MATHEMATICS

A. I. Logunov, Z. B. Tsalyuk

ON THE QUESTION OF UNIQUENESS OF SOLUTIONS OF VOLTERRA INTEGRAL EQUATIONS WITH RETARDED ARGUMENT

(Presented by Academician A. N. Kolmogorov on 23 VIII 1964)

As is known, the equation \(y' = f(t,y)\), \(y(0)=y_0\), has a unique solution if
\[ |f(t,y_1)-f(t,y_2)|\leq \omega(|y_1-y_2|) \quad\text{and}\quad \int_0^\varepsilon \frac{dx}{\omega(x)}=\infty, \]
i.e., if \(|\Delta f|=|f(t,y+\Delta y)-f(t,y)|\) is a slowly increasing function of \(\Delta y\) in a right neighborhood of zero.

A. D. Myshkis showed \((^1)\) that for the equation
\[ y'(t)=f(t,y(t-g(t))) \tag{1} \]
one can weaken the requirement on the growth of \(|\Delta f|\), if as \(t\to 0\) the argument \(t-g(t)\) tends to zero sufficiently rapidly. More precisely, if \(t-g(t)\leq bt^{1/\alpha}\), \(0<\alpha<1\), \(b>0\), then one may put \(\omega(x)=x^\alpha\).

The present note is devoted to a further development and generalization of A. D. Myshkis’ result; namely, it will be shown that whatever the order of growth of \(|\Delta f|\), with a corresponding delay equation (1) has a unique solution.

Consider the system
\[ x(t)=\int_a^t G(t,s,x(s-g(s)))\,ds+f(t),\qquad t\geq a, \tag{2} \]
\[ x(t)=\psi(t),\qquad t\leq a. \]

Here
\[ G(t,s,x)=\{G_1(t,s,x_1,\ldots,x_n),\ldots,G_n(t,s,x_1,\ldots,x_n)\} \]
and
\[ f(t)=\{f_1(t),\ldots,f_n(t)\} \]
are vector-functions continuous for \(a\leq s\leq t\leq T\), \(\|x\|<C\); \(g(t)\) is continuous and nonnegative for \(t\in[a,T)\); the vector-function
\[ \psi(t)=\{\psi_1(t),\ldots,\psi_n(t)\} \]
is continuous for \(t\leq a\) and \(\psi(a)=f(a)\).

Denote by \(P\) the set of such points \(\tau\) that \(t-g(t)\leq \tau\) for \(t\in[\tau,\tau+\varepsilon]\) for some \(\varepsilon>0\).

Theorem 1. Let, for each point \(t_0\notin P\), there exist continuous increasing functions \(\omega(\tau)\geq \tau\) and \(\varphi(\tau)\) \((\omega(0)=\varphi(0)=0)\), for \(\tau\in[0,\varepsilon]\), such that:

1. \[ \|G(t,s,x)-G(t,s,y)\|\leq K\omega(\|x-y\|) \]
   \[ (t_0\leq s\leq t\leq t_0+\varepsilon,\ \|x\|<C,\ \|y\|<C,\ \|x-y\|\leq \varepsilon). \]

2. \[ t-g(t)\leq t_0+\omega_{-1}[\varphi(t-t_0)],\qquad t\in[t_0,t_0+\varepsilon], \]
   where \(\omega_{-1}(\tau)\) is the function inverse to \(\omega(\tau)\).

3. \[ \omega_{-1}[\varphi(\tau)]\geq \varphi[\omega_{-1}(\tau)],\qquad \varphi(\tau)<\tau\quad\text{for }\tau>0. \]

Then system [2] has on \([a,T)\) not more than one solution.

Condition 3 is satisfied, for example, for \(\varphi(t)=\omega_{-1}(t)\). However, in concrete cases, as \(\varphi(t)\) one can usually choose a more rapidly increasing function. Thus, for \(\omega(t)=|\ln t|^{-1}\) one may set \(\varphi(t)=\exp\{-t^{-\nu}\}\), \(0<\nu\leq 1\), and we obtain

Corollary. Suppose that for \(t_0\leq s\leq t\leq t_0+\varepsilon\), \(t_0\notin P\), \(\varepsilon>0\), and arbitrary \(x\) and \(y\) \((\|x\|<C,\ \|y\|<C,\ \|x-y\|\leq \varepsilon)\):

1. \(\|G(t,s,x)-G(t,s,y)\|\leq K|\ln\|x-y\||^{-1}\).
2. \(t-g(t)\leq t_0+\exp\{-\exp(t-t_0)^{-\nu}\}\), \(0<\nu\leq 1\).

Then system (2) has no more than one solution on \([a,T]\).

For \(\omega(t)=t^\alpha\), \(0<\alpha<1\), and \(t-g(t)\leq t_0+b(t-t_0)^{\gamma(t)}\), by virtue of condition 3, Theorem 1 guarantees uniqueness of the solution only if \(\alpha\gamma>1\). The theorem of A. D. Myshkis (¹) shows that, if the function \(G\) does not depend on \(t\), system (2) has a unique solution also in the case \(\gamma=1/\alpha\) (see also (²)). Moreover, A. D. Myshkis showed that, whatever \(\varepsilon>0\) is, for \(\gamma=1/\alpha-\varepsilon\) there is always a system (2) having at least two solutions. The assertion given below somewhat refines these results: it turns out that one may set \(\gamma=1/\alpha-\varphi(t-t_0)\), if \(\varphi(t)\) tends to zero sufficiently rapidly as \(t\to 0\).

Theorem 2. Suppose that for \(t_0\leq s\leq t\leq t_0+\varepsilon\), \(t_0\notin P\), \(\varepsilon>0\), the following conditions are satisfied:

1. \(\|G(t,s,x)-G(t,s,y)\|\leq K\|x-y\|^\alpha\) \((\|x\|<C,\ \|y\|<C,\ \|x-y\|\leq \varepsilon,\ 0<\alpha<1)\).
2. \(t-g(t)\leq t_0+(t-t_0)^{1/\alpha-\varphi(t-t_0)}\), where \(\varphi(t)\) is a nonnegative and nonincreasing function such that, for some positive \(\beta<1/\alpha\) and \(\tau\in[0,\varepsilon]\),
   \[ 0<r-\alpha\sum_{k=1}^{r-1}(r-k)\varphi(\tau^{\beta k-1})\to\infty \quad \text{as } r\to\infty . \]

Then system (2) has a unique solution on \([a,T]\).

Corollary. Suppose that for \(t_0\leq s\leq t\leq t_0+\varepsilon\), \(t_0\notin P\), \(\varepsilon>0\), and arbitrary \(x\) and \(y\) \((\|x\|<C,\ \|y\|<C,\ \|x-y\|\leq \varepsilon)\):

1. \(\|G(t,s,x)-G(t,s,y)\|\leq K\|x-y\|^\alpha\).
2. \(t-g(t)\leq t_0+b(t-t_0)^{1/\alpha}|\ln(t-t_0)|^{|\ln(t-t_0)|^\mu}\), where \(0<\alpha<1\), \(b>0\), \(0\leq \mu<1\).

Then system (2) has a unique solution on \([a,T]\).

We note that the assertions formulated above can be extended to systems with a finite number of delays.

The authors express their gratitude to the members of the Izhevsk mathematical seminar and to the seminar on equations with deviating argument of Prof. L. E. Elsgolts for their attention to the present work.

Received
27 III 1963

CITED LITERATURE

¹ A. D. Myshkis, UMN, 4, No. 5, 99 (1949).
² A. I. Logunov, DAN, 151, No. 2, 256 (1963).