V. M. MILLIONSHCHIKOV

ON THE THEORY OF DIFFERENTIAL EQUATIONS

(\dfrac{dx}{dt}=f(x,t)) IN LOCALLY CONVEX SPACES

(Presented by Academician S. L. Sobolev on 23 XI 1959)

In this paper, some theorems known for the equation (dx/dt=f(x,t)) in Banach spaces ((^{1,12})) are generalized to the case of complete locally convex spaces.

Let (E) be a complete locally convex space and let ({p(x)}) be a sufficient set of seminorms in (E) ((^2)). Let (\mathfrak{F}(S,E)) be the set of mappings of a measurable set (S\subseteq E^n) ((\operatorname{mes} S<\infty)) into (E) ((E^n) is an (n)-dimensional Euclidean space).

A function (x(\alpha)\in\mathfrak{F}(S,E)) is called integrable on (S) if there exists a directed sequence of functions (x_\beta(\alpha)\in\mathfrak{F}(S,E)) (where (\beta\in B); (B) is a directed set) such that:

1) each (x_\beta(\alpha)) is a countably-valued function ((^3));

2) for each seminorm (p(x)\in{p(x)}),
[
I_S(x_\beta(\alpha))=\sum_{i=1}^{\infty} p\bigl[x_\beta(\alpha_{i\beta})\bigr]\operatorname{mes} S_{i,\beta}<\infty,
]
where (S_{i,\beta}) ((i=1,\ldots,n,\ldots)) are sets on which (x_\beta(\alpha)) is constant, and (\alpha_{i\beta}\in S_{i,\beta});

3) for every (\varepsilon>0) and (p(x)\in{p(x)}) there exists (\beta\in B) such that
[
I_S(x_{\beta'}(\alpha)-x_{\beta''}(\alpha))<\varepsilon
]
for all (\beta',\beta''>\beta); (\beta',\beta''\in B);

4) for every (\varepsilon>0) there exists (S_\varepsilon\subset S), (\operatorname{mes} S_\varepsilon<\varepsilon), such that ({x_\beta(\alpha)}) converges uniformly on (S\setminus S_\varepsilon) to (x(\alpha)).

Then
[
I=\lim_{\beta\in B}\sum_{i=1}^{\infty}x_\beta(\alpha_{i\beta})\operatorname{mes} S_{i,\beta}
]
exists and does not depend on the choice of ({x_\beta(\alpha)}) with properties 1)—4).

We call (I) the integral
[
\int_S x(\alpha)\,d\alpha
]
(a more general integral was introduced in ((^4))).

The integral introduced has the following properties:

(1^\circ.) (I) is a completely additive set function.

(2^\circ.) (I) is an absolutely continuous set function.

(3^\circ.) If (x(\alpha)) and (y(\alpha)) are integrable, then
[
\int_S [\mu x(\alpha)+\nu y(\alpha)]\,d\alpha
=
\mu\int_S x(\alpha)\,d\alpha+\nu\int_S y(\alpha)\,d\alpha .
]

(4^\circ.) If (x(\alpha)) is integrable, then (p(x(\alpha))) is summable and
[
p\left(\int_S x(\alpha)\,d\alpha\right)\leq
\int_S p(x(\alpha))\,d\alpha .
]

(5^\circ.) Let (x(\alpha)) be defined on a compact set (S\subset E^n), and let the set (\mathfrak{X}) of its values be bounded ((^5)). Suppose that for every (\varepsilon>0) there exists (S_\varepsilon\subset S),

(\operatorname{mes} S_\varepsilon<\varepsilon), such that (x(\alpha)) is continuous on (S\setminus S_\varepsilon). Then (x(\alpha)) is integrable on (S) and (\int_S x(\alpha)\,d\alpha\in \operatorname{mes} S\cdot[\mathfrak X]), where ([\mathfrak X]) is the closed convex hull of the set (\mathfrak X).

(6^\circ). If (x(\alpha)) is integrable on (S) and continuous at (\alpha_0\in S) (with respect to (S)), then
[
\lim_{d(Q)\to 0,\ \alpha_0\in Q}\frac{1}{\operatorname{mes} Q}\int_Q x(\alpha)\,d\alpha
]
exists and is equal to (x(\alpha_0)).

(7^\circ). The system
[
x(t_0)=x_0,\qquad \frac{dx}{dt}=f(x,t),\qquad x(t)\in M\subseteq E\quad \text{for } |t-t_0|\le a,
]
where (f(x,t)) maps (M\times E^1) continuously into (E), is equivalent to the equation
[
x(t)=x_0+\int_{t_0}^{t} f(x(\tau),\tau)\,d\tau.
]

Definition. An operator (A) ((A(E)\subseteq E)) is called contracting if there exists (0\le q<1) such that for all (p(x)\in{p(x)}), (x,y\in E),
[
p(A(x)-A(y))\le q p(x-y).
\tag{1}
]

Theorem 1. Let (A) be a contracting operator, (\varnothing\ne T=\overline T\subseteq E), (A(T)\subseteq T).

Then there exists, and moreover is unique in (E), an (x\in T) such that (A(x)=x).

Theorem 2 ((^6)). Let (T) be a closed convex set (\subseteq E). Suppose operators (A_i) ((A_i(E)\subseteq E,\ i=1,2)) are given on (T), with: 1) (A_1) a contracting operator; 2) (A_2(T)) bicompact; 3) (A_2) continuous; 4) if (x,y\in T), then (A_1(x)+A_2(y)\in T).

Then there exists (x\in T) such that (A_1(x)+A_2(x)=x).

Proof. Let (x\in T). Then (conditions 1), 4) and Theorem 1) there exists, and moreover is unique, (y\in T) such that (y=A_1(y)+A_2(x)), i.e., an operator (y=C(x)) is defined on (T), for which
[
C(x)=A_1C(x)+A_2(x),\qquad C(T)\subseteq T.
\tag{2}
]
Let (z,v\in T). Then from (2) and condition 1) we obtain
[
p(C(z)-C(v))\le \frac{1}{1-q}p(A_2(z)-A_2(v))
]
for every seminorm (p(x)\in{p(x)}). Hence, by conditions 2), 3), (C) is continuous and (\overline{C(T)}) is bicompact (the latter with the aid of ((^8))). Considering (C) only on the closed convex hull of the set (\overline{C(T)}) ((^7)) and applying Tikhonov’s principle ((^9)), we obtain the assertion of the theorem.

Let (\widetilde E_S) be the space of uniform convergence on compacta of continuous mappings of the set (S\subseteq E^1) into (E). Then (\widetilde E_S) is a complete locally convex space with a sufficient set of seminorms
[
{p_{p,B}(\tilde x)=\sup_{t\in B}p(x(t))},\qquad [\tilde x=x(t)\in \widetilde E_S],
]
where (p(x)) ranges over ({p(x)}), and (B) ranges over some covering of (S) by compacta. (\widetilde M_S) denotes the set of mappings of (S) into (M\subseteq E).

Lemma 1. Let (f(x,t)) map (M\times S) continuously into (E), where (M\subseteq E), (S=[t_0-h,t_0+h]\subset E^1). Let (K(t)\ge 0) be a real function such that
[
(L)\left|\int_{t_0}^{t}K(\tau)\,d\tau\right|\le q<1\quad (t\in S)
]
and for all (x,y\in M), (p(x)\in{p(x)}),
[
p(f(x,t)-f(y,t))\le K(t)p(x-y).
]

Then

[
A(\tilde{x})=x_0+\int_{t_0}^{t} f(x(\tau),\tau)\,d\tau
]

is a contraction operator defined on (\tilde{M}_S).

The lemma follows from properties (5^\circ, 2^\circ, 4^\circ) of the integral.

Theorem 3. Let (f(x,t)) continuously map (U\times [t_0-a,t_0+a]) into (E), where
[
U=U\bigl(x:p_i(x-x_0)\le \varepsilon\bigr), \quad p_i(x)\in{p(x)}\quad (i=1,\ldots,n),
]
and suppose
[
\sup_{\substack{x\in U;\ |t-t_0|\le a}} p_i(f(x,t))<\infty \quad (i=1,\ldots,n).
]
Let (K(t)\ge 0) be summable on ([t_0-a,t_0+a]), and for all (x,y\in U), (p(x)\in{p(x)}),

[
p(f(x,t)-f(y,t))\le K(t)p(x-y).
]

Then there exists (h_0), (0<h_0\le a), such that for every (h), (0<h\le h_0), there exists, and moreover is unique, a solution (x(t)) of the initial-value problem
[
dx/dt=f(x,t),\qquad x(t_0)=x_0,
]
defined on ([t_0-h,t_0+h]).

Proof. Put (h_0=\min(h_1,h_2)), where (h_1) is such that

[
\max\left(\int_{t_0}^{t_0+h_1}K(\tau)\,d\tau,\ \int_{t_0-h_1}^{t_0}K(\tau)\,d\tau\right)\le q<1,
\qquad
h_2=\frac{\varepsilon}{\displaystyle \sup_{\substack{x\in U;\ |t-t_0|\le a\ i=1,\ldots,n}} p_i(f(x,t))}.
]

Let (hh), and Theorem 1, the operator

[
A(\tilde{x})=x_0+\int_{t_0}^{t} f(x(\tau),\tau)\,d\tau
]

has a unique fixed point in the closed set
[
\tilde{U}{[t_0-h,t_0+h]}\subset \tilde{E}.
]
Since for every solution (x(t)) of the system (dx/dt=f(x,t)), (x(t_0)=x_0), there exists (\delta>0) such that (x(t)\in U) for (|t-t_0|<\delta), the theorem is proved.

Remark. If in the hypothesis of Theorem 3 one replaces (U) by all of (E) and

[
\int_{-\infty}^{+\infty}K(\tau)\,d\tau<1,
]

then there exists a solution defined on the entire line.

Lemma 2. Let (f(x,t)) continuously map (M\times S) into (E), where (M\subseteq E), (S\subseteq E^1).

Then the operator (F(\tilde{x})=f(x(t),t)): 1) is defined on (\tilde{M}_S) and takes values in (\tilde{E}_S); 2) is continuous.

Proof. 1) See ((10)). The idea of the proof of 2) is that a continuous operator (f(x,t)) is uniformly continuous with respect to each bicompact set (\mathfrak{X}\times B), where (B\subseteq S) is an arbitrary compact set, and (\mathfrak{X}) is the set of values on it of an arbitrary (\tilde{x}\in\tilde{E}_S).

Lemma 3. Let (f(x,t)) continuously map (M\times S) into (E) ((M\subseteq E), (S\subseteq E^1)), and suppose that for each compact (B\subseteq S) there exists a bicompact (F_B\subseteq E) such that
[
f(M\times B)\subseteq F_B.
]

Then the operator

[
A(\tilde{x})=\int_{t_0}^{t} f(x(\tau),\tau)\,d\tau:
]

1) is continuous on (\tilde{M}_S) and takes values in (\tilde{E}_S); 2) (A(\tilde{M}_S)) is bicompact.

Proof. 1) follows from Lemma 2 and property (4^\circ) of the integral. 2) follows from property (5^\circ) of the integral and Ascoli’s theorem ((11)).

Theorem 4. Let (f(x,t)=f_1(x,t)+f_2(x,t)), where (f_1(x,t)), (f_2(x,t)) continuously map (U\times [t_0-a,t_0+a]) into (E)
[
\bigl(U=U(x:p_i(x-x_0)\le \varepsilon,\ p_i(x)\in{p(x)},\ i=1,\ldots,n)\bigr).
]

Suppose (\overline{f_2(U\times [t_0-a,t_0+a])}) is bicompact, and (f_1(x,t)) satisfies the hypotheses of Theorem 3.

Then there exists (h>0) such that there is a solution of the initial-value problem
[
x(t_0)=x_0,\qquad dx/dt=f(x,t),
]
defined on ([t_0-a,t_0+a]).

Theorem 5({}^{(12)}). Let (f(x,t)) satisfy the conditions of Lemma 3, where (M=E), (S=[t_0,+\infty)), and let there exist (p_0(x)\in{p(x)}) and a continuous function (G(r,t)), nondecreasing in (r) ((t\geqslant 0,\ r\geqslant 0)), such that for (x\in E,\ t\in S)

[
p_0(f(x,t))\leqslant G(p_0(x),t).
\tag{3}
]

Suppose that for every (r_0>0) there exists a function (g(t)), defined on (S), such that

[
\frac{dg}{dt}\geqslant G(g(t),t),\qquad g(t_0)=r_0 .
\tag{4}
]

Then for every (x_0\in E) there exists a solution of the initial-value problem
[
dx/dt=f(x,t),\qquad x(t_0)=x_0,
]
defined on all of (S).

Proof. On the closed convex set
[
T=T(\widetilde{x}=x(t):\ p_0(x(t))\leqslant g(t))\quad (T\subset \widetilde{E}S)
]
define the operator
[
A(\widetilde{x})=x_0+\int f(x(\tau),\tau)\,d\tau .}^{t
]
By Lemma 3, (\overline{A(T)}) is bicompact (\subset \widetilde{E}_S), and (A) is continuous on (T). From (3) and (4) we infer (A(T)\subseteq T). Applying Tikhonov’s principle, we complete the proof.

Theorems 6 and 7({}^{(12)}) are proved analogously.

Theorem 6. Let the conditions of Theorem 5 be fulfilled for every (p_0(x)\in{p(x)}), and let (g(t)) be bounded.

Then the solution (x(t)) of the initial-value problem is bounded (i.e. the set of values of the function (x(t)) is bounded({}^{(5)})).

Theorem 7. Let the conditions of Theorem 5 be fulfilled for every (p_0(x)\in{p(x)}), where (3) may be fulfilled only for (x) in some neighborhood of zero, and in (4) there is strict equality. Let (f(0,t)=0), (G(0,t)=0), and let the point (g=0) for the equation (dg/dt=G(g(t),t)) be stable (asymptotically stable).

Then (x=0) is a stable (respectively, asymptotically stable) point for the equation (dx/dt=f(x,t)).

I express my gratitude to V. V. Nemytskii for posing the problem and for his guidance.

Moscow State University
named after M. V. Lomonosov

Received
20 XI 1959

CITED LITERATURE

({}^{1}) M. A. Krasnosel’skii, S. G. Krein, Proceedings of the Seminar on Functional Analysis, Voronezh State Univ., no. 2 (1956).
({}^{2}) M. A. Naimark, Normed Rings, Moscow, 1956, pp. 46–52.
({}^{3}) E. Hille, Functional Analysis and Semigroups, IL, 1951, p. 54, Definition 3.2.2 (3).
({}^{4}) R. S. Phillips, Trans. Am. Math. Soc., 47, no. 1, 114 (1940).
({}^{5}) I. M. Gel’fand, G. E. Shilov, Generalized Functions, vol. 2, Spaces of Basic and Generalized Functions, Moscow, 1958, p. 45.
({}^{6}) M. A. Krasnosel’skii, Uspekhi Mat. Nauk, 10, no. 1 (63), 123 (1955).
({}^{7}) N. Bourbaki, Vector Topological Spaces, IL, 1959, Ch. II, § 4, item 1.
({}^{8}) N. Bourbaki, General Topology, Moscow, 1958, Ch. II, § 4, Theorem 3.
({}^{9}) A. Tychonoff, Math. Ann., 111, 767 (1935).
({}^{10}) N. Bourbaki, General Topology, Moscow, 1958, Ch. I, § 4, Theorem 1.
({}^{11}) N. Bourbaki, Topologie générale, 1953, Ch. X, § 4, théorème 1.
({}^{12}) A. Stokes, Proc. Nat. Acad. Sci. USA, 45, no. 2, 231 (1959).