\) satisfies the following conditions:

\(C_1.\) For each \(t\in J\), \(A(t)\) satisfies the condition
\[ \|(I-\alpha A(t))^{-1}\|\le 1 \]
for \(\alpha>0\).

\(C_2.\) 1) The domain of definition of \(A(t)\) does not depend on \(t\); 2) \(B(t,s)=(I-A(t))(I-A(s))^{-1}\) is uniformly bounded for all \(s,t\); 3) \(B(t,s)\) has bounded variation with respect to \(t\), at least for some \(s\).

As T. Kato \((^1)\) showed, under conditions \(C_1\) and \(C_2\) there exists \(U(t,s)\) with certain properties, whence our conditions \(1^\circ\) and \(2^\circ\) follow.

Under conditions \(C_1\) and \(C_2\), we associate with the Cauchy problem (5) the nonlinear integral equation (1), whose solution we shall call a generalized solution of the Cauchy problem (5). Consequently, Theorem 1 establishes boundedness, and then stability, of generalized solutions of the Cauchy problem (5).

If the operator \(A(t)\) also satisfies the condition that
\[ A(t)\frac{d}{dt}A^{-1}(t) \]
is a strongly continuous operator in \(t\); if \(h(x(t),t)\) is a strongly differentiable function in \(t\), and \(x_0\in D(A(t))\), then the generalized solution is strongly differentiable and is a solution of the Cauchy problem (5) \((^{2-4})\).

Since nonstationary problems for a parabolic equation, in a certain sense, are reducible to the Cauchy problem (5), Theorem 1 gives boundedness (stability) conditions for solutions of the indicated nonstationary problems.

The question of stability of solutions of equation (5) was first investigated by M. G. Krein \((^5)\) in the case of bounded \(A(t)\). Theorem 1 is a generalization of the theorem of Massera and Schäffer \((^6)\) for almost bounded \(A(t)\), which in turn is a generalization of the well-known theorem of Perron \((^7)\) for finite-dimensional space.

Theorem 1 can also be formulated and proved for \(B\) distinct from \(C\).

Institute of Mathematics and Mechanics
Academy of Sciences of the Azerbaijan SSR

Received
3 XII 1960

REFERENCES

\(^1\) T. Kato, J. Math. Soc. Japan, 5 (1953); Matematika, 2, 4 (1958).
\(^2\) M. A. Krasnosel’skii, S. G. Krein, Tr. 3 Vsesoyuzn. matem. s”ezda, 3, 1958, p. 73.
\(^3\) T. Kato, Div. Electromag. Res. Inst. Math. Sci., N. Y. Univ., Res. Rep. No. BK-11 (1955).
\(^4\) M. A. Krasnosel’skii, S. G. Krein, P. E. Sobolevskii, DAN, 111, No. 1 (1956).
\(^5\) M. G. Krein, UMN, 3, No. 3 (1948).
\(^6\) J. L. Massera, J. J. Schäffer, Ann. Math., 67, No. 3, 517 (1958).
\(^7\) O. Perron, Math. Zs., 32 (1930).