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 \(h<h_0\). Then, by Lemma 1, the choice \(h_2>h\), 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
\[ \tilde{U}_{[t_0-h,t_0+h]}\subset \tilde{E}_{[t_0-h,t_0+h]}. \]
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
\[ A(\widetilde{x})=x_0+\int_{t_0}^{t} f(x(\tau),\tau)\,d\tau . \]
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

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