Mathematics

Ch. Foiaș, G. Gussi, and B. Poénaru

GENERALIZED SOLUTIONS OF A QUASILINEAR DIFFERENTIAL EQUATION IN A BANACH SPACE

(Presented by Academician S. L. Sobolev on 26 XI 1957)

1. Consider the equation

\[ \frac{dx(t)}{dt}=A(t)x(t) \tag{1} \]

in a Banach space \(X\), where \(A(t)\) are linear closed operators with domains \(D_{A(t)}\) dense in \(X\), satisfying the condition:

C I. There exist sequences \(J_n\) of idempotent operators such that \(J_nJ_m=0\) for \(n\ne m\) and, moreover:

1) \(I=J_1+J_2+\cdots+J_n+\cdots\) (the series converges strongly);

2) \(J_nA(t)\subseteq A(t)J_n\) \((t\in(a,b))\), \(n=1,2,\ldots\);

3) \(A(t)J_n\) is bounded in \(X_{(n)}=J_n(X)\).

4) \(A(t)J_n\) is continuous in norm in \(X_{(n)}\) for \(t\in(a,b)\), \(n=1,2,\ldots\).

It is possible that the Cauchy problem for equation (1) has no solution for all initial conditions \(y_0\in X\). Consider the space \(\mathfrak X\) of sequences \(x=\{x_n\}\), where \(x_n\in X_{(n)}\), with the family of seminorms

\[ p_{n_1n_2\ldots n_k}(x)=\sup(\|x_{n_1}\|,\ldots,\|x_{n_k}\|), \]

where \((n_1,\ldots,n_k)\) is any finite set of natural numbers; the correspondence \(x\to\{J_nx\}=\tilde x\) is a continuous isomorphism from \(X\) onto \(\widetilde X\subset\mathfrak X\) (\(\mathfrak X\) is the closure of \(\widetilde X\)).

Associate with equation (1) the equation

\[ \frac{dx}{dt}=\widetilde A(t)x(t), \tag{1′} \]

where \(\widetilde A(t)\{x_n\}=\{A(t)x_n\}\) is the extension of \(A(t)\) in \(\mathfrak X\) by continuity.

Theorem 1. 1) The Cauchy problem for equation (1′) has a solution, and moreover a unique one in \((a,b)\), for arbitrary initial data from \(\mathfrak X\).

2) The Cauchy problem is well posed in \(\mathfrak X\), i.e., on any closed interval contained in \((a,b)\) the solution depends continuously on the initial value uniformly with respect to \(t\).

3) If \(x(t)\) is a solution of the Cauchy problem for (1), \(x(c)=x_0\), then \(\tilde x(t)\) is a solution of the Cauchy problem for (1′), \(\tilde x(c)=\tilde x_0\).

It follows from this that the Cauchy problem for (1) has at most one solution and that this solution depends continuously on the initial data, if it is considered in \(\mathfrak X\). If the initial value belongs to \(\widetilde X\), then the solution of the Cauchy problem for (1′) is a generalized solution for (1).

We introduce the following condition:

C II. There exists a function \(\gamma(t)\), summable on any interval \([a_1,b_1]\subset(a,b)\), such that for any \(\varepsilon>\gamma(t)\) the operator \([\varepsilon I-A(t)]^{-1}\) exists,

defined on all of \(X\) and satisfies the inequality*

\[ \left\|[\varepsilon I-A(t)]^{-1}\right\|<\frac{1}{\varepsilon-\gamma(t)},\qquad \varepsilon>\gamma(t),\quad t\in(a,b). \]

Theorem 2. If \(A(t)\) satisfies condition C II, then:

1) The solution of the Cauchy problem for \((1')\) takes its values in \(\widetilde X\) for initial values from \(\widetilde X\).

2) The solution is continuous in \(X\) for \(t\geq c\).

3) If \(\widetilde x(t)\) is a solution of the Cauchy problem for \((1')\) such that \(\widetilde x(c)=\widetilde x_0\), \(x_0\in X\), then for \(t\geq c\)

\[ \|x(t)\|\leq \|x_0\|\exp\left[\int_c^t \gamma(\tau)\,d\tau\right]. \]

Corollary 1. 1) The solution of the Cauchy problem for \((1')\) with initial conditions in \(X\) and considered in \(X\) is a generalized solution of the Cauchy problem for (1) in the sense of S. L. Sobolev \((^2)\).

2) The generalized Cauchy problem is well posed.

Denote by \(T(t,s)x\) (\(t\geq s\)) the generalized solution of the Cauchy problem such that \(T(t,s)x\to x\) as \(t\to s+0\).

Corollary 2. 1) \(T(t,s)T(s,\gamma)=T(t,\gamma)\), \(t\geq s\geq \gamma\).

2) \(T(t,s)\) is strongly continuous on the triangle \(t,s\in(a,b)\), \(t\geq s\).

3)

\[ \|T(t,s)\|\leq \exp\left[\int_s^t \gamma(\tau)\,d\tau\right]. \]

4) If

\[ x_0\in \bigcup_{N=1}^{\infty}\sum_{n=1}^{N}X_{(n)}, \]

then \(T(t,s)x_0\) is a solution of the Cauchy problem for (1).

In Kato’s construction \((^1)\), the set \(\{x\}\) for which \(T(t,s)x\) is a solution of (1) coincides with \(D_{A(0)}=D_{A(t)}\). In our case it is possible that \(D_{A(t)}\neq D_{A(0)}\), but always

\[ \bigcup_{N=1}^{\infty}\sum_{n=1}^{N}X_{(n)}\subset D_{A(0)}, \]

and the inclusion is often strict.

We introduce the following condition:

C III. Let condition C II be satisfied and let \(\gamma(t)\leq\gamma<\infty\), and let \(\gamma_n\) be the least number such that:

1) \([\varepsilon I-A_{(n)}(t)]^{-1}\) exists in \(X_{(n)}\) and is defined on all \(X_{(n)}\) for \(\varepsilon>\gamma_n\) (\(A_n(t)\) is the restriction of \(A(t)\) to \(X_{(n)}\)).

2)

\[ \left\|[\varepsilon I-A_{(n)}(t)]^{-1}\right\|\leq \frac{1}{\varepsilon-\gamma_n},\qquad t\in(a,b). \]

Let, moreover,

\[ M_{(n)}=\max_{t\in(a,b)}\|A_{(n)}(t)\|. \]

Then

\[ \sum_{n=1}^{\infty} M_{(n)}e^{t\gamma_n}<\infty. \]

There exist operators satisfying these conditions. For example, if \(X\) is a Hilbert space and \(A(t)=B(t)N\), where \(N\) is a normal unbounded operator whose spectrum lies in an angle smaller than \(\pi\), with the negative half-axis \(Ox\) serving as the bisector of this angle, and \(B(t)\) is a strictly positive operator, continuous in norm and such that \(B(t)N\subset NB(t)\), then condition C III is satisfied \((^3)\).

Theorem \(1'\). If condition C III is satisfied, \(T(t,s)x\) is a solution of the Cauchy problem for (1) for \(x\in X\), \(t>s\).

This result is analogous to a property that holds for certain parabolic equations of I. G. Petrovskii \((^4)\).

1. Consider the equation

\[ \frac{dx}{dt}=A(t)x+f(t,x). \tag{2} \]

* The case \(\gamma(t)=\gamma\), see (1).

Suppose that \(A(t)\) satisfies condition C II and that \(f(t,x)\) is a continuous mapping of \((a,b)\times X\) into \(X\). Then every solution of the Cauchy problem for (2), \(x(s)=x_0,\ x_0\in X\), will be a solution* of the equation

\[ x(t)=T(t,s)x_0+\int_s^t T(t,\tau)f(\tau,x(\tau))\,d\tau, \tag{3} \]

and, conversely, every solution of this equation is a solution of the Cauchy problem for the equation

\[ \frac{d\widetilde{x}(t)}{dt}=\widetilde{A}(t)\widetilde{x}(t)+\widetilde{f}(t,\widetilde{x}) \]

from \(\mathfrak X\) (obviously, \(\widetilde f\) is defined only on \((a,b)\times\widetilde X\)). It is natural to regard a solution of (3) as a generalized solution for (2).

Theorem 3. Let \(s\in(a,b)\), and let \(S\) be a ball in \(X\) with center at \(x_0\). If \(f(t,x)\) is defined and continuous on \((a,b)\times S\) and \(f(t,x)=f_1(t,x)+f_2(t,x)\), where \(f_1(t,x)\) is compact and
\[ \|f_2(t,x)-f_2(t,y)\|\leq \overline K(t)\omega(\|x-y\|); \]
\(\overline K(t)\) is summable on \((a,b)\) and \(\omega(z)\) is an Osgood-type function \((^{6,7})\), then there exists an interval \([s,b)\subset(a,b)\) on which (3) has at least one solution \(x(s)=x_0\). If \(f_1\equiv0\), this solution is unique.

In the proofs the following lemma was used.

Lemma 1. Let \(C\) be the space of continuous functions on \([0,T]\) with values in \(X\), and let \(F\) be a mapping of a bounded closed set \(M\subset C\) into itself, satisfying, for \(u,v\in M\), the condition

\[ \|Fu(t)-Fv(t)\|\leq \int_0^t K(\tau)\omega(\|u(\tau)-v(\tau)\|)\,d\tau, \]

where \(K(t)\) is summable on \([0,T]\), and \(\omega(z)\) is an Osgood-type function. Then the equation \(u=Fu\) has a solution in \(M\), and moreover a unique one**.

Theorem 4. Under the hypotheses of Theorem 3, if \(f(t,u)\) satisfies on \((a,b)\times X\) the condition
\[ \|f(t,x)\|\leq \overline K(t)\overline\omega(\|x\|), \]
where \(\overline K(t)\) is summable on every interval \([a_1,b_1]\subset(a,b)\) and \(\overline\omega(z)\) is a Wintner-type function, then the solution of equation (3) can be continued to all of \([c,b)\) (\(b\) may be infinite).

We note that Theorems 3 and 4 also hold when \(T(t,s)\) satisfies conditions 1) and 2) of Corollary 2, and therefore in this case they also include the generalized solutions from \((^5)\).

For simplicity suppose that in condition C II \(\gamma(t)=\gamma\). Then:

Corollary 3. If, under the hypotheses of Theorem 4, \(\overline K(t)\) is summable on \((a,b)\), then:

1) If \(b<\infty\), the solution is uniformly bounded on \([c,b]\).
2) If \(b=\infty\) and \(\gamma\leq0\), the solution is uniformly bounded on \([c,\infty)\).
3) If \(b=\infty\) and \(\gamma<0\), the solution tends to zero as \(t\to\infty\) (whatever the initial conditions may be).

Corollary 4. 1) If, under the hypotheses of Theorem 4, \(f_1\equiv0\), then the Cauchy problem is well posed.

2) If, in addition, \(K(t)\) is summable on all of \((a,b)\), then the generalized Cauchy problem is well posed uniformly with respect to \(c\in[a_1,b_1]\subset(a,b)\).

3) If \(b=\infty\) and \(\gamma\leq0\), the solution of the generalized Cauchy problem is Lyapunov stable.

* This does not always hold under Kato’s conditions \((^5)\).

** This lemma makes it possible to extend Krasnosel’skii’s fixed-point theorem \((^8)\) and is proved with the aid of Lemma I. 1 from \((^6)\); the latter, in a more particular form, was also obtained by Krasnosel’skii \((^9)\).

1. Suppose that \(A(t)=A\).

Theorem 5. Let the hypotheses of Theorem 3 and the following conditions be satisfied:

1) For \(u\in D_A\), \(f(t,u)\in D_A\).
2) \(Af(t,u(t))\) is continuous for every continuous \(u(t)\in D_A\).
3) \(\|Af(t,u)-Af(t,v)\|\leq K_1(t)\omega_1(\|Au-Av\|)\), where \(u,v\in D_A\), \(K_1(t)\) is summable, and \(\omega_1(z)\) is a function of Osgood type.

Then every solution of the generalized Cauchy problem for (2) is a solution in the usual sense.

With slight changes the theorem remains valid also if \(A(t)\) depends on \(t\), but the Kato conditions \((^1)\) are satisfied; however, as is easy to see \((^5)\), such a theorem is ineffective.

Theorem 6. Let, under the hypotheses of Theorem 2:

1) \(f_t(t,u)\) exist and be continuous.
2) \(f(t,u)\) have a continuous Fréchet differential \(f_u(t,u)\).

Then the Cauchy problem for (2) has a unique solution on a sufficiently small interval for initial conditions from \(X\).

This theorem is an improvement of the theorem from \((^5)\) in the case where \(A(t)\) does not depend on \(t\). Theorems 5 and 6 also hold in the case where \(T(t)\) is a strongly continuous subgroup and \(T(t)\to I\), consequently, if \(A\) satisfies the general condition of Miyadera–Phillips \((^{10})\).

Received
7 X 1957

REFERENCES

\(^1\) T. Kato, J. Math. Soc. Japan, 5, No. 2 (1953).
\(^2\) S. L. Sobolev, Some Applications of Functional Analysis in Mathematical Physics, L., 1950.
\(^3\) S. Foiąş, G. Gussi, V. Poenaru, Trans. Am. Math. Soc. 89, No. 2 (1957).
\(^4\) I. G. Petrovskii, Bull. Moscow State Univ., Sect. A, 1, No. 7 (1938).
\(^5\) M. A. Krasnosel’skii, S. P. Krein, P. E. Sobolevskii, DAN, 111, No. 1 (1956).
\(^6\) S. Foiąş, G. Gussi, V. Poenaru, Math. Nachr., 15, H. 2 (1956).
\(^7\) G. Gussi, V. Poenaru, K. Foiąş, DAN, 112, No. 3 (1957).
\(^8\) M. A. Krasnosel’skii, Uspekhi Mat. Nauk, 10, issue 1 (63) (1955).
\(^9\) M. A. Krasnosel’skii, Proc. Seminar on Functional Analysis, Voronezh State Univ., 1957.
\(^ {10}\) I. Miyadera, Tôhoku Math. J., (2), 4 (1952).