UDC 917.911

ON UNIQUENESS THEOREMS FOR ORDINARY DIFFERENTIAL EQUATIONS

P. P. ZABREIKO

In recent years various authors (see, for example, [1—10]) have proposed numerous conditions for the uniqueness of the solution of the Cauchy problem

\[ \frac{d x}{d t}=f(t,x),\quad x(t_0)=x_0 \]

both for a scalar differential equation and for equations in Banach spaces.

In the present article a unified scheme is proposed for proving uniqueness theorems (for the case of a scalar equation). This scheme makes it possible not only to prove previously known uniqueness assertions by a uniform method, but also to establish new ones.

1. We first present auxiliary assertions on differential and integral inequalities.

Let the function \(\Phi(t,z)\) be defined for \(0<t\leq a,\ |z|\leq b\), satisfy the Carathéodory conditions [11] and the inequality

\[ |\Phi(t,z)|\leq m(t), \]

where \(m(t)\) is a function summable on each interval \([\varepsilon,a]\) \((0<\varepsilon<a)\). Let \(\omega(t)\) \((0<t\leq a)\) be some continuous function, and suppose that the ordinary differential equation

\[ \frac{d z}{d t}=\Phi(t,z) \tag{1} \]

has at least one solution \(z_0(t)\), defined on \((0,a]\), satisfying the conditions

\[ \lim_{t\to 0} z_0(t)=0,\quad z_0(t)\leq \omega(t). \]

Introduce for consideration the function

\[ \omega^*(t_0)=\sup z, \tag{2} \]

where the supremum is taken over all numbers \(z\in[-b,\omega(t_0)]\) such that equation (1) has at least one solution \(z(t)\) satisfying the conditions \(z(0)=0,\ z(t_0)=z\) and the inequality

\[ z(t)\leq \omega(t)\quad (0<t<t_0). \]

It is clear that the function \(\omega^*(t)\) is defined for all \(t\in(0,a]\) and satisfies the inequalities

\[ z_0(t)\leq \omega^*(t)\leq \omega(t)\quad (0<t\leq a). \]

We shall call the function \(\omega^*(t)\) the \(\omega\)-separator of equation (1).

Let \(t_0 \in (0,a]\). Consider the set \(\mathfrak M_{t_0}\) of all solutions \(z(t)\) of equation (1) satisfying the conditions

\[ z_0(t) \leq z(t) \leq w(t) \quad (0<t<t_0), \qquad z(0)=0,\qquad z(t_0)=w^*(t_0). \tag{3} \]

This set is, obviously, nonempty and compact in the uniform metric on each interval \([\varepsilon,a]\) \((0<\varepsilon<a)\). Hence the exact upper bound \(z_{t_0}(t)\) \((0<t<t_0)\) of this set on \((0,t_0)\) is a solution of equation (1); this solution satisfies conditions (3). Extend the function \(z_{t_0}(t)\) to values \(t>t_0\), taking it equal to the upper solution of equation (1) passing through the point \(\{t_0,w^*(t_0)\}\).

The function \(z_{t_0}(t)\), obviously, has the following properties:

a) the inequalities

\[ z_{t_0}(t)\leq w^*(t) \quad (0<t\leq t_0), \qquad z_{t_0}(t)\geq w^*(t) \quad (t_0\leq t); \]

b) if \(w^*(t_0)<w(t_0)\), then \(z_{t_0}(t)=w^*(t)\) on every interval \((\alpha,\beta)\subset[0,a]\) containing the point \(t_0\), on which the inequality \(w^*(t)<w(t)\) holds.

From these properties it follows immediately that the function \(w^*(t)\) is a solution of equation (1) on every interval \((\alpha,\beta)\subset[0,a]\) where the inequality \(w^*(t)<w(t)\) holds. In particular, the function \(w^*(t)\) is continuous at the point \(t_0\) if \(w^*(t_0)<w(t_0)\). Obviously, the function \(w^*(t)\) is also continuous at those points at which \(w^*(t_0)=w(t_0)\).

We give examples of the construction of separating functions.

Theorem 1. Let

\[ \Phi(t,z)=c(t)\omega(z), \]

where \(c(t)\) is a function summable on each interval \([\varepsilon,a]\) \((0<\varepsilon<a)\), and \(\omega(z)\) \((\omega(0)=0)\) is a continuous function positive for \(z>0\). Suppose that one of the following conditions is satisfied: a) the function \(c(t)\) is nonnegative; b) the function \(\omega(u)\) is of Osgood type

\[ \int_0^{c_0}\frac{du}{\omega(u)}=\infty. \]

Let \(w(t)\) \((0<t\leq a)\) be a nonnegative function. Then the \(w\)-separating function \(w^*(t)\) of equation (1) is determined by the formula

\[ w^*(t)=\inf_{0<\tau<t} L^{-1}\left\{L[w(\tau)]-\int_\tau^t c(s)\,ds\right\}, \tag{4} \]

where

\[ L(z)=\int_z^{c_0}\frac{du}{\omega(u)}. \]

Proof. Under the conditions of the theorem, equation (1) has the solution \(z_0(t)\equiv 0\), satisfying the conditions \(z_0(0)=0,\ z_0(t)\leq w(t)\). Therefore, for equation (1) there exists a \(w\)-separating function.

Denote now the right-hand side of equality (4) by \(\overline w(t)\) and show that \(w^*(t)=\overline w(t)\).

Let \(z(t)\) be a solution of equation (1) satisfying the conditions

\[ z(0)=0,\qquad z(t_0)=c,\qquad z(t)\leq w(t) \quad (0<t<t_0). \]

where \(c>0\). Without loss of generality one may assume that \(z(t)=0\) for \(t\leq t_1\) and \(z(t)>0\) for \(t>t_1\), where \(t_1\) is some number in \([0,t_0)\); moreover, if condition b) is satisfied, the number \(t_1\) is equal to zero. On the interval \([t_1,t_0]\), clearly, the equality holds

\[ L[z(t)]-L(c)=\int_t^{t_0} c(s)\,ds, \]

and on the interval \([0,t_1]\) the inequality

\[ L[z(t)]-L(c)\leq \int_t^{t_0} c(s)\,ds. \]

From these relations it follows that

\[ c\leq \inf_{0<\tau<t_0} L^{-1}\left\{L[w(t)]-\int_t^{t_0} c(s)\,ds\right\}=\overline w(t_0) \tag{5} \]

and, consequently, \(w^*(t_0)\leq \overline w(t_0)\).

Conversely, let a number \(c>0\) satisfy inequality (5). In this case the solution of equation (1), determined from the equality

\[ \int_{z(t)}^c \frac{du}{\omega(u)}=\int_t^{t_0} c(s)\,ds, \]

satisfies, evidently, the inequality \(z(t)\leq w(t)\) (for those values of \(t\) for which \(z(t)>0\)). Continuing this solution \(z(t)\) to the whole interval \([0,t_0]\) by zero, we obtain a solution of equation (1) satisfying the conditions \(z(0)=0\), \(z(t_0)=c\), \(0\leq z(t)\leq w(t)\). Therefore \(c\leq w^*(t_0)\), and hence \(w^*(t_0)\geq \overline w(t_0)\).

Thus, \(w^*(t_0)=\overline w(t_0)\). Since \(t_0\) is an arbitrary number from \((0,a]\), it follows that

\[ w^*(t)=\overline w(t)\qquad (0<t\leq a). \]

The theorem is proved.

In particular, from Theorem 1 it follows that

\[ w^*(t)= \begin{cases} \displaystyle \inf_{0<\tau<t} w(\tau)e^{\int_\tau^t c(s)\,ds}, & (k=1),\\[1.2em] \displaystyle \inf_{0<\tau<t}\left\{w(\tau)^{1-k}+(1-k)\int_\tau^t c(s)\,ds\right\}^{\frac{1}{1-k}}, & (k\neq 1), \end{cases} \tag{6} \]

if \(\Phi(t,z)=c(t)z^k\).

By means of a simple change of variable, Theorem 1 implies

Theorem 2. Let

\[ \Phi(t,z)=k(t)z+r(t)\omega[\lambda(t)z], \]

where \(k(t)\) and \(r(t)\) are functions summable on every interval \([\varepsilon,a]\) \((0<\varepsilon<a)\);

\[ \lambda(t)=e^{\int_t^a k(s)\,ds}; \]

\(\omega(z)\) \((\omega(0)=0)\) is a continuous function positive for \(z>0\). Suppose that one of the following conditions is satisfied: a) the function \(r(t)\) is nonnegative; b) the function \(\omega(u)\) is of Osgood type

\[ \int_0^{c_0}\frac{du}{\omega(u)}=\infty. \]

Let \(w(t)\) \((0<t\leq a)\) be a continuous nonnegative function. Then the \(w\)-separatrix \(w^*(t)\) of equation (1) is determined by the formula

\[ w^*(t)=\frac{1}{\lambda(t)}\inf_{0<\tau<t} L^{-1}\left\{L[\lambda(\tau)w(\tau)]-\int_\tau^t \lambda(s)r(s)\,ds\right\}, \tag{7} \]

where

\[ L(z)=\int_z^{c_0}\frac{du}{\omega(u)}. \]

Let us return to the general equation of the form (1).

Theorem 3. Let a function \(\varphi(t)\), defined on \([0,a]\) and absolutely continuous, \(\varphi(0)=0\), satisfy the inequality \(\varphi(t)\leq w(t)\) and, for almost all \(t\in(0,a]\), the inequality

\[ \varphi'(t)\leq \Phi[t,\varphi(t)]. \tag{8} \]

Then \(\varphi(t)\leq w^*(t)\).

Proof. Suppose that at some point \(t_0\in(0,a]\) the inequality \(\varphi(t_0)>w^*(t_0)\) holds. Consider, to the left of \(t_0\), the solution \(z^*(t)\) of equation (1) passing through the point \(\{t_0,\varphi(t_0)\}\). This solution, obviously, for all \(t\) for which it is defined, will satisfy the inequality \(z^*(t)>w^*(t)\). On the other hand, by the conditions of the theorem and the usual theorem on differential inequalities, it passes no higher than \(\varphi(t)\), i.e. \(z^*(t)\leq \varphi(t)\). Therefore it can be continued to all values \(t>0\). Since \(\varphi(0)=0\), we have \(z^*(0)=0\). Thus equation (1) has a solution \(z^*(t)\) satisfying the conditions

\[ z^*(0)=0,\quad z^*(t_0)=\varphi(t_0)>w^*(t_0),\quad z^*(t)\leq \varphi(t)\leq w(t)\quad (0<t<t_0). \]

This contradicts the definition of the function \(w^*(t)\).

The theorem is proved.

The assumption of absolute continuity of the function \(\varphi(t)\) may be replaced by the assumption of its continuity, if it is required that inequality (8) (or one of the more general inequalities

\[ D_*\varphi(t)\leq \Phi[t,\varphi(t)],\quad {}_*D\varphi(t)\leq \Phi[t,\varphi(t)]) \]

hold for \(t\in(0,a]\), except for a countable set of values of \(t\).

From Theorem 3, by a known scheme, there follows

Theorem 4. Let the function \(\Phi(t,z)\), for all \(t\in(0,a]\), be a nondecreasing function of \(z\), and let the function \(\Phi[t,w(t)]\) be summable on \([0,a]\). Let the continuous function \(\varphi(t)\), defined for \(0\leq t\leq a\), satisfy the inequality \(\varphi(t)\leq w(t)\) and the inequality

\[ \varphi(t)\leq \int_0^t \Phi[s,\varphi(s)]\,ds. \tag{9} \]

Then \(\varphi(t)\leq v^*(t)\), where \(v^*(t)\) is the \(v\)-separatrix of equation (1),

\[ v(t)=\int_0^t \Phi[s,w(s)]\,ds. \]

Combining the conditions of Theorems 1 and 2 with the conditions of Theorems 3 and 4, one can obtain various assertions on estimates of functions satisfying differential or integral inequalities. In particular, the lemma on integral inequalities from [3] can thus be obtained.

1. We apply the theorems established in the preceding section to the study of uniqueness of solutions of the Cauchy problem for an ordinary differential equation. We shall restrict ourselves to the case of the scalar Cauchy problem

\[ \frac{dx}{dt}=f(t,x), \qquad x(0)=0 . \tag{10} \]

Here \(f(t,u)\) is a function defined for \(0\leq t\leq a\), \(|x|\leq b\), and satisfying the Carathéodory conditions.

Theorem 5. Let the functions \(\Phi_i(t,z)\) \((i=1,\ldots,n)\) be defined for \(0<t\leq a\), \(|z|\leq b\), satisfy the Carathéodory conditions and the inequalities \(|\Phi_i(t,z)|\leq m(t)\), where \(m(t)\) is a function summable on every interval \([\varepsilon,a]\) \((0<\varepsilon<a)\), and let \(\Phi_i(t,0)=0\). Let \(w_0(t)=b\), and let \(w_i(t)\) \((i=1,\ldots,n)\) be \(w_{i-1}\)-determining for the equation

\[ \frac{dz}{dt}=\Phi_i(t,z) \quad (i=1,\ldots,n), \tag{11} \]

with \(w_n(t)=0\). Let the function \(f(t,u)\) satisfy the inequalities

\[ f(t,u)-f(t,v)\leq \Phi_i(t,u-v) \quad (0<t\leq a,\; 0\leq u-v\leq w_{i-1}(t)). \tag{12} \]

Then the Cauchy problem (10) has no more than one solution.

The proof is obvious.

Theorem 5 is readily generalized to the case where a countable set of estimates of type (12) is known. For \(n=2\) a close assertion was proved in [10].

We give some particular uniqueness assertions that follow from Theorem 5 (and Theorems 1 and 2).

Let \(c(t)\) be a nonnegative summable function, let \(k_0\) be some number in \([0,1)\), and

\[ w(t)=\left\{(1-k_0)\int_0^t c(s)\,ds\right\}^{\frac{1}{1-k_0}} . \tag{13} \]

Below we consider functions \(f(t,u)\) satisfying either condition (A):

\[ f(t,u)-f(t,v)\leq c(t)(u-v)^{k_0} \quad (0\leq t\leq a,\; 0\leq u-v\leq b), \tag{14} \]

or condition (B):

\[ f(t,u)-f(t,v)\leq \varepsilon(t_0,r_0)c(t)(u-v)^{k_0} \]

\[ (0\leq t\leq t_0\leq a,\quad 0\leq u-v\leq r_0\leq b), \tag{15} \]

where \(\lim_{t_0,r_0\to 0}\varepsilon(t_0,r_0)=0\). We note that neither of these conditions guarantees uniqueness for solutions of the Cauchy problem (10).

Theorem 6. Let the function \(f(t,u)\) satisfy the inequality

\[ f(t,u)-f(t,v)\leq k(t)(u-v) \quad (0<t\leq a,\; 0\leq u-v\leq w(t)), \tag{16} \]

where \(k(t)\) is a function summable on every interval \([\varepsilon,a]\) \((0<\varepsilon<a)\). Suppose that either condition (A) is fulfilled and

\[ \lim_{\tau\to 0}\left\{\int_{\tau}^{a} k(s)\,ds +\frac{1}{1-k_0}\ln\int_0^{\tau} c(s)\,ds\right\}=-\infty, \tag{17} \]

or condition (B) and

\[ \lim_{\tau\to 0}\left\{\int_\tau^a k(s)\,ds+\frac{1}{1-k_0}\ln\int_0^\tau c(s)\,ds\right\}<\infty . \tag{18} \]

Then the Cauchy problem (10) has at most one solution.

The assertion of this theorem follows from Theorem 5 and from (6). Theorem 5 contains the theorem of M. A. Krasnosel’skii and S. G. Krein, the Nagumo–Rosenblatt–Perron theorem, the Rosenblatt–Scorza–Dragoni theorem, and others (see, for example, [2, 3, 9]).

Theorem 7. Suppose that the function \(f(t,u)\) satisfies the inequality

\[ f(t,u)-f(t,v)\leq k(t)(u-v)+r(t)\omega[\lambda(t)(u-v)] \]
\[ (0<t\leq a,\quad 0\leq u-v\leq w(t)), \tag{19} \]

where \(k(t)\) and \(r(t)\) are functions summable on every interval \([\varepsilon,a]\) \((0<\varepsilon<a)\),

\[ \lambda(t)=e^{\int_t^a k(s)\,ds}, \]

and \(\omega(z)\) \((\omega(0)=0)\) is a continuous positive function satisfying the Osgood condition \(\lim_{z\to 0}L(z)=\infty\); here

\[ L(z)=\int_z^{c_0}\frac{du}{\omega(u)} . \]

Suppose that either condition (A) holds and

\[ \overline{\lim_{\tau\to 0}}\left\{L[\lambda(\tau)w(\tau)]-\int_\tau^a \lambda(s)r(s)\,ds\right\}=\infty, \tag{20} \]

or condition (B) holds and

\[ \overline{\lim_{\tau\to 0}}\left\{L[\varepsilon(\tau)\lambda(\tau)w(\tau)]-\int_\tau^a \lambda(s)r(s)\,ds\right\}=\infty \tag{21} \]

for every function \(\varepsilon(\tau)\), positive for \(\tau>0\), satisfying the condition \(\lim_{\tau\to 0}\varepsilon(\tau)=0\). Then the Cauchy problem (10) has at most one solution.

Theorem 7 generalizes the Scorza-Dragoni theorem (see, for example, [2]), the theorem of A. I. Perov [5, 9], and others.

Theorems 5–7 are carried over, by the well-known scheme (see, for example, [8]), to equations in Banach spaces.

The work was carried out at the Voronezh seminar on differential equations, directed by M. A. Krasnosel’skii. The author expresses sincere gratitude to M. A. Krasnosel’skii, A. I. Perov, A. V. Kibenko, and B. N. Sadovskii for discussion of the article and valuable advice.

References

1. I. G. Petrovskii, Lectures on the Theory of Ordinary Differential Equations. Nauka, 1964.

2. J. Sansone, Ordinary Differential Equations. IL, 1954.

3. M. A. Krasnosel’skii and S. G. Krein, On a class of uniqueness theorems. UMN, 11, no. 1, 1956, pp. 209–213.

4. M. A. Krasnosel’skii and S. G. Krein, On the theory of ordinary differential equations in Banach spaces. Proceedings of the Seminar on Functional Analysis, vol. 2, Vo-

Voronezh, 1956, pp. 3–23.

1. Perov A. I. DAN SSSR, 130, No. 4, 704–707, 1958.
2. Coddington E. A., Levinson N. Theory of Ordinary Differential Equations. IL, 1959.
3. Borůvka O. Acta Fac. rerum Natur. Univ. Comenjanae Math., 1, No. 4–6, 114–127, 1956.
4. Kibenko A. V. DAN SSSR, 136, No. 5, 1019–1021, 1961.
5. Perov A. I. Some questions in the qualitative theory of differential equations. Dissertation, 1959.
6. Brauer F. Canad. J. Math., 11, No. 4, 248–253, 1959.
7. Carathéodory C. Vorlesungen über reele Funktionen. Leipzig und Berlin, 1918.
8. Perov A. I. Izv. vuzov, No. 4, 104–112. Kazan, 1965.

Received by the editors
December 10, 1965

Voronezh State University